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.)

[edit] Definition

Given n samples of m-dimensional data, represented as the m-by-n matrix, X=[\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_n], the sample mean is

\overline\mathbf{x} = \frac{1}{n}\sum_{j=1}^n \mathbf{x}_j

where \mathbf{x}_j is the jth column of X\,.

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

S = \sum_{j=1}^n (\mathbf{x}_j-\overline\mathbf{x})(\mathbf{x}_j-\overline\mathbf{x})'

where {\,}' denotes matrix transpose. The scatter matrix may be expressed more succinctly as

S = X\,C_n\,X\,'

where \,C_n is the n-by-n centering matrix.

[edit] 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

C_{ML}=\frac{1}{n}S.

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

[edit] See also

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