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   -> Volume 3, Issue 9


Question: Wavelets and time series
 
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jim@gatsibm.larc.nasa.gov
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PostPosted: Mon Dec 02, 2002 1:08 pm    
Subject: Question: Wavelets and time series
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Question: Wavelets and time series

Hey gang!

I need some basic information concerning the variety of wavelet transforms
that are commonly used. I want to analyze time series and I'm having
trouble interpreting my results.

I have seen two or three methods for generating wavelet coefficients:

1. Continuous wavelet transform
2. Discrete wavelet transform
3. Multiresolution Analysis

In (1), we basically convolve our function (signal) with a wavelet, say
for example the Mexican Hat function. We can specify the scaling and
translation parameters to generate a mapping from $Re mapsto Re^2$.

From there one can look at the phase and modulus and try to locate
features that may be of interest, such as singularities.

In (2) we are essentially performing the same steps as in (1), again
generating the mapping.

Given these coefficients, I have seen methods to generate wavelet
syntheses of our function [Percival] as well as power spectrums [Hudgins].

Now in (3), one typically generates coefficients by {it S. Mallat's}
pyramidal decomposition scheme.

I understand the mechanics of all the above but the interpretation of
results eludes me. For example, how do the coefficients in (1) and (2)
relate to those generated in (3)? Once I have generated and displayed
$W(a,b)$ how do we relate scale to frequency. I'm aware of the
inverse relationship between scale and frequency but relating a particular
scale number with frequency (like we typically see in Fourier Power
Spectra).

Any basic help with the above is welcome. One thing to note is that I
have read most of 'popular' literature on wavelets, thus my understanding
of the mechanics. I have yet to find something equitable dealing with
interpretation. Perhaps I'm not seeing the trees!?

References:

Percival, D.. 'An Introduction fo Spectral Analysis and Wavelets'

Hudgins, L.. 'Wavelet Transforms and Spectral Estimation'

Jawerth, B., Sweldens, W.. 'An Overview of Wavelet Transforms and
Multiresolution Analysis'

Thanks in Advance and Apologies for the length!

Jim Craft
jim@gatsibm.larc.nasa.gov
All times are GMT + 1 Hour
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