sklearn.linear_model.Ridge

class sklearn.linear_model.Ridge(alpha=1.0, fit_intercept=True, normalize=False, copy_X=True, max_iter=None, tol=0.001, solver='auto', random_state=None)[source]

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 User Guide.

Parameters:
alpha : {float, array-like}, 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 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.

fit_intercept : boolean

Whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (e.g. data is expected to be already centered).

normalize : boolean, optional, 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 sklearn.preprocessing.StandardScaler before calling fit on an estimator with normalize=False.

copy_X : boolean, optional, default True

If True, X will be copied; else, it may be overwritten.

max_iter : int, optional

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

Precision of the solution.

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.
  • ‘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 when fit_intercept is True.

New in version 0.17: Stochastic Average Gradient descent solver.

New in version 0.19: SAGA solver.

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’.

New in version 0.17: random_state to support Stochastic Average Gradient.
Attributes:
coef_ : array, shape (n_features,) or (n_targets, n_features)

Weight vector(s).

intercept_ : float | array, shape = (n_targets,)

Independent term in decision function. Set to 0.0 if fit_intercept = False.

n_iter_ : array or None, shape (n_targets,)

Actual number of iterations for each target. Available only for sag and lsqr solvers. Other solvers will return None.

New in version 0.17.

See also

RidgeClassifier
Ridge classifier
RidgeCV
Ridge regression with built-in cross validation
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
>>> np.random.seed(0)
>>> y = np.random.randn(n_samples)
>>> X = np.random.randn(n_samples, n_features)
>>> clf = Ridge(alpha=1.0)
>>> clf.fit(X, y) # doctest: +NORMALIZE_WHITESPACE
Ridge(alpha=1.0, copy_X=True, fit_intercept=True, max_iter=None,
      normalize=False, random_state=None, solver='auto', tol=0.001)

Methods

fit(X, y[, sample_weight]) Fit Ridge regression model
get_params([deep]) Get parameters for this estimator.
predict(X) Predict using the linear 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=1.0, fit_intercept=True, normalize=False, copy_X=True, max_iter=None, tol=0.001, solver='auto', random_state=None)[source]

Initialize self. See help(type(self)) for accurate signature.

fit(X, y, sample_weight=None)[source]

Fit 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
Returns:
self : returns an instance of self.
get_params(deep=True)[source]

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)[source]

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.
score(X, y, sample_weight=None)[source]

Returns 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, shape = (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix 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, 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)[source]

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.

Returns:
self