Page 100 - Introduction to Statistical Pattern Recognition

82 Introduction to Statistical Pattern Recognition

ALR(t), resulting in a reduction of R(t) and an increase of &(t). However, as
Fig. 3-12 suggests, the change of ~(t) occurs in two different ways, depending
on the right and left sides of L&). That is, on the right side the ol-error
increases, while on the left side the w;?-error increases. Therefore, defining
AL and AL2 as the change of the reject region in the right and left sides,
I
A& = &(t+At) - &(t) = P Ip I(X)dX + P2p2(X) dX . (3.85)
MI ALL2
On the other hand,
-M=R(t)-R(t+At)=I p(X)dX+I p(X)dX. (3.86)
ml m2
Since t E PIpI(X)/p(X) in ALI and t zP2p2(X)/p(X) in AL2, (3.86) can be
modified to

=I, Plpl(X) dX + P2p2(X)dX = A&. (3.87)
I AL?
Therefore, integrating (3.87) from 0 to t,

E(t) = -[@R(<) . (3.88)

Thus, once we know R(t), E@) can be computed by (3.88) [12].
Model validity tests: In pattern recognition, we have a set of data, and
often assume a system model (the mathematical form of distributions) from
which the data were drawn. A typical example is the normality assumption.
Then, we need a procedure to test the model validity in order to assure a rea-
sonable fit of the model with the data. Since the description of the model is a
specification of probability distributions in n dimensions, it at first appears that
we face the difficult problem of multivariate goodness of fit tests. We avoid
this problem by using a transformation of the data to univariate statistics and
apply goodness of fit tests in one dimension. The reject probability is one of
the transformations.
The reject probability of (3.81) reveals that 1-R(t) is the distribution
function of a random variable I’ (X), P,.(r).

P,.(t) = PI’ { r(X) < t 1 = 1 - R (t) . (3.89)
Also, we know that &(t) is determined from R(t) by (3.88). These facts