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

60 Image Statistics

In the multivariate case we have a similar situation. The sample mean

vector

¯

Z =

1

m

m

X

ν=1

Z(ν)

is multivariate normally distributed with mean µ and covariance ma trix Σ/m

for samples Z(ν) ∼ N(µ, Σ), ν = 1 . . . m. The sample covariance matrix

S is described by a Wishart distribution in the following sense. Suppose

Z(ν) ∼ N(0, Σ), ν = 1 . . . m. We then have, with Equation (2.48),

(m − 1)S =

m

X

ν=1

Z(ν)Z(ν)

⊤

=: X. (2.55)

Realizations x of the random sample matrix X, namely

x =

m

X

ν=1

z(ν)z(ν)

⊤

,

are symmetric and, for s ufficie ntly large m, positive definite.

THEOREM 2.8

(Anderson, 2 003) The probability density function of X given by Equation

(2.55) ...