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Cross-spectral Analysis
Investigates the dependence or relationship between two time series.
yyxxN
LC Lk
LN
kkxy
11
1
Begin by reviewing cross-correlation function – how similar two time series x , y are
L = lagN = # of elements in time series
Covariance or cross-covariance
N
kk
N
kk
Lk
LN
kk
xy
yyN
xxN
yyxxN
L
1
2
1
2
1
11
11
11
yx
xyC
Cross-correlation
-1 < ψxy < 1
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Ensenada de la Paz
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Year Day
Current velocity in Ensenada de La Paz
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Current velocity and water level (red) in Ensenada de La Paz
Year Day
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Current velocity and water level (red) in Ensenada de La Paz
Year Day
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N
kk
N
kk
Lk
LN
kk
xy
yyN
xxN
yyxxN
L
1
2
1
2
1
11
11
11
Cross-correlation between current velocity and water level in Ensenada de La Paz
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N
kk
N
kk
Lk
LN
kk
xy
yyN
xxN
yyxxN
L
1
2
1
2
1
11
11
11
Cross-correlation between current velocity and water level in Ensenada de La Paz
![Page 8: Cross-spectral Analysis Investigates the dependence or relationship between two time series. Begin by reviewing cross-correlation function – how similar](https://reader030.vdocuments.net/reader030/viewer/2022032604/56649e5d5503460f94b554b9/html5/thumbnails/8.jpg)
N
kk
N
kk
Lk
LN
kk
xy
yyN
xxN
yyxxN
L
1
2
1
2
1
11
11
11
A generalization of:
ivuw
N
kkk
N
kkk
Lk
LN
kk
ww
w*wN
w*wN
w*wN
L
122
111
21
1
11
11
11
21
* is the complex conjugate
Complex Cross-correlation
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N
kkk
N
kkk
Lk
LN
kk
ww
w*wN
w*wN
w*wN
L
122
111
21
1
11
11
11
21
L
w1w
2
Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz
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N
kkk
N
kkk
Lk
LN
kk
ww
w*wN
w*wN
w*wN
L
122
111
21
1
11
11
11
21
Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz
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N
kkk
N
kkk
Lk
LN
kk
ww
w*wN
w*wN
w*wN
L
122
111
21
1
11
11
11
21
Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz
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Coherence spectrum (coherency)
kyykxx
kxykxy
fGfG
fGf
2
2
kykxkxy fXf*XtN
fG2
kxkxkxx fXf*XtN
fG2
kykykyy fXf*XtN
fG2
221
2 1 = 0.1, 0.05, 0.01
X is the Fourier Transform
< > = running mean
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Current velocity and water level (red) in Ensenada de La Paz
Year Day
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cpd
221
2 1
2
kyykxx
kxykxy
fGfG
fGf
2
2
= 0.1, 0.05, 0.01
Coh
eren
ce s
quar
ed
window size= 2
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221
2 1
kyykxx
kxykxy
fGfG
fGf
2
2
= 0.1, 0.05, 0.01
xy
xyk GRe
GImtanaf
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JAMES RIVER DATA
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kyykxx
kxykxy
fGfG
fGf
2
2
xy
xyk GRe
GImtanaf
Coh
eren
ce 2
Pha
se (
deg)
95%
99%
Window size= 10
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kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%99%
Window size= 24
xy
xyk GRe
GImtanaf
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kyykxx
kxykxy
fGfG
fGf
2
2
kykxkxy fXf*XtN
fG2
kxkxkxx fXf*XtN
fG2
kykykyy fXf*XtN
fG2
221
2 1
kyykxx
kxykxy
fGfG
fGf
2
2
kykx fXfX
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kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%
99%
Window size= 10
xy
xyk GRe
GImtanaf
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kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%
99%
Window size= 10
xy
xyk GRe
GImtanaf
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kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%99%
Window size= 24
xy
xyk GRe
GImtanaf
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kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%99%
Window size= 24
xy
xyk GRe
GImtanaf
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