package sklearn

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).

Read more in the :ref:`User Guide <kernel_ridge>`.

Parameters ---------- alpha : float or array-like of shape (n_targets,) Regularization strength; must be a positive float. Regularization improves the conditioning of the problem and reduces the variance of the estimates. Larger values specify stronger regularization. Alpha corresponds to ``1 / (2C)`` in other linear models such as :class:`~sklearn.linear_model.LogisticRegression` or :class:`sklearn.svm.LinearSVC`. If an array is passed, penalties are assumed to be specific to the targets. Hence they must correspond in number. See :ref:`ridge_regression` for formula.

kernel : string or callable, default='linear' Kernel mapping used internally. This parameter is directly passed to :class:`sklearn.metrics.pairwise.pairwise_kernel`. If `kernel` is a string, it must be one of the metrics in `pairwise.PAIRWISE_KERNEL_FUNCTIONS`. If `kernel` is 'precomputed', X is assumed to be a kernel matrix. Alternatively, if `kernel` is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two rows from X as input and return the corresponding kernel value as a single number. This means that callables from :mod:`sklearn.metrics.pairwise` are not allowed, as they operate on matrices, not single samples. Use the string identifying the kernel instead.

gamma : float, default=None Gamma parameter for the RBF, laplacian, 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_ : ndarray of shape (n_samples,) or (n_samples, n_targets) Representation of weight vector(s) in kernel space

X_fit_ : ndarray, sparse matrix of shape (n_samples, n_features) Training data, which is also required for prediction. If kernel == 'precomputed' this is instead the precomputed training matrix, of shape (n_samples, n_samples).

References ---------- * Kevin P. Murphy 'Machine Learning: A Probabilistic Perspective', The MIT Press chapter 14.4.3, pp. 492-493

See also -------- sklearn.linear_model.Ridge: Linear ridge regression. sklearn.svm.SVR: Support Vector Regression implemented using libsvm.

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)

Fit Kernel Ridge regression model

Parameters ---------- X : array-like, sparse matrix of shape (n_samples, n_features) Training data. If kernel == 'precomputed' this is instead a precomputed kernel matrix, of shape (n_samples, n_samples).

y : array-like of shape (n_samples,) or (n_samples, n_targets) Target values

sample_weight : float or array-like of shape n_samples Individual weights for each sample, ignored if None is passed.

Returns ------- self : returns an instance of self.

Get parameters for this estimator.

Parameters ---------- deep : bool, default=True 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 using the kernel ridge model

Parameters ---------- X : array-like, sparse matrix of shape (n_samples, n_features) Samples. If kernel == 'precomputed' this is instead a precomputed kernel matrix, shape = n_samples, n_samples_fitted, where n_samples_fitted is the number of samples used in the fitting for this estimator.

Returns ------- C : ndarray of shape (n_samples,) or (n_samples, n_targets) Returns predicted values.

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

The coefficient R^2 is defined as (1 - u/v), where u is the residual sum of squares ((y_true - y_pred) ** 2).sum() and v is the total sum of squares ((y_true - y_true.mean()) ** 2).sum(). The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a R^2 score of 0.0.

Parameters ---------- X : array-like of shape (n_samples, n_features) Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead, shape = (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.

y : array-like of shape (n_samples,) or (n_samples, n_outputs) True values for X.

sample_weight : array-like of shape (n_samples,), default=None Sample weights.

Returns ------- score : float R^2 of self.predict(X) wrt. y.

Notes ----- The R2 score used when calling ``score`` on a regressor uses ``multioutput='uniform_average'`` from version 0.23 to keep consistent with default value of :func:`~sklearn.metrics.r2_score`. This influences the ``score`` method of all the multioutput regressors (except for :class:`~sklearn.multioutput.MultiOutputRegressor`).

Set the parameters of this estimator.

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

Parameters ---------- **params : dict Estimator parameters.

Returns ------- self : object Estimator instance.

Attribute dual_coef_: get value or raise Not_found if None.

Attribute dual_coef_: get value as an option.

Attribute X_fit_: get value or raise Not_found if None.

Attribute X_fit_: get value as an option.

Print the object to a human-readable representation.

Print the object to a human-readable representation.

Pretty-print the object to a formatter.