RBF#
- classsklearn.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 scaleparameter\(l>0\), which can either be a scalar (isotropic variantof the kernel) or a vector with the same number of dimensions as the inputsX (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 thiskernel 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 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.
References
Examples
>>>fromsklearn.datasetsimportload_iris>>>fromsklearn.gaussian_processimportGaussianProcessClassifier>>>fromsklearn.gaussian_process.kernelsimportRBF>>>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]])
- __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 when
eval_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.
- 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#
Comparison of kernel ridge and Gaussian process regression
Probabilistic predictions with Gaussian process classification (GPC)
Gaussian process classification (GPC) on iris dataset
Illustration of Gaussian process classification (GPC) on the XOR dataset
Forecasting of CO2 level on Mona Loa dataset using Gaussian process regression (GPR)
Ability of Gaussian process regression (GPR) to estimate data noise-level
Gaussian Processes regression: basic introductory example
Illustration of prior and posterior Gaussian process for different kernels