Page 178 - Introduction to Statistical Pattern Recognition

160 Introduction to Statistical Pattern Recognition

Approximation of Covariance Matrices

Most of the difficulty in designing quadratic classifiers comes from the
covariance matrices. If we could impose some structure on the covariance
matrix, it would become much easier to design a quadratic classifier. Also, the
required design sample size would be reduced, and the classifier would become
more insensitive to variations in the test distributions. One possible structure is
the toeplitz form, based on the stationarity assumption. An example is seen in
(3.13). However, the stationarity assumption is too restrictive and is not well
suited to most applications in pattern recognition.

Toeplitz approximation of a correlation matrix: Another possibility is
to assume the toeplitz form only for the correlation matrices, allowing each
individual variable to have its own mean and variance. That is, departing from
(4.1 13) and (4.1 14)

where 0: = Var(x(k)), and pl~.*~ the correlation coefficient between x(k)
is
and x(U,) which depends only on I k4 I. Expressing the covariance matrix as
C = TRT from (2.18), the inverse matrix and the determinant are

r -I
11/01 1/01 0 0 1 PI . . Pn-1 1/01 0
PI 1
, (4.124)
PI
0 ' 0
1 10, Pn-1 PI 1 I /on