An Introduction to Wavelet Analysis with SAS
Michael Lane, Consultant, Watertown, MA
Wavelet analysis is a mathematical technique used to represent data or functions.
The wavelets used in the analysis are func-
tions that possess certain mathematical properties, and break the data down into different scales or resolutions.
better able to handle spikes and discontinuities than traditional Fourier analysis making them a perfect tool to de-noise noisy
Traditional applications of wavelets have focused on image compression and analysis, but they are also being used to analyze
time series, biological processes, spectroscopic data of chemical compounds, seismic signals for earthquake prediction, and
atmospheric data for weather prediction.
began introducing tools to perform wavelet analysis in version 8.2.
introduces some of the basic concepts of wavelet analysis, and how to perform wavelet analysis with SAS IML
In this paper, signal data refer to data with some type of time or spatial relationship.
The majority of signal data we encounter
in practical situations are a combination of low and high frequency components.
The low frequency component is somewhat
stationary over the length of the signal data.
An example of a low frequency component is a moving average in a time series,
or a constant background color in a photograph.
High frequency components are jump discontinuities or noisy pieces of the
signal, such as outliers in a time series or a shift from the background to a person’s face in the photograph.
Wavelet analysis employs two functions, often referred to as the father and mother wavelets, to generate a family of functions
that break up and reconstruct a signal.
The father wavelet is similar in concept to a moving average function, while the mother
wavelet quantifies the differences between the original signal and the average generated by the father wavelet.
tion of the two functions allows wavelet analysis to analyze both the low and high frequency components in a signal simultane-
Image analysis has been the application of choice for wavelets.
In 1995 the FBI had roughly 200 million fingerprint records
stored as inked impressions on paper cards, which they wanted to convert to digital form.
Each card required 10 MB of stor-
age space based on a resolution of 500 pixels per inch with 256 levels of gray-scale, which meant that the FBI needed
2,000,000,000 MB or 2,000,000 gigabytes worth of storage space.
An algorithm based on wavelets was chosen as the FBI
standard for fingerprint image compression, because it yielded compression ratios between 15:1 and 20:1 with minimal loss of
A compression ratio of 20:1 meant that only 100,000 gigabytes of space was needed, and while this was still a
large amount of memory it was a substantial savings.
The new JPEG-2000 image compression standard is also based on an
algorithm that uses wavelets, and has a compression ratio of 20:1.