Wavelet Transform and Wavelet Transform and Some Applications in Time Some Applications in Time

Series Analysis and Series Analysis and ForecastingForecasting

A little bit of history….A little bit of history….

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 and mechanics at École 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 Académie des Sciences in and served as secretary to the mathematical section. Published his prize winning essay Théorie analytique de la chaleur.

1824: Credited with the discovery that gases in the atmosphere might increase the surface temperature of the Earth (sur les températures du globe terrestre et des espaces planétaires ). He established the concept of planetary energy balance. Fourier called infrared radiation "chaleur obscure" or "dark heat“.

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

Windowed (Short-Time) Fourier Transform (1946)

James W. Cooley and John W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Math. Comput. 19, 297–301 (1965).

Independently re-invented an algorithm known to Carl Friedrich 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 principles underlying the science of holography, published his now-famous paper “Theory of Communication.”2

C. F. Gauss

Stephane Mallat, Yves Meyer

Jean Morlet

Presented the concept of wavelets (ondelettes) in its present theoretical form when he was working at the Marseille Theoretical Physics Center (France). (Continuous Wavelet Transform)

(Discrete Wavelet Transform) The main algorithm dates back to the work of Stephane Mallat in 1988. Then joined Y. Meyer.

Motivation….

Earthquake

Fourier Transform

1

0

/21ˆN

k

Ninkin ef

Nf

2,.....,1

2NNn

dfdttf22 ˆ

dtetff ti 2ˆ

deftf ti2ˆ

Fourier Transform

Inverse Fourier Transform

Parseval Theorem

Discrete Fourier Transform

Phase!!!

Limitations???Non-Stationary Signals…Fourier does not provide information about when different periods(frequencies) where important: No localization in time

has the same support for every and , but the number of cycles varies with frequency.

dttgtfuGf u,,

Windowed (Short-Time) Fourier Transform

tiu eutgtg

2

,

tg u, u

Estimates locally around , the amplitude of a sinusoidal wave of frequency

241 2ueug D. Gabor

u ug Function with local support.

Limitations??

Fixed resolution.

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

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

Selection of determines and . ug g g

ggt ˆ00 Localization:

Example….x(t) = cos(2π10t) for               x(t) = cos(2π25t) for                  x(t) = cos(2π50t) for                  x(t) = cos(2π100t) for                 

Wavelet Transform

dtttfuW u, , 0

uttu

1,

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

0dtt

dtt

12dttMother wavelet properties

0

2

,

0

2 ,

0ˆ

ˆ

ˆ

,

d

d

t

t

t

0,10,1

0,1

ˆ0ˆ

0t

t,

t,ˆ

t,ˆ

0,1ˆ

0ˆ0

ˆ0,1

,

t

.

Wavelet Transform

Some Continuous Wavelets

241

2

0

ee i

Morlet

uttu

1,

tiu eutgtg

2

,

241 2ueug Gabor

Torrence and Compo (1998)

Continuous Wavelet TransformFor Discrete Data

Time series

WaveletDefined as the convolution with a scaled and translated version of

DFT (FFT) of the time series

N times for each s: Slow!

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

Mallat's multiresolution framework Design method of most of the practically relevant discrete wavelet transforms (DWT)

Doppler Signal

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

Earthquake

Sun Spots

Power 9-12 years

Length of Day

Filtering (Inverse Wavelet Transform…)

Wavelet Coherency

Wavelet Cross-Spectrum Wavelet Coherency

Forecasting South-East AsiaForecasting South-East AsiaIntraseasonalIntraseasonal VariabilityVariability

Webster, P. J, and C. Hoyos, 2004: Prediction of Monsoon Rainfall and River Discharge on 15-30 day Time Scales. Bull. Amer. Met. Soc., 85 (11), 1745-1765.

Indian Monsoon: Spatial-Temporal Variability

Active and Break PeriodsActive and Break Periods1.1. Strong annual cycle. Strong spatial variability.Strong annual cycle. Strong spatial variability.2.2. Intraseasonal Variability >>> Interannual VariabilityIntraseasonal Variability >>> Interannual Variability3.3. Strong impact in India’s economyStrong impact in India’s economy

OLR Composites based on active periods.

Selection of Active phases

Regional Structure of the Monsoon Intraseasonal Variability – MISORegional Structure of the Monsoon Intraseasonal Variability – MISO

OLR Composites

Development of an empirical scheme

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

Time series are separated through identification of significant bands from wavelet analysis of the predictand (Same separation made for predictors).

Coefficients of the Multi-linear regression change are time-dependent.

PredictorsPredictorsOLR 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

PredictandsPredictands

1. Central India Precipitation. 2. Regional Precipitation 3. River Discharge

Statistical Scheme: Wavelet BandingStatistical 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

Statistical Scheme: Regression SchemeStatistical Scheme: Regression Scheme

Linear regression sets are formed between predictand and predictor and advanced in time.

20-day forecasts for Central India

Error Estimation Error Estimation

All schemes use identical predictors

Only the WB method appears to capture the intraseasonal variability

So why does WB appear to work?

Comparison of SchemesComparison of Schemes

Consider predictand made up of two 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:

With simple regressiontechnique, the wavesin the predictors (noise)that do not match the harmonics of the predictand introduce errors

Compare blue and red curves. Correlation isreasonable but signalis degraded

Regression Analysis

Filtering the predictors relative to the signature of the predictands eliminates noise.

In this simple case the forecast is perfect.

In complicated geophysical time series where coefficientsvary with time, spuriousmodes are eliminated and Bayesian statistical schemesare less ‘confused”.

Wavelet Banding