# 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, positive=False, random_state=None, return_n_iter=False, return_intercept=False, check_input=True)[source]

Solve the ridge equation by the method of normal equations.

Read more in the User Guide.

Parameters
X{ndarray, sparse matrix, LinearOperator} of shape (n_samples, n_features)

Training data

yndarray of shape (n_samples,) or (n_samples, n_targets)

Target values

alphafloat 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 LogisticRegression or LinearSVC. If an array is passed, penalties are assumed to be specific to the targets. Hence they must correspond in number.

sample_weightfloat or array-like of shape (n_samples,), default=None

Individual weights for each sample. If given a float, every sample will have the same weight. 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’, ‘lbfgs’}, 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 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.

• ‘lbfgs’ uses L-BFGS-B algorithm implemented in scipy.optimize.minimize. It can be used only when positive is True.

All last six solvers support both dense and sparse data. However, only ‘sag’, ‘sparse_cg’, and ‘lbfgs’ support sparse input when fit_intercept is True.

New in version 0.17: Stochastic Average Gradient descent solver.

New in version 0.19: SAGA solver.

max_iterint, default=None

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. For ‘lbfgs’ solver, the default value is 15000.

tolfloat, default=1e-3

Precision of the solution.

verboseint, default=0

Verbosity level. Setting verbose > 0 will display additional information depending on the solver used.

positivebool, default=False

When set to True, forces the coefficients to be positive. Only ‘lbfgs’ solver is supported in this case.

random_stateint, RandomState instance, default=None

Used when solver == ‘sag’ or ‘saga’ to shuffle the data. See Glossary for details.

return_n_iterbool, 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_interceptbool, 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.

check_inputbool, default=True

If False, the input arrays X and y will not be checked.

New in version 0.21.

Returns
coefndarray of shape (n_features,) or (n_targets, n_features)

Weight vector(s).

n_iterint, optional

The actual number of iteration performed by the solver. Only returned if return_n_iter is True.

interceptfloat or ndarray of 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.