Linear least squares with l2 regularization.

Minimizes the objective function::

||y - Xw||^2_2 + alpha * ||w||^2_2

This model solves a regression model where the loss function is the linear least squares function and regularization is given by the l2-norm. Also known as Ridge Regression or Tikhonov regularization. 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 <ridge_regression>`.

Parameters ---------- alpha : float, ndarray of shape (n_targets,), default=1.0 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.

fit_intercept : bool, default=True Whether to fit the intercept for this model. If set to false, no intercept will be used in calculations (i.e. ``X`` and ``y`` are expected to be centered).

normalize : bool, default=False This parameter is ignored when ``fit_intercept`` is set to False. If True, the regressors X will be normalized before regression by subtracting the mean and dividing by the l2-norm. If you wish to standardize, please use :class:`sklearn.preprocessing.StandardScaler` before calling ``fit`` on an estimator with ``normalize=False``.

copy_X : bool, default=True If True, X will be copied; else, it may be overwritten.

max_iter : int, default=None Maximum number of iterations for conjugate gradient solver. For 'sparse_cg' and 'lsqr' solvers, the default value is determined by scipy.sparse.linalg. For 'sag' solver, the default value is 1000.

tol : float, default=1e-3 Precision of the solution.

solver : 'auto', 'svd', 'cholesky', 'lsqr', 'sparse_cg', 'sag', 'saga', default='auto' Solver to use in the computational routines:

• 'auto' chooses the solver automatically based on the type of data.

• 'svd' uses a Singular Value Decomposition of X to compute the Ridge coefficients. More stable for singular matrices than 'cholesky'.

• 'cholesky' uses the standard scipy.linalg.solve function to obtain a closed-form solution.

• 'sparse_cg' uses the conjugate gradient solver as found in scipy.sparse.linalg.cg. As an iterative algorithm, this solver is more appropriate than 'cholesky' for large-scale data (possibility to set `tol` and `max_iter`).

• 'lsqr' uses the dedicated regularized least-squares routine scipy.sparse.linalg.lsqr. It is the fastest and uses an iterative procedure.

• 'sag' uses a Stochastic Average Gradient descent, and 'saga' uses its improved, unbiased version named SAGA. Both methods also use an iterative procedure, and are often faster than other solvers when both n_samples and n_features are large. Note that 'sag' and 'saga' fast convergence is only guaranteed on features with approximately the same scale. You can preprocess the data with a scaler from sklearn.preprocessing.

All last five solvers support both dense and sparse data. However, only 'sag' and 'sparse_cg' supports sparse input when `fit_intercept` is True.

.. versionadded:: 0.17 Stochastic Average Gradient descent solver. .. versionadded:: 0.19 SAGA solver.

random_state : int, RandomState instance, default=None Used when ``solver`` == 'sag' or 'saga' to shuffle the data. See :term:`Glossary <random_state>` for details.

.. versionadded:: 0.17 `random_state` to support Stochastic Average Gradient.

Attributes ---------- coef_ : ndarray of shape (n_features,) or (n_targets, n_features) Weight vector(s).

intercept_ : float or ndarray of shape (n_targets,) Independent term in decision function. Set to 0.0 if ``fit_intercept = False``.

n_iter_ : None or ndarray of shape (n_targets,) Actual number of iterations for each target. Available only for sag and lsqr solvers. Other solvers will return None.

.. versionadded:: 0.17

See also -------- RidgeClassifier : Ridge classifier RidgeCV : Ridge regression with built-in cross validation :class:`sklearn.kernel_ridge.KernelRidge` : Kernel ridge regression combines ridge regression with the kernel trick

Examples -------- >>> from sklearn.linear_model import Ridge >>> 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 = Ridge(alpha=1.0) >>> clf.fit(X, y) Ridge()

Fit Ridge regression model.

Parameters ---------- X : ndarray, sparse matrix of shape (n_samples, n_features) Training data

y : ndarray of shape (n_samples,) or (n_samples, n_targets) Target values

sample_weight : float or ndarray of shape (n_samples,), default=None Individual weights for each sample. If given a float, every sample will have the same weight.

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 linear model.

Parameters ---------- X : array_like or sparse matrix, shape (n_samples, n_features) Samples.

Returns ------- C : array, shape (n_samples,) 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 coef_: get value or raise Not_found if None.

Attribute coef_: get value as an option.

Attribute intercept_: get value or raise Not_found if None.

Attribute intercept_: get value as an option.

Attribute n_iter_: get value or raise Not_found if None.

Attribute n_iter_: 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.