- 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, , the sample mean is
where is the j-th column of .[1]
The scatter matrix is the m-by-m positive semi-definite matrix
where denotes matrix transpose,[2] and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as
where 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
When the columns of are independently sampled from a multivariate normal distribution, then has a Wishart distribution.
See also
- Estimation of covariance matrices
- Sample covariance matrix
- Wishart distribution
- Outer product—or X⊗X is the outer product of X with itself.
- Gram matrix
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.