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. between the points at which they are taken. 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 measurements taken at two points separated by d distance units is given by

is the gamma function is the modified Bessel function of the second kind, and ρ are positive parameters of the covariance.

A Gaussian process times differentiable in the mean-square sense.

Spectral density

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

, the Matérn covariance. The modified Bessel function of a fractional order is given by Equations 10.1.9 and 10.2.15 as

This allows for the Matérn covariance of half-integer to be expressed as

which gives:

The Gaussian case in the limit of infinite ν

, the Matérn covariance converges to the squared exponential covariance function

Taylor series at zero and spectral moments

and the basic relation satisfied by the Modified Bessel Function of the second

can be obtained by the following Taylor series is not an integer and bigger than 2):

When defined, the following spectral moments can be derived from the Taylor series:

, similar Taylor series is an integer, limiting values should be taken (see ).

See also

• Radial basis function

References