Page 40 - Introduction to Statistical Pattern Recognition

22 Introduction to Statistical Pattern Recognition

,.
Variances and covariances of cij: The variances and covariances of cij
(the i, j component of i) are hard to commte exactly. However, approxima-
tions may be obtained easily by using ?a = (1/N) c” (Xk - M)(XI - M)T in
I=I
place of of (2.42). The i, j component of ia as an approximation of iij is
then given by
IN
cij E -Z(x;k - mi)(x,* - mj) , (2.46)
A
Nk=,
where xik is the ith component of the kth sample Xg. The right hand side of
(2.46) is the sample estimate of E { (xi - mi)(xj - m,)]. Therefore, the argu-
ments used to derive (2.28). (2.29), and (2.30) can be applied without
modification, resulting in

E { cjj } Z cij , (2.47)

I
A
Var(cij } z -Var( (xi - mi)@ - mj) t , (2.48)
N
and

Note that the approximations are due to the use of mi on the left side and mi
on the right side. Both sides are practically the same for a large N.

Normal case with approximation: Let us assume that samples are
drawn from a normal distribution, Nx(O,A), where A is a diagonal matrix with
components A,,. . . , A,,. Since the covariance matrix is diagonal, xi and x,~ for
ii’j are mutually independent. Therefore, (2.48) and (2.49) are further
simplified to

,. I Aih,
Var(cij} = -Var(x,)Var(xjl = - (2.50)
N N ’

(2.51)