ExpQuad¶
-
class
probnum.kernels.
ExpQuad
(input_dim, lengthscale=1.0)¶ Bases:
probnum.kernels._kernel.Kernel
[numpy.ndarray
]Exponentiated quadratic / RBF kernel.
Covariance function defined by \(k(x_0, x_1) = \exp \big(-\frac{\lVert x_0 - x_1 \rVert^2}{2l^2}\big)\). This kernel is also known as the squared exponential or radial basis function kernel.
- Parameters
Examples
>>> import numpy as np >>> from probnum.kernels import ExpQuad >>> K = ExpQuad(input_dim=1, lengthscale=0.1) >>> K(np.linspace(0, 1, 3)[:, None]) 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
Dimension of arguments of the covariance function.
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
-
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
Methods Documentation
-
__call__
(x0, x1=None)[source]¶ Evaluate the kernel.
Computes the covariance function at
x0
andx1
. If the inputs have more than one dimension the covariance function is evaluated pairwise for all observations determined by the first dimension ofx0
andx1
. If onlyx0
is given the kernel matrix \(K=k(X_0, X_0)\) is computed.- Parameters
- Returns
shape=(), (output_dim, output_dim) or (n0, n1) or (n0, n1, output_dim, output_dim) – Kernel evaluated at
x0
andx1
or kernel matrix containing pairwise evaluations for all observations inx0
(andx1
).- Return type
cov