3.2.4.1.1. `sklearn.linear_model`.ElasticNetCV¶

class sklearn.linear_model.ElasticNetCV(l1_ratio=0.5, eps=0.001, n_alphas=100, alphas=None, fit_intercept=True, normalize=False, precompute='auto', max_iter=1000, tol=0.0001, cv='warn', copy_X=True, verbose=0, n_jobs=None, positive=False, random_state=None, selection='cyclic')[source]¶

Elastic Net model with iterative fitting along a regularization path.

See glossary entry for cross-validation estimator.

Read more in the User Guide.

Parameters:

l1_ratio : float or array of floats, optional

float between 0 and 1 passed to ElasticNet (scaling between l1 and l2 penalties). For l1_ratio = 0 the penalty is an L2 penalty. For l1_ratio = 1 it is an L1 penalty. For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2 This parameter can be a list, in which case the different values are tested by cross-validation and the one giving the best prediction score is used. Note that a good choice of list of values for l1_ratio is often to put more values close to 1 (i.e. Lasso) and less close to 0 (i.e. Ridge), as in [.1, .5, .7, .9, .95, .99, 1]

eps : float, optional

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3.

n_alphas : int, optional

Number of alphas along the regularization path, used for each l1_ratio.

alphas : numpy array, optional

List of alphas where to compute the models. If None alphas are set automatically

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.

precompute : True | False | ‘auto’ | array-like

Whether to use a precomputed Gram matrix to speed up calculations. If set to 'auto' let us decide. The Gram matrix can also be passed as argument.

max_iter : int, optional

The maximum number of iterations

tol : float, optional

The tolerance for the optimization: if the updates are smaller than tol, the optimization code checks the dual gap for optimality and continues until it is smaller than tol.

cv : int, cross-validation generator or an iterable, optional

Determines the cross-validation splitting strategy. Possible inputs for cv are:

None, to use the default 3-fold cross-validation,
integer, to specify the number of folds.
CV splitter,
An iterable yielding (train, test) splits as arrays of indices.

For integer/None inputs, KFold is used.

Refer User Guide for the various cross-validation strategies that can be used here.

Changed in version 0.20: cv default value if None will change from 3-fold to 5-fold in v0.22.

copy_X : boolean, optional, default True

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

verbose : bool or integer

Amount of verbosity.

n_jobs : int or None, optional (default=None)

Number of CPUs to use during the cross validation. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors. See Glossary for more details.

positive : bool, optional

When set to True, forces the coefficients to be positive.

random_state : int, RandomState instance or None, optional, default None

The seed of the pseudo random number generator that selects a random feature to update. 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 selection == ‘random’.

selection : str, default ‘cyclic’

If set to ‘random’, a random coefficient is updated every iteration rather than looping over features sequentially by default. This (setting to ‘random’) often leads to significantly faster convergence especially when tol is higher than 1e-4.

Attributes:

alpha_ : float: The amount of penalization chosen by cross validation
l1_ratio_ : float: The compromise between l1 and l2 penalization chosen by cross validation
coef_ : array, shape (n_features,) | (n_targets, n_features): Parameter vector (w in the cost function formula),
intercept_ : float | array, shape (n_targets, n_features): Independent term in the decision function.
mse_path_ : array, shape (n_l1_ratio, n_alpha, n_folds): Mean square error for the test set on each fold, varying l1_ratio and alpha.
alphas_ : numpy array, shape (n_alphas,) or (n_l1_ratio, n_alphas): The grid of alphas used for fitting, for each l1_ratio.
n_iter_ : int: number of iterations run by the coordinate descent solver to reach the specified tolerance for the optimal alpha.

See also

enet_path, ElasticNet

Notes

For an example, see examples/linear_model/plot_lasso_model_selection.py.

To avoid unnecessary memory duplication the X argument of the fit method should be directly passed as a Fortran-contiguous numpy array.

The parameter l1_ratio corresponds to alpha in the glmnet R package while alpha corresponds to the lambda parameter in glmnet. More specifically, the optimization objective is:

1 / (2 * n_samples) * ||y - Xw||^2_2
+ alpha * l1_ratio * ||w||_1
+ 0.5 * alpha * (1 - l1_ratio) * ||w||^2_2

If you are interested in controlling the L1 and L2 penalty separately, keep in mind that this is equivalent to:

a * L1 + b * L2

for:

alpha = a + b and l1_ratio = a / (a + b).

Examples

>>> from sklearn.linear_model import ElasticNetCV
>>> from sklearn.datasets import make_regression

>>> X, y = make_regression(n_features=2, random_state=0)
>>> regr = ElasticNetCV(cv=5, random_state=0)
>>> regr.fit(X, y)
ElasticNetCV(alphas=None, copy_X=True, cv=5, eps=0.001, fit_intercept=True,
       l1_ratio=0.5, max_iter=1000, n_alphas=100, n_jobs=None,
       normalize=False, positive=False, precompute='auto', random_state=0,
       selection='cyclic', tol=0.0001, verbose=0)
>>> print(regr.alpha_) 
0.1994727942696716
>>> print(regr.intercept_) 
0.398...
>>> print(regr.predict([[0, 0]])) 
[0.398...]

Methods

`fit`(X, y)	Fit linear model with coordinate descent
`get_params`([deep])	Get parameters for this estimator.
`path`(X, y[, l1_ratio, eps, n_alphas, …])	Compute elastic net path with coordinate descent
`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__(l1_ratio=0.5, eps=0.001, n_alphas=100, alphas=None, fit_intercept=True, normalize=False, precompute='auto', max_iter=1000, tol=0.0001, cv='warn', copy_X=True, verbose=0, n_jobs=None, positive=False, random_state=None, selection='cyclic')[source]¶

fit(X, y)[source]¶

Fit linear model with coordinate descent

Fit is on grid of alphas and best alpha estimated by cross-validation.

Parameters:	X : {array-like}, shape (n_samples, n_features) Training data. Pass directly as Fortran-contiguous data to avoid unnecessary memory duplication. If y is mono-output, X can be sparse. y : array-like, shape (n_samples,) or (n_samples, n_targets) Target values

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.

static path(X, y, l1_ratio=0.5, eps=0.001, n_alphas=100, alphas=None, precompute='auto', Xy=None, copy_X=True, coef_init=None, verbose=False, return_n_iter=False, positive=False, check_input=True, **params)[source]¶

Compute elastic net path with coordinate descent

The elastic net optimization function varies for mono and multi-outputs.

For mono-output tasks it is:

1 / (2 * n_samples) * ||y - Xw||^2_2
+ alpha * l1_ratio * ||w||_1
+ 0.5 * alpha * (1 - l1_ratio) * ||w||^2_2

For multi-output tasks it is:

(1 / (2 * n_samples)) * ||Y - XW||^Fro_2
+ alpha * l1_ratio * ||W||_21
+ 0.5 * alpha * (1 - l1_ratio) * ||W||_Fro^2

Where:

||W||_21 = \sum_i \sqrt{\sum_j w_{ij}^2}

i.e. the sum of norm of each row.

Parameters:	X : array_like or sparse matrix, shape (n_samples, n_features) Samples.
Returns:	C : array, shape (n_samples,) Returns predicted values.

3.2.4.1.1. sklearn.linear_model.ElasticNetCV¶

3.2.4.1.1. `sklearn.linear_model`.ElasticNetCV¶