glearn.kernels.Matern#

class glearn.kernels.Matern#

Matern kernel.

The kernel object is used as input argument to the instants of glearn.Covariance class.

Note

For the methods of this class, see the base class glearn.kernels.Kernel.

Parameters:
nufloat, default=0.5

The parameter \(\nu\) of the Matern function (see Notes below).

See also

glearn.Covariance

Notes

Matern Kernel Function:

Matern correlation (set \(\nu\) by nu) is defined as

\[k(x | \nu) = \frac{2^{1-\nu}}{\Gamma(\nu)} \left( \sqrt{2 \nu} x \right)^{\nu} K_{\nu}\left( \sqrt{2 \nu} x \right),\]

where \(K_{\nu}\) is the modified Bessel function of the second kind and \(\Gamma\) is the Gamma function. Both \(K_{\nu}\) and \(\Gamma\) are computed efficiently using the special_functions.besselk() and special_functions.lngamma() functions.

The Matern kernel for specific values of \(\nu\) has simplified representation:

  • The Matern kernel with \(\nu=\frac{1}{2}\) is equivalent to the exponential kernel, see glearn.kernels.Exponential.

  • \(\nu = \infty\) is equivalent to the square exponential kernel, see glearn.kernels.SquareExponential. If \(\nu > 100\), it is assumed that \(\nu\) is infinity.

  • If \(\nu = \frac{3}{2}\) the following expression of Matern kernel is used:

    \[k(x | \textstyle{\frac{3}{2}}) = \left(1+ \sqrt{3} x \right) e^{-\sqrt{3} x}.\]

  • If \(\nu = \frac{5}{2}\), the Matern kernel is computed with:

    \[k(x | \textstyle{\frac{5}{2}}) = \left(1+ \sqrt{5} x + \frac{5}{3} x^2 \right) e^{-\sqrt{5} x}.\]

First Derivative:

The first derivative of the kernel is computed as follows:

\[\frac{\mathrm{d} k(x)}{\mathrm{d}x} = c z^{\nu - 1} \left( \nu K_{\nu}(z) + z K'_{\nu}(z) \right),\]

where \(K'_{\nu}(z)\) is the first derivative of the modified Bessel function of the second kind with respect to \(z\), and

\[\begin{split}z = \begin{cases} \sqrt{2 \nu} \epsilon, & \vert x \vert < \epsilon, \\ \sqrt{2 \nu} x, & \vert x \vert \geq \epsilon, \end{cases}\end{split}\]

and

\[c = \frac{2^{1-\nu}}{\Gamma(\nu)} \sqrt{2 \nu}.\]

Second Derivative:

The second derivative of the kernel is computed as follows:

\[\frac{\mathrm{d} k(x)}{\mathrm{d}x} = c z^{\nu-2} \left( \nu (\nu-1) K_{\nu}(z) + 2 z K'_{\nu}(z) + z^2 K''_{\nu}(z) \right),\]

where \(K''_{\nu}(z)\) is the second derivative of the modified Bessel function of the second kind with respect to \(z\).

Examples

Create Kernel Object:

>>> from glearn import kernels

>>> # Create an exponential kernel
>>> kernel = kernels.Matern(nu=1.5)

>>> # Evaluate kernel at the point x=0.5
>>> x = 0.5
>>> kernel(x)
0.7848876539574506

>>> # Evaluate first derivative of kernel at the point x=0.5
>>> kernel(x, derivative=1)
0.6309300390811722

>>> # Evaluate second derivative of kernel at the point x=0.5
>>> kernel(x, derivative=2)
0.16905719445233686

>>> # Plot kernel and its first and second derivative
>>> kernel.plot()
../_images/kernel_matern.png

Where to Use Kernel Object:

Use the kernel object to define a covariance object:

>>> # Generate a set of sample points
>>> from glearn.sample_data import generate_points
>>> points = generate_points(num_points=50)

>>> # Create covariance object of the points with the above kernel
>>> from glearn import covariance
>>> cov = glearn.Covariance(points, kernel=kernel)