RationalQuadratic#

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

Rational Quadratic kernel.

The RationalQuadratic kernel can be seen as a scale mixture (an infinite sum) of RBF kernels with different characteristic length scales. It is parameterized by a length scale parameter l>0 and a scale mixture parameter α>0. Only the isotropic variant where length_scale l is a scalar is supported at the moment. The kernel is given by:

k(xi,xj)=(1+d(xi,xj)22αl2)α

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

Read more in the User Guide.

Added in version 0.18.

Parameters:
length_scalefloat > 0, default=1.0

The length scale of the kernel.

alphafloat > 0, default=1.0

Scale mixture parameter

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.

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

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

References

Examples

>>> from sklearn.datasets import load_iris
>>> from sklearn.gaussian_process import GaussianProcessClassifier
>>> from sklearn.gaussian_process.kernels import RationalQuadratic
>>> X, y = load_iris(return_X_y=True)
>>> kernel = RationalQuadratic(length_scale=1.0, alpha=1.5)
>>> gpc = GaussianProcessClassifier(kernel=kernel,
...         random_state=0).fit(X, y)
>>> gpc.score(X, y)
0.9733...
>>> gpc.predict_proba(X[:2,:])
array([[0.8881..., 0.0566..., 0.05518...],
        [0.8678..., 0.0707... , 0.0614...]])
__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 log of the kernel hyperparameter is computed. 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)

The gradient of the kernel k(X, X) with respect to the log of 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