# Scatter matrix

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

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

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

where $T$ denotes matrix transpose, and multiplication is with regards to the outer product. 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

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.