Tool in multivariate statistical analysis
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 specifies the covariance between two measurements as a function of the distance formula_1 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 
formula_2
where formula_3 is the gamma function, formula_4 is the modified Bessel function of the second kind, and "ρ" and formula_5 are positive parameters of the covariance.
A Gaussian process with Matérn covariance is formula_6 times differentiable in the mean-square sense.
Spectral density.
The power spectrum of a process with Matérn covariance defined on formula_7 is the ("n"-dimensional) Fourier transform of the Matérn covariance function (see Wiener–Khinchin theorem). Explicitly, this is given by
formula_8
Simplification for specific values of "ν".
Simplification for "ν" half integer.
When formula_9 , the Matérn covariance can be written as a product of an exponential and a polynomial of degree formula_10. The modified Bessel function of a fractional order is given by Equations 10.1.9 and 10.2.15 as
formula_11 .
This allows for the Matérn covariance of half-integer values of formula_5 to be expressed as
formula_13
which gives:
The Gaussian case in the limit of infinite "ν".
As formula_20, the Matérn covariance converges to the squared exponential covariance function
formula_21
Taylor series at zero and spectral moments.
From the basic relation satisfied by the Gamma function formula_22 
and the basic relation satisfied by the Modified Bessel Function of the second
formula_23 
and the definition of the modified Bessel functions of the first 
formula_24 
the behavior for formula_25 can be obtained by the following Taylor series (when formula_26 is not an integer and bigger than 2):
formula_27 
When defined, the following spectral moments can be derived from the Taylor series:
 formula_28
For the case of formula_29, similar Taylor series can be obtained: 
formula_30
When formula_26 is an integer limiting values should be taken, (see ).