// Workers AI · dad joke modeWhat did the scatter matrix say? "I'm all over the place".

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 j-th 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 $(\cdot )^{T}$ 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.

See also

• Estimation of covariance matrices
• Sample covariance matrix
• Wishart distribution
• Outer product—$XX^{\top }$or X⊗X is the outer product of X with itself.
• Gram matrix