class sklearn.linear_model.LarsCV(*, fit_intercept=True, verbose=False, max_iter=500, normalize=True, precompute='auto', cv=None, max_n_alphas=1000, n_jobs=None, eps=2.220446049250313e-16, copy_X=True) [source]

Cross-validated Least Angle Regression model. See glossary entry for cross-validation estimator. Read more in the User Guide.  Parameters

fit_interceptbool, default=True

whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (i.e. data is expected to be centered).
verbosebool or int, default=False

Sets the verbosity amount.
max_iterint, default=500

Maximum number of iterations to perform.
normalizebool, default=True

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 StandardScaler before calling fit on an estimator with normalize=False.
precomputebool, ‘auto’ or array-like , default=’auto’

Whether to use a precomputed Gram matrix to speed up calculations. If set to 'auto' let us decide. The Gram matrix cannot be passed as argument since we will use only subsets of X.
cvint, cross-validation generator or an iterable, default=None

Determines the cross-validation splitting strategy. Possible inputs for cv are:  None, to use the default 5-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 0.22: cv default value if None changed from 3-fold to 5-fold.
max_n_alphasint, default=1000

The maximum number of points on the path used to compute the residuals in the cross-validation
n_jobsint or None, 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.
epsfloat, default=np.finfo(float).eps

The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the tol parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization.
copy_Xbool, default=True

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

active_list of length n_alphas or list of such lists

Indices of active variables at the end of the path. If this is a list of lists, the outer list length is n_targets.
coef_array-like of shape (n_features,)

parameter vector (w in the formulation formula)
intercept_float

independent term in decision function
coef_path_array-like of shape (n_features, n_alphas)

the varying values of the coefficients along the path
alpha_float

the estimated regularization parameter alpha
alphas_array-like of shape (n_alphas,)

the different values of alpha along the path
cv_alphas_array-like of shape (n_cv_alphas,)

all the values of alpha along the path for the different folds
mse_path_array-like of shape (n_folds, n_cv_alphas)

the mean square error on left-out for each fold along the path (alpha values given by cv_alphas)
n_iter_array-like or int

the number of iterations run by Lars with the optimal alpha.      See also
lars_path, LassoLars, LassoLarsCV

  Examples >>> from sklearn.linear_model import LarsCV
>>> from sklearn.datasets import make_regression
>>> X, y = make_regression(n_samples=200, noise=4.0, random_state=0)
>>> reg = LarsCV(cv=5).fit(X, y)
>>> reg.score(X, y)
0.9996...
>>> reg.alpha_
0.0254...
>>> reg.predict(X[:1,])
array([154.0842...])
 Methods
fit(X, y) Fit the model using X, y as training data.
get_params([deep]) Get parameters for this estimator.
predict(X) Predict using the linear model.
score(X, y[, sample_weight]) Return the coefficient of determination \(R^2\) of the prediction.
set_params(**params) Set the parameters of this estimator.
fit(X, y) [source]

Fit the model using X, y as training data.  Parameters

Xarray-like of shape (n_samples, n_features)

Training data.
yarray-like of shape (n_samples,)

Target values.    Returns

selfobject

returns an instance of self.

get_params(deep=True) [source]

Get parameters for this estimator.  Parameters

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.

predict(X) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.

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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.


sklearn.linear_model.LarsCV
class sklearn.linear_model.LarsCV(*, fit_intercept=True, verbose=False, max_iter=500, normalize=True, precompute='auto', cv=None, max_n_alphas=1000, n_jobs=None, eps=2.220446049250313e-16, copy_X=True) [source]

Cross-validated Least Angle Regression model. See glossary entry for cross-validation estimator. Read more in the User Guide.  Parameters

fit_interceptbool, default=True

whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (i.e. data is expected to be centered).
verbosebool or int, default=False

Sets the verbosity amount.
max_iterint, default=500

Maximum number of iterations to perform.
normalizebool, default=True

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 StandardScaler before calling fit on an estimator with normalize=False.
precomputebool, ‘auto’ or array-like , default=’auto’

Whether to use a precomputed Gram matrix to speed up calculations. If set to 'auto' let us decide. The Gram matrix cannot be passed as argument since we will use only subsets of X.
cvint, cross-validation generator or an iterable, default=None

Determines the cross-validation splitting strategy. Possible inputs for cv are:  None, to use the default 5-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 0.22: cv default value if None changed from 3-fold to 5-fold.
max_n_alphasint, default=1000

The maximum number of points on the path used to compute the residuals in the cross-validation
n_jobsint or None, 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.
epsfloat, default=np.finfo(float).eps

The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the tol parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization.
copy_Xbool, default=True

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

active_list of length n_alphas or list of such lists

Indices of active variables at the end of the path. If this is a list of lists, the outer list length is n_targets.
coef_array-like of shape (n_features,)

parameter vector (w in the formulation formula)
intercept_float

independent term in decision function
coef_path_array-like of shape (n_features, n_alphas)

the varying values of the coefficients along the path
alpha_float

the estimated regularization parameter alpha
alphas_array-like of shape (n_alphas,)

the different values of alpha along the path
cv_alphas_array-like of shape (n_cv_alphas,)

all the values of alpha along the path for the different folds
mse_path_array-like of shape (n_folds, n_cv_alphas)

the mean square error on left-out for each fold along the path (alpha values given by cv_alphas)
n_iter_array-like or int

the number of iterations run by Lars with the optimal alpha.      See also
lars_path, LassoLars, LassoLarsCV

  Examples >>> from sklearn.linear_model import LarsCV
>>> from sklearn.datasets import make_regression
>>> X, y = make_regression(n_samples=200, noise=4.0, random_state=0)
>>> reg = LarsCV(cv=5).fit(X, y)
>>> reg.score(X, y)
0.9996...
>>> reg.alpha_
0.0254...
>>> reg.predict(X[:1,])
array([154.0842...])
 Methods
fit(X, y) Fit the model using X, y as training data.
get_params([deep]) Get parameters for this estimator.
predict(X) Predict using the linear model.
score(X, y[, sample_weight]) Return the coefficient of determination \(R^2\) of the prediction.
set_params(**params) Set the parameters of this estimator.
fit(X, y) [source]

Fit the model using X, y as training data.  Parameters

Xarray-like of shape (n_samples, n_features)

Training data.
yarray-like of shape (n_samples,)

Target values.    Returns

selfobject

returns an instance of self.

get_params(deep=True) [source]

Get parameters for this estimator.  Parameters

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.

predict(X) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.

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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.


fit(X, y) [source]

Fit the model using X, y as training data.  Parameters

Xarray-like of shape (n_samples, n_features)

Training data.
yarray-like of shape (n_samples,)

Target values.    Returns

selfobject

returns an instance of self.


get_params(deep=True) [source]

Get parameters for this estimator.  Parameters

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.


predict(X) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.


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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).


set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.


sklearn.linear_model.lars_path(X, y, Xy=None, *, Gram=None, max_iter=500, alpha_min=0, method='lar', copy_X=True, eps=2.220446049250313e-16, copy_Gram=True, verbose=0, return_path=True, return_n_iter=False, positive=False) [source]

Compute Least Angle Regression or Lasso path using LARS algorithm [1] The optimization objective for the case method=’lasso’ is: (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
 in the case of method=’lars’, the objective function is only known in the form of an implicit equation (see discussion in [1]) Read more in the User Guide.  Parameters

XNone or array-like of shape (n_samples, n_features)

Input data. Note that if X is None then the Gram matrix must be specified, i.e., cannot be None or False.
yNone or array-like of shape (n_samples,)

Input targets.
Xyarray-like of shape (n_samples,) or (n_samples, n_targets), default=None

Xy = np.dot(X.T, y) that can be precomputed. It is useful only when the Gram matrix is precomputed.
GramNone, ‘auto’, array-like of shape (n_features, n_features), default=None

Precomputed Gram matrix (X’ * X), if 'auto', the Gram matrix is precomputed from the given X, if there are more samples than features.
max_iterint, default=500

Maximum number of iterations to perform, set to infinity for no limit.
alpha_minfloat, default=0

Minimum correlation along the path. It corresponds to the regularization parameter alpha parameter in the Lasso.
method{‘lar’, ‘lasso’}, default=’lar’

Specifies the returned model. Select 'lar' for Least Angle Regression, 'lasso' for the Lasso.
copy_Xbool, default=True

If False, X is overwritten.
epsfloat, default=np.finfo(float).eps

The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the tol parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization.
copy_Grambool, default=True

If False, Gram is overwritten.
verboseint, default=0

Controls output verbosity.
return_pathbool, default=True

If return_path==True returns the entire path, else returns only the last point of the path.
return_n_iterbool, default=False

Whether to return the number of iterations.
positivebool, default=False

Restrict coefficients to be >= 0. This option is only allowed with method ‘lasso’. Note that the model coefficients will not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (alphas_[alphas_ > 0.].min() when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent lasso_path function.    Returns

alphasarray-like of shape (n_alphas + 1,)

Maximum of covariances (in absolute value) at each iteration. n_alphas is either max_iter, n_features or the number of nodes in the path with alpha >= alpha_min, whichever is smaller.
activearray-like of shape (n_alphas,)

Indices of active variables at the end of the path.
coefsarray-like of shape (n_features, n_alphas + 1)

Coefficients along the path
n_iterint

Number of iterations run. Returned only if return_n_iter is set to True.      See also
lars_path_gram

lasso_path

lasso_path_gram

LassoLars

Lars

LassoLarsCV

LarsCV

sklearn.decomposition.sparse_encode

  References
1
“Least Angle Regression”, Efron et al. http://statweb.stanford.edu/~tibs/ftp/lars.pdf
2
Wikipedia entry on the Least-angle regression
3
Wikipedia entry on the Lasso


sklearn.linear_model.lars_path_gram(Xy, Gram, *, n_samples, max_iter=500, alpha_min=0, method='lar', copy_X=True, eps=2.220446049250313e-16, copy_Gram=True, verbose=0, return_path=True, return_n_iter=False, positive=False) [source]

lars_path in the sufficient stats mode [1] The optimization objective for the case method=’lasso’ is: (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
 in the case of method=’lars’, the objective function is only known in the form of an implicit equation (see discussion in [1]) Read more in the User Guide.  Parameters

Xyarray-like of shape (n_samples,) or (n_samples, n_targets)

Xy = np.dot(X.T, y).
Gramarray-like of shape (n_features, n_features)

Gram = np.dot(X.T * X).
n_samplesint or float

Equivalent size of sample.
max_iterint, default=500

Maximum number of iterations to perform, set to infinity for no limit.
alpha_minfloat, default=0

Minimum correlation along the path. It corresponds to the regularization parameter alpha parameter in the Lasso.
method{‘lar’, ‘lasso’}, default=’lar’

Specifies the returned model. Select 'lar' for Least Angle Regression, 'lasso' for the Lasso.
copy_Xbool, default=True

If False, X is overwritten.
epsfloat, default=np.finfo(float).eps

The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the tol parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization.
copy_Grambool, default=True

If False, Gram is overwritten.
verboseint, default=0

Controls output verbosity.
return_pathbool, default=True

If return_path==True returns the entire path, else returns only the last point of the path.
return_n_iterbool, default=False

Whether to return the number of iterations.
positivebool, default=False

Restrict coefficients to be >= 0. This option is only allowed with method ‘lasso’. Note that the model coefficients will not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (alphas_[alphas_ > 0.].min() when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent lasso_path function.    Returns

alphasarray-like of shape (n_alphas + 1,)

Maximum of covariances (in absolute value) at each iteration. n_alphas is either max_iter, n_features or the number of nodes in the path with alpha >= alpha_min, whichever is smaller.
activearray-like of shape (n_alphas,)

Indices of active variables at the end of the path.
coefsarray-like of shape (n_features, n_alphas + 1)

Coefficients along the path
n_iterint

Number of iterations run. Returned only if return_n_iter is set to True.      See also
lars_path

lasso_path

lasso_path_gram

LassoLars

Lars

LassoLarsCV

LarsCV

sklearn.decomposition.sparse_encode

  References
1
“Least Angle Regression”, Efron et al. http://statweb.stanford.edu/~tibs/ftp/lars.pdf
2
Wikipedia entry on the Least-angle regression
3
Wikipedia entry on the 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

alphafloat, default=1.0

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 with the Lasso object is not advised. Given this, you should use the LinearRegression object.
fit_interceptbool, default=True

Whether to calculate the intercept for this model. If set to False, no intercept will be used in calculations (i.e. data is expected to be centered).
normalizebool, 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 StandardScaler before calling fit on an estimator with normalize=False.
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=False

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.
copy_Xbool, default=True

If True, X will be copied; else, it may be overwritten.
max_iterint, default=1000

The maximum number of iterations.
tolfloat, default=1e-4

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_startbool, default=False

When set to True, reuse the solution of the previous call to fit as initialization, otherwise, just erase the previous solution. See the Glossary.
positivebool, default=False

When set to True, forces the coefficients to be positive.
random_stateint, RandomState instance, default=None

The seed of the pseudo random number generator that selects a random feature to update. Used when selection == ‘random’. Pass an int for reproducible output across multiple function calls. See Glossary.
selection{‘cyclic’, ‘random’}, 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

coef_ndarray of shape (n_features,) or (n_targets, n_features)

Parameter vector (w in the cost function formula).
dual_gap_float or ndarray of shape (n_targets,)

Given param alpha, the dual gaps at the end of the optimization, same shape as each observation of y.
sparse_coef_sparse matrix of shape (n_features, 1) or (n_targets, n_features)

Sparse representation of the fitted coef_.
intercept_float or ndarray of shape (n_targets,)

Independent term in decision function.
n_iter_int or list of int

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)
>>> print(clf.coef_)
[0.85 0.  ]
>>> print(clf.intercept_)
0.15...
 Methods
fit(X, y[, sample_weight, check_input]) Fit model with coordinate descent.
get_params([deep]) Get parameters for this estimator.
path(*args, **kwargs) Compute elastic net path with coordinate descent.
predict(X) Predict using the linear model.
score(X, y[, sample_weight]) Return the coefficient of determination \(R^2\) of the prediction.
set_params(**params) Set the parameters of this estimator.
fit(X, y, sample_weight=None, check_input=True) [source]

Fit model with coordinate descent.  Parameters

X{ndarray, sparse matrix} of (n_samples, n_features)

Data.
y{ndarray, sparse matrix} of shape (n_samples,) or (n_samples, n_targets)

Target. Will be cast to X’s dtype if necessary.
sample_weightfloat or array-like of shape (n_samples,), default=None

Sample weight.  New in version 0.23.
check_inputbool, default=True

Allow to bypass several input checking. Don’t use this parameter unless you know what you do.     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

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.

static path(*args, **kwargs) [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, sparse matrix} of 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{array-like, sparse matrix} of shape (n_samples,) or (n_samples, n_outputs)

Target values.
l1_ratiofloat, default=0.5

Number between 0 and 1 passed to elastic net (scaling between l1 and l2 penalties). l1_ratio=1 corresponds to the Lasso.
epsfloat, default=1e-3

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3.
n_alphasint, default=100

Number of alphas along the regularization path.
alphasndarray, default=None

List of alphas where to compute the models. If None alphas are set automatically.
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’

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.
Xyarray-like of shape (n_features,) or (n_features, n_outputs), default=None

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

If True, X will be copied; else, it may be overwritten.
coef_initndarray of shape (n_features, ), default=None

The initial values of the coefficients.
verbosebool or int, default=False

Amount of verbosity.
return_n_iterbool, default=False

Whether to return the number of iterations or not.
positivebool, default=False

If set to True, forces coefficients to be positive. (Only allowed when y.ndim == 1).
check_inputbool, default=True

If set to False, the input validation checks are skipped (including the Gram matrix when provided). It is assumed that they are handled by the caller.
**paramskwargs

Keyword arguments passed to the coordinate descent solver.    Returns

alphasndarray of shape (n_alphas,)

The alphas along the path where models are computed.
coefsndarray of shape (n_features, n_alphas) or (n_outputs, n_features, n_alphas)

Coefficients along the path.
dual_gapsndarray of shape (n_alphas,)

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

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

predict(X) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.

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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.

property sparse_coef_
Sparse representation of the fitted coef_.


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

alphafloat, default=1.0

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 with the Lasso object is not advised. Given this, you should use the LinearRegression object.
fit_interceptbool, default=True

Whether to calculate the intercept for this model. If set to False, no intercept will be used in calculations (i.e. data is expected to be centered).
normalizebool, 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 StandardScaler before calling fit on an estimator with normalize=False.
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=False

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.
copy_Xbool, default=True

If True, X will be copied; else, it may be overwritten.
max_iterint, default=1000

The maximum number of iterations.
tolfloat, default=1e-4

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_startbool, default=False

When set to True, reuse the solution of the previous call to fit as initialization, otherwise, just erase the previous solution. See the Glossary.
positivebool, default=False

When set to True, forces the coefficients to be positive.
random_stateint, RandomState instance, default=None

The seed of the pseudo random number generator that selects a random feature to update. Used when selection == ‘random’. Pass an int for reproducible output across multiple function calls. See Glossary.
selection{‘cyclic’, ‘random’}, 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

coef_ndarray of shape (n_features,) or (n_targets, n_features)

Parameter vector (w in the cost function formula).
dual_gap_float or ndarray of shape (n_targets,)

Given param alpha, the dual gaps at the end of the optimization, same shape as each observation of y.
sparse_coef_sparse matrix of shape (n_features, 1) or (n_targets, n_features)

Sparse representation of the fitted coef_.
intercept_float or ndarray of shape (n_targets,)

Independent term in decision function.
n_iter_int or list of int

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)
>>> print(clf.coef_)
[0.85 0.  ]
>>> print(clf.intercept_)
0.15...
 Methods
fit(X, y[, sample_weight, check_input]) Fit model with coordinate descent.
get_params([deep]) Get parameters for this estimator.
path(*args, **kwargs) Compute elastic net path with coordinate descent.
predict(X) Predict using the linear model.
score(X, y[, sample_weight]) Return the coefficient of determination \(R^2\) of the prediction.
set_params(**params) Set the parameters of this estimator.
fit(X, y, sample_weight=None, check_input=True) [source]

Fit model with coordinate descent.  Parameters

X{ndarray, sparse matrix} of (n_samples, n_features)

Data.
y{ndarray, sparse matrix} of shape (n_samples,) or (n_samples, n_targets)

Target. Will be cast to X’s dtype if necessary.
sample_weightfloat or array-like of shape (n_samples,), default=None

Sample weight.  New in version 0.23.
check_inputbool, default=True

Allow to bypass several input checking. Don’t use this parameter unless you know what you do.     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

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.

static path(*args, **kwargs) [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, sparse matrix} of 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{array-like, sparse matrix} of shape (n_samples,) or (n_samples, n_outputs)

Target values.
l1_ratiofloat, default=0.5

Number between 0 and 1 passed to elastic net (scaling between l1 and l2 penalties). l1_ratio=1 corresponds to the Lasso.
epsfloat, default=1e-3

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3.
n_alphasint, default=100

Number of alphas along the regularization path.
alphasndarray, default=None

List of alphas where to compute the models. If None alphas are set automatically.
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’

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.
Xyarray-like of shape (n_features,) or (n_features, n_outputs), default=None

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

If True, X will be copied; else, it may be overwritten.
coef_initndarray of shape (n_features, ), default=None

The initial values of the coefficients.
verbosebool or int, default=False

Amount of verbosity.
return_n_iterbool, default=False

Whether to return the number of iterations or not.
positivebool, default=False

If set to True, forces coefficients to be positive. (Only allowed when y.ndim == 1).
check_inputbool, default=True

If set to False, the input validation checks are skipped (including the Gram matrix when provided). It is assumed that they are handled by the caller.
**paramskwargs

Keyword arguments passed to the coordinate descent solver.    Returns

alphasndarray of shape (n_alphas,)

The alphas along the path where models are computed.
coefsndarray of shape (n_features, n_alphas) or (n_outputs, n_features, n_alphas)

Coefficients along the path.
dual_gapsndarray of shape (n_alphas,)

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

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

predict(X) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.

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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.

property sparse_coef_
Sparse representation of the fitted coef_.

 Examples using sklearn.linear_model.Lasso

  Release Highlights for scikit-learn 0.23

  Compressive sensing: tomography reconstruction with L1 prior (Lasso)

  Lasso on dense and sparse data

  Joint feature selection with multi-task Lasso

  Lasso and Elastic Net for Sparse Signals

  Cross-validation on diabetes Dataset Exercise


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

Fit model with coordinate descent.  Parameters

X{ndarray, sparse matrix} of (n_samples, n_features)

Data.
y{ndarray, sparse matrix} of shape (n_samples,) or (n_samples, n_targets)

Target. Will be cast to X’s dtype if necessary.
sample_weightfloat or array-like of shape (n_samples,), default=None

Sample weight.  New in version 0.23.
check_inputbool, default=True

Allow to bypass several input checking. Don’t use this parameter unless you know what you do.     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.


static path(*args, **kwargs) [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, sparse matrix} of 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{array-like, sparse matrix} of shape (n_samples,) or (n_samples, n_outputs)

Target values.
l1_ratiofloat, default=0.5

Number between 0 and 1 passed to elastic net (scaling between l1 and l2 penalties). l1_ratio=1 corresponds to the Lasso.
epsfloat, default=1e-3

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3.
n_alphasint, default=100

Number of alphas along the regularization path.
alphasndarray, default=None

List of alphas where to compute the models. If None alphas are set automatically.
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’

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.
Xyarray-like of shape (n_features,) or (n_features, n_outputs), default=None

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

If True, X will be copied; else, it may be overwritten.
coef_initndarray of shape (n_features, ), default=None

The initial values of the coefficients.
verbosebool or int, default=False

Amount of verbosity.
return_n_iterbool, default=False

Whether to return the number of iterations or not.
positivebool, default=False

If set to True, forces coefficients to be positive. (Only allowed when y.ndim == 1).
check_inputbool, default=True

If set to False, the input validation checks are skipped (including the Gram matrix when provided). It is assumed that they are handled by the caller.
**paramskwargs

Keyword arguments passed to the coordinate descent solver.    Returns

alphasndarray of shape (n_alphas,)

The alphas along the path where models are computed.
coefsndarray of shape (n_features, n_alphas) or (n_outputs, n_features, n_alphas)

Coefficients along the path.
dual_gapsndarray of shape (n_alphas,)

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

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


property sparse_coef_
Sparse representation of the fitted coef_.


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=None, positive=False, random_state=None, selection='cyclic') [source]

Lasso linear model with iterative fitting along a regularization path. See glossary entry for cross-validation estimator. 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
 Read more in the User Guide.  Parameters

epsfloat, default=1e-3

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3.
n_alphasint, default=100

Number of alphas along the regularization path.
alphasndarray, default=None

List of alphas where to compute the models. If None alphas are set automatically.
fit_interceptbool, default=True

Whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (i.e. data is expected to be centered).
normalizebool, 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 StandardScaler before calling fit on an estimator with normalize=False.
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’

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_iterint, default=1000

The maximum number of iterations.
tolfloat, default=1e-4

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.
copy_Xbool, default=True

If True, X will be copied; else, it may be overwritten.
cvint, cross-validation generator or iterable, default=None

Determines the cross-validation splitting strategy. Possible inputs for cv are:  None, to use the default 5-fold cross-validation, int, to specify the number of folds.
CV splitter, An iterable yielding (train, test) splits as arrays of indices.  For int/None inputs, KFold is used. Refer User Guide for the various cross-validation strategies that can be used here.  Changed in version 0.22: cv default value if None changed from 3-fold to 5-fold.
verbosebool or int, default=False

Amount of verbosity.
n_jobsint, 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.
positivebool, default=False

If positive, restrict regression coefficients to be positive.
random_stateint, RandomState instance, default=None

The seed of the pseudo random number generator that selects a random feature to update. Used when selection == ‘random’. Pass an int for reproducible output across multiple function calls. See Glossary.
selection{‘cyclic’, ‘random’}, 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.
coef_ndarray of shape (n_features,) or (n_targets, n_features)

Parameter vector (w in the cost function formula).
intercept_float or ndarray of shape (n_targets,)

Independent term in decision function.
mse_path_ndarray of shape (n_alphas, n_folds)

Mean square error for the test set on each fold, varying alpha.
alphas_ndarray of shape (n_alphas,)

The grid of alphas used for fitting.
dual_gap_float or ndarray of shape (n_targets,)

The dual gap at the end of the optimization for the optimal alpha (alpha_).
n_iter_int

Number of iterations run by the coordinate descent solver to reach the specified tolerance for the optimal alpha.      See also
lars_path

lasso_path

LassoLars

Lasso

LassoLarsCV

  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. Examples >>> from sklearn.linear_model import LassoCV
>>> from sklearn.datasets import make_regression
>>> X, y = make_regression(noise=4, random_state=0)
>>> reg = LassoCV(cv=5, random_state=0).fit(X, y)
>>> reg.score(X, y)
0.9993...
>>> reg.predict(X[:1,])
array([-78.4951...])
 Methods
fit(X, y) Fit linear model with coordinate descent.
get_params([deep]) Get parameters for this estimator.
path(*args, **kwargs) Compute Lasso path with coordinate descent
predict(X) Predict using the linear model.
score(X, y[, sample_weight]) Return the coefficient of determination \(R^2\) of the prediction.
set_params(**params) Set the parameters of this estimator.
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, sparse matrix} of 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.
yarray-like of shape (n_samples,) or (n_samples, n_targets)

Target values.

get_params(deep=True) [source]

Get parameters for this estimator.  Parameters

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.

static path(*args, **kwargs) [source]

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. Read more in the User Guide.  Parameters

X{array-like, sparse matrix} of 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{array-like, sparse matrix} of shape (n_samples,) or (n_samples, n_outputs)

Target values
epsfloat, default=1e-3

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3
n_alphasint, default=100

Number of alphas along the regularization path
alphasndarray, default=None

List of alphas where to compute the models. If None alphas are set automatically
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’

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.
Xyarray-like of shape (n_features,) or (n_features, n_outputs), default=None

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

If True, X will be copied; else, it may be overwritten.
coef_initndarray of shape (n_features, ), default=None

The initial values of the coefficients.
verbosebool or int, default=False

Amount of verbosity.
return_n_iterbool, default=False

whether to return the number of iterations or not.
positivebool, default=False

If set to True, forces coefficients to be positive. (Only allowed when y.ndim == 1).
**paramskwargs

keyword arguments passed to the coordinate descent solver.    Returns

alphasndarray of shape (n_alphas,)

The alphas along the path where models are computed.
coefsndarray of shape (n_features, n_alphas) or (n_outputs, n_features, n_alphas)

Coefficients along the path.
dual_gapsndarray of shape (n_alphas,)

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

The number of iterations taken by the coordinate descent optimizer to reach the specified tolerance for each alpha.      See also
lars_path

Lasso

LassoLars

LassoCV

LassoLarsCV

sklearn.decomposition.sparse_encode

  Notes For an example, see examples/linear_model/plot_lasso_coordinate_descent_path.py. 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 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])
>>> 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) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.

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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.


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=None, positive=False, random_state=None, selection='cyclic') [source]

Lasso linear model with iterative fitting along a regularization path. See glossary entry for cross-validation estimator. 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
 Read more in the User Guide.  Parameters

epsfloat, default=1e-3

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3.
n_alphasint, default=100

Number of alphas along the regularization path.
alphasndarray, default=None

List of alphas where to compute the models. If None alphas are set automatically.
fit_interceptbool, default=True

Whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (i.e. data is expected to be centered).
normalizebool, 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 StandardScaler before calling fit on an estimator with normalize=False.
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’

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_iterint, default=1000

The maximum number of iterations.
tolfloat, default=1e-4

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.
copy_Xbool, default=True

If True, X will be copied; else, it may be overwritten.
cvint, cross-validation generator or iterable, default=None

Determines the cross-validation splitting strategy. Possible inputs for cv are:  None, to use the default 5-fold cross-validation, int, to specify the number of folds.
CV splitter, An iterable yielding (train, test) splits as arrays of indices.  For int/None inputs, KFold is used. Refer User Guide for the various cross-validation strategies that can be used here.  Changed in version 0.22: cv default value if None changed from 3-fold to 5-fold.
verbosebool or int, default=False

Amount of verbosity.
n_jobsint, 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.
positivebool, default=False

If positive, restrict regression coefficients to be positive.
random_stateint, RandomState instance, default=None

The seed of the pseudo random number generator that selects a random feature to update. Used when selection == ‘random’. Pass an int for reproducible output across multiple function calls. See Glossary.
selection{‘cyclic’, ‘random’}, 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.
coef_ndarray of shape (n_features,) or (n_targets, n_features)

Parameter vector (w in the cost function formula).
intercept_float or ndarray of shape (n_targets,)

Independent term in decision function.
mse_path_ndarray of shape (n_alphas, n_folds)

Mean square error for the test set on each fold, varying alpha.
alphas_ndarray of shape (n_alphas,)

The grid of alphas used for fitting.
dual_gap_float or ndarray of shape (n_targets,)

The dual gap at the end of the optimization for the optimal alpha (alpha_).
n_iter_int

Number of iterations run by the coordinate descent solver to reach the specified tolerance for the optimal alpha.      See also
lars_path

lasso_path

LassoLars

Lasso

LassoLarsCV

  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. Examples >>> from sklearn.linear_model import LassoCV
>>> from sklearn.datasets import make_regression
>>> X, y = make_regression(noise=4, random_state=0)
>>> reg = LassoCV(cv=5, random_state=0).fit(X, y)
>>> reg.score(X, y)
0.9993...
>>> reg.predict(X[:1,])
array([-78.4951...])
 Methods
fit(X, y) Fit linear model with coordinate descent.
get_params([deep]) Get parameters for this estimator.
path(*args, **kwargs) Compute Lasso path with coordinate descent
predict(X) Predict using the linear model.
score(X, y[, sample_weight]) Return the coefficient of determination \(R^2\) of the prediction.
set_params(**params) Set the parameters of this estimator.
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, sparse matrix} of 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.
yarray-like of shape (n_samples,) or (n_samples, n_targets)

Target values.

get_params(deep=True) [source]

Get parameters for this estimator.  Parameters

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.

static path(*args, **kwargs) [source]

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. Read more in the User Guide.  Parameters

X{array-like, sparse matrix} of 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{array-like, sparse matrix} of shape (n_samples,) or (n_samples, n_outputs)

Target values
epsfloat, default=1e-3

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3
n_alphasint, default=100

Number of alphas along the regularization path
alphasndarray, default=None

List of alphas where to compute the models. If None alphas are set automatically
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’

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.
Xyarray-like of shape (n_features,) or (n_features, n_outputs), default=None

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

If True, X will be copied; else, it may be overwritten.
coef_initndarray of shape (n_features, ), default=None

The initial values of the coefficients.
verbosebool or int, default=False

Amount of verbosity.
return_n_iterbool, default=False

whether to return the number of iterations or not.
positivebool, default=False

If set to True, forces coefficients to be positive. (Only allowed when y.ndim == 1).
**paramskwargs

keyword arguments passed to the coordinate descent solver.    Returns

alphasndarray of shape (n_alphas,)

The alphas along the path where models are computed.
coefsndarray of shape (n_features, n_alphas) or (n_outputs, n_features, n_alphas)

Coefficients along the path.
dual_gapsndarray of shape (n_alphas,)

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

The number of iterations taken by the coordinate descent optimizer to reach the specified tolerance for each alpha.      See also
lars_path

Lasso

LassoLars

LassoCV

LassoLarsCV

sklearn.decomposition.sparse_encode

  Notes For an example, see examples/linear_model/plot_lasso_coordinate_descent_path.py. 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 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])
>>> 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) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.

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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.

 Examples using sklearn.linear_model.LassoCV

  Combine predictors using stacking

  Model-based and sequential feature selection

  Lasso model selection: Cross-Validation / AIC / BIC

  Common pitfalls in interpretation of coefficients of linear models

  Cross-validation on diabetes Dataset Exercise


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, sparse matrix} of 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.
yarray-like of shape (n_samples,) or (n_samples, n_targets)

Target values.


static path(*args, **kwargs) [source]

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. Read more in the User Guide.  Parameters

X{array-like, sparse matrix} of 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{array-like, sparse matrix} of shape (n_samples,) or (n_samples, n_outputs)

Target values
epsfloat, default=1e-3

Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3
n_alphasint, default=100

Number of alphas along the regularization path
alphasndarray, default=None

List of alphas where to compute the models. If None alphas are set automatically
precompute‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’

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.
Xyarray-like of shape (n_features,) or (n_features, n_outputs), default=None

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

If True, X will be copied; else, it may be overwritten.
coef_initndarray of shape (n_features, ), default=None

The initial values of the coefficients.
verbosebool or int, default=False

Amount of verbosity.
return_n_iterbool, default=False

whether to return the number of iterations or not.
positivebool, default=False

If set to True, forces coefficients to be positive. (Only allowed when y.ndim == 1).
**paramskwargs

keyword arguments passed to the coordinate descent solver.    Returns

alphasndarray of shape (n_alphas,)

The alphas along the path where models are computed.
coefsndarray of shape (n_features, n_alphas) or (n_outputs, n_features, n_alphas)

Coefficients along the path.
dual_gapsndarray of shape (n_alphas,)

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

The number of iterations taken by the coordinate descent optimizer to reach the specified tolerance for each alpha.      See also
lars_path

Lasso

LassoLars

LassoCV

LassoLarsCV

sklearn.decomposition.sparse_encode

  Notes For an example, see examples/linear_model/plot_lasso_coordinate_descent_path.py. 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 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])
>>> 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]]


class sklearn.linear_model.LassoLars(alpha=1.0, *, fit_intercept=True, verbose=False, normalize=True, precompute='auto', max_iter=500, eps=2.220446049250313e-16, copy_X=True, fit_path=True, positive=False, jitter=None, random_state=None) [source]

Lasso model fit with Least Angle Regression a.k.a. Lars It is a Linear Model trained with an L1 prior as regularizer. The optimization objective for Lasso is: (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
 Read more in the User Guide.  Parameters

alphafloat, default=1.0

Constant that multiplies the penalty term. Defaults to 1.0. alpha = 0 is equivalent to an ordinary least square, solved by LinearRegression. For numerical reasons, using alpha = 0 with the LassoLars object is not advised and you should prefer the LinearRegression object.
fit_interceptbool, default=True

whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (i.e. data is expected to be centered).
verbosebool or int, default=False

Sets the verbosity amount.
normalizebool, default=True

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 StandardScaler before calling fit on an estimator with normalize=False.
precomputebool, ‘auto’ or array-like, default=’auto’

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_iterint, default=500

Maximum number of iterations to perform.
epsfloat, default=np.finfo(float).eps

The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the tol parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization.
copy_Xbool, default=True

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

If True the full path is stored in the coef_path_ attribute. If you compute the solution for a large problem or many targets, setting fit_path to False will lead to a speedup, especially with a small alpha.
positivebool, default=False

Restrict coefficients to be >= 0. Be aware that you might want to remove fit_intercept which is set True by default. Under the positive restriction the model coefficients will not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (alphas_[alphas_ >
0.].min() when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent Lasso estimator.
jitterfloat, default=None

Upper bound on a uniform noise parameter to be added to the y values, to satisfy the model’s assumption of one-at-a-time computations. Might help with stability.  New in version 0.23.
random_stateint, RandomState instance or None, default=None

Determines random number generation for jittering. Pass an int for reproducible output across multiple function calls. See Glossary. Ignored if jitter is None.  New in version 0.23.     Attributes

alphas_array-like of shape (n_alphas + 1,) or list of such arrays

Maximum of covariances (in absolute value) at each iteration. n_alphas is either max_iter, n_features or the number of nodes in the path with alpha >= alpha_min, whichever is smaller. If this is a list of array-like, the length of the outer list is n_targets.
active_list of length n_alphas or list of such lists

Indices of active variables at the end of the path. If this is a list of list, the length of the outer list is n_targets.
coef_path_array-like of shape (n_features, n_alphas + 1) or list of such arrays

If a list is passed it’s expected to be one of n_targets such arrays. The varying values of the coefficients along the path. It is not present if the fit_path parameter is False. If this is a list of array-like, the length of the outer list is n_targets.
coef_array-like of shape (n_features,) or (n_targets, n_features)

Parameter vector (w in the formulation formula).
intercept_float or array-like of shape (n_targets,)

Independent term in decision function.
n_iter_array-like or int

The number of iterations taken by lars_path to find the grid of alphas for each target.      See also
lars_path

lasso_path

Lasso

LassoCV

LassoLarsCV

LassoLarsIC

sklearn.decomposition.sparse_encode

  Examples >>> from sklearn import linear_model
>>> reg = linear_model.LassoLars(alpha=0.01)
>>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1, 0, -1])
LassoLars(alpha=0.01)
>>> print(reg.coef_)
[ 0.         -0.963257...]
 Methods
fit(X, y[, Xy]) Fit the model using X, y as training data.
get_params([deep]) Get parameters for this estimator.
predict(X) Predict using the linear model.
score(X, y[, sample_weight]) Return the coefficient of determination \(R^2\) of the prediction.
set_params(**params) Set the parameters of this estimator.
fit(X, y, Xy=None) [source]

Fit the model using X, y as training data.  Parameters

Xarray-like of shape (n_samples, n_features)

Training data.
yarray-like of shape (n_samples,) or (n_samples, n_targets)

Target values.
Xyarray-like of shape (n_samples,) or (n_samples, n_targets), default=None

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

selfobject

returns an instance of self.

get_params(deep=True) [source]

Get parameters for this estimator.  Parameters

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.

predict(X) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.

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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.


sklearn.linear_model.LassoLars
class sklearn.linear_model.LassoLars(alpha=1.0, *, fit_intercept=True, verbose=False, normalize=True, precompute='auto', max_iter=500, eps=2.220446049250313e-16, copy_X=True, fit_path=True, positive=False, jitter=None, random_state=None) [source]

Lasso model fit with Least Angle Regression a.k.a. Lars It is a Linear Model trained with an L1 prior as regularizer. The optimization objective for Lasso is: (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1
 Read more in the User Guide.  Parameters

alphafloat, default=1.0

Constant that multiplies the penalty term. Defaults to 1.0. alpha = 0 is equivalent to an ordinary least square, solved by LinearRegression. For numerical reasons, using alpha = 0 with the LassoLars object is not advised and you should prefer the LinearRegression object.
fit_interceptbool, default=True

whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (i.e. data is expected to be centered).
verbosebool or int, default=False

Sets the verbosity amount.
normalizebool, default=True

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 StandardScaler before calling fit on an estimator with normalize=False.
precomputebool, ‘auto’ or array-like, default=’auto’

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_iterint, default=500

Maximum number of iterations to perform.
epsfloat, default=np.finfo(float).eps

The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the tol parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization.
copy_Xbool, default=True

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

If True the full path is stored in the coef_path_ attribute. If you compute the solution for a large problem or many targets, setting fit_path to False will lead to a speedup, especially with a small alpha.
positivebool, default=False

Restrict coefficients to be >= 0. Be aware that you might want to remove fit_intercept which is set True by default. Under the positive restriction the model coefficients will not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (alphas_[alphas_ >
0.].min() when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent Lasso estimator.
jitterfloat, default=None

Upper bound on a uniform noise parameter to be added to the y values, to satisfy the model’s assumption of one-at-a-time computations. Might help with stability.  New in version 0.23.
random_stateint, RandomState instance or None, default=None

Determines random number generation for jittering. Pass an int for reproducible output across multiple function calls. See Glossary. Ignored if jitter is None.  New in version 0.23.     Attributes

alphas_array-like of shape (n_alphas + 1,) or list of such arrays

Maximum of covariances (in absolute value) at each iteration. n_alphas is either max_iter, n_features or the number of nodes in the path with alpha >= alpha_min, whichever is smaller. If this is a list of array-like, the length of the outer list is n_targets.
active_list of length n_alphas or list of such lists

Indices of active variables at the end of the path. If this is a list of list, the length of the outer list is n_targets.
coef_path_array-like of shape (n_features, n_alphas + 1) or list of such arrays

If a list is passed it’s expected to be one of n_targets such arrays. The varying values of the coefficients along the path. It is not present if the fit_path parameter is False. If this is a list of array-like, the length of the outer list is n_targets.
coef_array-like of shape (n_features,) or (n_targets, n_features)

Parameter vector (w in the formulation formula).
intercept_float or array-like of shape (n_targets,)

Independent term in decision function.
n_iter_array-like or int

The number of iterations taken by lars_path to find the grid of alphas for each target.      See also
lars_path

lasso_path

Lasso

LassoCV

LassoLarsCV

LassoLarsIC

sklearn.decomposition.sparse_encode

  Examples >>> from sklearn import linear_model
>>> reg = linear_model.LassoLars(alpha=0.01)
>>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1, 0, -1])
LassoLars(alpha=0.01)
>>> print(reg.coef_)
[ 0.         -0.963257...]
 Methods
fit(X, y[, Xy]) Fit the model using X, y as training data.
get_params([deep]) Get parameters for this estimator.
predict(X) Predict using the linear model.
score(X, y[, sample_weight]) Return the coefficient of determination \(R^2\) of the prediction.
set_params(**params) Set the parameters of this estimator.
fit(X, y, Xy=None) [source]

Fit the model using X, y as training data.  Parameters

Xarray-like of shape (n_samples, n_features)

Training data.
yarray-like of shape (n_samples,) or (n_samples, n_targets)

Target values.
Xyarray-like of shape (n_samples,) or (n_samples, n_targets), default=None

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

selfobject

returns an instance of self.

get_params(deep=True) [source]

Get parameters for this estimator.  Parameters

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.

predict(X) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.

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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).

set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.


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

Fit the model using X, y as training data.  Parameters

Xarray-like of shape (n_samples, n_features)

Training data.
yarray-like of shape (n_samples,) or (n_samples, n_targets)

Target values.
Xyarray-like of shape (n_samples,) or (n_samples, n_targets), default=None

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

selfobject

returns an instance of self.


get_params(deep=True) [source]

Get parameters for this estimator.  Parameters

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.    Returns

paramsdict

Parameter names mapped to their values.


predict(X) [source]

Predict using the linear model.  Parameters

Xarray-like or sparse matrix, shape (n_samples, n_features)

Samples.    Returns

Carray, shape (n_samples,)

Returns predicted values.


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

Return the coefficient of determination \(R^2\) of the prediction. The coefficient \(R^2\) is defined as \((1 - \frac{u}{v})\), 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

Xarray-like of shape (n_samples, n_features)

Test samples. For some estimators this may be a precomputed kernel matrix or a list of generic objects instead with shape (n_samples, n_samples_fitted), where n_samples_fitted is the number of samples used in the fitting for the estimator.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)

True values for X.
sample_weightarray-like of shape (n_samples,), default=None

Sample weights.    Returns

scorefloat

\(R^2\) of self.predict(X) wrt. y.     Notes The \(R^2\) score used when calling score on a regressor uses multioutput='uniform_average' from version 0.23 to keep consistent with default value of r2_score. This influences the score method of all the multioutput regressors (except for MultiOutputRegressor).


set_params(**params) [source]

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

**paramsdict

Estimator parameters.    Returns

selfestimator instance

Estimator instance.