multivariate time series analysis
DESCRIPTION
Multivariate Time Series Analysis. Definition :. Let { x t : t T } be a Multivariate time series. m ( t ) = mean value function of { x t : t T } = E [ x t ] for t T . - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/1.jpg)
Multivariate Time Series Analysis
![Page 2: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/2.jpg)
Let {xt : t T} be a Multivariate time series.
Definition:
(t) = mean value function of {xt : t T}
= E[xt] for t T.
(t,s) = Lagged covariance matrix of {xt : t T} = E{[ xt - (t)][ xs - (s)]'} for t,s T
![Page 3: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/3.jpg)
Definition:
The time series {xt : t T} is stationary if the joint distribution of
is the same as the joint distribution of
for all finite subsets t1, t2, ... , tk of T and all choices of h.
kttt xxx ,,,21
hththt k xxx ,,,21
![Page 4: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/4.jpg)
In this case then for t T.
and(t,s) = E{[ xt - ][ xs - ]'}
= E{[ xt+h - ][ xs+h - ]'}
= E{[ xt-s - ][ x0 - ]'}
= (t - s) for t,s T.
μμ )()( itxEt
![Page 5: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/5.jpg)
Definition:The time series {xt : t T} is weakly stationary if :
for t T.and
(t,s) = (t - s) for t, s T.
μμ )(t
![Page 6: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/6.jpg)
In this case
(h) = E{[ xt+h - ][ xs - ]'}
= Cov(xt+h,xt )
is called the Lagged covariance matrix of the process {xt : t T}
![Page 7: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/7.jpg)
The Cross Correlation Function and the Cross Spectrum
![Page 8: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/8.jpg)
Note: ij(h) = (i,j)th element of (h),
and is called the cross covariance function of
jht
it xx ,cov
. and js
it xx
00 jjii
ijij
hh
is called the cross correlation function of . and j
si
t xx
![Page 9: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/9.jpg)
Definitions:
. and js
it xxis called the cross spectrum of
ki
kijij ekf
21i)
Note: since ij(k) ≠ ij(-k) then fij() is complex.
If fij() = cij() - i qij() then cij() is called the Cospectrum (Coincident spectral density) and qij() is called the quadrature spectrum
ii)
![Page 10: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/10.jpg)
If fij() = Aij() exp{iij()} then Aij() is called the Cross Amplitude Spectrum and ij() is called the Phase Spectrum.
iii)
![Page 11: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/11.jpg)
Definition:
is called the Spectral Matrix
pppp
p
p
ijpp
fff
ffffff
f
21
22221
11211
F
![Page 12: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/12.jpg)
The Multivariate Wiener-Khinchin Relations (p-variate)
and
h
hi
ppppeh
ΣF
21
dehpp
hi
ppFΣ
![Page 13: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/13.jpg)
Lemma:
i) Positive semidefinite:a*F()a ≥ 0 if a*a ≥ 0, where a is any complex vector.
ii) Hermitian:F() = F*() = the Adjoint of F() = the complex conjugate transpose of F(). i.e.fij() = .
Assume that
||
hij h
Then F() is:
![Page 14: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/14.jpg)
Corrollary:The fact that F() is positive semidefinite also means that all square submatrices along the diagonal have a positive determinant
0
jjji
ijii
ffffHence
jiijjjii ffff and
jjiiijijij fffff 2*or
![Page 15: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/15.jpg)
Definition:
= Squared Coherency function
jjii
ijij ff
fK
2
2
12 ijKNote:
![Page 16: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/16.jpg)
Definition:
ii
ijij f
f
. and with associated jt
it xxfunctionTransfer
![Page 17: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/17.jpg)
Applications and Examples of Multivariate Spectral Analysis
![Page 18: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/18.jpg)
Example I - Linear Filters
![Page 19: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/19.jpg)
denote a bivariate time series with zero mean.
Let
t = ..., -2, -1, 0, 1, 2, ...
Tt
yx
t
t :
sstst xay
Suppose that the time series {yt : t T} is constructed as follows:
![Page 20: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/20.jpg)
The time series {yt : t T} is said to be constructed from {xt : t T} by means of a Linear Filter.
httyy yyEh
'''
sshts
ssts xaxaE
ssht
sstss xxaaE '
''
s s
shtstss xxEaa'
''
![Page 21: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/21.jpg)
continuing hyy
s sss sshaa
'' '
s s
xxsshi
ss dfeaa'
''
s s
xxsshi
ss dfeaa'
''
s s
xxsisi
sshi dfeeaae
'
''
dfeaeae xxs
sis
s
sis
hi
'
''
![Page 22: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/22.jpg)
continuing hyy
dfeae xxs
sis
hi2
dfAe xxhi
2
Thus the spectral density of the time series {yt : t T} is:
xxxx
s
sisyy fAfeaf 2
2
![Page 23: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/23.jpg)
Comment A:
is called the Transfer function of the linear filter.
is called the Gain of the filter while
is called the Phase Shift of the filter.
s
siseaA
A
Aarg
![Page 24: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/24.jpg)
Also httxy yxEh
sshtst xaxE
s
shtts xxEa
sxxs sha
![Page 25: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/25.jpg)
continuing
hxy
dfeas
xxshi
s
dfea xxs
shis
dfAe xxhi
![Page 26: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/26.jpg)
Thus cross spectrum of the bivariate time series
Tt
yx
t
t :
is:
xx
sxx
sisxy fAfeaf
![Page 27: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/27.jpg)
Comment B:
= Squared Coherency function.
yyxx
xyxy ff
fK
2
2
1 2
22
xxxx
xx
fAf
fA
![Page 28: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/28.jpg)
Example II - Linear Filterswith additive noise at the output
![Page 29: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/29.jpg)
denote a bivariate time series with zero mean.
Let
t = ..., -2, -1, 0, 1, 2, ...
Tt
yx
t
t :
Suppose that the time series {yt : t T} is constructed as follows:
ts
stst vxay
The noise {vt : t T} is independent of the series {xt : t T} (may be white)
![Page 30: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/30.jpg)
httyy yyEh
shtshts
ststs vxavxaE
s
htstss s
shtstss vxEaxxEaa'
''
thts
tshts vvEvxEa
'''
hsshaa vvs s
xxss
'' '
dfedfeae vvhi
xxs
sis
hi
2
![Page 31: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/31.jpg)
continuing
hyy
dffAe vvxxhi 2
s
siseaA where
Thus the spectral density of the time series {yt : t T} is:
vvxxyy ffAf 2
![Page 32: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/32.jpg)
Also httxy yxEh
shtshtst vxaxE
htts
shtts vxExxEa
sxxs sha
![Page 33: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/33.jpg)
continuing
hxy
dfeas
xxshi
s
dfea xxs
shis
dfAe xxhi
![Page 34: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/34.jpg)
Thus cross spectrum of the bivariate time series
Tt
yx
t
t :
is:
xx
sxx
sisxy fAfeaf
![Page 35: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/35.jpg)
Thus
= Squared Coherency function.
yyxx
xyxy ff
fK
2
2
vvxxxx
xx
ffAf
fA
2
22
11
1
1 2
xx
vv
fAf
Noise to Signal Ratio
![Page 36: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/36.jpg)
Estimation of the Cross Spectrum
![Page 37: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/37.jpg)
Let
T
T
yx
yx
yx
,,,2
2
1
1
denote T observations on a bivariate time series with zero mean.If the series has non-zero mean one uses
in place of
yyxx
t
t
t
t
yx
Again assume that T = 2m +1 is odd.
![Page 38: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/38.jpg)
Then define:
and
with k = 2k/T and k = 0, 1, 2, ... , m.
T
tkt
xk
T
tkt
xk tx
Tbtx
Ta
11
)sin(2,)cos(2
T
tkt
yk
T
tkt
yk ty
Tbty
Ta
11
)sin(2,)cos(2
![Page 39: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/39.jpg)
Also
and
for k = 0, 1, 2, ... , m.
T
tkt
xk
xkk tix
TibaX
1
exp2
T
tkt
yk
ykk tiy
TibaY
1
exp2
![Page 40: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/40.jpg)
The Periodogram &
the Cross-Periodogram
![Page 41: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/41.jpg)
Also
and
for k = 0, 1, 2, ... , m.
2
1
2
1
)cos()sin(2 T
tkt
T
tktk
xxT txtx
TI
222
222 kkkxk
xk XTXXTbaT
2
1
2
1
)cos()sin(2 T
tkt
T
tktk
yyT tyty
TI
222
222 kkkyk
yk YTYYTbaT
![Page 42: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/42.jpg)
Finally
yk
yk
xk
xkkkk
xyT ibaibaTYXTI
22
yk
xk
yk
xk
yk
xk
yk
xk baabibbaaT
2
xk
xk
yk
ykkkk
yxT ibaibaTXYTI
22
yk
xk
yk
xk
yk
xk
yk
xk baabibbaaT
2
kxy
TI of conjugatecomplex
![Page 43: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/43.jpg)
Note:
and
1
1 1
)exp(2 T
Thkht
hT
ttk
xxT hixx
TI
1
1
)exp(2T
Thkxx hihC
1
1 1
)exp(2 T
Thkht
hT
ttk
yyT hiyy
TI
1
1
)exp(2T
Thkyy hihC
![Page 44: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/44.jpg)
Also
and
1
1 1
)exp(2 T
Thkht
hT
ttk
xyT hiyx
TI
1
1
)exp(2T
Thkxy hihC
1
1 1
)exp(2 T
Thkht
hT
ttk
yxT hixy
TI
1
1
)exp(2T
Thkyx hihC
![Page 45: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/45.jpg)
The sample cross-spectrum, cospectrum
& quadrature spectrum
![Page 46: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/46.jpg)
Recall that the periodogram
2
1
2
1
)cos()sin(2 T
tkt
T
tktk
xxT txtx
TI
has asymptotic expectation 4fxx().
kxy
TI Similarly the asymptotic expectation of
is 4fxy().
An asymptotic unbiased estimator of fxy() can be obtained by dividing by 4. k
xyTI
![Page 47: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/47.jpg)
The sample cross spectrum
1
1
)exp(21
21ˆ
T
Thkxyk
xyTkxy hihCIf
![Page 48: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/48.jpg)
The sample cospectrum
1
1
)cos(21ˆReˆ
T
Thkxykxykxy hihCfc
![Page 49: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/49.jpg)
The sample quadrature spectrum
1
1
)sin(21ˆImˆ
T
Thkxykxykxy hihCfq
![Page 50: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/50.jpg)
The sample Cross amplitude spectrum,
Phase spectrum &
Squared Coherency
![Page 51: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/51.jpg)
Recall
22 xyxyxy qcSpectrumAmplitudeCrossA
xy
xyxy c
qSpectrumPhase 1tan
functionCoherencySquared
ffqc
ff
fK
yyxx
xyxy
yyxx
xyxy
222
2
![Page 52: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/52.jpg)
Thus their sample counter parts can be defined in a similar manner. Namely
\ sample ˆ SpectrumAmplitudeCrossAxy
xy
xyxy c
qSpectrumPhase
ˆˆ
tan sample ˆ 1
functionCoherencySquared
ff
qc
ff
fK
yyxx
xyxy
yyxx
xy
xy
sample
ˆˆˆˆ
ˆˆ
ˆˆ
222
2
22 ˆˆ xyxy qc
![Page 53: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/53.jpg)
Consistent Estimation of the Cross-spectrum fxy()
![Page 54: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/54.jpg)
Daniell Estimator
![Page 55: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/55.jpg)
= The Daniell Estimator of the Cospectrum
d
drrkk
xyTd c
dc ˆ
121ˆ ,
= The Daniell Estimator of the quadrature spectrum
d
drrkk
xyTd q
dq ˆ
121ˆ ,
![Page 56: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/56.jpg)
Weighted Covariance Estimator
![Page 57: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/57.jpg)
hhChwc kh
xymkxy
mTw
cos21ˆ ,,
hhChwq kh
xymkxy
mTw
sin21ˆ ,,
of sequence a are ,2,1,0: where hhwm
such that weights
100 i) mm whw
hwhw mm ii)
.for 0 iii) mhhwm
![Page 58: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/58.jpg)
Again once the Cospectrum and Quadrature Spectrum have been estimated,The Cross spectrum, Amplitude Spectrum, Phase Spectrum and Coherency can be estimated generally as follows using either the
a) Daniell Estimator or b) the weighted covariance estimator
of cxy() and qxy():
![Page 59: Multivariate Time Series Analysis](https://reader033.vdocuments.site/reader033/viewer/2022061615/56816387550346895dd47455/html5/thumbnails/59.jpg)
Namely
22 ˆˆˆxyxyxy qcA
xy
xyxy c
qˆˆ
tanˆ 1
yyxx
xyxy
yyxx
xy
xy ff
qc
ff
fK ˆˆ
ˆˆˆˆ
ˆˆ
222
2
xyxyxy qicf ˆˆˆ