Page 172 - Introduction to Statistical Pattern Recognition

154 Introduction to Statistical Pattern Recognition

of design samples are relatively small. This problem will be addressed in
Chapter 5.

Design Procedure of a Quadratic Classifier

The general quadratic classifier may be expressed as

WI
h(X)=XTQX+VTX+v, >< 0, (4.103)
02
where Q, V, and v,, are a matrix, vector, and scalar, respectively. Therefore,
we can optimizef(ql,l12,0T,0~) (qi =E(h(X)lwi) and O’ =Var(h(X)Iw;])
with respect to Q, V, and v, as was done in linear classifier design. Unfor-
tunately, the number of parameters, [n (n +1)/2]+n +1, is too large, and O’ is the
function of the third and fourth order moments of X. Therefore, it is not prac-
tical to optimize f(ql ,q2,0:,0$).

Linearization: Another possibility is to interpret (4.103) as a linear
equation as

ll(Il+l)
2 It
= c. aiyi + Cvixi + v,) , (4.104)
,=I i=l

where qij and itt are the components of Q and V. Each of the new variables, yi,
represents the product of two x’s, and a is the corresponding q. Since (4.104)
is a linear discriminant function, we can apply the optimum design procedure
for a linear classifier, resulting in
.
[ai. . .a,(,,+1)/2~~1 . .~’,,IT = [S KI + (~-S)K~I-*(D~ (4.105)
-01).
where Dj and Kj are the expected vector and covariance matrix of Z = [YTXTIT
with [n(n+1)/2]+n variables. Since the y’s are the product of two x’s, Ki
includes the third and fourth order moments of X. Again, the number of vari-
ables is too large to compute (4.105) in practice.

Data display: A practical solution to improve the quadratic classifier of
(4.1) is to plot samples in a coordinate system, where d!(X) =