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

>Table of contents

**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.,

m

_{x}= 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 C

_{x}in terms of x and m

_{x}is given as

C

_{x}= E{(x-m_{x}) (x-m_{x})^{T}}T denotes the transpose operation. Since, x is n dimensional, {(x-m

_{x}) (x-m

_{x})

^{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, c

_{ij}= c

_{ji}= 0.

For M vector samples from a random population, the mean vector and covariance matrix

can be approximated from the samples by

and

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