estimation of the spectral density function. the spectral density function, f( ) the spectral...
TRANSCRIPT
Estimation of the spectral density function
cos( )h h f d
0
1 10 cos( )
2 h
f h h
The spectral density function, f() The spectral density function, f(x), is a symmetric function defined on the interval [-,] satisfying
and
The spectral density function, f(x), can be calculated from the autocovariance function and vice versa.
1cos( )
2 h
h h
2. cos sinixe x i x
Some complex number results:
Use
1. where 1z x iy i
2 3
12! 3!
u u ue u
2 4
cos 12! 4!
u uu
3 5
sin3! 5!
u uu u
3. (polar representation)iz x iy Re
2 2where and tanx
R x yy
4. cos2
ix ixe ex
5. sin2
ix ixe ex
i
Expectations of Linear and Quadratic forms of a weakly
stationary Time Series
Expectations, Variances and Covariances of Linear forms
Theorem Let {xt:t T} be a weakly stationary time series.Let
Then
and
where
and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.
T
ttt xcL
1
T
ttcLE
1
1
11 1
)()(T
Tr Sssrs
T
s
T
tts
r
ccrstccLVar
dfCdfecT
t
tit
22
1
)(
T
t
tit ecC
1
)(
Proof
T
tt
T
ttt
T
ttt cxEcxcELE
111
T
t
T
ssstt xcxcELVar
1 1
T
t
T
sst stcc
1 1
T
t
T
sstst xxEcc
1 1
1
1
T
Tt
T
Sssrs
r
ccr
Also since
Q.E.D.
dfedfhh hi )()()cos(
T
t
T
s
stist dfeccLVar
1 1
dfececT
t
sit
T
t
tit
11
dfC
2
dfeecc sitiT
t
T
sst
1 1
dfecT
t
tit
2
1
Theorem Let {xt:t T} be a weakly stationary time series.
Let
and
T
tttc xcL
1
T
tttb xbL
1
Expectations, Variances and Covariances of Linear forms
Summary
Theorem Let {xt:t T} be a weakly stationary time series.Let
Then
and
where
and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.
T
ttt xcL
1
T
ttcLE
1
1
11 1
)()(T
Tr Sssrs
T
s
T
tts
r
ccrstccLVar
2( )C f d
T
t
tit ecC
1
)(
Theorem Let {xt:t T} be a weakly stationary time series.
Let and
T
tttc xcL
1
T
tttb xbL
1
Then
T
s
T
ttsbc stbcLLCov
1 1
)(,
1
1
)(T
Tr Sssrs
r
bcr
dfBC )()(
T
t
tit ecC
1
)(
T
t
tit ebB
1
)( where and
Then
T
s
T
ttsbc stbcLLCov
1 1
)(,
dfebecT
s
sis
T
t
tit
11
T
t
tit ecC
1
)(
T
t
tit ebB
1
)(
1
1
)(T
Tr Sssrs
r
bcr
dfBC )()(
where and
Also Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.
Expectations, Variances and Covariances of Quadratic forms
Theorem Let {xt:t T} be a weakly stationary time series.
Let
Then
T
s
T
ttsst xxaQ
1 1
T
s
T
t
T
Tr Strssst
r
arstaQE1 1
1
1,
dfAdfeaT
s
T
t
stist ),(
1 1
and
T
s
T
t
T
s
T
ttsst ttssaaQVar
1 1 1' 1''' ''2
1
1
1
1,2
T
Tr
T
Tp Ss Stptrsst
r r
aapr
1
1
1
1,, ,,
T
Tr
T
Tp Ss Stptrsrsst pstrstaa
r r
stssst ',',
ddffeaT
s
T
t
stist
2
1 1
2
T
s
T
t
T
s
T
ttsst stssstaa
1 1 1' 1''' ',',
ddffA2
),(2
T
s
T
t
T
s
T
ttsst stssstaa
1 1 1' 1''' ',',
T
s
T
t
stisteaA
1 1
, where
and Sr = {1,2, ..., T-r}, if r ≥ 0,
Sr = {1- r, 2 - r, ..., T} if r ≤ 0,
(h,r,s) = the fourth order cumulant
= E[(xt - )(xt+h - )(xt+r - )(xt+s - )]
- [(h)(r-s)+(r)(h-s)+(s)(h-r)]
Note (h,r,s) = 0 if {xt:t T}is Normal.
Theorem Let {xt:t T} be a weakly stationary time series.
Let
Then
T
s
T
ttsst xxaQ
1 1
T
s
T
ttsst xxbP
1 1
T
s
T
t
T
s
T
ttsst ttssbaPQCov
1 1 1' 1''' ''2,
stssst ',',
1
1
1
1,2
T
Tr
T
Tp Ss Stptrsst
r r
bapr
1
1
1
1,, ,,
T
Tr
T
Tp Ss Stptrsrsst pstrstba
r r
ddffebeaT
s
T
t
stits
T
s
T
t
stist
1' 1'
''''
1 1
2
T
s
T
t
T
s
T
ttsst stssstba
1 1 1' 1''' ',',
ddffBA ),(),(2
T
s
T
t
T
s
T
ttsst stssstba
1 1 1' 1''' ',',
and
where
T
s
T
t
stist eaA
1 1
,
T
s
T
t
stist eaB
1 1
,
ExamplesThe sample mean
1
11
T
sss
T
ss xcx
Tx
TsT
cs ,.,2,1for 1
where
and
Thus
1
11
T
s
T
ss T
cxE
T
s
T
t
T
s
T
tts st
TTstccxVar
1 11 1
)(11
)(
1
1
1
1
)()(2)0(1 T
Ts
T
s
rT
rTr
T
rT
T
Also
1
111
11
i
Tii
T
t
tiT
t
tit e
ee
Te
TecC
2/2/
2/2/2/11
ii
TiTiTi
ee
eee
T
2/sin
2/sin1 2/1
T
eT
Ti
and
2
1
2
T
t
titecC
2/sin
2/sin1
2/sin
2/sin1 2/12/1
T
eT
Te
TTiTi
T
HT
TT
22/sin
2/sin12
2
2
2
kernelFejer the
2/sin
2/sin22
22
T
THT
where
Thus
dfHT
xVar T
2
2
1
1
1
( )T
s T
T rVar x r
T
Compare with
Basic Property of the Fejer kernel:
If g(•) is a continuous function then :
0
2
0 4lim
gdgHTT
Thus
1
2002limk kT
kkfxVarT
43210-1-2-3-40
10
20
T = 2
The "Fejer " Kernel
43210-1-2-3-40
10
20
T = 5
43210-1-2-3-40
10
20
T =10
The sample autocovariance function
The sample autocovariance function is defined by:
hT
thttx xxxx
hThC
1
1
where
hT
thttx xx
hThC
1
1
T
s
T
ttsst xxa
1 1
otherwise0
,,1 and ,,1 if2
1ThthtshTthts
hTast
or if is known
where
hT
thttx xx
hThC
1
1
T
s
T
ttsst xxa
1 1
otherwise0
,,1 and ,,1 if2
1ThthtshTthts
hTast
or if is known
Theorem Assume is known and the time series is normal, then:
E(Cx(h))= (h),
hCVar x
12
1
11
T h
r T h
rr r h r h
T h T h
ddffhHhT hT
2cos
1 22
and
kChCCov xx ,
2sin
2))((
sin2
))((sin
))((
1
2
kThT
kThT
ddff
kh
2
)(cos
2
)(cos
Proof
Assume is known and the the time series is normal, then:
hxxEhT
hCEhT
thttx
1
1
and
T
s
T
ttsstx xxaVarhCVar
1 1
T
s
T
t
T
s
T
ttsst ttssaa
1 1 1' 1''' ''2
stssst ',',
T
s
T
t
T
s
T
ttsst ttssaa
1 1 1' 1''' ''2
0',', since stssst
hT
t
hT
t
tttthT 1 1'
2)]'()'(2[
)(4
1
hT
t
T
ht
tthththT 1 1'
2)]'()'(2[
)(4
1
T
ht
hT
t
tthththT 1 1'
2)]'()'(2[
)(4
1
T
ht
T
ht
tttthT 1 1'
2)]'()'(2[
)(4
1
)]()(][[)(
1 1
)1(
22
hrhrrrhThT
hT
hTr
1
)1(
2 )()(||
1)(
1 hT
hTr
hrhrrhT
r
hT
and
ddffAhCVar x
2),(2
T
s
T
t
T
s
T
ttsst stssstaa
1 1 1' 1''' ',',
ddffA2
),(2
.0',', since stssst
where
T
s
T
t
stisteaA
1 1
,
T
ht
httihT
t
htti ehT
ehT 11 2
1
2
1
hT
t
tihihi
ehT
ee
12
2sin
2sin
22/1
hT
ehT
ee hTihihi
since
1
1
1
i
Tii
T
t
ti
e
eee
2/2/
2/2/2/1
ii
TTiTi
ee
eee
2/sin
2/sin2/1
T
e Ti
hence
2sin
2sin
2, 2/1
hT
ehT
eeA hTi
hihi
2sin
2sin
22/1
2/2/
hT
ehT
ee Tihhihi
Thus
2sin
2sin
2/cos2/1
hT
hT
he Ti
2sin
2sin
2/cos,
2
2
2
22
hT
hT
hA
2
2
2
2/cos
hTHhT
h
and
hCVar x
hhddffhHhT hT
2/cos
1 22
Finally
hhddffBAkChCCov xx
,,,
Where
2sin
2sin
2/cos2/1
kT
kT
ke Ti
T
s
T
t
stistebB
1 1
,
Thus
,, BA
kThT
kh 2/cos2/cos
2sin
2sin
2sin
2
kThT
Expectations, Variances and Covariances of Linear forms
Summary
Theorem Let {xt:t T} be a weakly stationary time series.Let
Then
and
where
and Sr = {1,2, ..., T-r}, if r ≥ 0, Sr = {1- r, 2 - r, ..., T} if r ≤ 0.
T
ttt xcL
1
T
ttcLE
1
1
11 1
)()(T
Tr Sssrs
T
s
T
tts
r
ccrstccLVar
2( )C f d
T
t
tit ecC
1
)(
Theorem Let {xt:t T} be a weakly stationary time series.
Let and
T
tttc xcL
1
T
tttb xbL
1
Then
T
s
T
ttsbc stbcLLCov
1 1
)(,
1
1
)(T
Tr Sssrs
r
bcr
dfBC )()(
T
t
tit ecC
1
)(
T
t
tit ebB
1
)( where and
Expectations, Variances and Covariances of Quadratic forms
Theorem Let {xt:t T} be a weakly stationary time series.
Let
Then
T
s
T
ttsst xxaQ
1 1
T
s
T
t
T
Tr Strssst
r
arstaQE1 1
1
1,
dfAdfeaT
s
T
t
stist ),(
1 1
and
T
s
T
t
T
s
T
ttsst ttssaaQVar
1 1 1' 1''' ''2
1
1
1
1,2
T
Tr
T
Tp Ss Stptrsst
r r
aapr
1
1
1
1,, ,,
T
Tr
T
Tp Ss Stptrsrsst pstrstaa
r r
stssst ',',
ddffeaT
s
T
t
stist
2
1 1
2
T
s
T
t
T
s
T
ttsst stssstaa
1 1 1' 1''' ',',
ddffA2
),(2
T
s
T
t
T
s
T
ttsst stssstaa
1 1 1' 1''' ',',
T
s
T
t
stisteaA
1 1
, where
and Sr = {1,2, ..., T-r}, if r ≥ 0,
Sr = {1- r, 2 - r, ..., T} if r ≤ 0,
(h,r,s) = the fourth order cumulant
= E[(xt - )(xt+h - )(xt+r - )(xt+s - )]
- [(h)(r-s)+(r)(h-s)+(s)(h-r)]
Note (h,r,s) = 0 if {xt:t T}is Normal.
Theorem Let {xt:t T} be a weakly stationary time series.
Let
Then
T
s
T
ttsst xxaQ
1 1
T
s
T
ttsst xxbP
1 1
T
s
T
t
T
s
T
ttsst ttssbaPQCov
1 1 1' 1''' ''2,
stssst ',',
ddffBA ),(),(2
T
s
T
t
T
s
T
ttsst stssstba
1 1 1' 1''' ',',
Estimation of the spectral density function
The Discrete Fourier Transform
Let x1,x2,x3, ...xT denote T observations on a univariate one-dimensional time series with zero mean (If the series has non-zero mean one uses in place of xt).
Also assume that T = 2m +1 is odd.
Then
xxt
m
kkkkkt tbta
ax
1
0 )sin()cos(2
m
mkkkkk tbta )sin()cos(
2
1
where
T
tt
T
tktk t
T
kx
Ttx
Ta
11
2cos2
)cos(2
T
tt
T
tktk t
T
kx
Ttx
Tb
11
2sin2
)sin(2
with k = 2k/T and k = 0, 1, 2, ... , m.
The Discrete Fourier transform:
T
tkkttk ibaxcX
1
tic kt expT
2 where
tit kk sincosT
2
k = 0, 1,2, ... ,m.
Note:
m
kkkkkt tbta
ax
1
0 )sin()cos(2
m
mkkkkk tbtai )cos()sin(
2
1
m
mkkkkk tbta )sin()cos(
2
1
m
mkkkkk tbta )sin()cos(
2
1
Since
m
mkkkkk tbta )cos()sin(
m
mkk
T
tkt ttx
T)sin()'cos(
2
1''
)cos()'sin(2
1'' ttx
T k
T
tkt
m
mk
T
tkt ttx
T 1'' )'sin(
2
0
'2sin
2
1''
T
t
m
mkt T
ttkx
T
Thus
m
kkkkk tbtai
1
)cos()sin(2
1
m
mkkkkkt tbtax )sin()cos(
2
1
m
mkkkkk tbta )sin()cos(
2
1
)sin()cos(2
1titiba kk
m
mkkk
m
mkkkk titX sincos
2
1
m
mkkk tiX exp
2
1
m
mkkk tiX exp
2
1
Summary:The Discrete Fourier transform
T
tkkttk ibaxcX
1
tikt
ketic T
2exp
T
2 where
tit kk sincosT
2 k = 0, 1,2, ... ,m.
m
mk
tik
m
mkkkt
keXtiXx 2
1exp
2
1 and
Theorem
with kk/T)
E[Xk] = 0
dfkXXCov hkThk ,,
2sin
2sin
2sin
2sin
2 and
2
TT
TkT
with kk/T) and hh/T)
dfHT
XVar kTk
22
kernelFejer the
2/sin
2/sin22
22
T
THT
where
Proof Note
titec kkti
tk sincos
T
2
T
2 where
T
tttk xcX
1
Thus
0 since 02
1
μeT
XET
t
tik
k
kXVar
dfCdfecT
t
tit
22
1
)(
T
t
ti keT 1
2
kii
TiTiTi
ee
eee
T
with
22/2/
2/2/2/1
kTi T
eT
with 2/sin
2/sin2 2/1
T
t
titiT
t
tit ee
TecC k
11
2
1
12
k
k
k
i
Tii
e
ee
T
Thus
2
1
2
T
t
titecC
2/sin
2/sin2
2/sin
2/sin2 2/12/1
T
eT
Te
TTiTi
T
HT
TkT
k
k
2
2
2
2
2 2
2/sin
2/sin2
kernelFejer the
2/sin
2/sin22
22
T
THT
where
Thus
dfHT
XVar kTk
22
Also
dfBCXXCov hk ,
with =2(k/T)+
T
t
tiT
t
tTki eT
eT
C11
])/(2[ 22
)2/sin(
]2/)sin[(2 2/1
T
eT
Ti
T
t
tiT
t
tThi eT
eT
B11
])/(2[ 22
2/sin
2/sin2 2/1
T
eT
Ti
with =2(h/T)+
Thus
BC)2/sin(
)2/sin(22
T
T
2/sin
2/sin
T
dfkXXCov hkThk ,,
2sin
2sin
2sin
2sin
2 where
2
TT
TkT
and
Defn: The Periodogram:
2
1
2
1
)cos()sin(2 T
tkt
T
tktkT txtx
TI
222
222 kkkkk XT
XXT
baT
k = 0,1,2, ..., m
with k = 2k/T and k = 0, 1, 2, ... , m.
Periodogram for the sunspot data
0
10000
20000
30000
0 0.5 1 1.5 2 2.5 3
note:
2
1
2
1
)cos()sin(2 T
tkt
T
tktkT txtx
TI
T
tk
T
skst stxx
T 1 1
)sin()sin(2
T
tk
T
skst stxx
1 1
)cos()cos(
T
t
T
skst stxx
T 1 1
)cos(2
1
1 1
)cos(2 T
Th
hT
tkhtt hxx
T
1
1
2 cos( )T
x kh T
C h h
1
1
2 0 2 cos( )T
x x kh
C C h h
1
1
2 k
Th
xh T
e C h
Theorem
dfHIE kTkT )(2
22
)(
dfHIVar kTkT
2
)(,
dfk kkT
2
)(,
dfk kkT
2kTIE
hTkT IICov ,
2
)(,4
dfk hkT
sin sin
2 2 2where ,
sin sin2 2
T
T T
kT
2
)(,4
dfk hkT
In addition: )(4lim kkT
TfIE
)(16lim 22kkT
TfIVar
0,lim hTkT
TIICov
If k ≠ 0
If k ≠ h
Proof Note
stistist
kk eea
T
1
T
1 where
Let
kTI
T
s
T
ttsst xxa
1 1
T
s
T
t
stisteaA
1 1
T
s
T
t
stistiT
s
T
t
stisti eeee kk
1 11 1 T
1
T
1
1
1
Because 2 cos( ) and cos2
ix ixT
T k x kh T
e eI C h h x
T
s
T
t
ststiT
s
T
t
ststi kk ee1 11 1 T
1
T
1
T
t
tiT
s
si kk ee11T
1
T
t
tiT
s
si kk ee11T
1
kkTkkTsTi kke ,,
T
1 2/1
A
kkTkkT kkA ,,T
1,
22
2
1
2
1kTkT HH
dfHH kTkT )(2
1 22
2( )T kH f d
dfAIE kT )(,
Recall
Basic Property of the Fejer kernel:
If g(•) is a continuous function then : 0
2
0 4lim
gdgHTT
2and ( )T k T kE I H f d
2
Thus lim lim ( )T k T kT T
E I H f d
4 ( ) 4 ( )k kf f
The remainder of the proof is similar
Consistent Estimation of the Spectral Density function f()
Smoothed Periodogram Estimators
Defn: The Periodogram:
2
1
2
1
)cos()sin(2 T
tkt
T
tktkT txtx
TI
222
222 kkkkk XT
XXT
baT
k = 0,1,2, ..., m
Properties: )(4lim kkT
TfIE
)(16lim 22kkT
TfIVar
0,lim hTkT
TIICov
If k ≠ 0
If k ≠ h
Spectral density Estimator
2
1
2
1
)cos()sin(2
1
4
1ˆT
tkt
T
tktkT txtx
TIf
Properties:
)(4
1lim kkTT
fIE
)(4
1lim 2
kkTT
fIVar
If k ≠ 0
The second properties states that:
is not a consistent estimator of f():
kTk If
4
1)(ˆ
Periodogram Spectral density Estimator
2
1
2
1
)cos()sin(2
1
4
1ˆT
tkt
T
tktkT txtx
TIf
Properties: )(4
1lim kkTT
fIE
)(4
1lim 2
kkTT
fIVar
If k ≠ 0
The second property states that:
is not a consistent estimator of f():
kTk If
4
1)(ˆ
Asymptotically unbiased
Examples of using packagesSPSS, Statistica
Example 1 – Sunspot data
0
50
100
150
1770 1790 1810 1830 1850 1870
Using SPSSOpen the Data
Select Graphs-> Time Series - > Spectral
The following window appears
Select the variable
Select the Window
Choose the periodogram and/or spectral density
Choose whether to plot by frequency or period
0.0 0.1 0.2 0.3 0.4 0.5
Frequency
-3.679E-1
2.203E4
5.987E4
Per
iod
og
ram
Periodogram of no by Frequency
2.718E0 7.389E0 2.009E1 5.46E1 1.484E2
Period
-3.679E-1
2.203E4
5.987E4
Per
iod
og
ram
Periodogram of no by Period
Periodogram Spectral density Estimator
2
1
2
1
)cos()sin(2
1
4
1ˆT
tkt
T
tktkT txtx
TIf
Properties: )(4
1lim kkTT
fIE
)(4
1lim 2
kkTT
fIVar
If k ≠ 0
The second property states that:
is not a consistent estimator of f():
kTk If
4
1)(ˆ
Asymptotically unbiased
Smoothed Estimators of the spectral density
The Daniell Estimator
d
drrkTk
dT I
df
]12[4
1)(ˆ )(
2
1
2
1
)cos()sin(2 T
tkt
T
tktkT txtx
TI
Properties
d
drrkk
dT f
dfE
]12[
1)(ˆ )(
d
drrkk
dT f
dfVar 2
2)(
]12[
1)(ˆ
hkkd
Tkd
T ffd
khdffCov
2)()(
]12[
12)(ˆ),(ˆ
12 if dkh
1.
2.
3.
• Now let T ∞, d ∞ such that d/T 0. Then we obtain asymptotically unbiased and consistent estimators, that is
ffE kd
TdT
)(ˆlim )(
,
.0)(ˆlim )(
,
kd
TdT
fVar
• Choosing the Daniell option in SPSS
0.0 0.1 0.2 0.3 0.4 0.5
Frequency
2.009E1
5.46E1
1.484E2
4.034E2
1.097E3
2.981E3
8.103E3
2.203E4
5.987E4
1.628E5
Den
sity
Window: Unit (5)
Spectral Density of no by Frequencyk = 5
2.718E0 7.389E0 2.009E1 5.46E1 1.484E2
Period
2.009E1
5.46E1
1.484E2
4.034E2
1.097E3
2.981E3
8.103E3
2.203E4
5.987E4
1.628E5
Den
sity
Window: Unit (5)
Spectral Density of no by Periodk = 5
0.0 0.1 0.2 0.3 0.4 0.5
Frequency
1.484E2
4.034E2
1.097E3
2.981E3
8.103E3
2.203E4
5.987E4
1.628E5
Den
sity
Window: Unit (9)
Spectral Density of no by Frequencyk = 9
2.718E0 7.389E0 2.009E1 5.46E1 1.484E2
Period
1.484E2
4.034E2
1.097E3
2.981E3
8.103E3
2.203E4
5.987E4
1.628E5
De
ns
ity
Window: Unit (9)
Spectral Density of no by Period
k = 5
Other smoothed estimators
More generally consider the Smoothed Periodogram
d
drrkTdTk
dT ITrWf 2/)(
~,
)(
and
d
drdT TrW
4
12/,
TrWTrW dTdT 2/2/ ,,
where
Theorem (Asymptotic behaviour of Smoothed periodogram Estimators )
and
Let
0sstst ux
24 tuE
T
Tdlim
0lim T
dT
T
where {ut} are independent random variables with mean 0 and variance 2 with
Let dT be an increasing sequence such that
and
Then
Proof (See Fuller Page 292)
ffE kd
TT
)(~
lim )(
)(~
2/lim )(
1
2, k
dT
d
drdT
TfVarTrW
T
or 02
or 02
2
f
f
Weighted Covariance Estimators
Recall that
where
1
1
)(2T
Th
hkT
kehCI
h
hkehC )(2
1||for 0
1||for 1
1
Th
ThxxThC
hT
ttht
The Weighted Covariance Estimator
where {wm(h): h = 0, ±1,±2, ...} are a sequence of weights such that:
i) 0 ≤ wm(h) ≤ wm(0) = 1
ii) wm(-h) = wm(h)
iii) wm(h) = 0 for |h| > m
h
hmk
wmT
kehChwf
)(
2
1)(ˆ
,
The Spectral Window for this estimator is defined by:
i) Wm() = Wm(-)
ii)
h
himm ehwW
2
1
Properties :
1
dWm
also (Using a Reimann-Sum Approximation)
= the Smoothed Periodogram Estimator
Note:
dIWf TkmkwmT
,
ˆ
2/
2/1,
2ˆT
TrTkmk
wmT IW
Tf
1.
Asymptotic behaviour for large T
dfWT
fE kmkwmT
2ˆ,
2.
dfWT
fVar kmkwmT
22
,
2ˆ
hwmTk
wmT ffCov ,,
ˆ,ˆ3.
2 2
dfWWT hmkm
hk for
1. Bartlett
Examples wm(h) = w(h/m)
Note:
1||0
1||||1
xfor
xifxxw
2/sin
2/sin
2
12
2
m
m
mWm
2. Parzen
w(x) = 1 -2 a + 2a cos(x)
otherwise0
1||2/1 if|]|1[2
2/1|| if||6613
32
xx
xxx
xw
3. Blackman-Tukey
with a = 0.23 (Hamming) , a = 0.25 (Hanning)
0.0 0.1 0.2 0.3 0.4 0.5
Frequency
2.009E1
5.46E1
1.484E2
4.034E2
1.097E3
2.981E3
8.103E3
2.203E4
5.987E4
1.628E5
Den
sity
Window: Unit (5)
Spectral Density of no by Frequency
Daniell
0.0 0.1 0.2 0.3 0.4 0.5
Frequency
5.46E1
1.484E2
4.034E2
1.097E3
2.981E3
8.103E3
2.203E4
5.987E4
1.628E5
Den
sity
Window: Tukey (5)
Spectral Density of no by Frequency
Tukey
0.0 0.1 0.2 0.3 0.4 0.5
Frequency
5.46E1
1.484E2
4.034E2
1.097E3
2.981E3
8.103E3
2.203E4
5.987E4
1.628E5
Den
sity
Window: Parzen (5)
Spectral Density of no by Frequency
Parzen
0.0 0.1 0.2 0.3 0.4 0.5
Frequency
5.46E1
1.484E2
4.034E2
1.097E3
2.981E3
8.103E3
2.203E4
5.987E4
1.628E5
Den
sity
Window: Bartlett (5)
Spectral Density of no by Frequency
Bartlett
1.
Approximate Distribution and Consistency
dfWT
fE kmkwmT
2ˆ,
2.
dfWT
fVar kmkwmT
22
,
2ˆ
hwmTk
wmT ffCov ,,
ˆ,ˆ3.
2 2
dfWWT hmkm
hk for
1.
Note: If Wm() is concentrated in a "peak" about = 0 and f() is nearly constant over its width, then
2.
kk
wmT ffE ,
ˆ
dWT
ffVar mkkwmT
22
,
2ˆ
and
Confidence Limits in Spectral Density Estimation
1.
Satterthwaites Approximation:
2.
where c and r are chosen so that
2,
ˆrk
wmT cf
crcEfE rkwmT
2
,ˆ
rccVarfVar rkwmT
22, 2ˆ
Thus
= The equivalent df (EDF)
k
wmT
kwmT
fE
fVarc
,
,
ˆ2
ˆ
k
wmT
kwmT
fVar
fEr
,
2
,
ˆ
ˆ2
and
kk
wmT ffE ,
ˆ
dWT
ffVar mkkwmT
22
,
2ˆand
Now
Thus
k
wmT
kwmT
fE
fVarc
,
,
ˆ2
ˆ
k
wmT
kwmT
fVar
fEr
,
2
,
ˆ
ˆ2
dW
T
m
2
rfdWT
f kmk /2
Then a [1- 100 % confidence interval for f() is:
Confidence Limits for The Spectral Density function f():
Let and denote the upper and lower critical values for the Chi-square distribution with r d.f. i.e.
22/, r
22/1, r
2/22/1,
222/,
2 rrrr PP
22/1,
,
22/,
, )(ˆ)(
)(ˆ
r
wmT
r
wmT fr
ffr
Estimation of the spectral density function
Summary
cos( ) i hh h f d e f d
0
1 10 cos( )
2 h
f h h
The spectral density function, f() The spectral density function, f(x), is a symmetric function defined on the interval [-,] satisfying
and
1cos( )
2
1
2
h
i h
h
h h
h e
Using cos cos
Using cos sin and sin sinie i
Periodogram Spectral density Estimator
2 2
1 1
1 1ˆ sin( ) cos( )4 2
T T
k T k t k t kt t
f I x t x tT
Properties: )(4
1lim kkTT
fIE
)(4
1lim 2
kkTT
fIVar
If k ≠ 0
The second property states that:
is not a consistent estimator of f():
kTk If
4
1)(ˆ
Asymptotically unbiased
1
1
1
1
Note 2 cos( )
2 k
T
T k x kh T
Ti h
xh T
I C h h
C h e
1
1
1
1
1ˆand 4
1 cos( )
2
1
2k
k T k
T
x kh T
Ti h
xh T
f I
C h h
C h e
Smoothed Estimators of the spectral density
Smoothed Periodogram Estimators
d
drrkTdTk
dT ITrWf 2/)(
~,
)(
and
d
drdT TrW
4
12/,
TrWTrW dTdT 2/2/ ,,
where
The Daniell Estimator
( ) 1ˆ ( )4 [2 1]
dd
T k T k rr d
f Id
,
1/ 2
4 [2 1]T dW r Td
The Weighted Covariance Estimator
where {wm(h): h = 0, ±1,±2, ...} are a sequence of weights such that:
i) 0 ≤ wm(h) ≤ wm(0) = 1
ii) wm(-h) = wm(h)
iii) wm(h) = 0 for |h| > m
,
1ˆ ( ) ( )2
kw i hT m k m
h
f w h C h e
1. Bartlett
Choices for wm(h) = w(h/m)
1||0
1||||1
xfor
xifxxw
2. Parzen
w(x) = 1 -2 a + 2a cos(x)
2 3
3
1 6 6 | | if | | 1/ 2
2[1 | |] if 1/ 2 | | 1
0 otherwise
x x x
w x x x
3. Blackman-Tukey
with a = 0.23 (Hamming) , a = 0.25 (Hanning)
The Spectral Window for this estimator is defined by:
i) Wm() = Wm(-)
ii)
h
himm ehwW
2
1
Properties :
1
dWm
also (Using a Reimann-Sum Approximation)
= the Smoothed Periodogram Estimator
Note:
dIWf TkmkwmT
,
ˆ
1 / 2
,1 / 2
2ˆT
wT m k m k T
r T
f W IT
1.
Approximate Distribution and Consistency
dfWT
fE kmkwmT
2ˆ,
2.
dfWT
fVar kmkwmT
22
,
2ˆ
hwmTk
wmT ffCov ,,
ˆ,ˆ3.
2 2
dfWWT hmkm
hk for
1.
Note: If Wm() is concentrated in a "peak" about = 0 and f() is nearly constant over its width, then
2.
kk
wmT ffE ,
ˆ
dWT
ffVar mkkwmT
22
,
2ˆ
and
Then a [1- 100 % confidence interval for f() is:
Confidence Limits for The Spectral Density function f():
Let and denote the upper and lower critical values for the Chi-square distribution with r d.f. i.e.
22/, r
22/1, r
2/22/1,
222/,
2 rrrr PP
22/1,
,
22/,
, )(ˆ)(
)(ˆ
r
wmT
r
wmT fr
ffr
and
kk
wmT ffE ,
ˆ
dWT
ffVar mkkwmT
22
,
2ˆand
Now
Thus
k
wmT
kwmT
fE
fVarc
,
,
ˆ2
ˆ
k
wmT
kwmT
fVar
fEr
,
2
,
ˆ
ˆ2
dW
T
m
2
rfdWT
f kmk /2
Then a [1- 100 % confidence interval for f() is:
Confidence Limits for The Spectral Density function f():
Let and denote the upper and lower critical values for the Chi-square distribution with r d.f. i.e.
22/, r
22/1, r
2/22/1,
222/,
2 rrrr PP
22/1,
,
22/,
, )(ˆ)(
)(ˆ
r
wmT
r
wmT fr
ffr
and
k
wmT
kwmT
fE
fVarc
,
,
ˆ2
ˆ
2m
Tr
W d
rfdWT
f kmk /2