Page 243 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 225

The R and L Methods for the Quadratic Classifier

Perturbation equation: The above discussion may be xtend d to the
more complex but more useful quadratic classifier [14]. In the quadratic
classifier of (5.54), we need a perturbation equation for the covariance matrix
in addition to the mean vector of (5.1 19) as follows.

c, = - - (xZ"-h;p)(Xi!'-hjk)

N,-2

=E,+- 1 - N, (Xf)-h,)(xp-h,y (5.124)
.
N,-2 " - (N,-1)(Nf-2)
The inverse matrix of & can be obtained from (2.160)
r

-2 -2
where d, (Xp)) = (Xt'-h,f ?L'(XV)-h,). The L distance, dfa(Xf'), is from
(5.120) and (5.125)
&Xf)) = (Xp-hJi; (Xf'-h,,)

-4
Njd, (Xf')
(N, - 1 )2-N, &xp,

-2
= d, (Xf') + (N,2-3Nf+1 )iz (Xf ))/(N, -l)+N,i:(X:')) (5.1 26)
(N, - 1 )2-N, 2; (Xp ),
Also, using (2.143), the determinant of can be calculated as follows.