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Matérn covariance function
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<h2 id="definition">Definition</h2> <p>The Matérn covariance between measurements taken at two points separated by <i>d</i> distance units is given by <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> 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 )},} <p>where Γ {\displaystyle \Gamma } is the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>, K ν {\displaystyle K_{\nu }} is the modified <a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a> of the second kind, and <i>ρ</i> and ν {\displaystyle \nu } are positive <a href="/facts/Parameter/zFvVFMv9">parameters</a> of the covariance. </p><p>A <a href="/facts/Gaussian_process/MrBq7kYW">Gaussian process</a> with Matérn covariance is ⌈ ν ⌉ − 1 {\displaystyle \lceil \nu \rceil -1} times differentiable in the mean-square sense.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h2 id="spectral-density">Spectral density</h2> <p>The power spectrum of a process with Matérn covariance defined on R n {\displaystyle \mathbb {R} ^{n}} is the (<i>n</i>-dimensional) <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of the Matérn covariance function (see <a href="/facts/Wiener%25E2%2580%2593Khinchin_theorem/CmmaWroa">Wiener–Khinchin theorem</a>). Explicitly, this is given by </p> 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)}.} <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> <h2 id="simplification-for-specific-values-of-">Simplification for specific values of <i>ν</i></h2> <h3>Simplification for <i>ν</i> half integer</h3> <p>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 degree p {\displaystyle p} .<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> The modified Bessel function of a fractional order is given by Equations 10.1.9 and 10.2.15<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> as </p><p> π 2 z K p + 1 / 2 ( z ) = π 2 z e − z ∑ k = 0 n ( n + k ) ! k ! Γ ( n − k + 1 ) ( 2 z ) − k {\displaystyle {\sqrt {\frac {\pi }{2z}}}K_{p+1/2}(z)={\frac {\pi }{2z}}e^{-z}\sum _{k=0}^{n}{\frac {(n+k)!}{k!\Gamma (n-k+1)}}\left(2z\right)^{-k}} . </p><p>This allows for the Matérn covariance of <a href="/facts/Half-integer/oJ29mf8G">half-integer</a> values of ν {\displaystyle \nu } to be expressed as </p><p> 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},} </p><p>which gives: </p> <ul><li>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),} </li> <li>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),} </li> <li>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).} </li></ul> <h3>The Gaussian case in the limit of infinite <i>ν</i></h3> <p>As ν → ∞ {\displaystyle \nu \rightarrow \infty } , the Matérn covariance converges to the squared exponential <a href="/facts/Covariance_function/pqUTwANG">covariance function</a> </p> 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).} <h2 id="taylor-series-at-zero-and-spectral-moments">Taylor series at zero and spectral moments</h2> <p>From the basic relation satisfied by the Gamma function Γ ( z ) Γ ( 1 − z ) = π sin ( π z ) {\displaystyle \Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin(\pi z)}}} and the basic relation satisfied by the Modified Bessel Function of the second </p><p> K ν ( x ) = π 2 I − ν ( x ) − I ν ( x ) sin ( π ν ) {\displaystyle K_{\nu }(x)={\frac {\pi }{2}}{\frac {I_{-\nu }(x)-I_{\nu }(x)}{\sin(\pi \nu )}}} </p><p>and the definition of the modified Bessel functions of the first I ν ( x ) = ∑ m = 0 ∞ 1 m ! Γ ( m + ν + 1 ) ( x 2 ) 2 m + ν , {\displaystyle I_{\nu }(x)=\sum _{m=0}^{\infty }{\frac {1}{m!\,\Gamma (m+\nu +1)}}\left({\frac {x}{2}}\right)^{2m+\nu },} </p><p>the behavior for d → 0 {\displaystyle d\rightarrow 0} can be obtained by the following <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> (when ν {\displaystyle \nu } is not an integer and bigger than 2): </p><p> 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 6 ∧ ( 2 ν ) ) ) , ν > 2. {\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^{6\wedge (2\nu )}\right)\right),\,\,\nu >2.} <a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>When defined, the following spectral moments can be derived from the Taylor series: </p> 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}}} <p>For the case of ν ∈ ( 0 , 1 ) ∪ ( 1 , 2 ) {\displaystyle \nu \in (0,1)\cup (1,2)} , similar <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> can be obtained: C ν ( d ) = σ 2 ( 1 + ν 2 ( 1 − ν ) ( d ρ ) 2 − Γ ( 1 − ν ) Γ ( 1 + ν ) ( ν 2 ) ν ( d ρ ) 2 ν + O ( d 4 ∧ ( 2 ν + 2 ) ) ) , ν ∈ ( 0 , 1 ) ∪ ( 1 , 2 ) . {\displaystyle C_{\nu }(d)=\sigma ^{2}\left(1+{\frac {\nu }{2(1-\nu )}}\left({\frac {d}{\rho }}\right)^{2}-{\frac {\Gamma (1-\nu )}{\Gamma (1+\nu )}}\left({\frac {\nu }{2}}\right)^{\nu }\left({\frac {d}{\rho }}\right)^{2\nu }+{\mathcal {O}}\left(d^{4\wedge (2\nu +2)}\right)\right),\,\,\nu \in (0,1)\cup (1,2).} When ν {\displaystyle \nu } is an integer limiting values should be taken, (see <a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>). </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Radial_basis_function/QnWM1mBN">Radial basis function</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Genton, Marc G. (1 March 2002). "Classes of kernels for machine learning: a statistics perspective". The Journal of Machine Learning Research. 2: 303–304. <a href="https://jmlr.org/papers/v2/genton01a.html" target="_blank">https://jmlr.org/papers/v2/genton01a.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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 href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Rasmussen, Carl Edward and Williams, Christopher K. I. (2006) Gaussian Processes for Machine Learning <a href="http://www.gaussianprocess.org/gpml/chapters/RW4.pdf" target="_blank">http://www.gaussianprocess.org/gpml/chapters/RW4.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Rasmussen, Carl Edward and Williams, Christopher K. I. (2006) Gaussian Processes for Machine Learning <a href="http://www.gaussianprocess.org/gpml/chapters/RW4.pdf" target="_blank">http://www.gaussianprocess.org/gpml/chapters/RW4.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Santner, T. J., Williams, B. J., & Notz, W. I. (2013). The design and analysis of computer experiments. Springer Science & Business Media. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Rasmussen, Carl Edward and Williams, Christopher K. I. (2006) Gaussian Processes for Machine Learning <a href="http://www.gaussianprocess.org/gpml/chapters/RW4.pdf" target="_blank">http://www.gaussianprocess.org/gpml/chapters/RW4.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Stein, M. L. (1999). Interpolation of spatial data: some theory for kriging. Springer Series in Statistics. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Peter Guttorp & Tilmann Gneiting, 2006. "Studies in the history of probability and statistics XLIX On the Matern correlation family," Biometrika, Biometrika Trust, vol. 93(4), pages 989-995, December. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Abramowitz and Stegun (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office. ISBN 0-486-61272-4. <a href="0-486-61272-4" target="_blank">0-486-61272-4</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Cheng, Dan (July 2024). "Smooth Matérn Gaussian random fields: Euler characteristic, expected number and height distribution of critical points". Statistics & Probability Letters. 210: 110116. arXiv:2307.01978. doi:10.1016/j.spl.2024.110116. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Cheng, Dan (July 2024). "Smooth Matérn Gaussian random fields: Euler characteristic, expected number and height distribution of critical points". Statistics & Probability Letters. 210: 110116. arXiv:2307.01978. doi:10.1016/j.spl.2024.110116. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> </ol>