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
x(t) = cos(225t) for
x(t) = cos(250t) for
x(t) = cos(2100t) 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