##
Wavelets vs. Simple Filters

How do wavelet filters compare to simple filters, like windowed mean
and windowed median?

The window filters operate by moving the window over the array, from
the start to the end. The mean filter takes the average across the
window and sets the element in the middle of the window to this
average. Then the window is advanced one step and the mean (average)
is calculated again.

The median is the middle of a sorted list of numbers. For example,
the median of 1 3 5 8 15 is 5. In contrast, the mean (a.k.a. average)
is 6.4. The median does not move as much in response to a single
large element. If the window size is even, the median is the average
of the middle two elements. So, for example, the median of 2 6 13 16
is 9.5 (9.5 = (6 + 13)/2).

As in the case with the mean filter, the window for the median filter
is moved over the array from the start to the end. The list is
initially sorted and the median number is found. The middle element
in the window is set to the median. The window is moved and a number
is added to the ordered list and the median is chosen again and the
element following the one previously changes is set to the median.

For a window size of four and the AMAT time series, the windowed mean
and windowed median filters produce virtually the same result. So
only the windowed median is shown below.

A windowed median filter with a window size of four. The filtered
data is shown in blue, the unfiltered close price is in red.

Note that the mean and median filters lose a more detail than the
wavelet filters. To the extent that this detail might be of interest,
these filters are not as powerful as the wavelet filters. By the same
argument, these windowed filters are probably better than a cubic
spline, since they preserve more detail. The AMAT time series, with
the high frequence spectrum removed is shown below for comparision.

AMAT close price time series (in blue) without the high frequency spectrum
(e.g., the time series was regenerated from coefficient spectrums 1,
2, 4, 8, 16, 32, 64, and 128). As with the median filter plot, the
unfiltered close price is shown in red.

Ian Kaplan, July 2001

Revised:

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