Wavelet Transform and Some Applications in Time Series Analysis and Forecasting

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Wavelet Transform and Some Applications in Time
Series Analysis and Forecasting
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A little bit of history.
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Jean Baptiste Joseph Fourier (1768 1830)
MGP Leibniz - Bernoulli - Bernoulli - Euler -
Lagrange - Fourier Dirichlet - .
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.
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Dennis Gabor
Windowed (Short-Time) Fourier Transform (1946)
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
James W. Cooley and John W. Tukey
Fast Fourier Transform
James W. Cooley and John W. Tukey, "An algorithm
for the machine calculation of complex Fourier
series," Math. Comput. 19, 297301 (1965).
Independently re-invented an algorithm known to
Carl Friedrich Gauss around 1805
C. F. Gauss
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)
Stephane Mallat, Yves Meyer
(Discrete Wavelet Transform) The main algorithm
dates back to the work of Stephane Mallat in
1988. Then joined Y. Meyer.
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Motivation.
Earthquake
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Fourier Transform
Fourier Transform
Inverse Fourier Transform
Parseval Theorem
Discrete Fourier Transform
Phase!!!
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Limitations???
Non-Stationary Signals Fourier does not provide
information about when different
periods(frequencies) where important No
localization in time
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Windowed (Short-Time) Fourier Transform
Estimates locally around , the amplitude of a
sinusoidal wave of frequency
Function with local support.
has the same support for every
and , but the number of cycles varies with
frequency.
D. Gabor
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Limitations??
Fixed resolution.
Selection of determines and .

Localization
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).
Related to the Heisenberg uncertainty principle.
The product of the standard deviation in time and
frequency is limited.
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Example.
x(t) cos(2p10t) for               x(t)
cos(2p25t) for                  x(t)
cos(2p50t) for                  x(t)
cos(2p100t) for                 
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Wavelet Transform
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.
Mother wavelet properties
.
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Wavelet Transform
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Some Continuous Wavelets
Gabor
Morlet
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Continuous Wavelet Transform For Discrete Data
Time series
Defined as the convolution with a scaled and
translated version of
Wavelet
N times for each s Slow!
Using the convolution theorem, the wavelet
transform is the inverse Fourier transform
DFT (FFT) of the time series
Torrence and Compo (1998)
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Mallat's multiresolution framework
Design method of most of the practically 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 Coherency
Wavelet Cross-Spectrum
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Forecasting South-East Asia Intraseasonal
Variability
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.
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Indian Monsoon Spatial-Temporal Variability

1. Strong annual cycle. Strong spatial variability.
2. Intraseasonal Variability gtgtgt Interannual
   Variability
3. Strong impact in Indias economy

Active and Break Periods
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Regional Structure of the Monsoon Intraseasonal
Variability MISO
OLR Composites based on active periods.
Selection of Active phases
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OLR Composites
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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.
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Predictors
Upper-tropospheric predictors
200mb U-comp
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Predictands

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

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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 between
predictand 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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Comparison of Schemes
All schemes use identical predictors Only the
WB method appears to capture the intraseasonal
variability So why does WB appear to work?
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The reason wavelet banding works can be seen from
a simple example
Consider predictand made up of two periodic
modes
Consider two predictors

• We can solve problem using
• A regression technique
• Or
• Wavelet banding then
• regression

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Regression Analysis
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 signal is degraded
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Wavelet Banding
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
coefficients vary with time, spurious modes are
eliminated and Bayesian statistical schemes are
less confused.