sklearn.kernel_ridge.KernelRidge

class sklearn.kernel_ridge.KernelRidge(alpha=1, kernel='linear', gamma=None, degree=3, coef0=1, kernel_params=None)

Kernel ridge regression.

Kernel ridge regression (KRR) combines ridge regression (linear least squares with l2-norm regularization) with the kernel trick. It thus learns a linear function in the space induced by the respective kernel and the data. For non-linear kernels, this corresponds to a non-linear function in the original space.

The form of the model learned by KRR is identical to support vector regression (SVR). However, different loss functions are used: KRR uses squared error loss while support vector regression uses epsilon-insensitive loss, both combined with l2 regularization. In contrast to SVR, fitting a KRR model can be done in closed-form and is typically faster for medium-sized datasets. On the other hand, the learned model is non-sparse and thus slower than SVR, which learns a sparse model for epsilon > 0, at prediction-time.

This estimator has built-in support for multi-variate regression (i.e., when y is a 2d-array of shape [n_samples, n_targets]).

Parameters:

alpha : {float, array-like}, shape = [n_targets]

Small positive values of alpha improve the conditioning of the problem and reduce the variance of the estimates. Alpha corresponds to (2*C)^-1 in other linear models such as LogisticRegression or LinearSVC. If an array is passed, penalties are assumed to be specific to the targets. Hence they must correspond in number.

kernel : string or callable, default=”linear”

Kernel mapping used internally. A callable should accept two arguments and the keyword arguments passed to this object as kernel_params, and should return a floating point number.

gamma : float, default=None

Gamma parameter for the RBF, polynomial, exponential chi2 and sigmoid kernels. Interpretation of the default value is left to the kernel; see the documentation for sklearn.metrics.pairwise. Ignored by other kernels.

degree : float, default=3

Degree of the polynomial kernel. Ignored by other kernels.

coef0 : float, default=1

Zero coefficient for polynomial and sigmoid kernels. Ignored by other kernels.

kernel_params : mapping of string to any, optional

Additional parameters (keyword arguments) for kernel function passed as callable object.

Attributes:

dual_coef_ : array, shape = [n_features] or [n_targets, n_features]

Weight vector(s) in kernel space

X_fit_ : {array-like, sparse matrix}, shape = [n_samples, n_features]

Training data, which is also required for prediction

See also

Ridge

Linear ridge regression.

SVR

Support Vector Regression implemented using libsvm.

References

• Kevin P. Murphy “Machine Learning: A Probabilistic Perspective”, The MIT Press chapter 14.4.3, pp. 492-493

Examples

>>> from sklearn.kernel_ridge import KernelRidge
>>> import numpy as np
>>> n_samples, n_features = 10, 5
>>> rng = np.random.RandomState(0)
>>> y = rng.randn(n_samples)
>>> X = rng.randn(n_samples, n_features)
>>> clf = KernelRidge(alpha=1.0)
>>> clf.fit(X, y) 
KernelRidge(alpha=1.0, coef0=1, degree=3, gamma=None, kernel='linear',
            kernel_params=None)

Methods

fit(X[, y, sample_weight])	Fit Kernel Ridge regression model
get_params([deep])	Get parameters for this estimator.
predict(X)	Predict using the the kernel ridge model
score(X, y[, sample_weight])	Returns the coefficient of determination R^2 of the prediction.
set_params(**params)	Set the parameters of this estimator.

__init__(alpha=1, kernel='linear', gamma=None, degree=3, coef0=1, kernel_params=None)

fit(X, y=None, sample_weight=None)

Fit Kernel Ridge regression model

Parameters:

X : {array-like, sparse matrix}, shape = [n_samples, n_features]

Training data

y : array-like, shape = [n_samples] or [n_samples, n_targets]

Target values

sample_weight : float or numpy array of shape [n_samples]

Individual weights for each sample, ignored if None is passed.

Returns:

self : returns an instance of self.

get_params(deep=True)

Get parameters for this estimator.

Parameters:

deep: boolean, optional :

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

Returns:

params : mapping of string to any

Parameter names mapped to their values.

predict(X)

Predict using the the kernel ridge model

Parameters:

X : {array-like, sparse matrix}, shape = [n_samples, n_features]

Samples.

Returns:

C : array, shape = [n_samples] or [n_samples, n_targets]

Returns predicted values.

score(X, y, sample_weight=None)

Returns the coefficient of determination R^2 of the prediction.

The coefficient R^2 is defined as (1 - u/v), where u is the regression sum of squares ((y_true - y_pred) ** 2).sum() and v is the residual sum of squares ((y_true - y_true.mean()) ** 2).sum(). Best possible score is 1.0, lower values are worse.

Parameters:

X : array-like, shape = (n_samples, n_features)

Test samples.

y : array-like, shape = (n_samples) or (n_samples, n_outputs)

True values for X.

sample_weight : array-like, shape = [n_samples], optional

Sample weights.

Returns:

score : float

R^2 of self.predict(X) wrt. y.

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The former have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns:self :

Examples using sklearn.kernel_ridge.KernelRidge

Comparison of kernel ridge regression and SVR