Matérn covariance function

In statistics, the Matérn covariance, also called the Matérn kernel,[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 named after the Swedish forestry statistician Bertil Matérn.[2] It specifies the covariance between two measurements as a function of the distance ${\displaystyle d}$ 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 [3]

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

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

A Gaussian process with Matérn covariance is ${\displaystyle \lceil \nu \rceil -1}$ times differentiable in the mean-square sense.[3][4]

Spectral density

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

${\displaystyle S(f)=\sigma ^{2}{\frac {2^{n}\pi ^{n/2}\Gamma (\nu +{\frac {n}{2}})(2\nu )^{\nu }}{\Gamma (\nu )\rho ^{2\nu }}}\left({\frac {2\nu }{\rho ^{2}}}+4\pi ^{2}f^{2}\right)^{-\left(\nu +{\frac {n}{2}}\right)}.}$[3]

Simplification for specific values of ν

Simplification for ν half integer

When ${\displaystyle \nu =p+1/2,\ p\in \mathbb {N} ^{+}}$ , the Matérn covariance can be written as a product of an exponential and a polynomial of degree ${\displaystyle p}$.[5][6] The modified Bessel function of a fractional order is given by Equations 10.1.9 and 10.2.15[7] as

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

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

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

which gives:

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

The Gaussian case in the limit of infinite ν

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

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

Taylor series at zero and spectral moments

The behavior for ${\displaystyle d\rightarrow 0}$ can be obtained by the following Taylor series (reference is needed, the formula below leads to division by zero in case ${\displaystyle \nu =1}$):

$${\displaystyle C_{\nu }(d)=\sigma ^{2}\left(1+{\frac {\nu }{2(1-\nu )}}\left({\frac {d}{\rho }}\right)^{2}+{\frac {\nu ^{2}}{8(2-3\nu +\nu ^{2})}}\left({\frac {d}{\rho }}\right)^{4}+{\mathcal {O}}\left(d^{5}\right)\right).}$$

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

${\displaystyle {\begin{aligned}\lambda _{0}&=C_{\nu }(0)=\sigma ^{2},\\[8pt]\lambda _{2}&=-\left.{\frac {\partial ^{2}C_{\nu }(d)}{\partial d^{2}}}\right|_{d=0}={\frac {\sigma ^{2}\nu }{\rho ^{2}(\nu -1)}}.\end{aligned}}}$

See also

• Radial basis function

References

1. Genton, Marc G. (1 March 2002). "Classes of kernels for machine learning: a statistics perspective". The Journal of Machine Learning Research. 2 (3/1/2002): 303–304.
2. 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.
3. Rasmussen, Carl Edward and Williams, Christopher K. I. (2006) Gaussian Processes for Machine Learning
4. Santner, T. J., Williams, B. J., & Notz, W. I. (2013). The design and analysis of computer experiments. Springer Science & Business Media.
5. Stein, M. L. (1999). Interpolation of spatial data: some theory for kriging. Springer Series in Statistics.
6. 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.
7. Abramowitz and Stegun (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. ISBN 0-486-61272-4.