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 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), optional 

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