forecasting using r - rob j. hyndman · class: “ts ” print and ... ## 1992 6362 6389 6433 6491...
TRANSCRIPT
Forecasting using R 1
Forecasting using R
Rob J Hyndman
11 Time series graphics
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time series in R 2
Time series in R
Time series consist of sequences of observationscollected over timeWe will assume the time periods are equally spaced
Time series examplesDaily IBM stock pricesMonthly rainfallAnnual Google profitsQuarterly Australian beer production
Forecasting using R Time series in R 3
Time series in R
ausgdp lt- ts(scan(gdpdat)frequency=4start=1971+24)
Class ldquotsrdquoPrint and plotting methods available
ausgdp
Qtr1 Qtr2 Qtr3 Qtr4 1971 4612 4651 1972 4645 4615 4645 4722 1973 4780 4830 4887 4933 1974 4921 4875 4867 4905 1975 4938 4934 4942 4979 1976 5028 5079 5112 5127 1977 5130 5101 5072 5069 1978 5100 5166 5244 5312 1979 5349 5370 5388 5396 1980 5388 5403 5442 5482 1981 5506 5531 5560 5583 1982 5568 5524 5452 5358 1983 5303 5320 5408 5531 1984 5624 5669 5697 5736 1985 5811 5894 5952 5965 1986 5943 5924 5935 5979 1987 6035 6097 6167 6227 1988 6256 6272 6295 6345 1989 6413 6468 6497 6511 1990 6514 6512 6490 6442 1991 6390 6346 6328 6340 1992 6362 6389 6433 6491 1993 6541 6566 6602 6671 1994 6765 6847 6890 6918 1995 6962 7018 7083 7134 1996 7173 7212 7242 7276 1997 7332 7400 7478 7550 1998 7618
Forecasting using R Time series in R 4
Time series in R
5000
6000
7000
1975 1980 1985 1990 1995Time
ausg
dp
Forecasting using R Time series in R 5
autoplot(ausgdp)
Time series in R
Residential electricity sales
elecsales
Time Series Start = 1989 End = 2008 Frequency = 1 [1] 235434 237971 231852 246899 238609 256947 257572 276272 [9] 284450 300070 310810 335750 307570 318060 322160 317620 [17] 343060 352748 363789 365500
Forecasting using R Time series in R 6
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time series in R 2
Time series in R
Time series consist of sequences of observationscollected over timeWe will assume the time periods are equally spaced
Time series examplesDaily IBM stock pricesMonthly rainfallAnnual Google profitsQuarterly Australian beer production
Forecasting using R Time series in R 3
Time series in R
ausgdp lt- ts(scan(gdpdat)frequency=4start=1971+24)
Class ldquotsrdquoPrint and plotting methods available
ausgdp
Qtr1 Qtr2 Qtr3 Qtr4 1971 4612 4651 1972 4645 4615 4645 4722 1973 4780 4830 4887 4933 1974 4921 4875 4867 4905 1975 4938 4934 4942 4979 1976 5028 5079 5112 5127 1977 5130 5101 5072 5069 1978 5100 5166 5244 5312 1979 5349 5370 5388 5396 1980 5388 5403 5442 5482 1981 5506 5531 5560 5583 1982 5568 5524 5452 5358 1983 5303 5320 5408 5531 1984 5624 5669 5697 5736 1985 5811 5894 5952 5965 1986 5943 5924 5935 5979 1987 6035 6097 6167 6227 1988 6256 6272 6295 6345 1989 6413 6468 6497 6511 1990 6514 6512 6490 6442 1991 6390 6346 6328 6340 1992 6362 6389 6433 6491 1993 6541 6566 6602 6671 1994 6765 6847 6890 6918 1995 6962 7018 7083 7134 1996 7173 7212 7242 7276 1997 7332 7400 7478 7550 1998 7618
Forecasting using R Time series in R 4
Time series in R
5000
6000
7000
1975 1980 1985 1990 1995Time
ausg
dp
Forecasting using R Time series in R 5
autoplot(ausgdp)
Time series in R
Residential electricity sales
elecsales
Time Series Start = 1989 End = 2008 Frequency = 1 [1] 235434 237971 231852 246899 238609 256947 257572 276272 [9] 284450 300070 310810 335750 307570 318060 322160 317620 [17] 343060 352748 363789 365500
Forecasting using R Time series in R 6
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series in R
Time series consist of sequences of observationscollected over timeWe will assume the time periods are equally spaced
Time series examplesDaily IBM stock pricesMonthly rainfallAnnual Google profitsQuarterly Australian beer production
Forecasting using R Time series in R 3
Time series in R
ausgdp lt- ts(scan(gdpdat)frequency=4start=1971+24)
Class ldquotsrdquoPrint and plotting methods available
ausgdp
Qtr1 Qtr2 Qtr3 Qtr4 1971 4612 4651 1972 4645 4615 4645 4722 1973 4780 4830 4887 4933 1974 4921 4875 4867 4905 1975 4938 4934 4942 4979 1976 5028 5079 5112 5127 1977 5130 5101 5072 5069 1978 5100 5166 5244 5312 1979 5349 5370 5388 5396 1980 5388 5403 5442 5482 1981 5506 5531 5560 5583 1982 5568 5524 5452 5358 1983 5303 5320 5408 5531 1984 5624 5669 5697 5736 1985 5811 5894 5952 5965 1986 5943 5924 5935 5979 1987 6035 6097 6167 6227 1988 6256 6272 6295 6345 1989 6413 6468 6497 6511 1990 6514 6512 6490 6442 1991 6390 6346 6328 6340 1992 6362 6389 6433 6491 1993 6541 6566 6602 6671 1994 6765 6847 6890 6918 1995 6962 7018 7083 7134 1996 7173 7212 7242 7276 1997 7332 7400 7478 7550 1998 7618
Forecasting using R Time series in R 4
Time series in R
5000
6000
7000
1975 1980 1985 1990 1995Time
ausg
dp
Forecasting using R Time series in R 5
autoplot(ausgdp)
Time series in R
Residential electricity sales
elecsales
Time Series Start = 1989 End = 2008 Frequency = 1 [1] 235434 237971 231852 246899 238609 256947 257572 276272 [9] 284450 300070 310810 335750 307570 318060 322160 317620 [17] 343060 352748 363789 365500
Forecasting using R Time series in R 6
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series in R
ausgdp lt- ts(scan(gdpdat)frequency=4start=1971+24)
Class ldquotsrdquoPrint and plotting methods available
ausgdp
Qtr1 Qtr2 Qtr3 Qtr4 1971 4612 4651 1972 4645 4615 4645 4722 1973 4780 4830 4887 4933 1974 4921 4875 4867 4905 1975 4938 4934 4942 4979 1976 5028 5079 5112 5127 1977 5130 5101 5072 5069 1978 5100 5166 5244 5312 1979 5349 5370 5388 5396 1980 5388 5403 5442 5482 1981 5506 5531 5560 5583 1982 5568 5524 5452 5358 1983 5303 5320 5408 5531 1984 5624 5669 5697 5736 1985 5811 5894 5952 5965 1986 5943 5924 5935 5979 1987 6035 6097 6167 6227 1988 6256 6272 6295 6345 1989 6413 6468 6497 6511 1990 6514 6512 6490 6442 1991 6390 6346 6328 6340 1992 6362 6389 6433 6491 1993 6541 6566 6602 6671 1994 6765 6847 6890 6918 1995 6962 7018 7083 7134 1996 7173 7212 7242 7276 1997 7332 7400 7478 7550 1998 7618
Forecasting using R Time series in R 4
Time series in R
5000
6000
7000
1975 1980 1985 1990 1995Time
ausg
dp
Forecasting using R Time series in R 5
autoplot(ausgdp)
Time series in R
Residential electricity sales
elecsales
Time Series Start = 1989 End = 2008 Frequency = 1 [1] 235434 237971 231852 246899 238609 256947 257572 276272 [9] 284450 300070 310810 335750 307570 318060 322160 317620 [17] 343060 352748 363789 365500
Forecasting using R Time series in R 6
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series in R
5000
6000
7000
1975 1980 1985 1990 1995Time
ausg
dp
Forecasting using R Time series in R 5
autoplot(ausgdp)
Time series in R
Residential electricity sales
elecsales
Time Series Start = 1989 End = 2008 Frequency = 1 [1] 235434 237971 231852 246899 238609 256947 257572 276272 [9] 284450 300070 310810 335750 307570 318060 322160 317620 [17] 343060 352748 363789 365500
Forecasting using R Time series in R 6
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series in R
Residential electricity sales
elecsales
Time Series Start = 1989 End = 2008 Frequency = 1 [1] 235434 237971 231852 246899 238609 256947 257572 276272 [9] 284450 300070 310810 335750 307570 318060 322160 317620 [17] 343060 352748 363789 365500
Forecasting using R Time series in R 6
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series in R
Main package used in this coursegt library(fpp)
This loads
some data for use in examples and exercisesforecast package (for forecasting functions)tseries package (for a few time series functions)fma package (for lots of time series data)expsmooth package (for more time series data)lmtest package (for some regression functions)
Forecasting using R Time series in R 7
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Time plots 8
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time plotsautoplot(melsyd[EconomyClass])
0
10
20
30
1988 1989 1990 1991 1992 1993Time
mel
syd[
E
cono
my
Cla
ss]
Forecasting using R Time plots 9
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time plots
autoplot(a10 ylab=$ million xlab=Yearmain=Antidiabetic drug sales)
10
20
30
1995 2000 2005Year
$ m
illio
n
Antidiabetic drug sales
Forecasting using R Time plots 10
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 1 11
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Lab Session 1
Forecasting using R Lab session 1 12
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal plots 13
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Seasonal plotsggseasonplot(a10 ylab=$ million
yearlabels=TRUE yearlabelsleft=TRUE) +ggtitle(Seasonal plot antidiabetic drug sales)
1991 19911992 1992199319931994 19941995 19951996 199619971997
1998199819991999
2000 2000
20012001
20022002
2003 20032004 20042005 2005
2006 2006
20072007
2008
2008
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal plot antidiabetic drug sales
Forecasting using R Seasonal plots 14
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Seasonal plots
Data plotted against the individual ldquoseasonsrdquo in whichthe data were observed (In this case a ldquoseasonrdquo is amonth)Something like a time plot except that the data fromeach season are overlappedEnables the underlying seasonal pattern to be seenmore clearly and also allows any substantialdepartures from the seasonal pattern to be easilyidentifiedIn R ggseasonplot or seasonplot
Forecasting using R Seasonal plots 15
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Seasonal subseries plots
ggmonthplot(a10) + ylab($ million) +ggtitle(Seasonal subseries plot antidiabetic drug sales)
10
20
30
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
$ m
illio
n
Seasonal subseries plot antidiabetic drug sales
Forecasting using R Seasonal plots 16
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Seasonal subseries plots
Data for each season collected together in time plotas separate time seriesEnables the underlying seasonal pattern to be seenclearly and changes in seasonality over time to bevisualizedIn R ggmonthplot or monthplot
Forecasting using R Seasonal plots 17
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Quarterly Australian Beer Production
beer lt- window(ausbeerstart=1992)
autoplot(beer)
ggseasonplot(beeryearlabels=TRUE)
ggmonthplot(beer)
Forecasting using R Seasonal plots 18
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Quarterly Australian Beer Production
400
450
500
1995 2000 2005Time
beer
Forecasting using R Seasonal plots 19
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Quarterly Australian Beer Production
1992
1993
1994
1995
1996
199719981999
20002001
2002
2003
2004
20052006
2007
2008400
450
500
Q1 Q2 Q3 Q4Quarter
Seasonal plot beer
Forecasting using R Seasonal plots 20
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Quarterly Australian Beer Production
400
450
500
Q1 Q2 Q3 Q4Quarter
beer
Forecasting using R Seasonal plots 21
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Seasonal or cyclic 22
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series patterns
Trend pattern exists when there is a long-termincrease or decrease in the data
Seasonal pattern exists when a series is influenced byseasonal factors (eg the quarter of the yearthe month or day of the week)
Cyclic pattern exists when data exhibit rises and fallsthat are not of fixed period (duration usually ofat least 2 years)
Forecasting using R Seasonal or cyclic 23
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series patterns
8000
10000
12000
14000
1980 1985 1990 1995Year
GW
h
Australian electricity production
Forecasting using R Seasonal or cyclic 24
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series patterns
200
300
400
500
600
1960 1970 1980 1990Year
mill
ion
units
Australian clay brick production
Forecasting using R Seasonal or cyclic 25
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series patterns
40
60
80
1975 1980 1985 1990 1995Year
Tota
l sal
es
Sales of new oneminusfamily houses USA
Forecasting using R Seasonal or cyclic 26
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series patterns
86
88
90
0 20 40 60 80 100Day
pric
e
US Treasury Bill Contracts
Forecasting using R Seasonal or cyclic 27
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Time series patterns
0
2000
4000
6000
1820 1840 1860 1880 1900 1920Year
Num
ber
trap
ped
Annual Canadian Lynx Trappings
Forecasting using R Seasonal or cyclic 28
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Seasonal or cyclic
Differences between seasonal and cyclic patterns
seasonal pattern constant length cyclic patternvariable lengthaverage length of cycle longer than length of seasonalpatternmagnitude of cycle more variable than magnitude ofseasonal pattern
The timing of peaks and troughs is predictable withseasonal data but unpredictable in the long term withcyclic data
Forecasting using R Seasonal or cyclic 29
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lag plots and autocorrelation 30
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example Beer production
beer lt- window(ausbeer start=1992)gglagplot(beer lags=9 dolines=FALSE continuous=FALSE)
Forecasting using R Lag plots and autocorrelation 31
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example Beer productionlag 1 lag 2 lag 3
lag 4 lag 5 lag 6
lag 7 lag 8 lag 9
400
450
500
400
450
500
400
450
500
400 450 500 400 450 500 400 450 500
season
1
2
3
4
Forecasting using R Lag plots and autocorrelation 32
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Lagged scatterplots
Each graph shows yt plotted against ytminusk for differentvalues of kThe autocorrelations are the correlations associatedwith these scatterplots
Forecasting using R Lag plots and autocorrelation 33
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Autocorrelation
Covariance and correlation measure extent of linearrelationship between two variables (y and X)Autocovariance and autocorrelation measure linearrelationship between lagged values of a time series y
We measure the relationship between yt and ytminus1yt and ytminus2yt and ytminus3etc
Forecasting using R Lag plots and autocorrelation 34
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
AutocorrelationWe denote the sample autocovariance at lag k by ck andthe sample autocorrelation at lag k by rk Then define
ck =1T
Tsumt=k+1
(yt minus y)(ytminusk minus y)
and rk = ckc0
r1 indicates how successive values of y relate to eachotherr2 indicates how y values two periods apart relate toeach otherrk is almost the same as the sample correlationbetween yt and ytminusk
Forecasting using R Lag plots and autocorrelation 35
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
AutocorrelationResults for first 9 lags for beer datafootnotesize
r1 r2 r3 r4 r5 r6 r7 r8 r9
-0097 -0651 -0072 0872 -0074 -0628 -0083 0825 -0088
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 36
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Autocorrelation
r4 higher than for the other lags This is due to theseasonal pattern in the data the peaks tend to be 4quarters apart and the troughs tend to be 2 quartersapartr2 is more negative than for the other lags becausetroughs tend to be 2 quarters behind peaksTogether the autocorrelations at lags 1 2 makeup the autocorrelation or ACFThe plot is known as a correlogram
Forecasting using R Lag plots and autocorrelation 37
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
ACF
ggAcf(beer)
minus05
00
05
4 8 12 16Lag
AC
F
Series beer
Forecasting using R Lag plots and autocorrelation 38
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Recognizing seasonality in a time series
If there is seasonality the ACF at the seasonal lag (eg 12for monthly data) will be large and positive
For seasonal monthly data a large ACF value will beseen at lag 12 and possibly also at lags 24 36 For seasonal quarterly data a large ACF value will beseen at lag 4 and possibly also at lags 8 12
Forecasting using R Lag plots and autocorrelation 39
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Aus monthly electricity production
8000
10000
12000
14000
1980 1985 1990 1995Time
elec
2
Forecasting using R Lag plots and autocorrelation 40
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Aus monthly electricity production
000
025
050
075
0 12 24 36 48Lag
AC
F
Series elec2
Forecasting using R Lag plots and autocorrelation 41
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Aus monthly electricity production
Time plot shows clear trend and seasonality
The same features are reflected in the ACF
The slowly decaying ACF indicates trendThe ACF peaks at lags 12 24 36 indicateseasonality of length 12
Forecasting using R Lag plots and autocorrelation 42
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Which is which
40
60
80
0 20 40 60
chir
ps p
er m
inut
e
1 Daily temperature of cow
7
8
9
10
11
1973 1974 1975 1976 1977 1978 1979
thou
sand
s
2 Monthly accidental deaths
200
400
600
1950 1952 1954 1956 1958 1960
thou
sand
s
3 Monthly air passengers
30
60
90
1850 1860 1870 1880 1890 1900 1910
thou
sand
s
4 Annual mink trappings
00
05
10
12 246 18
AC
F
A
00
05
10
5 10 15
AC
F
B
00
05
10
5 10 15
AC
F
C
00
05
10
12 246 18
AC
F
D
Forecasting using R Lag plots and autocorrelation 43
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R White noise 44
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example White noise
minus3
minus2
minus1
0
1
2
0 5 10 15 20 25 30 35Time
wn
White noise data is uncorrelated across time with zeromean and constant variance(Technically we require independence as well)
Forecasting using R White noise 45
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example White noise
r1 -011r2 -023r3 -031r4 015r5 021r6 004r7 -017r8 -020r9 026r10 007
Sample autocorrelations for white noise series
For uncorrelated data we would expect eachautocorrelation to be close to zero
Forecasting using R White noise 46
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wn
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Sampling distribution of autocorrelations
Sampling distribution of rk for white noise data isasymptotically N(01T)
95 of all rk for white noise must lie withinplusmn196
radicT
If this is not the case the series is probably not WNCommon to plot lines atplusmn196
radicT when plotting
ACF These are the critical values
Forecasting using R White noise 47
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Autocorrelation
Forecasting using R White noise 48
minus02
00
02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Lag
AC
F
Series wnExampleT = 36 and so criticalvalues atplusmn196
radic36 = plusmn0327
All autocorrelationcoefficients lie withinthese limits confirmingthat the data are whitenoise (More preciselythe data cannot bedistinguished from white noise)
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example Pigs slaughtered
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Year
thou
sand
s
Number of pigs slaughtered in Victoria
Forecasting using R White noise 49
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example Pigs slaughtered
minus02
00
02
12 246 18Lag
AC
F
Series pigs2
Forecasting using R White noise 50
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Example Pigs slaughtered
Monthly total number of pigs slaughtered in the state ofVictoria Australia from January 1990 through August1995 (Source Australian Bureau of Statistics)
Difficult to detect pattern in time plotACF shows some significant autocorrelation at lags 12 and 3r12 relatively large although not significant This mayindicate some slight seasonality
These show the series is not a white noise series
Forecasting using R White noise 51
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Combination graph
ggtsdisplay(pigs2 plottype=scatter)
80000
90000
100000
110000
1990 1991 1992 1993 1994 1995Time
x
pigs2
minus02
00
02
0 5 10 15 20Lag
AC
F
80000
90000
100000
110000
80000 90000 100000 110000Ytminus1
Yt
Forecasting using R White noise 52
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Outline1 Time series in R
2 Time plots
3 Lab session 1
4 Seasonal plots
5 Seasonal or cyclic
6 Lag plots and autocorrelation
7 White noise
8 Lab session 2
Forecasting using R Lab session 2 53
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-
Lab Session 2
Forecasting using R Lab session 2 54
- Time series in R
- Time plots
- Lab session 1
- Seasonal plots
- Seasonal or cyclic
- Lag plots and autocorrelation
- White noise
- Lab session 2
-