Page 216 - Introduction to Statistical Pattern Recognition
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198 Introduction to Statistical Pattern Recognition
(5.39)
where Xy), . . . ,X$! are Ni test samples drawn from pi(X), and 6(.) is a unit
impulse function.
Thus, the estimate of the error probability is
(5.40)
where
Since af) is the inverse Fourier transform of l/jw, it becomes sign(h(Xy)))/2.
That is, aft) is either +1/2 or -1/2, depending on h(X7)) > 0 or h(X5;)) < 0.
For i = 1, the a5I)’s are +1/2 for misclassified Xsl)’s and -1/2 for correctly
classified X:”’s. Thus, summing up these 4)’)’s
1 1
= -(# of wl-errors) - - , (5.42)
NI 2
where (# of ul -corrects) = N I - (# of u1 -errors) is used to obtain the second
line from the first. Likewise, for i = 2,
(5.43)
Substituting (5.42) and (5.43) into (5.40),
