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New techniques for climate data homogenizationNew techniques for climate data homogenization
Xiaolan L. WangXiaolan L. Wang
Climate Research Division, ASTD, STB, Environment CanadaClimate Research Division, ASTD, STB, Environment Canada
CMC Seminar, 19 September 2008CMC Seminar, 19 September 2008
-6
-4
-2
0
2
4
6
2000199519901985198019751970196519601955
Mon
thly
win
d sp
eed
anom
alie
s (k
m/h
r)
Time of observation
adjustedraw
Monthly mean surface wind speed series (Nanaimo, BC)
different signs!-0.0015100.002231
-8
-6
-4
-2
0
2
4
6
199519901985198019751970196519601955
Tem
pera
ture
ano
mal
ies
(deg
ree
C)
Time of obervation
-0.000324-0.001287
De-seasonalized
monthly mean series of daily max. temperatures (Collegeville, NS)1. Temperature:
2. Wind speed:
-1.54°CEffects of mean-shifton trendestimation
Mean, var., &extreme indices
biased!
Some examples of discontinuity in climate data series
3. Precipitation -
discontinuity due to joining of observations from nearby stations
Annual total rainfall for Moose Jaw Sask
0
100
200
300
400
500
600
700
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
mm
Moose Jaw Chab Moose Jaw A Moose Jaw CS
Use caution before adjusting to short interval
Courtesy of Lucie Vincent and Eva Mekis
See Vincent and Mekis
(2008), Discontinuities due to joining precipitation station observations in Canada,
J. App. Meteor. Climatol., in press.
Courtesy of Lucie Vincent
See Vincent et al. (2007), Surface temperature and humidity trends in Canada for 1953-2005. J. Clim., 20, 5100-5113.
4. Relative humidity –
discontinuity due to introduction of dewcel
in June 1978
60
65
70
75
80
85
90
1950 1960 1970 1980 1990 2000 2010
%
60
65
70
75
80
85
90
1950 1960 1970 1980 1990 2000 2010
%
60
65
70
75
80
85
90
1950 1960 1970 1980 1990 2000 2010
%
60
65
70
75
80
85
90
1950 1960 1970 1980 1990 2000 2010
%
Original valuesAdjusted values
Kuujjuaq, Québec
Winter Spring
Summer Fall
step= -8.0% step= -7.1%
step= -2.8%
step= -3.3%
Fog occurrence
5. Fog & low ceiling occurrence frequency -
Effects of mean-shift on trend estimation
Fog occurrence
occurrence offrequency theis
month in soccurrence ofnumber theis month in nsobservatio ofnumber theis
5.05.0
log odds Log
t
tt
t
t
tt
tt
MS
f
tStM
SMS
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−
+=η
Low ceiling occurrence
6. Cloudiness frequency data
Clear sky occurrence
Overcast occurrence
-
examples of
hourly
station pressure values
900
1000
1100
1200
1300
2002200120001999199819971996199519941993
pres
sure
(hPa
)
Year
P0Pz
Cape Hooper, NULong run of obviously wrong station pressure values
(with some correct ones in between)
Δ
= 294.6
hPa
920
960
1000
1040
Nov. 2000 Apr. 2001 Jan. 2002
pres
sure
(hPa
)
Year
PzP0
Dease Lake LWIS, BC
Long run of obviously wrong station pressure values (with a few correct ones in between)Δ
= 91.4
hPa
Dashed lines -
corrected values
-20
-15
-10
-5
0
5
10
15
199519901985198019751970
Mon
thly
pre
ssur
e an
omal
ies
(hPa
)
Time of obervation
0.009574-0.002386
De-seasonalized
monthly
mean station pressure series (Burgeo, NFLD)
Ignore 10.6 m elev.
7. Pressure data:
-2.956 hPa
Relatively small shift but it changes the sign
of trend estimate
Huge errors!
SLPStation pressure
Station pressureSLP
∆
= 3.4 hPa
940
960
980
1000
1020
19901985198019751970
Pz
Year
Lytton, BC
Original Corrected
Rz
series of original pressure data
Rz
series of corrected pressure data
27.4
m
960
990
1020
1970 1975 1980 1985 1990
P z(hPa
) ∆
= 3.4 hPa
- Another problematic hourly pressure series (relocation with elev. drop of 27.4 m)
Periodic feature -
reflects a problem in sealevel pressure reduction for elevated sites
It affects results of extreme (e.g., storminess) analysis!
We use a physical equation combined with a statistical method to tackle this kind of problems
See Wan et al. (2007), A Quality Assurance System for Canadian Hourly Pressure Data.
J. App. Meteor. Climatol., 46 (No. 11), 1804-1817.
Data discontinuities are inevitable due to inevitable changes in
observing instrument/location/environment… and evolving technology
Huge amount of $ spent on collecting climate data
big waste of $ if there were no effort to clean up the data
more and more effort devoted to climate data homogenization
Such discontinuities not only affect trend
assessment, but also affect
-
calibration of statistical relationships
for use in statistical weather forecast,
-
estimates of other statistics
that are used to study
climate variability
and
climate extremes, and to validate
model simulations
-
many other aspects of our study and understanding of the climate system
Special cases:
(a trend-change without anaccompanying mean-shift)
Data Homogenization Tools:1. Use physical relationships to correct some known
shifts
2. Use statistical methods to detect shifts
and estimate their magnitudeMetadata
e.g., hydrostatic equation to adjust pressure data for elevation
changes,log wind profile for anemometer height adjustments for wind speeds
2) TPR3 test for detecting mean-shifts
in constant trend
series (Wang 2003):
3) TPR4 test for detecting mean-shifts and/or
trend-changes
(Lund & Reeves 2002):
1) SNH test for detecting mean-shifts
in zero-trend
series (Alexandersson
1986):
Most commonly used statistical methods include
e.g., urbanization effects on T
Use a variant of TPR3!
changepoint
Problems/drawbacks of these changepoint detection methods:
Our recent studiesOur recent studies1. Propose two penalized tests to 1. Propose two penalized tests to even outeven out distributiondistribution of false alarm rate & detection powerof false alarm rate & detection power2. Extend these penalized tests to 2. Extend these penalized tests to account foraccount for the first order the first order autocorrelationautocorrelation3. Propose a stepwise testing algorithm for detecting 3. Propose a stepwise testing algorithm for detecting multiple changepointsmultiple changepointsNow, Now, developingdeveloping methods to deal with nonmethods to deal with non--Gaussian data (e.g. daily Gaussian data (e.g. daily precipprecip.),.), and and
multimulti--categorical data (e.g., frequencies of cloudiness conditions)categorical data (e.g., frequencies of cloudiness conditions)
1. Uneven distribution
of false alarm rate and detection power (details next)2. Model errors are assumed IID Gaussian3. For a series that contains at most one changepoint
-
The RHtestV2 software package-
Examples of application
-
The new methods and their detection power aspects (vs. those of the old methods)-
The uneven distribution problem of the old methods
The remainder of this presentation -
outline
Detection of a changepoint in a homogeneous series
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0 50 100 150 200 250 300 350 400 450 500
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 150N = 200N = 300N = 500
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Fals
e al
arm
rat
e (F
AR
)Changepoint position k
N = 8N = 10N = 15N = 20N = 25N = 30N = 40
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 50N = 60N = 70N = 80N = 90N = 100
αα
α
SNH test :(or its equivalentmaximal t test)
SNH test: U-shape FAR(k) curves!
SNH & TPR3 suffer from the effect of unequal sample sizes
(N1
= k
≠
N2
= N-k, generally)FAR(k) =αe(k) ≇α, pe(k) ≇ p !
Problems!
TPR3: W-shape FAR(k) curves!
0.03
0.04
0.05
0.06
0.07
0.08
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 10N = 15N = 20N = 25N = 30N = 40
α
0.03
0.05
0.07
0.09
0.11
0.13
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 50N = 60N = 70N = 80N = 90N = 100
α
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 50 100 150 200 250 300 350 400 450 500
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 150N = 200N = 250N = 300N = 400N = 500
α
Smaller effectsThe effective level of confidence on the identified changepoints that are
near the ends of series is much lower than the specified level p = (1 - α)
Not all thus-identified changepoints are actually significant at the nominal level,
while some actually-significant changepoints might go undetected – cheating!
Detection power depends on changepoint location (near the ends or middle of the series);
it has a much higher chance to give you a false alarm near the ends of a homogeneous series
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
0 50 100 150 200 250 300 350 400 450 500
Scal
ing
fact
or P
k
Changepoint position k
N = 150N = 200N = 300N = 500
0.8
0.85
0.9
0.95
1
1.05
1.1
0 5 10 15 20 25 30 35 40
Scal
ing
fact
or P
k
Changepoint position k
N = 8N = 10N = 15N = 20N = 25N = 30N = 40
0.8
0.85
0.9
0.95
1
1.05
1.1
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Scal
ing
fact
or P
k
Changepoint position k
N = 50N = 60N = 70N = 80N = 90N = 100
Wang et al. (2007) propose a Penalized Maximal t (PMT) testto even out the U-shaped FAR(k) curves of the SNH type tests:
[ ]
⎟⎟⎠
⎞⎜⎜⎝
⎛−+−
−=
−==
−⎟⎠⎞
⎜⎝⎛ −
=
=
∑ ∑
∑∑
= +=
+==
−≤≤
k
t
N
ktttk
N
ktt
k
tt
k
Nk
XXXXN
XkN
XXk
X
XXN
kNkkT
kTkPPT
1 1
22
21
2
12
11
2121
11max
)()(2
1ˆ
1 ;1
;)(ˆ1)( where
)( )(max
σ
σ
∩-shape penalty P(k):(thick curves)
)05.0())((
max
maxT
kTR e
kα
=☓, ╋, □, ○, …:
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 50
SNHTPMT
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 100
SNHTPMT
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0 5 10 15 20 25 30 35 40 45 50 55 60
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 60
SNHTPMT
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 80
SNHTPMT
0.01
0.05
0.09
0.13
0.17
0.21
0.25
0.29
0.33
0.37
0.41
0.45
0.49
0.53
0 40 80 120 160 200 240 280 320 360 400 440 480
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 500
SNHTPMT
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.21
0.23
0.25
0 20 40 60 80 100 120 140
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 150
SNHTPMT
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0 20 40 60 80 100 120 140 160 180 200
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 200
SNHTPMT
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0.31
0.33
0.35
0.37
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 300
SNHTPMT
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0 2 4 6 8 10 12 14 16 18 20Fa
lse
alar
m r
ate
(FA
R)
Changepoint position k
N = 20
SNHTPMT
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0 2 4 6 8 10
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 10
SNHTPMT
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 30
SNHTPMT
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 40
SNHTPMT
FAR(k): SNH test vs
PMT test
PMT makes FAR(k) =αe(k) ≋α, p e(k
) ≋ p !
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
0 50 100 150 200 250 300 350 400 450 500
Rat
io o
f m
ean
sign
ific
ance
rat
e
Time series length
RSS = 0.25RSS = 0.5 RSS = 1.0 RSS = 1.5 RSS = 2.0
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
1.2
1.22
1.24
1.26
0 50 100 150 200 250 300 350 400 450 500
Rat
io o
f m
ean
hit r
ate
Time series length
RSS = 0.25RSS = 0.5 RSS = 1.0 RSS = 1.5 RSS = 2.0
0.98
1
1.02
1.04
1.06
1.08
1.1
0 50 100 150 200 250 300 350 400 450 500
Rat
io o
f m
ean
posi
tion
rate
Time series length
RSS = 0.25RSS = 0.5 RSS = 1.0 RSS = 1.5 RSS = 2.0
Power of detection: PMT vs
SNH(real mean-shift Δ
at t = k)σ/Δ=RSS
Mean
Hit Rate:(PMT/SNH)
Mean
Position Rate:(PMT/SNH)
Mean
Significance Rate:(PMT/SNH)
Detect it in [k-2, k+2]& with p ≥
0.95
Detect it in [k-2, k+2]regardless of p
Detect it with p ≥
0.95regardless of position
averaged over2 ≤
k
≤
N-2
Each value from 1000 simulations
Wang (2008a) proposes a Penalized Maximal F test (PMF testto even out the W-shaped FAR(k) curves of TPR3 test
[ ]
∑ ∑
∑
= +=
=
−≤≤
−−+−−=
−−=
−−
=
=
k
t
N
ktttA
N
tt
A
A
fNk
tXtXSSE
tXSSE
NSSESSESSEkF
kFkPPF
1 1
22
21
1
2000
0
11max
)ˆˆ()ˆˆ(
;)ˆˆ(
;)3/(
1/)()( where
)( )(max
βμβμ
βμ
M-shape penalty Pf (k):(thick curves)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
0 50 100 150 200 250 300 350 400 450 500
Scal
ing
fact
or P
k
Changepoint position k
N = 150N = 200N = 300N = 400N = 500
0.75
0.8
0.85
0.9
0.95
1
1.05
0 5 10 15 20 25 30 35 40
Scal
ing
fact
or P
k
Changepoint position k
N = 15N = 20N = 25N = 30N = 40
0.75
0.8
0.85
0.9
0.95
1
1.05
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Scal
ing
fact
or P
k
Changepoint position k
N = 50N = 60N = 70N = 80N = 90N = 100
)05.0())((
max
maxF
kFR e
kα
=☓, ╋, □, ○, …:
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 40
TPR3PMFT
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 2 4 6 8 10 12 14
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 15
TPR3PMFT
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 2 4 6 8 10 12 14 16 18 20
Fals
e al
arm
rat
e (F
AR
)Changepoint position k
N = 20
TPR3PMFT
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 30
TPR3PMFT
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 50
TPR3PMFT
0.03
0.05
0.07
0.09
0.11
0 5 10 15 20 25 30 35 40 45 50 55 60
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 60
TPR3PMFT
0.03
0.05
0.07
0.09
0.11
0.13
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 80
TPR3PMFT
0.03
0.05
0.07
0.09
0.11
0.13
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 100
TPR3PMFT
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0 20 40 60 80 100 120 140
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 150
TPR3PMFT
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0 20 40 60 80 100 120 140 160 180 200
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 200
TPR3PMFT
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.21
0.23
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 300
TPR3PMFT
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0 40 80 120 160 200 240 280 320 360 400
Fals
e al
arm
rat
e (F
AR
)
Changepoint position k
N = 400
TPR3PMFT
FAR(k): TPR3 vs
PMF test
PMF test makes FAR(k) =αe(k) ≋α, p e(k
) ≋ p !
Power of detection: PMF vs
TPR3(real mean-shift Δ
at t = k)σ/Δ=RSS
Mean
Hit Rate:(PMF/TPR3)
Mean
Position Rate:(PMF/TPR3)
Mean
Significance Rate:(PMF/TPR3)
Detect it in [k-3, k+3]& with p ≥
0.95
Detect it in [k-3, k+3]regardless of p
Detect it with p ≥
0.95regardless of position
averaged over2 ≤
k
≤
N-2
Each value from 10,000 simulations
0.98
1
1.02
1.04
1.06
1.08
1.1
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700
Rat
io o
f m
ean
sign
ific
ance
rat
e
Time series length
RSS = 0.25RSS = 0.5 RSS = 1.0 RSS = 1.5 RSS = 2.0
N
1
1.02
1.04
1.06
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700
Rat
io o
f m
ean
posi
tion
rate
Time series length
RSS = 0.25RSS = 0.5 RSS = 1.0 RSS = 1.5 RSS = 2.0
N
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700
Rat
io o
f m
ean
hit r
ate
Time series length
RSS = 0.25RSS = 0.5 RSS = 1.0 RSS = 1.5 RSS = 2.0
N
PMT and PMF tests: still assume IID Gaussian errors!
Autocorrelation and Periodicity-
inherent features of most climate data series
Common practice:
-
use ref series to diminish periodicity (and trend)
> problem: ref series are not always available or homogeneous!
-
apply tests to annual series to reduce autocorr.
(usually autocorr. can not be diminished by using ref series! example)
sample size: only N/12
diff. shift-years identified in diff. month series for the same shift!
Some people ignore the effects, unaware of them
Consider AR(1) errors:
ttt z+= −1φεε
IID Gaussian noise(constant var.)
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
17.5
18
18.5
19
19.5
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
95-t
h pe
rcen
tiles
of
PTm
ax
AR(1) autocorrelation
N=10
N=15
N=20
N=25
N=30
N=35
N=40
N=45
N=50
N=55
N=60
N=65
N=1200
),(),0( max,max, NPTNPT φαα <
Ignore apositive autocorr.
increase # offalse alarms!
),(),0( max,max, NPTNPT φαα >
Ignore a negative
autocorr.let changepoints
go undetected!
PTmax
percentiles as a function of autocor. Φ
for each N
Effects of autocor. Φ
on PTmax
Each value estimated from 10 million simulations
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6
6.8
7
7.2
7.4
7.6
7.8
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200
95-t
h pe
rcen
tiles
of
PTm
ax
Sample size (series length) N
-0.200-0.150-0.100
-0.050
0 0.025
0.0500.075
0.1000.125
0.1500.175
0.2000.225
0.2500.275
0.3000.325
0.3500.375
0.4000.4250.450
0.475
0.500
0.525
0.550
0.575
0.600
0.625
0.650
0.675
0.700
0.725
0.750
0.775
0.800
PTmax percentiles as a function of sample size N for each φ
The PMTredalgorithm
(RHtestV2)
),(max, NPT φα
Table of critical values
)],ˆ( ),,ˆ([ ),ˆ(
series residual model full thefrom ]ˆ,ˆ[ ˆ
maxmaxmax NPTNPTNPT UL
UL
φφφ
φφφ
⇒
Each value estimated from 10 million simulations
Effects of autocor. Φ
on PFmax
Consider AR(1) errors:
ttt z+= −1φεε
IID Gaussian noise(constant var.)
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180
185
190
195
200
205
210
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
95-t
h pe
rcen
tiles
of
PFm
ax
AR(1) autocorrelation
N=20
N=25
N=30
N=35
N=40
N=45
N=50
N=60
N=70
N=80
N=90
N=100N=1200
),(),0( max,max, NPFNPF φαα >
Ignore negative
autocorr.let changepoints
go undetected!
PFmax
percentiles as a function of autocor. Φ
),(),0( max,max, NPFNPF φαα <
Ignore apositive autocorr.
increase # offalse alarms!
Each value estimated from 10 million simulations
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
92
94
96
98
0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 960 1020 1080 1140 1200
95-t
h pe
rcen
tiles
of
PFm
ax
Sample size (series length) N
-0.200-0.150-0.100-0.0500 0.0250.0500.075
0.1000.125
0.1500.175
0.2000.225
0.2500.275
0.3000.325
0.3500.375
0.400
0.425
0.450
0.475
0.500
0.525
0.550
0.575
0.600
0.625
0.650
0.675
0.700
0.725
0.750
0.775
0.800
Table of critical values
PFmax
percentiles as a function of sample size N
for each ɸ
The PMFredalgorithm
(RHtestV2)
),(max, NPF φα
)],ˆ( ),,ˆ([ ),ˆ(
series residual model full thefrom ]ˆ,ˆ[ ̂
maxmaxmax NPFNPFNPF UL
UL
φφφ
φφφ
⇒
Each value estimated from 10 million simulations
PMTred algorithm (for zero-trend series): PMFred algorithm (for constant trend series):
# of shifts
Random order
Power aspects: False Alarm Rate (FAR) and Hit Rate (HR; k±18)
FAR ⋍α=0.05, regardless of ɸ value
Ignore positive ɸ FAR » α
Power of accurate detection is satisfactory, Nc = 1 or Nc > 1!
φ̂in y uncertaint - rangey uncertaint 95%
(Each value estimated from 10,000 simulations)
Power depends on shift magnitude &
average segment length
No shift
Singleshift
Multipleshifts;AR(1)errors
IID
AR(1)
A large sample size is needed for a reasonably accurate estimate of ɸ!
The RHtestV2 software packageThe RHtestV2 software package --
recognized by WMO at COP-13 (Bali, Dec. 2007)
(R & FORTRAN)(R & FORTRAN)
1. PMT1. PMTredred
algorithm algorithm --
for zerofor zero--trend series with IID or AR(1) Gaussian errors trend series with IID or AR(1) Gaussian errors --
for use with reference seriesfor use with reference series
--
FindU.wRefFindU.wRef, , FindUD.wRefFindUD.wRef, , StepSize.wRefStepSize.wRef
2. PMF2. PMFredred
algorithmalgorithm
--
for constant trend series with IID or AR(1) Gaussian errors for constant trend series with IID or AR(1) Gaussian errors --
can be used can be used withoutwithout
reference seriesreference series
--
FindUFindU, , FindUDFindUD, , StepSizeStepSize
Input data format for the RHtestV2(Annual series) OR
(Monthly series)
OR
(Daily series)
1974 00 00
-4.87
1967 7 00 1015.70
1994 1 27 8.11975 00 00
-3.88
1967 8 00 1015.95
1994 1 28 5.3 1976 00 00
-5.65
1967 9 00 1016.10
1994 1 29 4.91977 00 00
-3.34
1967 10 00 -999.99
1994 1 30 4.91978 00 00
-5.72
1967 11 00 1010.71
1994 1 31 4.01979 00 00
-4.08
1967 12 00 1011.58
1994 2 1 3.91980 00 00
-4.97
1968 1 00 1009.37
1994 2 2 7.21981 00 00
-2.58
1968 2 00 1003.07
1994 2 3 8.71982 00 00
-4.48
1968 3 00 1011.94
1994 2 4 6.31983 00 00
-999.
1968 4 00 1014.74
1994 2 5 -999.1984 00 00
-3.40
1968 5 00 1009.59
1994 2 6 -999.1985 00 00
-4.58
1968 6 00 1011.77
1994 2 7 -999.1986 00 00
-4.25
1968 7 00 1014.35
1994 2 8 -999. 1987 00 00
-2.25
1968 8 00 1010.87
1994 2 9 9.01988 00 00
-3.94
1968 9 00 1016.45
1994 2 10 6.0…
…
…
YYYY MM DD data YYYY MM DD data
MissingValuecode
Missingvaluecode
A Ref series should be in a separate file of the same format!Base and Ref series
can have data for different periods or missing valuesat different times, but they must have the same missing value code!
YYYY MM DD data
MissingValuecode
Stepwise testing algorithm-
Possible
multiple change-points in one single time series
-
Most methods developed assuming at most one change-point
1st change-point: might be false or inaccurate (“contaminated”)
NC0
1st
check
S1 S2
C03C01
2nd
check
[1] Find the most probable changepoint in the series being tested:
S43rd
check S5
S3
=S4
+S5(re-assessment of change-point C0 C02)
Is the largest among C01
, C02
, C03
significant at the nominal level α?No (End!)Yes, …
[3] Estimate the significance of each changepoint Ki in the presence of{Cj
} (j=1,2,…Nc
) [4] Find the most significant
one among {Ki } (i=1,2,…Nc
+1) . Is it significant at the nominal level α?
series whitened thefrom ˆ and ,),ˆ(or ),ˆ( ,ˆ max,max, pNPFNPT ii φφφ αα
[1] Find the most probable changepoint in the series being tested C1
Stepwise testing algorithm for detecting multiple changepoints
[2] Mc =Nc ≥1 changepoints identified so far:
{Cj
} (j=1,2…,Nc ) Nc+1 segments. Identify the most probable changepoint in each segment {Ki } (i=1,2,…Nc+1)
No (Nc=Mc
)
[5] Find the smallest
among the Nc
shifts, re-assess its significance in thepresence of others. Is it significant at the nomial
level α?
Yes, repeat [2]-[4]
[6] output the list {Cj
} (j=1,2,…Nc
) andobtain the final estimates of -
shift significance-
shift size (Nc+1)-phase model fit
Yes
Delete it from the list {Cj
} (j=1,2,…Nc
),set Nc
= Nc
-1, re-assess significanceof the remaining shifts, and repeat [5]
No
add it to list {Cj
} (j=1,2,…Nc
), anddelete it from list {Ki } (i=1,2,…Nc
+1); then, set Nc
=Nc
+1, & repeat [3]-[4]
YesIs Nc
> Mc
?No
Main points:(1) Significant changepoints are added to
the list of changepoints, one by one:the most
significant, the next most…2) Significance of a changepoint is always
assessed in the presence of others listed,and re-assessed if the list changes.
(3) In the resulting list, each changepoint is significant in the presence of all others in the list.
(4) Final estimates (Nc+1)-phase model fitto de-seasonalized
base series, andto the base-minus-reference series (β≡0)if a reference series is used
Examples of applicationExamples of application 1. Single changepoint case1. Single changepoint case
--
PMTPMTredred BaseBase--RefRef seriesseries
((BurgeoBurgeo--Yarmouth)Yarmouth)
Burgeo
monthly station pressure series
-20
-15
-10
-5
0
5
10
15
199519901985198019751970
Mon
thly
pre
ssur
e an
omal
ies
(hPa
) De-seasonalized
pressure series
990
995
1000
1005
1010
1015
1020
1025
199519901985198019751970M
onth
ly p
ress
ure
anom
alie
s (h
Pa)
detectedmetadata
Actual shift: Dec. 1976 (10.6m 0)Detected shift: Oct. 1976
990
995
1000
1005
1010
1015
1020
1025
199519901985198019751970
Mon
thly
mea
n pr
essu
res
(hPa
)
Adjusted pressure series
0.117] [-0.098, 0.010ˆmonthper hPa 009574.0ˆ
=
=
φ
β
-2.956 hPa
Plots like theseare included
in the RHtestV2 output
Examples of applicationExamples of application 2. Two changepoints2. Two changepoints
--
PMFPMFredred dede--seasonalizedseasonalizedtemp. seriestemp. series
(shifts & seasonal means(shifts & seasonal meansestimated via iteration)estimated via iteration)
-30
-20
-10
0
10
20
19951990198519801975197019651960195519501945194019351930192519201915
Tem
pera
ture
(de
gree
C)
Time of obervation
Amos monthly means of daily minimum temperatures
DetectedPMFred Metadata
(1) Oct 1953
Aug 1948-Oct 1958: Stn
elevation change(990 feet 1002 feet)
(2) Aug 1958
Oct 1958: Thermometer screen &stand replaced
-10
-5
0
5
10
19951990198519801975197019651960195519501945194019351930192519201915
Tem
pera
ture
ano
mal
ies
(deg
ree
C)
Time of obervation
De-seasonalized
series
(2)1.97°C
(1)-1.23°C
0.233] [0.112, 0.173ˆmonthper C 001385.0ˆ
=
=
φ
β
Type-1: in boldType-0: in italic
DetectedPMTred_gRef PMFred Metadata( 1) Feb 1960 Oct 1960 May 1957-Feb 1961: anemometer 45B U2A( 2) Jul 1965 Dec 1964 Feb 1961-Mar 1967: Ane. height change( 3) Feb 1974 Jan 1974 Early 1974: WSD detector has a rusted U-arm ( 4) Jan 1976 Nov 1975 Dec 1975: WSD detector replaced( 5) Mar 1980 X Mar 1981: U2A had bearing jam, was replaced ( 6) Aug 1982 Jan 1984 Jan 1984: WSD detector replaced (was sticking)( 7) Feb 1985 X
Mar 1985: Wind system relocated( 8) Nov 1985 X
Jun 1986: Wind system completely recalibrated( 9) Feb 1993 Feb 1993 Jan 1993: WSD detector & tilt pole replaced (10) Jun 1994 Apr 1994 Jun 1994: WSD detector replaced, new tilt pole(11) Apr 1997 Apr 1997 No metadata from Jul. 1994 to Apr. 1997
Nmin
=5 Nmin
=10
Examples of application Examples of application 3.3.
Multiple changepoints caseMultiple changepoints case
--
PMTPMTredred BaseBase--RefRef seriesseriesRef = Ref = gRefgRef
= geostrophic winds (= geostrophic winds (homo. SLPhomo. SLP) ) or or Ref = Ref = mRefmRef
= 5= 5--stns mean surface windsstns mean surface winds--
PMFPMFredred dede--seasonalizedseasonalized base seriesbase series
Nanaimo monthly wind speed series
0
2
4
6
8
10
12
14
16
2000199519901985198019751970196519601955
Mon
thly
mea
n w
ind
spee
ds (
km/h
r)
Time of observation
BaseDiffResults of PMTred
0
2
4
6
8
10
12
14
16
20052000199519901985198019751970196519601955
Win
d sp
eed
(km
/hr)
Time of observation
PMF basePMT gRef diffResults of PMFred
(solid line)&PMTred
(dashed line)
Nseg <10
0
2
4
6
8
10
12
14
16
2000199519901985198019751970196519601955
Mon
thly
mea
n w
ind
spee
ds (
km/h
r)
Time of observation
BaseDiff
Results of PMTred (using mRef)
(1)(2)
(3)(4)
(5)(6)
(7) (8)(9)
(11)(10)
37.1ˆ03.0ˆ
−=Δ
−=Δ
dNot a shift in the base series (Δ=-0.03)but a problem in the mRef
series!
Two estimates of shift-size - Any notable inconsistency
between them indicates …
Aug. 1971
Apply PMFred
to a Ref series to eliminate large artificial shifts (if any) before it is used!
It is always necessary to check the veracity
of changepoints identified statistically!Use of good reference series increases detection power!
For more details about using the RHtestV2, please refer to For more details about using the RHtestV2, please refer to
““Wang, X. L. and Y. Feng, 2007: Wang, X. L. and Y. Feng, 2007: RHtestV2 User Manual.RHtestV2 User Manual.””
Available atAvailable at
http://http://cccma.seos.uvic.ca/ETCCDMI/software.shtmlcccma.seos.uvic.ca/ETCCDMI/software.shtml,,
along with the open source softwarealong with the open source software
Thank you very much!Thank you very much!
References:Wang, X. L., 2008a: Penalized maximal F test for detecting undocumented mean-shift without trend change.
J. Atmos. Oceanic Technol., 25 (No. 3), 368-384. DOI:10.1175/2007/JTECHA982.1Wang, X. L., 2008b: Accounting for autocorrelation in detecting mean-shifts in climate data series
using the penalized maximal t or F test. J. App. Meteor. Climatol, 47, 2423–2444.Wang, X. L., Q. H. Wen, and Y. Wu, 2007: Penalized Maximal t-test for Detecting Undocumented Mean Change
in Climate Data Series. J. App. Meteor. Climatol., 46 (No. 6), 916-931. DOI:10.1175/JAM2504.1Wan, H., X. L. Wang, and V. R. Swail, 2007: A Quality Assurance System for Canadian Hourly Pressure Data.
J. App. Meteor. Climatol., 46 (No. 11), 1804-1817.
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