Matérn Covariance Function

In statistics, the Matérn covariance, also called the Matérn kernel, 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 named after the Swedish forestry statistician Bertil Matérn. It specifies the covariance between two measurements as a function of the distance $d$ between the points at which they are taken. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the Matérn covariance is also isotropic.

Definition

The Matérn covariance between measurements taken at two points separated by ''d'' distance units is given by Rasmussen, Carl Edward and Williams, Christopher K. I. (2006
Gaussian Processes for Machine Learning
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:$C_\nu(d) = \sigma^2\frac^\nu K_\nu\Bigg(\sqrt\frac\Bigg),$

where $\Gamma$ is the gamma function, $K_\nu$ is the modified Bessel function of the second kind, and ''ρ'' and $\nu$ are positive parameters of the covariance.

A Gaussian process with Matérn covariance is $\lceil \nu \rceil-1$ times differentiable in the mean-square sense.Santner, T. J., Williams, B. J., & Notz, W. I. (2013). ''The design and analysis of computer experiments.'' Springer Science & Business Media.

Spectral density

The power spectrum of a process with Matérn covariance defined on $\mathbb^n$ is the (''n''-dimensional) Fourier transform of the Matérn covariance function (see Wiener–Khinchin theorem). Explicitly, this is given by

:$S(f)=\sigma^2\frac\left(\frac + 4\pi^2f^2\right)^.$

Simplification for specific values of ''ν''

Simplification for ''ν'' half integer

When $\nu = p+1/2,\ p\in \mathbb^+$ , the Matérn covariance can be written as a product of an exponential and a polynomial of degree $p$.Stein, M. L. (1999). ''Interpolation of spatial data: some theory for kriging.'' Springer Series in Statistics.Peter Guttorp & Tilmann Gneiting, 2006. "Studies in the history of probability and statistics XLIX On the Matern correlation family," Biometrika, Biometrika Trust, vol. 93(4), pages 989-995, December. The modified Bessel function of a fractional order is given by Equations 10.1.9 and 10.2.15 as

$\sqrt K_(z) = \frace^\sum_^n \frac \left( 2z \right) ^$ .

This allows for the Matérn covariance of half-integer values of $\nu$ to be expressed as

$C_(d) = \sigma^2\exp\left(-\frac\right)\frac\sum_^p\frac\left(\frac\right)^,$

which gives:
* for $\nu = 1/2\ (p=0)$: $C_(d) = \sigma^2\exp\left(-\frac\right),$
* for $\nu = 3/2\ (p=1)$: $C_(d) = \sigma^2\left(1+\frac\right)\exp\left(-\frac\right),$
* for $\nu = 5/2\ (p=2)$: $C_(d) = \sigma^2\left(1+\frac+\frac\right)\exp\left(-\frac\right).$

The Gaussian case in the limit of infinite ''ν''

As $\nu\rightarrow\infty$, the Matérn covariance converges to the squared exponential covariance function

:$\lim_C_\nu(d) = \sigma^2\exp\left(-\frac\right).$

Taylor series at zero and spectral moments

From the basic relation satisfied by the Gamma function $\Gamma(z)\Gamma(1-z)=\frac$
and the basic relation satisfied by the Modified Bessel Function of the second

$K_(x) = \frac \frac$

and the definition of the modified Bessel functions of the first
$I_(x)= \sum_^\infty \frac\left(\frac\right)^,$

the behavior for $d\rightarrow0$ can be obtained by the following Taylor series (when $\nu$ is not an integer and bigger than 2):

$C_\nu(d) = \sigma^2\left(1 + \frac\left(\frac\right)^2 + \frac\left(\frac\right)^4 + \mathcal\left(d^\right)\right),\,\, \nu>2 .$

When defined, the following spectral moments can be derived from the Taylor series:
:
\begin
\lambda_0 & = C_\nu(0) = \sigma^2, \\ pt\lambda_2 & = -\left.\frac\_ = \frac.
\end

For the case of $\nu\in(0,1)\cup(1,2)$, similar Taylor series can be obtained:
$C_\nu(d) = \sigma^2\left(1 + \frac\left(\frac\right)^2 - \frac\left(\frac\right)^ \left(\frac\right)^ + \mathcal\left(d^\right)\right),\,\, \nu\in (0,1)\cup(1,2) .$
When $\nu$ is an integer limiting values should be taken, (see ).

See also

* Radial basis function

References

{{DEFAULTSORT:Matern covariance function
Geostatistics
Spatial analysis
Covariance and correlation