maraun wavelet spectral analysis developments and examples

7/31/2019 Maraun Wavelet Spectral Analysis Developments and Examples

1/89


2/89


3/89

Introduction

Dec. 13th 2005 p.3/5


4/89


5/89

Motivation

Wavelet Spectrum of White Noise Realization

Dec. 13th 2005 p.5/5


6/89


7/89


8/89


9/89


10/89


11/89

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

Dec. 13th 2005 p.11/5


12/89


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.

Dec. 13th 2005 p.12/5


13/89


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

Dec. 13th 2005 p.13/5


14/89


Internal Correlations

Wavelet Spectrum of White Noise Realization

Dec. 13th 2005 p.13/5


15/89


Properties of the Continuous Wavelet Transformation

Reproducing kernel of the morlet wavelet at scales 4, 16, 64.

Dec. 13th 2005 p.14/5


16/89


17/89

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.

Dec. 13th 2005 p.16/5


18/89

Measures I

Wavelet Cross Spectrum



g s2(t) = A1,2g (b, a)e

i1,2g

(b,a)

WCO1,2g (b, a) =

|WCP1,2g (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.

Dec. 13th 2005 p.16/5


19/89

Measures I

Wavelet Coherency



g s2(t) = A1,2g (b, a)e

i1,2g

(b,a)

WCO1,2g (b, a) =

|WCP1,2g (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.

Dec. 13th 2005 p.16/5


20/89

Measures I

Wavelet Coherency



g s2(t) = A1,2g (b, a)e

i1,2g

(b,a)

WCO1,2g (b, a) =

|WCP1,2g (b, a)|


1/2

These measures

depend on the wavelet used for the analysis,

are defined by time series, not by processes.

Dec. 13th 2005 p.16/5


21/89

Estimation I

Ideal setting to study an estimator:

Process with knownproperties

direct problem

inverse problem

Realization(i.e. time series)

Dec. 13th 2005 p.17/5


22/89

Estimation I

Actual setting in wavelet spectral analysis:

Process with unknownproperties

direct problem

inverse problem

Realization(i.e. time series)

Dec. 13th 2005 p.17/5


23/89


24/89

Part II

Dec. 13th 2005 p.18/5

M II


25/89

Measures II

For the study of wavelet spectralestimators, we suggest to utilize

nonstationary stochastic processes

with known wavelet spectra

Dec. 13th 2005 p.19/5

M II


26/89

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).

Dec. 13th 2005 p.20/5

M II


27/89

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.

Dec. 13th 2005 p.21/5

Meas res II


28/89

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)

Dec. 13th 2005 p.22/5

Measures II : Example


29/89

Measures II : Example

Example: Stochastic Chirp

Spectrum S(b, a) = |m(a, b)|2

Dec. 13th 2005 p.23/5


30/89


31/89


32/89


33/89


34/89

Estimation II : Significance Testing


35/89

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.

Dec. 13th 2005 p.28/5


36/89


37/89

Estimation II : Significance Testing : Pointwise


38/89

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:

Dec. 13th 2005 p.29/5


39/89



40/89

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

Dec. 13th 2005 p.29/5


41/89


42/89


43/89



44/89

Wavelet Spectrum of NINO3 time series [Gu & Philander 1995]

Dec. 13th 2005 p.30/5


45/89


46/89



47/89

Wavelet cross spectrumm1(b, a)m

2(b, a)

No significance test is possible!

(Maraun & Kurths 2004)

Dec. 13th 2005 p.32/5


48/89



49/89

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.

Dec. 13th 2005 p.32/5



50/89

Wavelet Cross Spectrum of White Noise Realization and Sine Wave

Dec. 13th 2005 p.33/5



51/89

Wavelet coherency

Given two processesX(t) and Y(t) and two realizations x(t) andy(t). A pointwise test might be performed as follows:

Dec. 13th 2005 p.34/5



52/89

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)!

Dec. 13th 2005 p.34/5


53/89


54/89

Estimation II : Significance Testing : Globally


55/89

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.

Dec. 13th 2005 p.35/5



56/89


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.

Dec. 13th 2005 p.35/5


57/89


58/89



59/89


1. Estimate the patchsize-distribution:

Dec. 13th 2005 p.37/5


60/89


61/89


62/89



63/89

Examples revised

Wavelet Spectrum of NINO3 time seriesDec. 13th 2005 p.38/5



64/89

Examples revised

Wavelet Spectrum of White NoiseDec. 13th 2005 p.39/5


65/89



66/89

A specificity study

Given a stationary white noise spectrum, isthe test specific to identify that all patches

are spurious?

Dec. 13th 2005 p.41/5



67/89

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.

Dec. 13th 2005 p.41/5


68/89



69/89

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

Dec. 13th 2005 p.42/5



70/89

m(a,b), bumpwidth = 2

Dec. 13th 2005 p.43/5


71/89



72/89


Dec. 13th 2005 p.43/5



73/89


Dec. 13th 2005 p.43/5


74/89


75/89



76/89

bumpwidth = 12, noiselevel=0.1Dec. 13th 2005 p.44/5


77/89


78/89


79/89


80/89

Thank you for your attention!

Dec. 13th 2005 p.47/5

Estimation II : Variance


81/89

The wavelet scalogram is 2-distributed with 2 degrees offreedom.

Dec. 13th 2005 p.48/5


82/89


83/89



84/89

Smoothing to reduce the variance

Recalling the reproducing kernel I.

Dec. 13th 2005 p.49/5



85/89

Smoothing to reduce the variance (Maraun & Kurths, 2004)

moving average overscale windows in

logarithmic scales

moving average overtime windows proportio-nal to scales

Dec. 13th 2005 p.50/5


86/89


S hi i i di i


87/89

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

Dec. 13th 2005 p.52/5

Discussion

H R i f ll NAO


88/89

Hannover Rainfall vs. NAO

Wavelet Cross Spectrum (after Markovic 2005)

Dec. 13th 2005 p.53/5

Discussion

H R i f ll NAO


89/89

Hannover Rainfall vs. NAO

Wavelet Coherency

Dec. 13th 2005 p.54/5

maraun wavelet spectral analysis developments and examples

Documents