Scatter matrix explained

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,\ldots,x} _{n]} , the sample mean is\overline{ x } = \frac\sum_^n \mathbf_j

where

x _{j} is the j-th column of X .^[1]

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

S = \sum ^{n} _{j =1} ( x _{j-\overline{x} })(\mathbf_j-\overline)^T = \sum_^n (\mathbf_j-\overline)\otimes(\mathbf_j-\overline) = \left(\sum_^n \mathbf_j \mathbf_j^T \right) - n \overline \overline^T

where

( ⋅ ) ^{T} denotes matrix transpose,^[2] and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly asS = XC ^{T} _{nX}

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 _{ML} = (1)/(n) S. ^[3]

When the columns of

X are independently sampled from a multivariate normal distribution, then S has a Wishart distribution.

See also

• Estimation of covariance matrices
• Sample covariance matrix
• Wishart distribution
• Outer product—

XX ^{\top} or X⊗X is the outer product of X with itself.

• Gram matrix

Notes and References

1. Web site: Raghavan . 2018-08-16 . Scatter matrix, Covariance and Correlation Explained . 2022-12-28 . Medium . en.
2. Web site: Raghavan . 2018-08-16 . Scatter matrix, Covariance and Correlation Explained . 2022-12-28 . Medium . en.
3. Liu . Zhedong . April 2019 . Robust Estimation of Scatter Matrix, Random Matrix Theory and an Application to Spectrum Sensing . Master of Science . King Abdullah University of Science and Technology.