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

RatQuad

Matern

Matern kernel.

Examples

>>> import numpy as np
>>> 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 Shape of single, i.e. non-batched, arguments of the covariance function. input_size Syntactic sugar for the product over the input size. lengthscale Deprecated. lengthscales Input lengthscales along the different input dimensions. 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.

input_size

Syntactic sugar for the product over the input size.

lengthscale

Deprecated.

lengthscales

Input lengthscales along the different input dimensions.

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

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= batch_shape + output_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.

__call__
See documentation of class Kernel.