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 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.