Matern#

classsklearn.gaussian_process.kernels.Matern(length_scale=1.0,length_scale_bounds=(1e-05,100000.0),nu=1.5)[source]#

Matern kernel.

The class of Matern kernels is a generalization of theRBF.It has an additional parameter\(\nu\) which controls thesmoothness of the resulting function. The smaller\(\nu\),the less smooth the approximated function is.As\(\nu\rightarrow\infty\), the kernel becomes equivalent totheRBF kernel. When\(\nu = 1/2\), the Matérn kernelbecomes identical to the absolute exponential kernel.Important intermediate values are\(\nu=1.5\) (once differentiable functions)and\(\nu=2.5\) (twice differentiable functions).

The kernel is given by:

\[k(x_i, x_j) = \frac{1}{\Gamma(\nu)2^{\nu-1}}\Bigg(\frac{\sqrt{2\nu}}{l} d(x_i , x_j )\Bigg)^\nu K_\nu\Bigg(\frac{\sqrt{2\nu}}{l} d(x_i , x_j )\Bigg)\]

where\(d(\cdot,\cdot)\) is the Euclidean distance,\(K_{\nu}(\cdot)\) is a modified Bessel function and\(\Gamma(\cdot)\) is the gamma function.See[1], Chapter 4, Section 4.2, for details regarding the differentvariants of the Matern kernel.

Read more in theUser Guide.

Added 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 isused. If an array, an anisotropic kernel is used where each dimensionof 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 duringhyperparameter tuning.

nufloat, default=1.5

The parameter nu controlling the smoothness of the learned function.The smaller nu, the less smooth the approximated function is.For nu=inf, the kernel becomes equivalent to the RBF kernel and fornu=0.5 to the absolute exponential kernel. Important intermediatevalues are nu=1.5 (once differentiable functions) and nu=2.5(twice differentiable functions). Note that values of nu not in[0.5, 1.5, 2.5, inf] incur a considerably higher computational cost(appr. 10 times higher) since they require to evaluate the modifiedBessel function. Furthermore, in contrast to l, nu is kept fixed toits initial value and not optimized.

References

Examples

>>>fromsklearn.datasetsimportload_iris>>>fromsklearn.gaussian_processimportGaussianProcessClassifier>>>fromsklearn.gaussian_process.kernelsimportMatern>>>X,y=load_iris(return_X_y=True)>>>kernel=1.0*Matern(length_scale=1.0,nu=1.5)>>>gpc=GaussianProcessClassifier(kernel=kernel,...random_state=0).fit(X,y)>>>gpc.score(X,y)0.9866>>>gpc.predict_proba(X[:2,:])array([[0.8513, 0.0368, 0.1117],        [0.8086, 0.0693, 0.1220]])
__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 ofthe 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), optional

The gradient of the kernel k(X, X) with respect to the log of thehyperparameter of the kernel. Only returned wheneval_gradientis True.

propertybounds#

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 isevaluated.

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 andcontained subobjects that are estimators.

Returns:
paramsdict

Parameter names mapped to their values.

propertyhyperparameters#

Returns a list of all hyperparameter specifications.

is_stationary()[source]#

Returns whether the kernel is stationary.

propertyn_dims#

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

propertyrequires_vector_input#

Returns whether the kernel is defined on fixed-length featurevectors or generic objects. Defaults to True for backwardcompatibility.

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
propertytheta#

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

Note that theta are typically the log-transformed values of thekernel’s hyperparameters as this representation of the search spaceis more amenable for hyperparameter search, as hyperparameters likelength-scales naturally live on a log-scale.

Returns:
thetandarray of shape (n_dims,)

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

Gallery examples#

Illustration of prior and posterior Gaussian process for different kernels

Illustration of prior and posterior Gaussian process for different kernels