Maraun Wavelet Spectral Analysis Developments and Examples
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Transcript of Maraun Wavelet Spectral Analysis Developments and Examples
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Introduction
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Motivation
Wavelet Spectrum of White Noise Realization
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Wavelet Transformation
Continuous Wavelet Transformation
Given a time series s(t), then its CWT with respect to the wavelet gat scale a and time b reads
Wgs(t)[b, a] =
dt
1
ag
t b
a
s(t)
Inverse Transformation
Given a wavelet transformation r(b, a), then a possible inverse
transformation with respect to the reconstruction wavelet h at time treads
Mhr(b, a)[t] =
H
db da
ar(b, a)
1
ah
t b
a
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Wavelet Transformation
Properties of the Continuous Wavelet Transformation
Reproducing kernel
r(b, a) is a wavelet transformation, if and only if
r(b, a) =
0
da
a
0
db1
aPg,h b b
a,a
a r(b, a)
with Pg,h(b, a) =Wgh(t)
Any time/frequency resolved methods are subject to a
time/frequency uncertainty relation. This causes internalcorrelations, given by the reproducing kernel.
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Wavelet Transformation
Internal Correlations
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0 1 2 3 4 5 6
Fourier Spectrum of White Noise Realization
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Wavelet Transformation
Internal Correlations
Wavelet Spectrum of White Noise Realization
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Wavelet Transformation
Properties of the Continuous Wavelet Transformation
Reproducing kernel of the morlet wavelet at scales 4, 16, 64.
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Measures I
Wavelet Spectrum
WPSg(b, a) = |Wgs(t)|2
WCP1,2g (b, a) = Wgs1(t)W
g s2(t) = A1,2g (b, a)e
i1,2g
(b,a)
WCO1,2g (b, a) =
|WCP1,2g (b, a)|
(WPS1g(b, a)WPS2g(b, a))
1/2
The spectrum WPS(b, a) denotes the variance at scale a and timeb.
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Measures I
Wavelet Cross Spectrum
WPSg(b, a) = |Wgs(t)|2
WCP1,2g (b, a) = Wgs1(t)W
g s2(t) = A1,2g (b, a)e
i1,2g
(b,a)
WCO1,2g (b, a) =
|WCP1,2g (b, a)|
(WPS1g(b, a)WPS2g(b, a))
1/2
The cross spectrum WCS(b, a) denotes the fraction of covarianceat scale a and time b. The phase spectrum (a, b) denotes thephase difference between the processes at scale a and time b.
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Measures I
Wavelet Coherency
WPSg(b, a) = |Wgs(t)|2
WCP1,2g (b, a) = Wgs1(t)W
g s2(t) = A1,2g (b, a)e
i1,2g
(b,a)
WCO1,2g (b, a) =
|WCP1,2g (b, a)|
(WPS1g(b, a)WPS2g(b, a))
1/2
The coherency is defined as the normalized modulus of the crossspectrum, i.e. a value between 0 and 1, giving the degree of a linearrelationship between the two processes at scale a and time b.
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Measures I
Wavelet Coherency
WPSg(b, a) = |Wgs(t)|2
WCP1,2g (b, a) = Wgs1(t)W
g s2(t) = A1,2g (b, a)e
i1,2g
(b,a)
WCO1,2g (b, a) =
|WCP1,2g (b, a)|
(WPS1g(b, a)WPS2g(b, a))
1/2
These measures
depend on the wavelet used for the analysis,
are defined by time series, not by processes.
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Estimation I
Ideal setting to study an estimator:
Process with knownproperties
direct problem
inverse problem
Realization(i.e. time series)
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Estimation I
Actual setting in wavelet spectral analysis:
Process with unknownproperties
direct problem
inverse problem
Realization(i.e. time series)
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Part II
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M II
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Measures II
For the study of wavelet spectralestimators, we suggest to utilize
nonstationary stochastic processes
with known wavelet spectra
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M II
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Measures II
A class of nonstationary stochastic processes
Characterize a process in wavelet domain by
wavelet multipliersm(b, a)
Realization in the time domain by filtering white noise:
s(t) =Mhm(b, a)Wg(t)
with (t) N(0, 1) and (t1)(t2) = (t1 t2).
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M II
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Measures II
Equivalence class of processes
Spectrum
S(b, a) = |m(b, a)|2
Cross Spectrum
CS(b, a) = m1(b, a)m
2(b, a)
Coherency
COH(b, a) =|m1(b, a)m2(b, a)|
[(|m1|2 + |m1,i|2)(|m2|2 + |m2,i|2)]1/2,
wherem1,i andm2,i denote fractions of independent power.
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Meas res II
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Measures II
Estimators
Spectrum
Sg(b, a) = A(a|Wgs(t)|2)
Cross Spectrum
CS1,2
g (b, a) = A(aWgs1(t)W
g s2(t))
Coherency
COH1,2
g (b, a) =|CS
1,2
g (b, a)|
(S1g(b, a)S2g(b, a))
1/2
(A(.) denotes an averaging operator)
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Measures II : Example
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Measures II : Example
Example: Stochastic Chirp
Spectrum S(b, a) = |m(a, b)|2
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Estimation II : Significance Testing
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g g
Sensitivity: low -error
Every time the null hypothesis is wrong,the test detects it.
Specificity: low -error
Only if the null hypothesis is wrong,
the test detects it.
A test can never be perfectly sensitive and specific.
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Estimation II : Significance Testing : Pointwise
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Wavelet spectrum |m(b, a)|2
Given a processX(t) and a realization x(t). A pointwise testagainst a red background spectrum can be performed as follows:
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Estimation II : Significance Testing : Pointwise
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Wavelet spectrum |m(b, a)|2
Given a processX(t) and a realization x(t). A pointwise testagainst a red background spectrum can be performed as follows:
define a significance level 1
fit AR1-model to the data
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Estimation II : Significance Testing : Pointwise
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Wavelet Spectrum of NINO3 time series [Gu & Philander 1995]
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Estimation II : Significance Testing : Pointwise
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Wavelet cross spectrumm1(b, a)m
2(b, a)
No significance test is possible!
(Maraun & Kurths 2004)
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Estimation II : Significance Testing : Pointwise
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Wavelet cross spectrumm1(b, a)m
2(b, a)
No significance test is possible!
(Maraun & Kurths 2004)
Given two processesX(t) and Y(t) and two realizations x(t) and
y(t).
The true wavelet cross spectrum (WCS) measures thecovarying power. For uncorrelated processes, the WCS is
zero, for correlated processes different from zero. The WCS-estimator is always different from zero. Since WCS
is a non-normalized measure, it is not possible to decide,whether a large deviation from zero is due to high power in one
or the other of the processes, or because of covarying power.
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Estimation II : Significance Testing : Pointwise
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Wavelet Cross Spectrum of White Noise Realization and Sine Wave
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Estimation II : Significance Testing : Pointwise
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Wavelet coherency
Given two processesX(t) and Y(t) and two realizations x(t) andy(t). A pointwise test might be performed as follows:
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Estimation II : Significance Testing : Pointwise
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Wavelet coherency
Given two processesX(t) and Y(t) and two realizations x(t) andy(t). A pointwise test might be performed as follows:
Similar to wavelet spectrum, but because of the normalizationthe critical values are constant in scale and independent of theprocesses to be analyzed (Maraun & Kurths, 2004)!
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Estimation II : Significance Testing : Globally
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Global tests for wavelet spectrum and coherency
Pointwise tests are highly unspecific
General problem of time series analysis:Due to multiple testing, spurious results occur.
Specific problem of continuous wavelet analysis:Due to internal correlations, the spurious results appear as patches.
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Estimation II : Significance Testing : Globally
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Global tests for wavelet spectrum and coherency
Pointwise tests are highly unspecific
General problem of time series analysis:Due to multiple testing, spurious results occur.
Specific problem of continuous wavelet analysis:Due to internal correlations, the spurious results appear as patches.
but:
Specific advantage of continuous wavelet analysis:
Due to internal correlations, spurious patches have characteristicsizes depending on scale.
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Estimation II : Significance Testing : Globally
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Global tests for wavelet spectrum and coherency
1. Estimate the patchsize-distribution:
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Estimation II : Significance Testing : Globally
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Examples revised
Wavelet Spectrum of NINO3 time seriesDec. 13th 2005 p.38/5
Estimation II : Significance Testing : Globally
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Examples revised
Wavelet Spectrum of White NoiseDec. 13th 2005 p.39/5
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Estimation II : Significance Testing : Globally
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A specificity study
Given a stationary white noise spectrum, isthe test specific to identify that all patches
are spurious?
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Estimation II : Significance Testing : Globally
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A specificity study
Given a stationary white noise spectrum, isthe test specific to identify that all patches
are spurious?
Idea:
simulateN realizations of gaussian white noise
count the number of false positive patches in relation to therejected patches.
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Estimation II : Significance Testing : Globally
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A sensitivity study
Given a spectrum af a certain size in thetime/frequency domain superimposed by
noise, is the significance test sensitive todetect it?
Idea:
simulateN realizations of a gaussian bump of different sizesand background noise levels
check whether the significance test detects the bump
the ratio between the number Np/N of positive tests versustotal realizations gives the power of the test
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Estimation II : Significance Testing : Globally
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m(a,b), bumpwidth = 2
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Estimation II : Significance Testing : Globally
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m(a,b), bumpwidth = 12
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Estimation II : Significance Testing : Globally
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m(a,b), bumpwidth = 20
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Estimation II : Significance Testing : Globally
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bumpwidth = 12, noiselevel=0.1Dec. 13th 2005 p.44/5
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Thank you for your attention!
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Estimation II : Variance
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The wavelet scalogram is 2-distributed with 2 degrees offreedom.
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Estimation II : Variance
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Smoothing to reduce the variance
Recalling the reproducing kernel I.
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Estimation II : Variance
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Smoothing to reduce the variance (Maraun & Kurths, 2004)
moving average overscale windows in
logarithmic scales
moving average overtime windows proportio-nal to scales
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Estimation II : Variance
S hi i i di i
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Smoothing in time direction
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
half window length / scale
varia
nce
Variance as a function of the smoothing length
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Discussion
H R i f ll NAO
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Hannover Rainfall vs. NAO
Wavelet Cross Spectrum (after Markovic 2005)
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Discussion
H R i f ll NAO
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Hannover Rainfall vs. NAO
Wavelet Coherency
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