Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

50 Image Statistics

Expanding the product of sums yields

(m − 1)S =

X

i

Z

2

i

−

1

m

X

i

Z

2

i

−

1

m

X

i6=i

′

Z

i

Z

i

′

.

But Z

i

and Z

i

′

are independent random variables; s ee Definition 2.4 and

Equation (2.19), so that

hZ

i

Z

i

′

i = hZi

2

.

Therefore, sinc e the double sum above has m(m − 1) terms ,

(m − 1)hSi = mhZ

2

i−

1

m

mhZ

2

i −

1

m

m(m − 1)hZi

2

or

hSi = hZ

2

i − hZi

2

= var(Z).

The denominator (m −1) in the de finitio n of the sample variance is thus seen

to be required for unbiased es tima tion of the covariance matrix and to be due

to the appearance of the sample mean

¯

Z r ather than the distribution mean

hZi in the definitio n. The maximum likelihood method, which we will meet

in Section 2.4,