- Scatter matrix
In
multivariate statistics andprobability theory , the scatter matrix is astatistic that is used to make estimates of thecovariance matrix of themultivariate normal distribution . (The scatter matrix is unrelated to thescattering matrix ofquantum mechanics .)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 .
The scatter matrix is the "m"-by-"m"
positive semi-definite matrix:
where denotes
matrix transpose . 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
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