Matérn covariance function

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.

Definition

The Matérn covariance between two points separated by d distance units is given by ^[2]

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 $\Gamma$ is the gamma function, $K_{\nu }$ 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 $\lceil \nu -1\rceil$ times differentiable.^[3]

Simplification for specific values of ν

Simplification for ν half integer

When $\nu =p+1/2,\ p\in \mathbb {N} ^{+}$ , the Matérn covariance can be written as a product of an exponential and a polynomial of order $p$ :^[4]

C_{p+1/2}(d)=\sigma ^{2}\exp \left(-{\frac {{\sqrt {2\nu }}d}{\rho }}\right){\frac {\Gamma (p+1)}{\Gamma (2p+1)}}\sum _{i=0}^{p}{\frac {(p+i)!}{i!(p-i)!}}\left({\frac {{\sqrt {8\nu }}d}{\rho }}\right)^{p-i},

which gives:

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

The Gaussian case in the limit of infinite ν

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

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

Taylor series at zero and spectral moments

The behavior for $d\rightarrow 0$ can be obtained by the following Taylor series:

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^{5}\right)\right).

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

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

References

↑ Minasny, B.; McBratney, A. B. (2005). "The Matérn function as a general model for soil variograms". Geoderma. 128 (3–4): 192–207. doi:10.1016/j.geoderma.2005.04.003.
↑ Rasmussen, Carl Edward (2006) Gaussian Processes for Machine Learning
↑ Rasmussen, Carl Edward (2006) Gaussian Processes Covariance Functions and Classification. Presentation at Gaussian Processes in Practice
↑ Abramowitz and Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. ISBN 0-486-61272-4.

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