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, [math]\displaystyle{ X=[\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_n] }[/math], the sample mean is
- [math]\displaystyle{ \overline{\mathbf{x}} = \frac{1}{n}\sum_{j=1}^n \mathbf{x}_j }[/math]
where [math]\displaystyle{ \mathbf{x}_j }[/math] is the j-th column of [math]\displaystyle{ X }[/math].[1]
The scatter matrix is the m-by-m positive semi-definite matrix
- [math]\displaystyle{ 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 }[/math]
where [math]\displaystyle{ (\cdot)^T }[/math] denotes matrix transpose,[2] and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as
- [math]\displaystyle{ S = X\,C_n\,X^T }[/math]
where [math]\displaystyle{ \,C_n }[/math] 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
- [math]\displaystyle{ C_{ML}=\frac{1}{n}S. }[/math][3]
When the columns of [math]\displaystyle{ X }[/math] are independently sampled from a multivariate normal distribution, then [math]\displaystyle{ S }[/math] has a Wishart distribution.
See also
- Estimation of covariance matrices
- Sample covariance matrix
- Wishart distribution
- Outer product—[math]\displaystyle{ XX^\top }[/math]or X⊗X is the outer product of X with itself.
- Gram matrix
References
- ↑ Raghavan (2018-08-16). "Scatter matrix, Covariance and Correlation Explained" (in en). https://medium.com/@raghavan99o/scatter-matrix-covariance-and-correlation-explained-14921741ca56.
- ↑ Raghavan (2018-08-16). "Scatter matrix, Covariance and Correlation Explained" (in en). https://medium.com/@raghavan99o/scatter-matrix-covariance-and-correlation-explained-14921741ca56.
- ↑ Liu, Zhedong (April 2019). Robust Estimation of Scatter Matrix, Random Matrix Theory and an Application to Spectrum Sensing (PDF) (Master of Science). King Abdullah University of Science and Technology.
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