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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]
A Gaussian process with Matérn covariance is times differentiable in the mean-square sense.[3][4]
Spectral density
The power spectrum of a process with Matérn covariance defined on is the (n-dimensional) Fourier transform of the Matérn covariance function (see Wiener–Khinchin theorem). Explicitly, this is given by
^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.