Page 320 - Introduction to Statistical Pattern Recognition

302 Introduction to Statistical Pattern Recognition

If < is satisfied, Xi!) is correctly classified, and if > is satisfied, Xi” is
misclassified. The R estimate of the q-error, cIR, is obtained by testing
Xi’), . . . ,Xyl, counting the number of misclassified samples, and dividing the
number by NI. Similarly, &2R is estimated by testing xi2), . . . ,x$!.

On the other hand, when the L method is applied to test Xi1), Xi’) must
be excluded from the design set. Therefore, the numerator of (7.2) must be
replaced by

Again, Xi!) (k=l, . . . ,N I) are tested and the misclassified samples are
counted. Note that the amount subtracted in (7.3), K~ (0), does not depend on k.
When an 02-sample is tested, the denominator of (7.2) is modified in the same
way.
Typical kernel functions, such as (6.3), generally satisfy ~~(0) 2 K;(Y)
(and subsequently ~~(0) pj(Y)). Then,
2

That is, the L density estimate is always smaller than the R density estimate.
Therefore, the left-hand side of (7.2) is larger in the L method than in the R
method, and consequently Xi’) has more of a chance to be misclassified. Also,
note that the L density estimate can be obtained from the R density estimate by
simple scalar operations - subtracting K~ (0) and dividing by (N -1). There-
fore, the computation time needed to obtain both the L and R density estimates
is almost the same as that needed for the R density estimate alone.