Matérn covariance function

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) [http://www.gaussianprocess.org/gpml/chapters/RW4.pdf Gaussian Processes for Machine Learning]

:$$

C_\nu(d) = \sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}{\Bigg(\sqrt{2\nu}\frac{d}{\rho}\Bigg)}^\nu K_\nu\Bigg(\sqrt{2\nu}\frac{d}{\rho}\Bigg),

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

A Gaussian process with Matérn covariance is $\backslash lceil\; \backslash nu\; \backslash rceil-1$ 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.

The power spectrum of a process with Matérn covariance defined on $\backslash mathbb\{R\}^n$ 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{2^n\pi^{n/2}\Gamma(\nu+\frac{n}2)(2\nu)^\nu}{\Gamma(\nu)\rho^{2\nu}}\left(\frac{2\nu}{\rho^2} + 4\pi^2f^2\right)^{-\left(\nu+\frac{n}2\right)}.

When $\backslash nu\; =\; p+1/2,\backslash \; p\backslash in\; \backslash mathbb\{N\}^+$ , 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{{Cite book|title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables|last=Abramowitz and Stegun|year=1965 |publisher=U.S. Government Printing Office |isbn=0-486-61272-4|url-access=registration|url=https://archive.org/details/handbookofmathe000abra}} as

$$\backslash sqrt\{\backslash frac\{\backslash pi\}\{2z\}\}\; K\_\{p+1/2\}(z)\; =\; \backslash frac\{\backslash pi\}\{2z\}e^\{-z\}\backslash sum\_\{k=0\}^n\; \backslash frac\{(n+k)!\}\{k!\backslash Gamma(n-k+1)\}\; \backslash left(\; 2z\; \backslash right)\; ^\{-k\}$$ .

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

$$C\_\{p+1/2\}(d)\; =\; \backslash sigma^2\backslash exp\backslash left(-\backslash frac\{\backslash sqrt\{2p+1\}d\}\{\backslash rho\}\backslash right)\backslash frac\{p!\}\{(2p)!\}\backslash sum\_\{i=0\}^p\backslash frac\{(p+i)!\}\{i!(p-i)!\}\backslash left(\backslash frac\{2\backslash sqrt\{2p+1\}d\}\{\backslash rho\}\backslash right)^\{p-i\},$$

which gives:

• for $\backslash nu\; =\; 1/2\backslash \; (p=0)$: $C\_\{1/2\}(d)\; =\; \backslash sigma^2\backslash exp\backslash left(-\backslash frac\{d\}\{\backslash rho\}\backslash right),$
• for $\backslash nu\; =\; 3/2\backslash \; (p=1)$: $C\_\{3/2\}(d)\; =\; \backslash sigma^2\backslash left(1+\backslash frac\{\backslash sqrt\{3\}d\}\{\backslash rho\}\backslash right)\backslash exp\backslash left(-\backslash frac\{\backslash sqrt\{3\}d\}\{\backslash rho\}\backslash right),$
• for $\backslash nu\; =\; 5/2\backslash \; (p=2)$: $C\_\{5/2\}(d)\; =\; \backslash sigma^2\backslash left(1+\backslash frac\{\backslash sqrt\{5\}d\}\{\backslash rho\}+\backslash frac\{5d^2\}\{3\backslash rho^2\}\backslash right)\backslash exp\backslash left(-\backslash frac\{\backslash sqrt\{5\}d\}\{\backslash rho\}\backslash right).$

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

:$$

\lim_{\nu\rightarrow\infty}C_\nu(d) = \sigma^2\exp\left(-\frac{d^2}{2\rho^2}\right).

From the basic relation satisfied by the Gamma function $$

\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}

and the basic relation satisfied by the Modified Bessel Function of the second

K_{\nu}(x) = \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin (\pi\nu )}

and the definition of the modified Bessel functions of the first

I_{\nu}(x)= \sum_{m=0}^\infty \frac{1}{m!\, \Gamma(m+\nu+1)}\left(\frac{x}{2}\right)^{2m+\nu},

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{\nu}{2(1-\nu)}\left(\frac{d}{\rho}\right)^2 + \frac{\nu^2}{8(2-3\nu+\nu^2)}\left(\frac{d}{\rho}\right)^4 + \mathcal{O}\left(d^{6\wedge(2\nu)}\right)\right),\,\, \nu>2

. {{cite journal |last1=Cheng |first1=Dan |title=Smooth Matérn Gaussian random fields: Euler characteristic, expected number and height distribution of critical points |journal=Statistics & Probability Letters |date=July 2024 |volume=210 |pages=110116 |doi=10.1016/j.spl.2024.110116 |arxiv=2307.01978 }}

When defined, the following spectral moments can be derived from the Taylor series:

: $$

\begin{align}

\lambda_0 & = C_\nu(0) = \sigma^2, \\[8pt]

\lambda_2 & = -\left.\frac{\partial^2C_\nu(d)}{\partial d^2}\right|_{d=0} = \frac{\sigma^2\nu}{\rho^2(\nu-1)}.

\end{align}

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{\nu}{2(1-\nu)}\left(\frac{d}{\rho}\right)^2 - \frac{\Gamma(1-\nu)}{ \Gamma(1+\nu)}\left(\frac{\nu}{2}\right)^{\nu} \left(\frac{d}{\rho}\right)^{2\nu} + \mathcal{O}\left(d^{4\wedge(2\nu+2)}\right)\right),\,\, \nu\in (0,1)\cup(1,2)

.

When $$

\nu

is an integer limiting values should be taken, (see ).

• Radial basis function

{{DEFAULTSORT:Matern covariance function}}

Category:Geostatistics

Category:Spatial analysis

Category:Covariance and correlation