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, ${\displaystyle X=[\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n}]}$, the sample mean is

${\displaystyle {\overline {\mathbf {x} ))={\frac {1}{n))\sum _{j=1}^{n}\mathbf {x} _{j))$

where ${\displaystyle \mathbf {x} _{j))$ is the j-th column of ${\displaystyle X}$.^[1]

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

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

where ${\displaystyle (\cdot )^{T))$ denotes matrix transpose,^[2] and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as

${\displaystyle S=X\,C_{n}\,X^{T))$

where ${\displaystyle \,C_{n))$ 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

${\displaystyle C_{ML}={\frac {1}{n))S.}$^[3]

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