sklearn.gaussian_process.kernels.RBF

class sklearn.gaussian_process.kernels.RBF(length_scale=1.0, length_scale_bounds=1e-05, 100000.0)[source]

Radial-basis function kernel (aka squared-exponential kernel).

The RBF kernel is a stationary kernel. It is also known as the “squared exponential” kernel. It is parameterized by a length scale parameter \(l>0\), which can either be a scalar (isotropic variant of the kernel) or a vector with the same number of dimensions as the inputs X (anisotropic variant of the kernel). The kernel is given by:

\[k(x_i, x_j) = \exp\left(- \frac{d(x_i, x_j)^2}{2l^2} \right)\]

where \(l\) is the length scale of the kernel and \(d(\cdot,\cdot)\) is the Euclidean distance. For advice on how to set the length scale parameter, see e.g. [1].

This kernel is infinitely differentiable, which implies that GPs with this kernel as covariance function have mean square derivatives of all orders, and are thus very smooth. See [2], Chapter 4, Section 4.2, for further details of the RBF kernel.

Read more in the User Guide.

New in version 0.18.

Parameters
length_scalefloat or ndarray of shape (n_features,), default=1.0

The length scale of the kernel. If a float, an isotropic kernel is used. If an array, an anisotropic kernel is used where each dimension of l defines the length-scale of the respective feature dimension.

length_scale_boundspair of floats >= 0 or “fixed”, default=(1e-5, 1e5)

The lower and upper bound on ‘length_scale’. If set to “fixed”, ‘length_scale’ cannot be changed during hyperparameter tuning.

Attributes
anisotropic
bounds

Returns the log-transformed bounds on the theta.

hyperparameter_length_scale
hyperparameters

Returns a list of all hyperparameter specifications.

n_dims

Returns the number of non-fixed hyperparameters of the kernel.

requires_vector_input

Returns whether the kernel is defined on fixed-length feature vectors or generic objects.

theta

Returns the (flattened, log-transformed) non-fixed hyperparameters.

References

1

David Duvenaud (2014). “The Kernel Cookbook: Advice on Covariance functions”.

2

Carl Edward Rasmussen, Christopher K. I. Williams (2006). “Gaussian Processes for Machine Learning”. The MIT Press.

Examples

>>> from sklearn.datasets import load_iris
>>> from sklearn.gaussian_process import GaussianProcessClassifier
>>> from sklearn.gaussian_process.kernels import RBF
>>> X, y = load_iris(return_X_y=True)
>>> kernel = 1.0 * RBF(1.0)
>>> gpc = GaussianProcessClassifier(kernel=kernel,
...         random_state=0).fit(X, y)
>>> gpc.score(X, y)
0.9866...
>>> gpc.predict_proba(X[:2,:])
array([[0.8354..., 0.03228..., 0.1322...],
       [0.7906..., 0.0652..., 0.1441...]])

Methods

__call__(X[, Y, eval_gradient])

Return the kernel k(X, Y) and optionally its gradient.

clone_with_theta(theta)

Returns a clone of self with given hyperparameters theta.

diag(X)

Returns the diagonal of the kernel k(X, X).

get_params([deep])

Get parameters of this kernel.

is_stationary()

Returns whether the kernel is stationary.

set_params(**params)

Set the parameters of this kernel.

__call__(X, Y=None, eval_gradient=False)[source]

Return the kernel k(X, Y) and optionally its gradient.

Parameters
Xndarray of shape (n_samples_X, n_features)

Left argument of the returned kernel k(X, Y)

Yndarray of shape (n_samples_Y, n_features), default=None

Right argument of the returned kernel k(X, Y). If None, k(X, X) if evaluated instead.

eval_gradientbool, default=False

Determines whether the gradient with respect to the kernel hyperparameter is determined. Only supported when Y is None.

Returns
Kndarray of shape (n_samples_X, n_samples_Y)

Kernel k(X, Y)

K_gradientndarray of shape (n_samples_X, n_samples_X, n_dims), optional

The gradient of the kernel k(X, X) with respect to the hyperparameter of the kernel. Only returned when eval_gradient is True.

property bounds

Returns the log-transformed bounds on the theta.

Returns
boundsndarray of shape (n_dims, 2)

The log-transformed bounds on the kernel’s hyperparameters theta

clone_with_theta(theta)[source]

Returns a clone of self with given hyperparameters theta.

Parameters
thetandarray of shape (n_dims,)

The hyperparameters

diag(X)[source]

Returns the diagonal of the kernel k(X, X).

The result of this method is identical to np.diag(self(X)); however, it can be evaluated more efficiently since only the diagonal is evaluated.

Parameters
Xndarray of shape (n_samples_X, n_features)

Left argument of the returned kernel k(X, Y)

Returns
K_diagndarray of shape (n_samples_X,)

Diagonal of kernel k(X, X)

get_params(deep=True)[source]

Get parameters of this kernel.

Parameters
deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns
paramsdict

Parameter names mapped to their values.

property hyperparameters

Returns a list of all hyperparameter specifications.

is_stationary()[source]

Returns whether the kernel is stationary.

property n_dims

Returns the number of non-fixed hyperparameters of the kernel.

property requires_vector_input

Returns whether the kernel is defined on fixed-length feature vectors or generic objects. Defaults to True for backward compatibility.

set_params(**params)[source]

Set the parameters of this kernel.

The method works on simple kernels as well as on nested kernels. The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns
self
property theta

Returns the (flattened, log-transformed) non-fixed hyperparameters.

Note that theta are typically the log-transformed values of the kernel’s hyperparameters as this representation of the search space is more amenable for hyperparameter search, as hyperparameters like length-scales naturally live on a log-scale.

Returns
thetandarray of shape (n_dims,)

The non-fixed, log-transformed hyperparameters of the kernel