Scatter matrix

Given n samples of m-dimensional data, represented as the m-by-n matrix, $X=[\backslash mathbf\{x\}\_1,\backslash mathbf\{x\}\_2,\backslash ldots,\backslash mathbf\{x\}\_n]$, the sample mean is

:$\backslash overline\{\backslash mathbf\{x\}\}\; =\; \backslash frac\{1\}\{n\}\backslash sum\_\{j=1\}^n\; \backslash mathbf\{x\}\_j$

where $\backslash mathbf\{x\}\_j$ is the j-th column of $X$.{{Cite web |last=Raghavan |date=2018-08-16 |title=Scatter matrix, Covariance and Correlation Explained |url=https://medium.com/@raghavan99o/scatter-matrix-covariance-and-correlation-explained-14921741ca56 |access-date=2022-12-28 |website=Medium |language=en}}

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

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

where $(\backslash cdot)^T$ denotes matrix transpose,{{Cite web |last=Raghavan |date=2018-08-16 |title=Scatter matrix, Covariance and Correlation Explained |url=https://medium.com/@raghavan99o/scatter-matrix-covariance-and-correlation-explained-14921741ca56 |access-date=2022-12-28 |website=Medium |language=en}} and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as

:$S\; =\; X\backslash ,C\_n\backslash ,X^T$

where $\backslash ,C\_n$ is the n-by-n centering matrix.

The maximum likelihood estimate, given n samples, for the covariance matrix of a multivariate normal distribution can be expressed as the normalized scatter matrix

:$C\_\{ML\}=\backslash frac\{1\}\{n\}S.${{cite thesis |last=Liu |first=Zhedong |date=April 2019 |title=Robust Estimation of Scatter Matrix, Random Matrix Theory and an Application to Spectrum Sensing |url=https://repository.kaust.edu.sa/bitstream/handle/10754/652444/Thesis.pdf?sequence=1&isAllowed=y |type=Master of Science |publisher=King Abdullah University of Science and Technology}}

When the columns of $X$ are independently sampled from a multivariate normal distribution, then $S$ has a Wishart distribution.

• Estimation of covariance matrices
• Sample covariance matrix
• Wishart distribution
• Outer product—$XX^\backslash top$or X⊗X is the outer product of X with itself.
• Gram matrix

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Category:Covariance and correlation

Category:Matrices (mathematics)

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