Scatter matrix

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In multivariate statistics and probability theory, the scatter matrix is a statistic that is used to make estimates of the covariance matrix of the multivariate normal distribution. (The scatter matrix is unrelated to the scattering matrix of quantum mechanics.)

Definition

Given n samples of m-dimensional data, represented as the m-by-n matrix, , the sample mean is

where is the jth column of .

The scatter matrix is the m-by-m positive semi-definite matrix

where denotes matrix transpose. The scatter matrix may be expressed more succinctly as

where is the n-by-n centering matrix.

Application

The maximum likelihood estimate, given n samples, for the covariance matrix of a multivariate normal distribution can be expressed as the normalized scatter matrix

When the columns of are independently sampled from a multivariate normal distribution, then has a Wishart distribution.

See also



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