Scatter matrix

Summary

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 edit

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 edit

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

 [3]

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

See also edit

References edit

  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.