Page 282 - Introduction to Statistical Pattern Recognition

264 Introduction to Statistical Pattern Recognition

1
1
+
= - -r4ja2(X)p2(X)dX . (6.39)
Nv 4
Again, by solving alMSElar = 0 [5],

The resulting criterion value is obtained by substituting (6.40) into (6.39),

Optimal Metric

Another important question in obtaining a good density estimate is how
to select the metric, A of (6.3). The discussion of the optimal A is very com-
plex unless the matrix is diagonalized. Therefore, we first need to study the
effect of linear transformations on the various functions used in the previous
sections.

Linear transformation: Let @ be a non-singular matrix used to define a
linear transformation. This transformation consists of a rotation and a scale
change of the coordinate system. Under the transformation, a vector and
metric become

z=aTx, (6.42)

AZ = @'Ax@ . (6.43)

The distance of (6.24) is invariant since

(Y-X)TAX'(Y-X) = (W-Z)'AZ'(W-Z), (6.44)
where W = @'Y. The following is the list of effects of this transformation on