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.

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]

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]

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

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

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

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

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

and the definition of the modified Bessel functions of the first ${\displaystyle I_{\nu }(x)=\sum _{m=0}^{\infty }{\frac {1}{m!\,\Gamma (m+\nu +1)}}\left({\frac {x}{2}}\right)^{2m+\nu },}$

the behavior for ${\displaystyle d\rightarrow 0}$ can be obtained by the following Taylor series (when ${\displaystyle \nu }$ is not an integer and bigger than 2):

$${\displaystyle C_{\nu }(d)=\sigma ^{2}\left(1+{\frac {\nu }{2(1-\nu )}}\left({\frac {d}{\rho }}\right)^{2}+{\frac {\nu ^{2}}{8(2-3\nu +\nu ^{2})}}\left({\frac {d}{\rho }}\right)^{4}+{\mathcal {O}}\left(d^{6\wedge (2\nu )}\right)\right),\,\,\nu >2.}$$ ^[8]

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

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

For the case of ${\displaystyle \nu \in (0,1)\cup (1,2)}$, similar Taylor series can be obtained: $${\displaystyle C_{\nu }(d)=\sigma ^{2}\left(1+{\frac {\nu }{2(1-\nu )}}\left({\frac {d}{\rho }}\right)^{2}-{\frac {\Gamma (1-\nu )}{\Gamma (1+\nu )}}\left({\frac {\nu }{2}}\right)^{\nu }\left({\frac {d}{\rho }}\right)^{2\nu }+{\mathcal {O}}\left(d^{4\wedge (2\nu +2)}\right)\right),\,\,\nu \in (0,1)\cup (1,2).}$$ When ${\displaystyle \nu }$ is an integer limiting values should be taken, (see ^[8]).

• Radial basis function