Feature selection in the wavelet domain: adaptive wavelets

Signals are all around us! Electromagnetic broadcasts permeating the air, stock market ticker-tape on the floor, Voyager space probe emissions streaming into the universe, and electrical pulses through human nerves – all are different types of signals. Signals are broadly defined as a sequence of measurements that can either be continuous or discrete. Often, sequences are captured at varying intervals in time, frequency, wave numbers, or distance. (While introducing ideas about wavelets and how they can be applied in chemical settings, for simplicity and generality let us consider a signal to be a function of time.) A signal of particular interest is the absorption or reflection of electromagnetic radiation measured in intervals of wavelength, as this signal is characteristic of the chemical and physical composition of the substance being measured. Another property of signals is if two or more signals overlap, then the resulting overlap is also a signal. This signal addition is usually an undesirable property as it is usually impossible to measure the individual signals independently of each other. A classic example is when a sample contains two molecules having similar absorbance signals – absorption signals with overlapping wavelength regions. When the absorption signal of the sample is measured, a combination of the two absorption signals is recorded simultaneously. The addition of signals poses a problem when information regarding only one of the two molecules is sought. Sequential measurement and signal addition are essential principles for the analysis of signals. First, the sequential principle gives a signal a quintessential ‘shape’ and, second, signal addition distorts a sought-after signal/shape by the presence of other signals. Ideally, any method for the analysis of signals should aim at extracting the components of a signal such as expressing the signal as a series of shapes. Wavelet analysis incorporates the principles of both signal addition and sequential shape representation, making wavelets a suitable method for signal analysis. Wavelet analysis represents signals by a series of orthogonal basis functions. In Fourier analysis, the basis function is the sinusoidal function, whereas in wavelet analysis, the basis function is largely undefined with the exception that the basis function is localized and orthogonal onto itself upon translation or dilation. This flexibility of wavelet basis functions enables a wide range of signal shapes to be investigated within the context of signal addition. There exist a large suite of functions that fit the wavelet description such as the derivatives of the Gaussian, and because of the popularity of wavelets in the 1990s and 2000s, there exist a large variety of standard wavelet basis to choose from. Some of these basis functions have been given names such as Daubechies wavelets, Coiflets, the Haar wavelet, and the Mexican wavelet. The choice of the wavelet basis is an important issue because the basis function is typically meant to mimic localized information embedded in the signal. The chemometricians can investigate the performance of some of the ‘famous’ wavelet basis functions, or they can design their own wavelet basis functions. The latter allows wavelet basis functions to be designed to suit both the data set and subsequent analysis method. The objective of this chapter is to demonstrate how wavelet basis functions can be computed for a range of multivariate statistical tasks such as unsupervised mapping (Section 3.23.3.1), discriminant analysis (Section 3.23.3.2), regression analysis (Section 3.23.3.3), and multivariate analysis of variance (MANOVA) (Section 3.23.3.4). Following a description of the wavelet theory in Section 3.23.2, the extension to adaptive wavelets is described in Section 3.23.2.7. Section 3.23.3 surveys the statistical methods that utilize the adaptive wavelet coefficients, and Section 3.23.4 puts all the above to practice by providing worked examples of the adaptive wavelet feature transformation procedure using near-infrared (NIR) spectra