Page 203 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 185

little more complex. In order to simplify the notation, A,. (r = 1,2) are used to
indicate the diagonalized class covariances, where AI = I and h2 = A.
The parameters Mi and Ci can be estimated without bias by the sample
mean and sample covariance matrix

,.
where Xp) is the jth sample vector from class I-. Thus, the parameter vector Y
of (1) consists of 2[n +n (n +1)/2] components

where mi”) is the ith component of M,., and (i 2j) is the ith row and jth
n
column component of X,..
The random variables of (5.10) satisfy the following statistical properties,.
where Amy-) = m:) - my) and Ac(r) = - ~(r):
IJ rJ IJ

(1) The sample mean and covariance matrix are unbiased:

E(Am:’) =O and E(Acjj)) =O. (5.1 1)

(2) Samples from different classes are independent:

E(Arnj1)Am)*’) =E(Amj”) E(Amj2)) =0,

(3) The covariance matrices of the sample means are diagonal [see (2.34)l: