Scatter 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

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

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.