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_1,\mathbf_2,\ldots,\mathbf_n/math>, the sample mean is

:$\overline = \frac\sum_^n \mathbf_j$

where $\mathbf_j$ is the ''j''-th column of $X$.

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

:$S = \sum_^n (\mathbf_j-\overline)(\mathbf_j-\overline)^T = \sum_^n (\mathbf_j-\overline)\otimes(\mathbf_j-\overline) = \left( \sum_^n \mathbf_j \mathbf_j^T \right) - n \overline \overline^T$

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
:$C_=\fracS.$

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

References

Covariance and correlation
Matrices

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