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.
The Matérn covariance between two points separated by d distance units is given by 
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
Simplification for specific values of νEdit
Simplification for ν half integerEdit
When , the Matérn covariance can be written as a product of an exponential and a polynomial of order :
- for :
- for :
- for :
The Gaussian case in the limit of infinite νEdit
As , the Matérn covariance converges to the squared exponential covariance function
Taylor series at zero and spectral momentsEdit
The behavior for can be obtained by the following Taylor series:
When defined, the following spectral moments can be derived from the Taylor series:
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