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 specifies the covariance between two measurements as a function of the distance
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 measurements taken at two points separated by d distance units is given by [3] where
is the gamma function,
ν
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
⌈ ν ⌉ − 1
times differentiable in the mean-square sense.
[3][4] 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).
ν = p + 1
, the Matérn covariance can be written as a product of an exponential and a polynomial of degree
[5][6] The modified Bessel function of a fractional order is given by Equations 10.1.9 and 10.2.15[7] as
This allows for the Matérn covariance of half-integer values of
, the Matérn covariance converges to the squared exponential covariance function From the basic relation satisfied by the Gamma function
sin ( π z )
and the basic relation satisfied by the Modified Bessel Function of the second
sin ( π ν )
and the definition of the modified Bessel functions of the first
can be obtained by the following Taylor series (when
is not an integer and bigger than 2):
[8] When defined, the following spectral moments can be derived from the Taylor series: For the case of
, similar Taylor series can be obtained:
is an integer limiting values should be taken, (see [8]).