Concept in probability theory
- 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
![{\displaystyle {\overline {\mathbf {x} }}={\frac {1}{n}}\sum _{j=1}^{n}\mathbf {x} _{j}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is the j-th column of
.[1]
The scatter matrix is the m-by-m positive semi-definite matrix
![{\displaystyle S=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x} }})(\mathbf {x} _{j}-{\overline {\mathbf {x} }})^{T}=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x} }})\otimes (\mathbf {x} _{j}-{\overline {\mathbf {x} }})=\left(\sum _{j=1}^{n}\mathbf {x} _{j}\mathbf {x} _{j}^{T}\right)-n{\overline {\mathbf {x} }}{\overline {\mathbf {x} }}^{T}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
denotes matrix transpose,[2] and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as
![{\displaystyle S=X\,C_{n}\,X^{T}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
[3]
When the columns of
are independently sampled from a multivariate normal distribution, then
has a Wishart distribution.
See also
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.