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, $X = [𝐱_{1}, 𝐱_{2}, \dots, 𝐱_{n}]$ , the sample mean is

\overline{𝐱} = \frac{1}{n} \sum_{j = 1}^{n} 𝐱_{j}

where $𝐱_{j}$ is the j-th column of $X$ .

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

S = \sum_{j = 1}^{n} (𝐱_{j} - \overline{𝐱}) (𝐱_{j} - \overline{𝐱})^{T} = \sum_{j = 1}^{n} (𝐱_{j} - \overline{𝐱}) \otimes (𝐱_{j} - \overline{𝐱}) = (\sum_{j = 1}^{n} 𝐱_{j} 𝐱_{j}^{T}) - n \overline{𝐱} {\overline{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_{M L} = \frac{1}{n} S .

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

References

↑ Raghavan (2018-08-16). "Scatter matrix, Covariance and Correlation Explained". Medium. Retrieved 2022-12-28.
↑ Raghavan (2018-08-16). "Scatter matrix, Covariance and Correlation Explained". Medium. Retrieved 2022-12-28.
↑ 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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[1] Raghavan (2018-08-16). "Scatter matrix, Covariance and Correlation Explained". Medium. Retrieved 2022-12-28.

[2] Raghavan (2018-08-16). "Scatter matrix, Covariance and Correlation Explained". Medium. Retrieved 2022-12-28.

[3] 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.

Scatter matrix

Definition

Application

See also

References