Matérn covariance function

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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 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 [3]

[math]\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), }[/math]

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

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

Spectral density

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

[math]\displaystyle{ S(f)=\sigma^2\frac{2^n\pi^{\frac{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)}. }[/math][3]

Simplification for specific values of ν

Simplification for ν half integer

When [math]\displaystyle{ \nu = p+1/2,\ p\in \mathbb{N}^+ }[/math] , the Matérn covariance can be written as a product of an exponential and a polynomial of order [math]\displaystyle{ p }[/math]:[5]

[math]\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}, }[/math]

which gives:

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

The Gaussian case in the limit of infinite ν

As [math]\displaystyle{ \nu\rightarrow\infty }[/math], the Matérn covariance converges to the squared exponential covariance function

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

Taylor series at zero and spectral moments

The behavior for [math]\displaystyle{ d\rightarrow0 }[/math] can be obtained by the following Taylor series (reference is needed, the formula below leads to division by zero in case [math]\displaystyle{ \nu = 1 }[/math]):

[math]\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) . }[/math]

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

[math]\displaystyle{ \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} }[/math]

See also

References

  1. Genton, Marc G. (1 March 2002). "Classes of kernels for machine learning: a statistics perspective" (in EN). The Journal of Machine Learning Research 2 (3/1/2002): 303-304. https://dl.acm.org/doi/10.5555/944790.944815. 
  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. 3.0 3.1 3.2 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. Abramowitz and Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. ISBN 0-486-61272-4. https://archive.org/details/handbookofmathe000abra.