Matérn covariance function

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]

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:

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

• Radial basis function