sklearn.linear_model.Lasso

class sklearn.linear_model.Lasso(alpha=1.0, fit_intercept=True, normalize=False, precompute=False, copy_X=True, max_iter=1000, tol=0.0001, warm_start=False, positive=False, random_state=None, selection='cyclic')[source]

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).

Read more in the User Guide.

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. WARNING : The 'auto' option is deprecated and will be removed in 0.18.

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.

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.

random_state : int, RandomState instance, or None (default)

The seed of the pseudo random number generator that selects a random feature to update. Useful only when selection is set to ‘random’.
Attributes:

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.

n_iter_ : int | array-like, shape (n_targets,)

number of iterations run by the coordinate descent solver to reach the specified tolerance.

See also

lars_path, lasso_path, LassoLars, LassoCV, LassoLarsCV, sklearn.decomposition.sparse_encode

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=False, random_state=None,
   selection='cyclic', tol=0.0001, warm_start=False)
>>> print(clf.coef_)
[ 0.85  0.  ]
>>> print(clf.intercept_)
0.15

Methods

decision_function(*args, **kwargs) DEPRECATED: and will be removed in 0.19
fit(X, y[, check_input]) 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=False, copy_X=True, max_iter=1000, tol=0.0001, warm_start=False, positive=False, random_state=None, selection='cyclic')[source]
decision_function(*args, **kwargs)[source]

DEPRECATED: and will be removed in 0.19

Decision function of the linear model

Parameters:

X : numpy array or scipy.sparse matrix of shape (n_samples, n_features)
Returns:

T : array, shape (n_samples,)

The predicted decision function
fit(X, y, check_input=True)[source]

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)[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.

Read more in the User Guide.

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.

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.

params : kwargs

keyword arguments passed to the coordinate descent solver.

return_n_iter : bool

whether to return the number of iterations or not.

positive : bool, default False

If set to True, forces coefficients to be positive.

check_input : bool, default True

Skip input validation checks, including the Gram matrix when provided assuming there are handled by the caller when check_input=False.
Returns:

alphas : array, shape (n_alphas,)

The alphas along the path where models are computed.

coefs : array, shape (n_features, n_alphas) or (n_outputs, n_features, n_alphas)

Coefficients along the path.

dual_gaps : array, shape (n_alphas,)

The dual gaps at the end of the optimization for each alpha.

n_iters : array-like, shape (n_alphas,)

The number of iterations taken by the coordinate descent optimizer to reach the specified tolerance for each alpha. (Is returned when return_n_iter is set to True).

See also

MultiTaskElasticNet, MultiTaskElasticNetCV, ElasticNet, ElasticNetCV

Notes

See examples/plot_lasso_coordinate_descent_path.py for an example.

predict(X)[source]

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)[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 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 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.

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 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