Page 89 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 71

2a

Fig. 3-9 Probability coverage.

On the other hand, for the outer ring, region B, with radius between a and 2a,
Pr (B ] = c [(2a)" - a"] p (X,) = c (2"-l)a"p (XB), where X, is located some-
where in B. Therefore, Pr { B )/Pr (A ) = (2"-l)p (X,)@ (XA). This becomes,
for example, 2 x 10'' for na4 and p(X,)/p(X,) = 10. That is, the probability
of having a sample in region A is so much smaller than the probability for
region B, that we would never see samples in A by drawing a resonable
number (say io4) of samples.

Performance of a single hypothesis test: Suppose that two classes are
distributed with expected vectors MI = 0 and M2 = M, and covariance matrices
El = I and C2 =A (a diagonal matrix with hi's as the components), respec-
tively. Without loss of generality, any two covariance matrices can be simul-
taneously diagonalized to I and A, and a coordinate shift can bring the
expected vector of wI to zero. As shown in (3.52) and (3.54), E (d2 I o1 } = n
and Var( d2 I wI = yn, and y = 2 if the o1 -distribution is normal. On the other
]
hand, the distance of an %-sample from the wI -expected vector, 0, is

d2 = XTX = (X - M +M)'(X - M + M)
= (X - M)T(X - M) + 2MT(X - M) + MTM

= tr [(X - M)(X - M)T] + 2M7(X - M) + M'M . (3.62)

Taking the expectation with respect to 02,