Page 343 - Introduction to Statistical Pattern Recognition

7 Nonparametric Classification and Error Estimation 325

E(A&) :-IIE(Ah(X) + -Ah jo 2 (X))
1
2n 2
eJMX) (7.45)
I
[PIP (XI - P2P2(X)ldWX '
From (7.44), Ah(X) and Ah2(X) are derived as follows.

,. (7.47)
A
where Api(X) = PAX) - pi(X), Af = t - t, and the expansions are terminated at
the second order terms. The bias of the error estimate may be obtained by tak-
ing the expectations of (7.46) and (7.47), inserting them into (7.45), and cay-
ing out the integration.

Parzen kernel: When the Parzen kernel approach is used, E{;,(X)]
and Var( if 1 are available in (6.18) and (6.19) respectively. Since
(X)
E { AP;(X> I = E { Pi(X) i - Pj(X) and E 1 Ap! (XI I = E ( [Pi (x) - p,(X)I2 1
= MSE{;i(X)l = Var{pi(X)} + E2{Api(X)),

(7.48)

(7.49)

where MI, of (6.15) is expressed by sir-" and si is given in (6.20) and (6.26) for