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Page 1: Cross-spectral Analysis

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

Page 2: Cross-spectral Analysis

Ensenada de la Paz

Page 3: Cross-spectral Analysis

Year Day

Current velocity in Ensenada de La Paz

Page 4: Cross-spectral Analysis

Current velocity and water level (red) in Ensenada de La Paz

Year Day

Page 5: Cross-spectral Analysis

Current velocity and water level (red) in Ensenada de La Paz

Year Day

Page 6: Cross-spectral Analysis

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 7: Cross-spectral Analysis

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

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

Page 9: Cross-spectral Analysis

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

Page 10: Cross-spectral Analysis

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

Page 11: Cross-spectral Analysis

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

Page 12: Cross-spectral Analysis

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

Page 13: Cross-spectral Analysis

Current velocity and water level (red) in Ensenada de La Paz

Year Day

Page 14: Cross-spectral Analysis

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

Page 15: Cross-spectral Analysis

221

2 1

kyykxx

kxykxy

fGfG

fGf

2

2

= 0.1, 0.05, 0.01

xy

xyk GRe

GImtanaf

Page 16: Cross-spectral Analysis

JAMES RIVER DATA

Page 17: Cross-spectral Analysis

kyykxx

kxykxy

fGfG

fGf

2

2

xy

xyk GRe

GImtanaf

Coh

eren

ce 2

Pha

se (

deg)

95%

99%

Window size= 10

Page 18: Cross-spectral Analysis

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%99%

Window size= 24

xy

xyk GRe

GImtanaf

Page 19: Cross-spectral Analysis

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

Page 20: Cross-spectral Analysis

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%

99%

Window size= 10

xy

xyk GRe

GImtanaf

Page 21: Cross-spectral Analysis

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%

99%

Window size= 10

xy

xyk GRe

GImtanaf

Page 22: Cross-spectral Analysis

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%99%

Window size= 24

xy

xyk GRe

GImtanaf

Page 23: Cross-spectral Analysis

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%99%

Window size= 24

xy

xyk GRe

GImtanaf

Page 24: Cross-spectral Analysis