Page 109 - Introduction to Statistical Pattern Recognition
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3 Hypothesis Testing 91
Quadratic Classifiers
For a normal test distribution: When the quadratic classifier of (3.1 1)
is designed and tested on a normal distribution pT(X), h (X) and p7(X) can be
given as
and
where t of (3.1 19) is a threshold. Applying simultaneous diagonalization and a
coordinate shift, Y = A T(X-M,),
ATXTA =I and A7(&'--Cs1)-'A =A. (3.121)
Then, M, and Zj are converted to
AT(Mj - MT) = Di and A7&A = Kj (i=1,2). (3.122)
Thus, in the Y-space
1
h(Y) = -YTA-'Y - V'Y + c , (3.123)
2
(3.124)
where
V = KT'D, - KT1D2 , (3.125)
1
IKlI
1
+
c = -(D~KT'D~ - D:K:~D~) 2 in - (3.126)
t.
-
2 IK,I
The characteristic function FT(a) can be computed now as
