Page 332 - Introduction to Statistical Pattern Recognition

3 14 Introduction to Statistical Pattern Recognition

The expectation of (7.28) can be computed in three stages as
ExE,Ex,,(61p,X) where p is d(X,XNN).
The first expectation is taken with respect to XNN given X = X and
p = p. That is, the bias is averaged over all XNN on the hyperellipsoidal sur-
face, S (p) = ( Y :p = d (Y,X) ), specified by a constant p. Thus,

(xNN
(p~p MXNN
Ex,(61P9X) =
(xNNMxNN

(7.29)

where p(X) is the mixture density function, P 'p,(X) + P2p2(X). In order to
obtain the second line of (7.291, the following formulas are used, along with 6
of (7.28) [see (B.7)-(B.9)1

(7.3 1)

I,,, S (7.32)
(Y-X)(Y-X)TdY = -r2A .
n

Note that all odd order terms of (XNN-X) disappear, since S(p) is symmetric
around X.
In order to take the expectation with respect to p, we can rewrite (7.29)
in terms of u, since the density function of u is known in (6.76). Using the
first order approximation of u Z pv,

I
UZ d"2 p (X) A I "*p" . (7.33)
n +2
UT)

Therefore,