16.Explain the basic principle of Hotelling transform.

16.Explain the basic principle of Hotelling transform.

   December 28, 2013       legend

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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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