Page 86 - Introduction to Statistical Pattern Recognition

68 Introduction to Statistical Pattern Recognition

Single hypothesis schemes have been proposed to solve this problem.
Typically, they involve measuring the distance of the object from the target
mean (normalized by the target covariance matrix), and applying a threshold to
determine if it is or is not a target. This technique works well when the dimen-
sionality of the data, n, is very low (such as 1 or 2). However, as n increases,
the error of this technique increases significantly. The mapping from the origi-
nal n-dimensional feature space to a one-dimensional distance space destroys
valuable classification information which existed in the original feature space.
In order to understand this phenomena, let us study here the statistics of the
distance.

Distribution of the distance: Let us consider a distribution of X with
the expected vector M and the covariance matrix X. Then, the normalized dis-
tance of X from M is
n
d2 = (X-M)TX-l(X-M) = ZTZ = ZZ? , (3.51)
i=l
where Z = AT(X-M) and A is the whitening transformation. Since the
expected vector and covariance matrix of Z are 0 and I respectively, the zi’s
are uncorrelated, and E { zi] = 0 and Var(zi ] = 1. Thus, the expected value and
variance of d2 are

E{d2] =n E{z?] =n (3.52)

Var{d2] =E{(d2)2] -E2(d2]

(3.53)

When the 2;’s are uncorrelated (this is satisfied when the zi’s are independent),
and E { 24 ] is independent of i, the variance of d2 can be further simplified to

Var{d2) =n y , (3.54)
where