Page 111 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 93

ATZIA =I, ArZ2A = p, and AT(M2-MI) = L . (3.129)

Then, (Z.;’-Z;’)-’ is also transformed to a diagonal matrix A by A as

A = AT(Z,T’-ZT1)-‘A = AT[A (l-p-’)AT]-’A

-I
= (I-p )-I . (3.130)
Since MT=MI, DI =O and D2 =AT(M2-MI)=L. from (3.122). Also,

KI =I and K2 = c1 from (3.122) and (3.129). Therefore, inserting these into
(3.125) and (3.126), Vand c are

v = -p-IL , (3.131)

(3.132)

That is, after computing p and L by (3.129), we replace hi and vi of (3.127)
by
(3.133)

where p, andi, are the components of p and L. Then is computed by the
first equation of (3.128).

MT = Mz and ZT = Z,: After applying the transformation of
Y = A T(X-M2) where A is determined by (3.129), a further transformation of
2 = p-”’Y is applied. Then, (3.129) is modified to

-1/2
p-1/2A T(M -Ma) = -p L . (3.134)
I
Also, (Zyl-Z;l)-l is diagonalized as
A = p-1/2(1-p-1)-1p-1’2 (3.135)
(p-[)-I
=
,
-112 -I
Since MT = M2 this time, D I = -p L, Dz = 0, K = p , and Kz = I from
(3.122) and (3.134). Therefore, inserting them into (3.125) and (3.126), V and