Matern

Matérn, or the Matérn covariance function, is a class of stationary covariance functions used in spatial statistics and Gaussian process modeling. Named after the statistician Bertil Matérn, it provides a flexible way to encode spatial dependency and smoothness of random fields.

The Matérn covariance between two points separated by distance h is given by:

C(h) = sigma^2 * (2^{1-nu} / Gamma(nu)) * (sqrt(2 nu) h / ell)^nu * K_nu (sqrt(2 nu) h / ell),

where h is the distance between points, ell > 0 is a length-scale parameter, nu > 0 is the smoothness parameter, sigma^2 > 0 is the variance, and K_nu is the modified Bessel function of the second kind of order nu.

Special cases and interpretation: when nu = 1/2, the Matérn function reduces to the exponential covariance C(h) = sigma^2 exp(-h/ell), representing a rough process. As nu increases, the associated random field becomes smoother; in the limit as nu approaches infinity, the Matérn covariance converges to the Gaussian (squared exponential) covariance. The parameter ell controls the range of spatial dependence, with larger ell implying correlations persist over longer distances.

Path properties: for nu > k + 0.5, the Gaussian process with Matérn covariance is k-times mean-square differentiable; for nu <= 0.5, sample paths are continuous but not differentiable.

Applications: Matérn covariances are widely used in geostatistics for kriging and spatial interpolation, as well as in Bayesian modeling and Gaussian-process-based machine learning, due to their interpretability and flexibility in modelling smoothness.

History: The function was introduced by Matérn in the 1960s and has since become a standard choice for modeling spatial covariance.