Page 33 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 1s

C=E[XXT} -E{X)MT-ME(XT} +MMT=S-MMT, (2.15)

(2.16)

Derivation of (2.15) is straightforward since M = E[ X). The matrix S of
(2.16) is called the autocorrelafion matri.r of X. Equation (2.15) gives the
relation between the covariance and autocorrelation matrices, and shows that
both essentially contain the same amount of information.
Sometimes it is convenient to express cii by
cII = (3, 2 and c,, = p,ioioj , (2.17)

where 0: is the variance of xi, Var(xi }, or (3/ is the standard deviation of xi,
SD[xi}, and pi, is the correlation coefficient between xi and xi. Then
z=rRr (2.18)

where

J, 0 ... 0
0 (32
r= (2.19)

and
Pi,,
1 1 PI2 ' ' ' Pi,,
PI2
'
'
'
1
PI2 1
PI2
R=
R= ' ' (2.20)
Pin
Pin ... 1
... 1
Thus, C can be expressed as the combination of two types of matrices: one is
the diagonal matrix of standard deviations and the other is the matrix of the