Abstract
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.
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.
| Original language | English |
|---|---|
| Title of host publication | Comprehensive Chemometrics : Chemical and Biochemical Data Analysis |
| Editors | S.d. Brown, R. Tauler, B. Walczak |
| Publisher | Elsevier |
| Pages | 647-679 |
| Number of pages | 33 |
| Volume | 4 |
| ISBN (Print) | 978-0-444-52702-8 |
| Publication status | Published - 1 Mar 2010 |
Publication series
| Name | |
|---|---|
| Number | 3 |
Bibliographical note
S.D. Brown, R. Tauler, B. WalczakKeywords
- Adaptive Wavelets Chemometrics