In statistics, the Matérn covariance (named after the Swedish forestry statistician Bertil Matérn) 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 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.
Contents
Definition
The Matérn covariance between two points separated by d distance units is given by
where
A Gaussian process with Matérn covariance has sample paths that are
Simplification for ν half integer
When
which gives:
The Gaussian case in the limit of infinite ν
As
Taylor series at zero and spectral moments
The behavior for
When defined, the following spectral moments can be derived from the Taylor series: