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, $X=[{\mathbf{𝐱}}_1,{\mathbf{𝐱}}_2,\dots ,{\mathbf{𝐱}}_n]$, 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

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

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—$X{X}^{\mathrm{\top }}$or X⊗X is the outer product of X with itself.
• Gram matrix

References

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