Matern¶

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

Bases: Kernel, IsotropicMixin

Matern kernel.

Covariance function defined by

\begin{equation} k(x_0, x_1) = \frac{1}{\Gamma(\nu) 2^{\nu - 1}} \left( \frac{\sqrt{2 \nu}}{l} \lVert x_0 - x_1 \rVert_2 \right)^\nu K_\nu \left( \frac{\sqrt{2 \nu}}{l} \lVert x_0 - x_1 \rVert_2 \right), \end{equation}

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_shape (Union[int, Integral, integer, Iterable[Union[int, Integral, integer]]]) – Shape of the kernel’s input.
lengthscale (Union[int, float, complex, Number, number]) – Lengthscale \(l\) of the kernel. Describes the input scale on which the process varies.
nu (Union[int, float, complex, Number, number]) – Hyperparameter \(\nu\) controlling differentiability.

See also

ExpQuad: Exponentiated Quadratic / RBF kernel.
ProductMatern: Product Matern kernel.

Examples

>>> import numpy as np
>>> from probnum.randprocs.kernels import Matern
>>> K = Matern(input_shape=(), lengthscale=0.1, nu=2.5)
>>> xs = np.linspace(0, 1, 3)
>>> K.matrix(xs)
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_ndim`	Syntactic sugar for `len(input_shape)`.
`input_shape`	Shape of single, i.e. non-batched, arguments of the covariance function.
`output_ndim`	Syntactic sugar for `len(output_shape)`.
`output_shape`	Shape of single, i.e. non-batched, return values of the covariance function.

Methods Summary

`__call__`(x0, x1)	Evaluate the (cross-)covariance function(s).
`matrix`(x0[, x1])	A convenience function for computing a kernel matrix for two sets of inputs.

Attributes Documentation

input_ndim¶: Syntactic sugar for len(input_shape).

input_shape¶: Shape of single, i.e. non-batched, arguments of the covariance function.

output_ndim¶: Syntactic sugar for len(output_shape).

output_shape¶

Shape of single, i.e. non-batched, return values of the covariance function.

If output_shape is (), the Kernel instance represents a single (cross-)covariance function. Otherwise, i.e. if output_shape is non-empty, the Kernel instance represents a tensor of (cross-)covariance functions whose shape is given by output_shape.

Methods Documentation

__call__(x0, x1)¶

Evaluate the (cross-)covariance function(s).

The evaluation of the (cross-covariance) function(s) is vectorized over the batch shapes of the arguments, applying standard NumPy broadcasting.

Parameters

x0 (ArrayLike) – shape= batch_shape_0 + input_shape – (Batch of) input(s) for the first argument of the Kernel.
x1 (Optional[ArrayLike]) – shape= batch_shape_1 + input_shape – (Batch of) input(s) for the second argument of the Kernel. Can also be set to None, in which case the function will behave as if x1 = x0 (but it is implemented more efficiently).

Returns

shape= bcast_batch_shape + output_shape – The (cross-)covariance function(s) evaluated at (x0, x1). Since the function is vectorized over the batch shapes of the inputs, the output array contains the following entries:

k_x0_x1[batch_idx + output_idx] = k[output_idx](
    x0[batch_idx, ...],
    x1[batch_idx, ...],
)

where we assume that x0 and x1 have been broadcast to a common shape bcast_batch_shape + input_shape, and where output_idx and batch_idx are indices compatible with output_shape and bcast_batch_shape, respectively. By k[output_idx] we refer to the covariance function at index output_idx in the tensor of covariance functions represented by the Kernel instance.

Return type

k_x0_x1

Raises

ValueError – If one of the input shapes is not of the form batch_shape_{0,1} + input_shape.
ValueError – If the inputs can not be broadcast to a common shape.