Page 107 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 89

1 1
= -j[x&o) + -]F,(o)do
27c JO
1 1
= -j[TC6(o) + -IF, (w)e@do I
2lt JW

= jomph(h lol) dh . (3.1 12)

Likewise, for 02, multiplying by [7c6(o) - l/jo] in the Fourier domain
corresponds to an integration in the time domain from t to +ob. Therefore,

Procedure to compute the error: Thus, when a classifier and test distri-
butions are given, the error of the classifier can be computed as follows [ 141:
(1) Compute the characteristic function, F,(o), by carrying out the n-
dimensional integration of (3.109). For quadratic classifiers with normal
test distributions, the explicit expression for F,(o) can be obtained, as
will be discussed later.
(2) Carry out the inverse operations of (3.105) and (3.106) as

(3.1 14)

where + and - are used for i = 1 and 2, respectively. The real and ima-
ginary parts of Fi(o) are even and odd. Therefore, the real and ima-
ginary parts of F,(o)lw are odd and even, which lead us to the second
line of (3.1 14). Note that this integration is one-dimensional. Therefore,