Page 106 - Introduction to Statistical Pattern Recognition

88 Introduction to Statistical Pattern Recognition

E = PIE, + P*E2

(3.107)

where

P(x)=plPl(x)-~2P2(x)~ (3.108)
That is, the error is a function of h(X) and p(X), which specify the classifier
and the test distributions, respectively.
Another interpretation of (3.105) is given as follows. Let us define the
F
characteristic function of h (X) for wl, (a), as
I

= jeJ'ph(h lal) dh . (3.1 09)

That is, F,(w) may be obtained through an n-dimensional integration using
p I (X) or through a one-dimensional integration using ph(h I al Since F I (a)
).
is the Fourier transform of ph(h lol), except for the sign of jo, the inverse
Fourier transform from F I (0) to ph(h 101 ) is given by

or

1
.
ph(-h I 0,) = -jF (o)eJwhdw (3.1 1 1)
2rc
Equation (3.1 11) indicates that F I (0) is the Fourier transform of ph(-h I wl).
The multiplication by [@a) + l/jw] in the Fourier domain corresponds to an
integration in the time domain from -00 to 1. Therefore, from the second line
of (3.105) and the first line of (3.109)