ExpQuad

class probnum.randprocs.kernels.ExpQuad(input_shape, lengthscale=1.0)

Bases: probnum.randprocs.kernels.Kernel, probnum.randprocs.kernels.IsotropicMixin

Exponentiated quadratic / RBF kernel.

Covariance function defined by

\[k(x_0, x_1) = \exp \left( -\frac{\lVert x_0 - x_1 \rVert_2^2}{2 l^2} \right).\]

This kernel is also known as the squared exponential or radial basis function kernel.

Parameters

See also

RatQuad

Rational quadratic kernel.

Matern

Matern kernel.

Examples

>>> import numpy as np
>>> from probnum.randprocs.kernels import ExpQuad
>>> K = ExpQuad(input_shape=(), lengthscale=0.1)
>>> xs = np.linspace(0, 1, 3)
>>> K.matrix(xs)
array([[1.00000000e+00, 3.72665317e-06, 1.92874985e-22],
       [3.72665317e-06, 1.00000000e+00, 3.72665317e-06],
       [1.92874985e-22, 3.72665317e-06, 1.00000000e+00]])

Attributes Summary

input_ndim

Syntactic sugar for len(input_shape).

input_shape

Dimension of single arguments of the covariance function.

shape

If shape is (), the Kernel instance represents a single (cross-)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).

Return type

int

input_shape

Dimension of single arguments of the covariance function.

Return type

int

shape

If shape is (), the Kernel instance represents a single (cross-)covariance function.

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

Return type

Tuple[int, ...]

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
Returns

shape= shape + bcast_batch_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[output_idx + batch_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 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.

See also

matrix

Convenience function to compute a kernel matrix, i.e. a matrix of pairwise evaluations of the kernel on two sets of points.

Examples

See documentation of class Kernel.

matrix(x0, x1=None)

A convenience function for computing a kernel matrix for two sets of inputs.

This is syntactic sugar for k(x0[:, None], x1[None, :]). Hence, it computes the matrix (stack) of pairwise covariances between two sets of input points. If k represents a single covariance function, then the resulting matrix will be symmetric positive-(semi)definite for x0 == x1.

Parameters
Returns

shape= shape + batch_shape – The matrix / stack of matrices containing the pairwise evaluations of the (cross-)covariance function(s) on x0 and x1. Depending on the shape of the inputs, batch_shape is either (M, N), (M,), (N,), or ().

Return type

kernmat

Raises

ValueError – If the shapes of the inputs don’t match the specification.

See also

__call__

Evaluate the kernel more flexibly.

Examples

See documentation of class Kernel.