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 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 [3]
C
ν
(
d
)
=
σ
2
2
1
−
ν
Γ
(
ν
)
(
2
ν
d
ρ
)
ν
K
ν
(
2
ν
d
ρ
)
,
{\displaystyle C_{\nu }(d)=\sigma ^{2}{\frac {2^{1-\nu }}{\Gamma (\nu )}}{\Bigg (}{\sqrt {2\nu }}{\frac {d}{\rho }}{\Bigg )}^{\nu }K_{\nu }{\Bigg (}{\sqrt {2\nu }}{\frac {d}{\rho }}{\Bigg )},}
where
Γ
{\displaystyle \Gamma }
is the gamma function ,
K
ν
{\displaystyle K_{\nu }}
is the modified Bessel function of the second kind, and ρ and
ν
{\displaystyle \nu }
are positive parameters of the covariance.
A Gaussian process with Matérn covariance is
⌈
ν
⌉
−
1
{\displaystyle \lceil \nu \rceil -1}
times differentiable in the mean-square sense.[3] [4]
Spectral density
The power spectrum of a process with Matérn covariance defined on
R
n
{\displaystyle \mathbb {R} ^{n}}
is the (n -dimensional) Fourier transform of the Matérn covariance function (see Wiener–Khinchin theorem ). Explicitly, this is given by
S
(
f
)
=
σ
2
2
n
π
n
/
2
Γ
(
ν
+
n
2
)
(
2
ν
)
ν
Γ
(
ν
)
ρ
2
ν
(
2
ν
ρ
2
+
4
π
2
f
2
)
−
(
ν
+
n
2
)
.
{\displaystyle S(f)=\sigma ^{2}{\frac {2^{n}\pi ^{n/2}\Gamma (\nu +{\frac {n}{2}})(2\nu )^{\nu }}{\Gamma (\nu )\rho ^{2\nu }}}\left({\frac {2\nu }{\rho ^{2}}}+4\pi ^{2}f^{2}\right)^{-\left(\nu +{\frac {n}{2}}\right)}.}
[3]
Simplification for specific values of ν
Simplification for ν half integer
When
ν
=
p
+
1
/
2
,
p
∈
N
+
{\displaystyle \nu =p+1/2,\ p\in \mathbb {N} ^{+}}
, the Matérn covariance can be written as a product of an exponential and a polynomial of order
p
{\displaystyle p}
:[5]
C
p
+
1
/
2
(
d
)
=
σ
2
exp
(
−
2
p
+
1
d
ρ
)
p
!
(
2
p
)
!
∑
i
=
0
p
(
p
+
i
)
!
i
!
(
p
−
i
)
!
(
2
2
p
+
1
d
ρ
)
p
−
i
,
{\displaystyle C_{p+1/2}(d)=\sigma ^{2}\exp \left(-{\frac {{\sqrt {2p+1}}d}{\rho }}\right){\frac {p!}{(2p)!}}\sum _{i=0}^{p}{\frac {(p+i)!}{i!(p-i)!}}\left({\frac {2{\sqrt {2p+1}}d}{\rho }}\right)^{p-i},}
which gives:
for
ν
=
1
/
2
(
p
=
0
)
{\displaystyle \nu =1/2\ (p=0)}
:
C
1
/
2
(
d
)
=
σ
2
exp
(
−
d
ρ
)
,
{\displaystyle C_{1/2}(d)=\sigma ^{2}\exp \left(-{\frac {d}{\rho }}\right),}
for
ν
=
3
/
2
(
p
=
1
)
{\displaystyle \nu =3/2\ (p=1)}
:
C
3
/
2
(
d
)
=
σ
2
(
1
+
3
d
ρ
)
exp
(
−
3
d
ρ
)
,
{\displaystyle C_{3/2}(d)=\sigma ^{2}\left(1+{\frac {{\sqrt {3}}d}{\rho }}\right)\exp \left(-{\frac {{\sqrt {3}}d}{\rho }}\right),}
for
ν
=
5
/
2
(
p
=
2
)
{\displaystyle \nu =5/2\ (p=2)}
:
C
5
/
2
(
d
)
=
σ
2
(
1
+
5
d
ρ
+
5
d
2
3
ρ
2
)
exp
(
−
5
d
ρ
)
.
{\displaystyle C_{5/2}(d)=\sigma ^{2}\left(1+{\frac {{\sqrt {5}}d}{\rho }}+{\frac {5d^{2}}{3\rho ^{2}}}\right)\exp \left(-{\frac {{\sqrt {5}}d}{\rho }}\right).}
The Gaussian case in the limit of infinite ν
As
ν
→
∞
{\displaystyle \nu \rightarrow \infty }
, the Matérn covariance converges to the squared exponential covariance function
lim
ν
→
∞
C
ν
(
d
)
=
σ
2
exp
(
−
d
2
2
ρ
2
)
.
{\displaystyle \lim _{\nu \rightarrow \infty }C_{\nu }(d)=\sigma ^{2}\exp \left(-{\frac {d^{2}}{2\rho ^{2}}}\right).}
Taylor series at zero and spectral moments
The behavior for
d
→
0
{\displaystyle d\rightarrow 0}
can be obtained by the following Taylor series (reference is needed, the formula below leads to division by zero in case
ν
=
1
{\displaystyle \nu =1}
):
C
ν
(
d
)
=
σ
2
(
1
+
ν
2
(
1
−
ν
)
(
d
ρ
)
2
+
ν
2
8
(
2
−
3
ν
+
ν
2
)
(
d
ρ
)
4
+
O
(
d
5
)
)
.
{\displaystyle C_{\nu }(d)=\sigma ^{2}\left(1+{\frac {\nu }{2(1-\nu )}}\left({\frac {d}{\rho }}\right)^{2}+{\frac {\nu ^{2}}{8(2-3\nu +\nu ^{2})}}\left({\frac {d}{\rho }}\right)^{4}+{\mathcal {O}}\left(d^{5}\right)\right).}
When defined, the following spectral moments can be derived from the Taylor series:
λ
0
=
C
ν
(
0
)
=
σ
2
,
λ
2
=
−
∂
2
C
ν
(
d
)
∂
d
2
|
d
=
0
=
σ
2
ν
ρ
2
(
ν
−
1
)
.
{\displaystyle {\begin{aligned}\lambda _{0}&=C_{\nu }(0)=\sigma ^{2},\\[8pt]\lambda _{2}&=-\left.{\frac {\partial ^{2}C_{\nu }(d)}{\partial d^{2}}}\right|_{d=0}={\frac {\sigma ^{2}\nu }{\rho ^{2}(\nu -1)}}.\end{aligned}}}
See also
References
^ Genton, Marc G. (1 March 2002). "Classes of kernels for machine learning: a statistics perspective" . The Journal of Machine Learning Research . 2 (3/1/2002): 303–304.
^ Minasny, B.; McBratney, A. B. (2005). "The Matérn function as a general model for soil variograms". Geoderma . 128 (3–4): 192–207. doi :10.1016/j.geoderma.2005.04.003 .
^ a b c Rasmussen, Carl Edward and Williams, Christopher K. I. (2006) Gaussian Processes for Machine Learning
^ Santner, T. J., Williams, B. J., & Notz, W. I. (2013). The design and analysis of computer experiments. Springer Science & Business Media.
^ Abramowitz and Stegun (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . ISBN 0-486-61272-4 .