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3.2.3.1.5. sklearn.linear_model.LassoCV

class sklearn.linear_model.LassoCV(eps=0.001, n_alphas=100, alphas=None, fit_intercept=True, normalize=False, precompute='auto', max_iter=1000, tol=0.0001, copy_X=True, cv=None, verbose=False, n_jobs=1, positive=False)

Lasso linear model with iterative fitting along a regularization path

The best model is selected by cross-validation.

The optimization objective for Lasso is:

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

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

alphas : numpy array, optional

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

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 : integer or cross-validation generator, optional

If an integer is passed, it is the number of fold (default 3). Specific cross-validation objects can be passed, see the sklearn.cross_validation module for the list of possible objects.

verbose : bool or integer

Amount of verbosity.

n_jobs : integer, optional

Number of CPUs to use during the cross validation. If -1, use all the CPUs. Note that this is used only if multiple values for l1_ratio are given.

positive : bool, optional

If positive, restrict regression coefficients to be positive

Notes

See examples/linear_model/lasso_path_with_crossvalidation.py for an example.

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

Attributes

alpha_ float The amount of 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,) independent term in decision function.
mse_path_ array, shape = (n_alphas, n_folds) mean square error for the test set on each fold, varying alpha
alphas_ numpy array, shape = (n_alphas,) The grid of alphas used for fitting
dual_gap_ numpy array, shape = (n_alphas,) The dual gap at the end of the optimization for the optimal alpha (alpha_).

Methods

decision_function(X) Decision function of the linear model.
fit(X, y) Fit linear model with coordinate descent
get_params([deep]) Get parameters for this estimator.
path(X, y[, eps, n_alphas, alphas, ...]) Compute Lasso 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__(eps=0.001, n_alphas=100, alphas=None, fit_intercept=True, normalize=False, precompute='auto', max_iter=1000, tol=0.0001, copy_X=True, cv=None, verbose=False, n_jobs=1, positive=False)
decision_function(X)

Decision function of the linear model.

Parameters:

X : {array-like, sparse matrix}, shape = (n_samples, n_features)

Samples.

Returns:

C : array, shape = (n_samples,)

Returns predicted values.

fit(X, y)

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 float64, 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)

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, eps=0.001, n_alphas=100, alphas=None, precompute='auto', Xy=None, fit_intercept=None, normalize=None, copy_X=True, coef_init=None, verbose=False, return_models=False, **params)

Compute Lasso path with coordinate descent

The Lasso optimization function varies for mono and multi-outputs.

For mono-output tasks it is:

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

For multi-output tasks it is:

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

Where:

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

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

Parameters:

X : {array-like, sparse matrix}, shape (n_samples, n_features)

Training data. Pass directly as Fortran-contiguous data to avoid unnecessary memory duplication. If y is mono-output then X can be sparse.

y : ndarray, shape = (n_samples,), or (n_samples, n_outputs)

Target values

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

alphas : ndarray, optional

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

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.

Xy : array-like, optional

Xy = np.dot(X.T, y) that can be precomputed. It is useful only when the Gram matrix is precomputed.

fit_intercept : bool

Fit or not an intercept. WARNING : deprecated, will be removed in 0.16.

normalize : boolean, optional, default False

If True, the regressors X will be normalized before regression. WARNING : deprecated, will be removed in 0.16.

copy_X : boolean, optional, default True

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

coef_init : array, shape (n_features, ) | None

The initial values of the coefficients.

verbose : bool or integer

Amount of verbosity.

return_models : boolean, optional, default True

If True, the function will return list of models. Setting it to False will change the function output returning the values of the alphas and the coefficients along the path. Returning the model list will be removed in version 0.16.

params : kwargs

keyword arguments passed to the coordinate descent solver.

Returns:

models : a list of models along the regularization path

(Is returned if return_models is set True (default).

alphas : array, shape (n_alphas,)

The alphas along the path where models are computed. (Is returned, along with coefs, when return_models is set to False)

coefs : array, shape (n_features, n_alphas) or

(n_outputs, n_features, n_alphas)

Coefficients along the path. (Is returned, along with alphas, when return_models is set to False).

dual_gaps : array, shape (n_alphas,)

The dual gaps at the end of the optimization for each alpha. (Is returned, along with alphas, when return_models is set to False).

Notes

See examples/linear_model/plot_lasso_coordinate_descent_path.py for an example.

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

Note that in certain cases, the Lars solver may be significantly faster to implement this functionality. In particular, linear interpolation can be used to retrieve model coefficients between the values output by lars_path

Deprecation Notice: Setting return_models to False will make the Lasso Path return an output in the style used by lars_path. This will be become the norm as of version 0.16. Leaving return_models set to True will let the function return a list of models as before.

Examples

Comparing lasso_path and lars_path with interpolation:

>>> X = np.array([[1, 2, 3.1], [2.3, 5.4, 4.3]]).T
>>> y = np.array([1, 2, 3.1])
>>> # Use lasso_path to compute a coefficient path
>>> _, coef_path, _ = lasso_path(X, y, alphas=[5., 1., .5],
...                              fit_intercept=False)
>>> print(coef_path)
[[ 0.          0.          0.46874778]
 [ 0.2159048   0.4425765   0.23689075]]
>>> # Now use lars_path and 1D linear interpolation to compute the
>>> # same path
>>> from sklearn.linear_model import lars_path
>>> alphas, active, coef_path_lars = lars_path(X, y, method='lasso')
>>> from scipy import interpolate
>>> coef_path_continuous = interpolate.interp1d(alphas[::-1],
...                                             coef_path_lars[:, ::-1])
>>> print(coef_path_continuous([5., 1., .5]))
[[ 0.          0.          0.46915237]
 [ 0.2159048   0.4425765   0.23668876]]
predict(X)

Predict using the linear model

Parameters:

X : {array-like, sparse matrix}, shape = (n_samples, n_features)

Samples.

Returns:

C : array, shape = (n_samples,)

Returns predicted values.

score(X, y, sample_weight=None)

Returns the coefficient of determination R^2 of the prediction.

The coefficient R^2 is defined as (1 - u/v), where u is the regression sum of squares ((y_true - y_pred) ** 2).sum() and v is the residual sum of squares ((y_true - y_true.mean()) ** 2).sum(). Best possible score is 1.0, lower values are worse.

Parameters:

X : array-like, shape = (n_samples, n_features)

Test samples.

y : array-like, shape = (n_samples,)

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)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The former have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns:self :
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