# 16.Explain the basic principle of Hotelling transform.

## 16.Explain the basic principle of Hotelling transform.

Hotelling transform:

The basic principle of hotelling transform is the statistical properties of vector
representation. Consider a population of random vectors of the form,

And the mean vector of the population is defined as the expected value of x i.e.,

mx = E{x}

The suffix m represents that the mean is associated with the population of x vectors. The
expected value of a vector or matrix is obtained by taking the expected value of each elememt.
The covariance matrix Cx in terms of x and mx is given as

Cx = E{(x-mx) (x-mx)T

T denotes the transpose operation. Since, x is n dimensional, {(x-mx) (x-mx)T} will be of
n x n dimension. The covariance matrix is real and symmetric. If elements xi and xj are
uncorrelated, their covariance is zero and, therefore, cij = cji = 0.

For M vector samples from a random population, the mean vector and covariance matrix
can be approximated from the samples by

and