Page 63 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 45

Rnrxn = BnixrCrxii . (2.169)

Since BTB is an r x rnonsingular matrix, C can be obtained by
c = (B~B)-IB~R . (2.170)

From (2.170), C has rank r so that CCT is also an I' x I' nonsingular matrix. The
pseudoinverse R * of R is defined by

R* = CT(CCT)-'(BTB)-'BT (2.171)
.
It can be shown that Rx satisfies (2.167) and is therefore a generalized inverse.
Further, Rx is unique. The pseudoinverse is the most often used generalized
inverse.

Standard Data and Experimental Procedure

Throughout this book the following data will be used:

Type of distribution: normal,
Dimension: n = 8 unless specified otherwise,
Number of classes: L = 2,
Distribution parameters:

M, =O=[O. 017, M*=M=[m1 ...

0

z, =I=

hR

Data 1-1:
ml =m, m2= ... =mx=O,

1, = . . . = h* = 1
In this data, both C, and C2 are 1. The value of m controls the overlap
between the two distributions. Unless m (or .In/lTM or IIMz -MI 1)) is specified