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In [[statistics]], the '''Matérn covariance''' (named after the Swedish forestry statistician Bertil Matérn<ref>{{cite doi|10.1016/j.geoderma.2005.04.003}}</ref>) is a [[covariance function]] used in [[spatial statistics]], [[geostatistics]], [[machine learning]], image analysis, and other applications of multivariate statistical analysis on [[metric space]]s. 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 process|stationary]]. If the distance is [[Euclidean distance]], the Matérn covariance is also [[isotropic]]. | |||
The Matérn covariance between two points separated by ''d'' distance units is given by | |||
:<math> | |||
C(d) = \sigma^2\frac{1}{\Gamma(\nu)2^{\nu-1}}\Bigg(\sqrt{2\nu}\frac{d}{\rho}\Bigg)^\nu K_\nu\Bigg(\sqrt{2\nu}\frac{d}{\rho}\Bigg), | |||
</math> | |||
where Γ is the [[gamma function]], ''K''<sub>ν</sub> is the modified [[Bessel function]] of the second kind, and ρ and ν are non-negative [[parameter]]s of the covariance. | |||
A [[Gaussian process]] with Matérn covariance has sample paths that are <math>\lfloor \nu-1 \rfloor</math> times differentiable.<ref name=R>Rasmussen, Carl Edward (2006) [http://ml.dcs.shef.ac.uk/gpip/slides/rasmussen.pdf Gaussian Processes Covariance Functions and Classification]. Presentation at ''Gaussian Processes in Practice''</ref> As <math>\nu\rightarrow\infty</math>, the '''Matérn covariance''' converges to the squared exponential covariance function | |||
:<math> | |||
C(d) = \sigma^2\exp(-d^2/2 \rho^2). | |||
</math> | |||
When ν = 0.5, the '''Matérn covariance''' is identical to the exponential covariance function. Other cases are:<ref name=R/> | |||
:<math>C(d) = \sigma^2 \Bigg(1 + \frac{ \sqrt{3}d }{\rho} \Bigg) \exp \Bigg(-\frac{\sqrt{3}d}{\rho} \Bigg) \quad \quad \nu= \tfrac{3}{2}, | |||
</math> | |||
:<math>C(d) = \sigma^2 \Bigg(1 + \frac{ \sqrt{5}d }{\rho} +\frac{ 5d^2}{3 \rho^2 } \Bigg) \exp \Bigg(-\frac{\sqrt{5}d}{\rho} \Bigg) \quad \quad \nu= \tfrac{5}{2}. | |||
</math> | |||
==See also== | |||
* [[Radial basis function]] | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Matern covariance function}} | |||
[[Category:Geostatistics]] | |||
[[Category:Spatial data analysis]] | |||
[[Category:Covariance and correlation]] | |||
Revision as of 07:06, 16 November 2013
In statistics, the Matérn covariance (named after the Swedish forestry statistician Bertil Matérn[1]) is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis 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 Matérn covariance is also isotropic.
The Matérn covariance between two points separated by d distance units is given by
where Γ is the gamma function, Kν is the modified Bessel function of the second kind, and ρ and ν are non-negative parameters of the covariance.
A Gaussian process with Matérn covariance has sample paths that are times differentiable.[2] As , the Matérn covariance converges to the squared exponential covariance function
When ν = 0.5, the Matérn covariance is identical to the exponential covariance function. Other cases are:[2]
See also
References
- ↑ Template:Cite doi
- ↑ 2.0 2.1 Rasmussen, Carl Edward (2006) Gaussian Processes Covariance Functions and Classification. Presentation at Gaussian Processes in Practice