Scatter matrix

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For the notion in quantum mechanics, see scattering matrix.

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.

Definition[edit]

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}})^T = \sum_{j=1}^n (\mathbf{x}_j-\overline{\mathbf{x}})\otimes(\mathbf{x}_j-\overline{\mathbf{x}})^T = \left( \sum_{j=1}^n \mathbf{x}_j \mathbf{x}_j^T \right) - n \overline{\mathbf{x}} \overline{\mathbf{x}}^T

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

S = X\,C_n\,X^T

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

Application[edit]

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.

See also[edit]