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, for instance of the multivariate normal distribution.

Definition[edit]

Given n samples of m-dimensional data, represented as the n-by-m 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, and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as

where 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

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

See also[edit]