Daily Speculations
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James Sogi
Philosopher, Juris Doctor, surfer,
trader, investor, musician, black belt, sailor,
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(address is not clickable)04/26/2006
Wavelet Methods for Time Series Analysis
Wavelet Methods for Time Series Analysis, by Donald B. Percival and Andrew T. Walden, is another wonderful book in the Cambridge Series in Statistical and Probabilistic Mathematics. This series really hits the sweet spot for the spec-listers by discussing the cutting edge subjects from the ground up in a self study format with exercises, (with the solutions provided) and a web site with reference to S-Plus code and computational approaches. The authors carefully cover the underlying basic material on Fourier analysis which is a real bonus in reference to the Minister of Non Predictive Studies Fourier analysis posted on list. This series speaks plain English and S-Plus in addition to the basic statistical and mathematical framework and is accessible to the unwashed with a bit of work.
A wavelet is a small wave set in a limited time that grows and decays, as distinguished from the big wave which is a sine wave that keeps on oscillating up and down. The wavelet analysis is a transform of the data allowing analysis and re-transformation back into the original data form for aid to analysis. The wavelets under the formula are the Haar wavelet which is related to the first derivative of the Gaussian probability density function, and the Mexican Hat wavelet, which is related to the second derivative of the Gaussian probability density function. Wavelets are related to Fourier transform. The changes of the wavelets average amplitude changes over changes in scale allow analysis.
Average values of signals over various scales are used in the physical sciences such as 1)1 second averages of temperature and vertical velocity over a forest, 2) ten minute and hourly rainfall rate in a severe storm, 3) monthly mean sea temperatures on the equatorial Pacific, 4)thirty minute average vertical wind velocity profiles, 5) Yearly average temperatures over England. In physical sciences often it is the changes in the averages that are of interest rather than the absolute values, which of course is of interest to the speculator, the changes, rather than absolute numbers. The analysis looks at non-overlapping discrete intervals. The analysis allows study of time series that have scale based characteristics that are not homogeneous over time...in other words, market time with ever changing cycles,. Statistical analysis is performed on the wavelets. The underlying series can be then recovered. I am quite excited about applying wavelet analysis to the markets with their ability to express an autocorrelated time series in a combination of uncorrelated variables. The variables depend on a limited portion of time series leading to the ability to dealing with time series whose statistical characteristics evolve over time.
An example, by way of illustration rather than prediction, of possible use of wavelets is the moon problem discussed on the List. Fourier analysis did not detect a regular sine wave over the years that coincided with the moon phase, however, the discrete period analysis might show some movement within certain windows that evolve, say over the seasons, that might otherwise throw a sine wave function off. A lunar effect would presumably carry an evolving seasonal component as well frustrating Fourier analysis.
The wavelets can detect signal hidden by noise in a time series. The Professor should have a field day with these ideas and hope we hear from others on the list who have the technical skills to understand these matters better than a mere surfer. Often in these esoteric tomes lay hidden a few gems with code that can be used by specs and it is well worth wading through the tall grass to find them. Finding one could make a career or a meal for life.
Jim Sogi, May 2005
(address is not clickable)