Page 39 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 21

n n
y = (xi - m,)(xj - mi) . (2.40)
Then
E(m,.} =E{y) f pippi. (2.41)

That is, the expectation of the sample estimate is still the same as the expected
value of y given by (2.40). However, the expectation of (2.40) is not equal to
that of (2.38) which we want to estimate.

Sample covariance matrix: In order to study the expectation of (2.40)
in a matrix form, let us define the sample estimate of a covariance matrix as
I N
c = ,C(X, - M)(X, - M)T (2.42)

Then

h
Thus, taking the expectation of C

E{ i] = c - E{ (M - M)(M - M)T )
1
=I:--E=- N-1 c.
N N

That is, (2.44) shows that C is a hiased estimate of C. This bias can be elim-
inated by using a modified estimate for the covariance matrix as

(2.45)

Both (2.42) and (2.45) are termed a sample co\qarianc.e muriiu. In this book,
we use (2.45) as the estimate of a covariance matrix unless otherwise stated,
because of its unbiasedness. When N is large, both are practically the same.