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
Ensenada de la Paz
Year Day
Current velocity in Ensenada de La Paz
Current velocity and water level (red) in Ensenada de La Paz
Year Day
Current velocity and water level (red) in Ensenada de La Paz
Year Day
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
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
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
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
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
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
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
Current velocity and water level (red) in Ensenada de La Paz
Year Day
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
221
2 1
kyykxx
kxykxy
fGfG
fGf
2
2
= 0.1, 0.05, 0.01
xy
xyk GRe
GImtanaf
JAMES RIVER DATA
kyykxx
kxykxy
fGfG
fGf
2
2
xy
xyk GRe
GImtanaf
Coh
eren
ce 2
Pha
se (
deg)
95%
99%
Window size= 10
kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%99%
Window size= 24
xy
xyk GRe
GImtanaf
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
kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%
99%
Window size= 10
xy
xyk GRe
GImtanaf
kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%
99%
Window size= 10
xy
xyk GRe
GImtanaf
kyykxx
kxykxy
fGfG
fGf
2
2 C
oher
ence
2P
hase
(de
g)
95%99%
Window size= 24
xy
xyk GRe
GImtanaf
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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