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Filtering Using Haar Wavelets

The coefficients that result from applying the Haar transform to a
time series can be ordered by frequency. The Java code published with
these web pages orders the coefficients in order of increasing
frequency. For example, the AMAT close price time series consists of
512 points. The in-place Haar tranform replaces the data with a data
point that represents the average over the time series (at index zero)
and the coefficients for each spectrum. The number of coefficients in
each spectrum is a power of two (e.g., 1, 2, 4, 8, 16, 32, 64, 128,
256).

A given spectrum can be removed from the time series by setting the
coefficients to zero and then applying the inverse wavelet transform.
The graphs below show the AMAT close price time series with various
spectrum removed. In all cases the spectrum 1, 2, 4, 8, 16, 32 are
included. The result of the inverse Haar transform is graphed in
blue. The AMAT close price time series is shown in red.

Coefficient spectrum 1, 2, 4, 8, 16, 32 and 256

This reconstruction of the time series shows the high frequency noise
spectrum (elements 256 to 511) on top of the time series "back bone".

Coefficient spectrum 1, 2, 4, 8, 16, 32 and 128

Coefficient spectrum 1, 2, 4, 8, 16, 32 and 64

The graphs below show the time series with one or more spectrum
removed. The first graph shows the time series without the high
frequency spectrum. The next two graphs show the time series without
the high frequency spectrum and one of the lower frequency spectrums.
The high frequency spectrum is removed because it tends to obscure the
details of the contribution of the lower frequency spectrum.

Time series without the 256 spectrum (e.g., coefficient spectrum 1, 2,
4, 8, 16, 32, 64, 128)

Time series without the 128 and 256 spectrum (e.g., coefficient spectrum 1, 2,
4, 8, 16, 32, 64)

Time series without the 64 and 256 spectrum (e.g., coefficient spectrum 1, 2,
4, 8, 16, 32, 128)

Ian Kaplan, July 2001

Revised:

back to *Applying the Haar
Wavelet Transform to Time Series Information*