Matern¶

class probnum.kernels.Matern(input_dim, lengthscale=1.0, nu=1.5)¶

Bases: probnum.kernels.Kernel

Matern kernel.

Covariance function defined by \(k(x_0, x_1) = \frac{1}{\Gamma(\nu)2^{ \nu-1}}\big(\frac{\sqrt{2\nu}}{l} \lVert x_0 , x_1\rVert \big)^\nu K_\nu\big(\frac{\sqrt{2\nu}}{l} \lVert x_0 , x_1 \rVert \big)\), where \(K_\nu\) is a modified Bessel function. The Matern kernel generalizes the ExpQuad kernel via its additional parameter \(\nu\) controlling the smoothness of the function. For \(\nu \rightarrow \infty\) the Matern kernel converges to the ExpQuad kernel. A Gaussian process with Matern covariance function is \(\lceil \nu \rceil - 1\) times differentiable.

Parameters

input_dim (Integral) – Input dimension of the kernel.
lengthscale (Number) – Lengthscale of the kernel. Describes the input scale on which the process varies.
nu (Number) – Hyperparameter controlling differentiability.

See also

ExpQuad: Exponentiated Quadratic / RBF kernel.

Examples

>>> import numpy as np
>>> from probnum.kernels import Matern
>>> K = Matern(input_dim=1, lengthscale=0.1, nu=2.5)
>>> K(np.linspace(0, 1, 3)[:, None])
array([[1.00000000e+00, 7.50933789e-04, 3.69569622e-08],
       [7.50933789e-04, 1.00000000e+00, 7.50933789e-04],
       [3.69569622e-08, 7.50933789e-04, 1.00000000e+00]])

Attributes Summary

`input_dim`	Dimension of arguments of the covariance function.
`output_dim`	Dimension of the evaluated covariance function.

Methods Summary

__call__(x0[, x1])

Evaluate the kernel.

Attributes Documentation

input_dim¶

Dimension of arguments of the covariance function.

The dimension of inputs to the covariance function \(k : \mathbb{R}^{ d_{in}} \times \mathbb{R}^{d_{in}} \rightarrow \mathbb{R}^{d_{out} \times d_{out}}\).

Return type: int

output_dim¶

Dimension of the evaluated covariance function.

The resulting evaluated kernel \(k(x_0, x_1) \in \mathbb{R}^{d_{out} \times d_{out}}\) has shape=(output_dim, output_dim).

Return type: int

Methods Documentation

__call__(x0, x1=None)[source]¶

Evaluate the kernel.

Computes the covariance function at x0 and x1. If the inputs have more than one dimension the covariance function is evaluated pairwise for all observations determined by the first dimension of x0 and x1. If only x0 is given the kernel matrix \(K=k(X_0, X_0)\) is computed.

Parameters

x0 (ndarray) – shape=(input_dim,) or (n0, input_dim) – First input.
x1 (Optional[ndarray]) – shape=(input_dim,) or (n1, input_dim) – Second input.

Returns

shape=(), (output_dim, output_dim) or (n0, n1) or (n0, n1, output_dim, output_dim) – Kernel evaluated at x0 and x1 or kernel matrix containing pairwise evaluations for all observations in x0 (and x1).

Return type

cov