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.

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