Matérn kernel

Gaussian processes model the correlation (or covariance) between two function values \(f(x), f(x')\) by means of a covariance function \(k(x,x') = k(||x-x'||)\), also called kernel. The kernel is monotonically decreasing for increasing distance \(d = ||x-x'||\) such that far apart function values are uncorrelated while close-by function values are strongly correlated.

The distance \(d\) between function values is defined as the scaled Euclidean distance

\[d = ||x-x'||=\sqrt{\sum_{i=1}^{D}\frac{(x_i−x_i^{'})^{2}}{l_i^{2}}},\]

where the hyperparameters \(l_1,\dots,l_D\) determine the characteristic length scales at which the covariance between separated function values becomes negligible.

The Matérn covariance function is defined as

\[k_{\nu}(x, x') = \frac{\sigma_0^2}{\Gamma(\nu)2^{\nu-1}} \left[\sqrt{2\nu} d \right]^\nu K_\nu\left( \sqrt{2\nu} d\right),\]

where \(\sigma_0\) is a hyperparameter that determines the standard deviation of possible function values, \(K_\nu(\cdot)\) is a modified Bessel function, and \(\Gamma(\cdot)\) is the gamma function.

matern_order (int)

For half-integer \(\nu\) the Matérn functions simplify e.g. to

  • \(k_{1/2}(x,x′)= \sigma^2 \exp(−d)\) ,

  • \(k_{3/2}(x,x′)= \sigma^2 (1+3-\sqrt{3}d)\exp(-\sqrt{3}d)\) ,

  • \(k_{5/2}(x,x′)= \sigma^2 (1+5-\sqrt{5}d + \frac{5}{3}d^2)\exp(-\sqrt{5}d)\) .

A Gaussian process with Matérn covariance is \(\lceil \nu \rceil\) times differentiable in the mean-square sense. The matern_order is defined as \(m = 2\nu\). Hence, \(m=5\) is best suited for describing two-times differentiable functions, \(m=3\) for one-times differentiable functions, and \(m=1\) for functions, that are not differentiable.

Default: 5 Choices: 1, 3, 5.

Note

For matern_order=1 it is not possible to add derivative information.