# sklearn.linear_model.ridge_regression¶

sklearn.linear_model.ridge_regression(X, y, alpha, sample_weight=None, solver='auto', max_iter=None, tol=0.001, verbose=0, random_state=None, return_n_iter=False, return_intercept=False)

Solve the ridge equation by the method of normal equations.

Read more in the User Guide.

Parameters: X : {array-like, sparse matrix, LinearOperator}, shape = [n_samples, n_features] Training data y : array-like, shape = [n_samples] or [n_samples, n_targets] Target values alpha : {float, array-like}, shape = [n_targets] if array-like 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 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. sample_weight : float or numpy array of shape [n_samples] Individual weights for each sample. If sample_weight is not None and solver=’auto’, the solver will be set to ‘cholesky’. New in version 0.17. solver : {‘auto’, ‘svd’, ‘cholesky’, ‘lsqr’, ‘sparse_cg’, ‘sag’, ‘saga’} 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 via a Cholesky decomposition of dot(X.T, X) ‘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 ‘saga’ supports sparse input whenfit_intercept is True. New in version 0.17: Stochastic Average Gradient descent solver. New in version 0.19: SAGA solver. max_iter : int, optional Maximum number of iterations for conjugate gradient solver. For the ‘sparse_cg’ and ‘lsqr’ solvers, the default value is determined by scipy.sparse.linalg. For ‘sag’ and saga solver, the default value is 1000. tol : float Precision of the solution. verbose : int Verbosity level. Setting verbose > 0 will display additional information depending on the solver used. random_state : int, RandomState instance or None, optional, default None The seed of the pseudo random number generator to use when shuffling the data. If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random. Used when solver == ‘sag’. return_n_iter : boolean, default False If True, the method also returns n_iter, the actual number of iteration performed by the solver. New in version 0.17. return_intercept : boolean, default False If True and if X is sparse, the method also returns the intercept, and the solver is automatically changed to ‘sag’. This is only a temporary fix for fitting the intercept with sparse data. For dense data, use sklearn.linear_model._preprocess_data before your regression. New in version 0.17. coef : array, shape = [n_features] or [n_targets, n_features] Weight vector(s). n_iter : int, optional The actual number of iteration performed by the solver. Only returned if return_n_iter is True. intercept : float or array, shape = [n_targets] The intercept of the model. Only returned if return_intercept is True and if X is a scipy sparse array.

Notes

This function won’t compute the intercept.