Page 34 - Introduction to Statistical Pattern Recognition

16 Introduction to Statistical Pattern Recognition

correlation coefficients. We will call R a correlation matrix. Since standard
deviations depend on the scales of the coordinate system, the correlation matrix
retains the essential information of the relation between random variables.

Normal Distributions

An explicit expression of p (X) for a normal distribution is

(2.21)

where Nx(M, C) is a shorthand notation for a normal distribution with the
expected vector M and covariance matrix X, and

(2.22)

where h, is the i, j component of C-'. The term trA is the trace of a matrix A
and is equal to the summation of the diagonal components of A. As shown in
(2.21), a normal distribution is a simple exponential function of a distance
function (2.22) that is a positive definite quadratic function of the x's. The
coefficient (2~)~"'~ is selected to satisfy the probability condition
IC I
lp(X)dX = 1 . (2.23)

Normal distributions are widely used because of their many important
properties. Some of these are listed below.
(1) Parameters that specify the distribution: The expected vector M and
covariance matrix C are sufficient to characterize a normal distribution
uniquely. All moments of a normal distribution can be calculated as functions
of these parameters.
(2) Wncorrelated-independent: If the xi's are mutually uncorrelated, then
they are also independent.
(3) Normal marginal densities and normal conditional densities: The
marginal densities and the conditional densities of a normal distribution are all
normal.
(4) Normal characteristic functions: The characteristic function of a nor-
mal distribution, Nx(M, C), has a normal form as