Matérn covariance function
In statistics, the Matérn covariance, also called the Matérn kernel, 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. 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 where formula_2 is the gamma function, formula_3 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 formula_4 times differentiable in the mean-square sense. Spectral density. The power spectrum of a process with Matérn covariance defined on formula_5 is the ("n"-dimensional) Fourier transform of the Matérn covariance function (see Wiener–Khinchin theorem). Explicitly, this is given by Simplification for specific values of "ν". Simplification for "ν" half integer. When formula_7 , the Matérn covariance can be  written as a product of an exponential and a polynomial of order formula_8: formula_9 which gives: The Gaussian case in the limit of infinite "ν". As formula_16, the Matérn covariance converges to the squared exponential covariance function Taylor series at zero and spectral moments. The behavior for formula_18 can be obtained by the following Taylor series (reference is needed, the formula below leads to division by zero in case formula_19): formula_20 When defined, the following spectral moments can be derived from the Taylor series: