wave let class

7/30/2019 Wave Let Class

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Wavelet Transform and

Some Applications in Time

Series Analysis andForecasting


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A little bit of history.


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Jean Baptiste Joseph Fourier (1768 1830)

1787: Train for priest (Left but Never married!!!).

1793: Involved in the local Revolutionary Committee.1974: Jailed for the first time.

1797: Succeeded Lagrange as chair of analysis andmechanics at cole Polytechnique.

1798: Joined Napoleon's army in its invasion of Egypt.

1804-1807: Political Appointment. Work on Heat.Expansion of functions as trigonometrical series.Objections made by Lagrange and Laplace.

1817: Elected to the Acadmie des Sciences in andserved as secretary to the mathematical section.Published his prize winning essay Thorie analytique de

la chaleur.1824: Credited with the discovery that gases in theatmosphere might increase the surface temperature ofthe Earth (sur les tempratures du globe terrestre et desespaces plantaires ). He established the concept ofplanetary energy balance. Fourier called infrared

radiation "chaleur obscure" or "dark heat.

MGP: Leibniz - Bernoulli - Bernoulli - Euler - Lagrange - Fourier Dirichlet - .


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Windowed (Short-Time) Fourier Transform (1946)

James W. Cooley and John W. Tukey, "An algorithmfor the machine calculation of complex Fourier series,"Math. Comput.19, 297301 (1965).

Independently re-invented an algorithm known to CarlFriedrich Gauss around 1805

Fast Fourier Transform

Dennis Gabor

James W. Cooley and John W. Tukey

Winner of the 1971 Nobel Prize for contributions to the principlesunderlying the science of holography, published his now-famous paperTheory of Communication.2

C. F. Gauss

Stephane Mallat, Yves Meyer

Jean Morlet

Presented the concept of wavelets (ondelettes) in its present theoretical formwhen he was working at the Marseille Theoretical Physics Center (France).

(Continuous Wavelet Transform)

(Discrete Wavelet Transform) The main algorithm datesback to the work of Stephane Mallat in 1988. Then joined Y.Meyer.
http://euler.ciens.ucv.ve/matematicos/images/gauss.gif


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Motivation.

Earthquake


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Fourier Transform

1

0

/21N

k

Nink

in efN

f

2,.....,1

2

NNn

dfdttf22

dtetff ti 2

deftf ti2

Fourier Transform

Inverse Fourier Transform

Parseval Theorem

Discrete Fourier Transform

Phase!!!


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Limitations???

Non-Stationary SignalsFourier does not provide information about when different periods(frequencies)where important: No localization in time


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has the same support for everyand , but the number of cycles varieswith frequency.

dttgtfuGf u,,

Windowed (Short-Time) Fourier Transform

tiu eutgtg

2

,

tg u, u

Estimates locally around , the amplitude ofa sinusoidal wave of frequency

2412ueug D. Gabor

u ug Function with local support.


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Limitations??

Fixed resolution.

Related to the Heisenberg uncertainty principle. The product of the standard deviationin time and frequency is limited.

The width of the windowing function relates to the how the signal is represented itdetermines whether there is good frequency resolution (frequency components closetogether can be separated) or good time resolution (the time at which frequencies change).

Selection of determines and . ug g g

ggt

00 Localization:


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Example.

x(t) = cos(210t) for





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Wavelet Transform

dtttfuW u,, 0

utt

u

1,

Gives good time resolution for high frequency events, and good frequency resolution for lowfrequency events, which is the type of analysis best suited for many real signals.

0dtt

dtt

12dttMother wavelet

properties

0

2

,

0

2

,

0

,

d

d

t

t

t

0,10,1

0,1

0

0

t

t,

t,

t,

0,1

0

0

0,1

,

t

.


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Wavelet Transform


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Some Continuous Wavelets

241

2

0

eei

Morlet

ut

tu

1,

tiu eutgtg

2

,

2412

ueug Gabor


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Torrence and Compo (1998)

Continuous Wavelet

Transform

For Discrete Data

Time series

Wavelet

Defined as the convolution with a scaledand translated version of

DFT (FFT) of the time series

N times for each s: Slow!

Using the convolution theorem,the wavelet transform is theinverse Fourier transform


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Mallat's multiresolution framework

Design method of most of thepractically relevant discrete

wavelet transforms (DWT)


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Doppler Signal


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sin(5t)+sin(10t) sin(5t) sin(10t)


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Earthquake


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Sun Spots

Power 9-12 years


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Length of Day


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Filtering (Inverse Wavelet Transform)


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Wavelet Coherency


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Wavelet Cross-SpectrumWavelet Coherency


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Forecasting South-East AsiaIntraseasonalVariability

Webster, P. J, and C. Hoyos, 2004: Prediction of Monsoon

Rainfall and River Discharge on 15-30 day Time Scales.

Bul l . Amer. Met. Soc., 85 (11), 1745-1765.


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Indian Monsoon: Spatial-Temporal Variability

Active and Break Periods1. Strong annual cycle. Strong spatial variability.2. Intraseasonal Variability >>> Interannual Variability

3. Strong impact in Indias economy


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OLR Composites based

on active periods.

Selection of Active phases

Regional Structure of the Monsoon Intraseasonal Variability MISO


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OLR Composites


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Development of an empirical scheme

Choice of the predictors: These are physicallybased and strongly related the MISO evolution(identified from diagnostic studies).

Time series are separated through identification ofsignificant bands from wavelet analysis of thepredictand (Same separation made for predictors).

Coefficients of the Multi-linear regression change

are time-dependent.


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Predictors

OLR Field Predictors

Central India

Central IO

Somali Jet Intensity

Tropical Easterly Jet IndexSea-level pressure

Central India

Surface Wind Predictors

U-comp

U, V-comp 200mb U-comp

Upper-tropospheric predictors


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Predictands

1. Central India Precipitation. 2. Regional Precipitation3. River Discharge

St ti ti l S h W l t B di


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Statistical Scheme: Wavelet Banding

Statistical scheme uses wavelets to determine

spectral structure of predictand.

Based on the definition of the bands in the

predictand, the predictors are also banded identically


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Statistical Scheme: Regression Scheme

Linear regression sets

are formed betweenpredictand and predictor

and advanced in time.


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20-day forecasts for Central India


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Error Estimation


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All schemes use identical

predictors

Only the WB method

appears to capture the

intraseasonal variability

So why does WB appear

to work?

Comparison of Schemes


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Consider predictand made up oftwo periodic modes:

F(t)2sin(t) sin(6t)

Consider two predictors:

G(t) sin(t 20) sin(3t)

H(t) sin(6t 20) sin(4t)

We can solve problem using:

A regression technique

Or

Wavelet banding then

regression

The reason wavelet banding works can be seen from a simple example:

R i A l i


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With simple regression

technique, the waves

in the predictors (noise)

that do not match the

harmonics of the

predictand introduce

errors

Compare blue and red

curves. Correlation is

reasonable but signalis degraded

Regression Analysis


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Filtering the predictors

relative to the signature ofthe predictands eliminates

noise.

In this simple case the

forecast is perfect.

In complicated geophysical

time series where coefficients

vary with time, spurious

modes are eliminated andBayesian statistical schemes

are less confused.

Wavelet Banding

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