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# sklearn.linear_model.Lasso¶

class sklearn.linear_model.Lasso(alpha=1.0, fit_intercept=True, normalize=False, precompute='auto', copy_X=True, max_iter=1000, tol=0.0001, warm_start=False, positive=False)

Linear Model trained with L1 prior as regularizer (aka the Lasso)

The optimization objective for Lasso is:

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


Technically the Lasso model is optimizing the same objective function as the Elastic Net with l1_ratio=1.0 (no L2 penalty).

Parameters: alpha : float, optional Constant that multiplies the L1 term. Defaults to 1.0. alpha = 0 is equivalent to an ordinary least square, solved by the LinearRegression object. For numerical reasons, using alpha = 0 is with the Lasso object is not advised and you should prefer the LinearRegression object. 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 If True, the regressors X will be normalized before regression. copy_X : boolean, optional, default True If True, X will be copied; else, it may be overwritten. 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. For sparse input this option is always True to preserve sparsity. 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. warm_start : bool, optional When set to True, reuse the solution of the previous call to fit as initialization, otherwise, just erase the previous solution. positive : bool, optional When set to True, forces the coefficients to be positive. coef_ : array, shape = (n_features,) | (n_targets, n_features) parameter vector (w in the cost function formula) sparse_coef_ : scipy.sparse matrix, shape = (n_features, 1) | (n_targets, n_features) sparse_coef_ is a readonly property derived from coef_ intercept_ : float | array, shape = (n_targets,) independent term in decision function.

Notes

The algorithm used to fit the model is coordinate descent.

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

Examples

>>> from sklearn import linear_model
>>> clf = linear_model.Lasso(alpha=0.1)
>>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2])
Lasso(alpha=0.1, copy_X=True, fit_intercept=True, max_iter=1000,
normalize=False, positive=False, precompute='auto', tol=0.0001,
warm_start=False)
>>> print(clf.coef_)
[ 0.85  0.  ]
>>> print(clf.intercept_)
0.15


Methods

 decision_function(X) Decision function of the linear model fit(X, y) Fit 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__(alpha=1.0, fit_intercept=True, normalize=False, precompute='auto', copy_X=True, max_iter=1000, tol=0.0001, warm_start=False, positive=False)
decision_function(X)

Decision function of the linear model

Parameters: X : numpy array or scipy.sparse matrix of shape (n_samples, n_features) T : array, shape = (n_samples,) The predicted decision function
fit(X, y)

Fit model with coordinate descent.

Parameters: X : ndarray or scipy.sparse matrix, (n_samples, n_features) Data y : ndarray, shape = (n_samples,) or (n_samples, n_targets) Target

Notes

Coordinate descent is an algorithm that considers each column of data at a time hence it will automatically convert the X input as a Fortran-contiguous numpy array if necessary.

To avoid memory re-allocation it is advised to allocate the initial data in memory directly using that format.

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. 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, fit_intercept=True, normalize=False, copy_X=True, coef_init=None, verbose=False, return_models=False, **params)

Compute elastic net path with coordinate descent

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

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


(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}, 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 l1_ratio : float, optional float between 0 and 1 passed to elastic net (scaling between l1 and l2 penalties). l1_ratio=1 corresponds to the Lasso eps : float 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 False 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. 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/plot_lasso_coordinate_descent_path.py for an example.

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 0.15. Leaving return_models set to True will let the function return a list of models as before.

predict(X)

Predict using the linear model

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

sparse representation of the fitted coef