Page 35 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 17

where SZ = [o, . . . o,]~ and O, is the ith frequency component.
(5) Linear- transformations: Under any nonsingular linear transformation,
the distance function of (2.22) keeps its quadratic form and also does not lose
its positive definiteness. Therefore, after a nonsingular linear transformation, a
normal distribution becomes another normal distribution with different parame-
ters.
Also, it is always possible to find a nonsingular linear transformation
which makes the new covariance matrix diagonal. Since a diagonal covariance
matrix means uncorrelated variables (independent variables for a normal distri-
bution), we can always find for a normal distribution a set of axes such that
random variables are independent in the new coordinate system. These sub-
jects will be discussed in detail in a later section.
(6) Physical jusfification: The assumption of normality is a reasonable
approximation for many real data sets. This is, in particular, true for processes
where random variables are sums of many variables and the central limit
theorem can be applied. However, normality should not be assumed without
good justification. More often than not this leads to meaningless conclusions.

2.2 Estimation of Parameters

Sample Estimates

Although the expected vector and autocorrelation matrix are important
parameters for characterizing a distribution, they are unknown in practice and
should be estimated from a set of available samples. This is normally done by
using the sample estimation technique [6,7]. In this section, we will discuss
the technique in a generalized form first, and later treat the estimations of the
expected vector and autocorrelation matrix as the special cases.

Sample estimates: Let y be a function of x, , . . . , x,, as
s=f'cx,,. . ., x,,) (2.25)

with the expcctcd value rn, and variance 0::