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

Multivariate distributions 55

Note that the order of transposition is reversed relative to Equation (2.26)

because the observations are stored as rows. The rules of matrix multiplica-

tion, Equation (1.14), take care of the sum over all o bservations, so it only

remains to divide by m −1 to obtain the desired estimate. If d is the diagonal

matrix having the diag onal elements of s,

(d)

ij

=



(s)

ij

for i = j

0 otherwise,

then an unbiase d estimate r of the correlation matrix R is (Exercise 11 )

r = d

−1/2

sd

−1/2

. (2.54)

In terms of a column-centered data matrix stored in the numerical Python

matrix Z, for instance, the covariance and correlation matrices ca