Rational quadratic covariance function

In statistics, the rational quadratic covariance function is used in spatial statistics, geostatistics, machine learning, image analysis, and other fields where multivariate statistical analysis is conducted on metric spaces. It is commonly used to define the statistical covariance between measurements made at two points that are d units distant from each other. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the rational quadratic covariance function is also isotropic.

The rational quadratic covariance between two points separated by d distance units is given by

C(d)={\Bigg (}1+{\frac {d^{2}}{2\alpha k^{2}}}{\Bigg )}^{-\alpha }

where α and k are non-negative parameters of the covariance.[1][2]

References

Williams, Christopher K.I, Rasmussen, Carl Edward (2006). Gaussian Processes for Machine Learning. United Kingdom: MIT Press. p. 86.{{cite book}}: CS1 maint: multiple names: authors list (link)
Kocijan, Juš (2015-11-22), "Control with GP Models", Modelling and Control of Dynamic Systems Using Gaussian Process Models, Advances in Industrial Control, Cham: Springer International Publishing, pp. 147–208, doi:10.1007/978-3-319-21021-6_4, ISBN 978-3-319-21020-9, retrieved 2022-11-25

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Williams, Christopher K.I, Rasmussen, Carl Edward (2006). Gaussian Processes for Machine Learning. United Kingdom: MIT Press. p. 86.{{cite book}}: CS1 maint: multiple names: authors list (link)

[2] Kocijan, Juš (2015-11-22), "Control with GP Models", Modelling and Control of Dynamic Systems Using Gaussian Process Models, Advances in Industrial Control, Cham: Springer International Publishing, pp. 147–208, doi:10.1007/978-3-319-21021-6_4, ISBN 978-3-319-21020-9, retrieved 2022-11-25