From vectors to functions

This section sets the stage for some of the ideas discussed later. Going from equations with discrete vectors and matrices to equations with functions in its basic form is not difficult. Let us demonstrate these ideas by looking at an example where PCA is applied to functions. Let X be a matrix of continuous spectra. This means that the N rows in X are really functions such that X = [xi(t) X2(t) ... xisi(t) ]. One way to find the principal components of X is to solve the eigenquation of the covariance matrix G = XX. For the discrete case G can be written in terms of vector inner products 

The bracket notation is similar to the one used in quantum mechanics [17]. Basically, the summation signs are replaced with the corresponding integration signs in the equations for PCA (and other similar multivariate algorithms). Thus, the covariance matrix G = XX has elements 

G will here be an N x N matrix whereas the dual R = X X is not a matrix, but a 2D function (also referred to as a kernel) 

It is very common to represent the smooth functions x (t) in a finite basis 

For some bases the calculation of Uy = (4 il t j) matrix will be easy. As we shall see below, the discrete wavelet transform from the Mallat algorithm produces an orthonormal basis which makes U equal to the identity matrix. For orthonormal bases no modification of the original multivariate algorithms is necessary and we can use the method directly on the basis of coefficients C. The conceptual relationship between function, sampled data and the coefficient space is shown in Fig. 1. 

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