Page 42 - Introduction to Statistical Pattern Recognition

24 Introduction to Statistical Pattern Recognition

E(C;;] = - =I;, (2.56)
*
ai

2hf
+I
Var{ ;.;; ) = pr = - (2.57)
ai N-1
On the other hand, the moments of ljj = l/(N-l)xr=, (xjk-m;)(xjk-mj) for
i#j can be computed as follows:

l N I I
E(c;,) = -xE(xik - mjlE(xjL- - mi) = 0, (2.58)
N-1 k=l

(2.59)

The expectations can be broken into the product of two expectations because x,
,. ,.
and x, are mutually independent. E ( (xlL-m,)(x,,-m,)] =h, S,, (N-l)/N, because
Xn and X, are independent. Similarly, the covariances are
,.A
COV{C,~,C~,) except {i=k and j=l), (2.60)
=O
A
because some of the (x..-m.) terms are independent from the others and
,.
E{x..-m.) =O.
Note that (2.59) and (2.60) are the same as (2.50) and (2.52) respectively.
Equation (2.51) may be obtained from (2.57) by using the approximation of
N-1 N. This confirms that the approximations are good for a large N.

2.3 Linear Transformation
Linear Transformation

When an n-dimensional vector X is transformed linearly to another n-
dimensional vector Y, Y is expressed as a function of X as