Wavelet analysis is an exciting and relatively new field of study that enables one to extract underlying patterns either from spatially varying or temporally varying data. Pixel values representing the relative brightness and color that constitute an image are an example of spatially varying data, and daily variations of financial market prices are examples of temporally varying data. By focusing on underlying trends and patterns in the data, wavelet analysis has been used to make significant advances in information and image storage and retrieval, medical diagnosis, and even speech and voice recognition. Wavelet analysis is accomplished in SAS by leveraging various built-in subroutines available in the SAS/IML® software.
You might ask what a wavelet is and why wavelets are important. As the name suggests, a wavelet is a short-lived ripple/oscillation that has a specific mathematical representation. An example of a wavelet is shown below:
Wavelet-based techniques employ wavelets to extract underlying patterns present in the data, remove noise from the data, and achieve data compression.
To understand how wavelet analysis works, let us consider its application in the context of electrocardiogram (ECG/EKG) time-series data of a patient with occasional arrhythmia, shown the figure below (in red). An EKG signal records the electrical activity of the heart and is used to determine whether or not the heart is functioning normally. The morphology and regularity of the heart-related features in an EKG signal are analyzed for detection of abnormalities in the heart. However these features are often embedded in noise and other spurious features that complicate the analysis. To make definitive diagnosis of a possible heart disease, it is crucial to remove these irrelevant features from the EKG signal as best as possible. Shown in the figure are two such irrelevant features in the EKG signal. First, the patient’s respiration causes the overall signal to slowly drift from a reference baseline; this drift, known as the baseline drift, is captured in the slowly undulating dashed blue line in the graphic. Second, the EKG signal is partially embedded in noise generated from artifacts unrelated to the heart activity, captured in the irregular fluctuations in the graphic. Both the noise and the baseline drift are required to be removed for accurate diagnosis.
To understand how wavelets are used in this type of analysis, we simply have to change our perspective from ‘the whole’ to the ‘sum of parts’ or from the signal itself to the components or building blocks that constitute the composite EKG signal. With the help of wavelets, a complex signal such as the EKG can be broken down or decomposed into individual components that help capture patterns repeating at distinct time intervals. For example, the baseline drift in the EKG signal has a pattern that is quite distinct from the noise pattern as well as from the heart rhythm pattern; these different patterns can be detected and isolated by wavelet analysis, leaving us with data that pertains to the heart activity alone, allowing for accurate diagnosis. Furthermore, reconstruction of the decomposed signal after removal of noise and other artifacts not only improve the fidelity of the patient’s heart recording but also compresses the size of the dataset by getting rid of extraneous data. In essence, wavelets allow us to convey pertinent information using fewer data samples.
So how does wavelet analysis work on image storage and retrieval? If you think of an image as a stream of pixelated information that varies in terms of its lightness and darkness, especially when reading from left to right and top to bottom, the information stream has characteristic oscillations that define content areas of the picture. If we capture only the most important aspects of the contrast and color through wavelet analysis, then we can reduce the amount of information saved. By reversing the process of converting the transformations back to regular data, we can accurately and precisely reconstruct the image.
Other areas of application of wavelets include:
- The bio-medical industry performs DNA/protein and blood-pressure analysis, cancer detection, and breathing pattern analysis in new-born babies using wavelets.
- In the government, wavelet-based techniques are being employed for facial recognition algorithms, fingerprint detection etc.
- In the finance industry, quick variation in market prices and trading patterns are being studied using wavelets.
- In the oil and gas industry, estimation of subsoil properties required for oil exploration employ wavelet-based analysis. This allows them to focus on underground features likely to hold the most oil or to map out the underlying rock structure.
Though wavelets were traditionally used for image analysis and compression, they are quickly gaining momentum in a variety of problems that require careful interpretation of complex patterns present in the data at different spatial/temporal resolutions and for isolating weak signals from the noise.
Get started today to leverage the value of wavelets in SAS/IML® software to reveal underlying patterns in the data for a better understanding of your business problem.