Multi-task L1/L2 ElasticNet with built-in cross-validation.
The optimization objective for MultiTaskElasticNet 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: |
-
l1_ratio : float or array of floats -
The ElasticNet mixing parameter, with 0 < l1_ratio <= 1. For l1_ratio = 1 the penalty is an L1/L2 penalty. For l1_ratio = 0 it is an L2 penalty. For 0 < l1_ratio < 1, the penalty is a combination of L1/L2 and L2. This parameter can be a list, in which case the different values are tested by cross-validation and the one giving the best prediction score is used. Note that a good choice of list of values for l1_ratio is often to put more values close to 1 (i.e. Lasso) and less close to 0 (i.e. Ridge), as in [.1, .5, .7,
.9, .95, .99, 1] -
eps : float, optional -
Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3. -
n_alphas : int, optional -
Number of alphas along the regularization path -
alphas : array-like, optional -
List of alphas where to compute the models. If not provided, set automatically. -
fit_intercept : boolean -
whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (e.g. data is expected to be already centered). -
normalize : boolean, optional, default False -
This parameter is ignored when fit_intercept is set to False. If True, the regressors X will be normalized before regression by subtracting the mean and dividing by the l2-norm. If you wish to standardize, please use sklearn.preprocessing.StandardScaler before calling fit on an estimator with normalize=False. -
max_iter : int, optional -
The maximum number of iterations -
tol : float, optional -
The tolerance for the optimization: if the updates are smaller than tol, the optimization code checks the dual gap for optimality and continues until it is smaller than tol. -
cv : int, cross-validation generator or an iterable, optional -
Determines the cross-validation splitting strategy. Possible inputs for cv are: - None, to use the default 3-fold cross-validation,
- integer, to specify the number of folds.
- An object to be used as a cross-validation generator.
- An iterable yielding train/test splits.
For integer/None inputs, KFold is used. Refer User Guide for the various cross-validation strategies that can be used here. Changed in version 0.20: cv default value if None will change from 3-fold to 5-fold in v0.22. -
copy_X : boolean, optional, default True -
If True, X will be copied; else, it may be overwritten. -
verbose : bool or integer -
Amount of verbosity. -
n_jobs : int or None, optional (default=None) -
Number of CPUs to use during the cross validation. Note that this is used only if multiple values for l1_ratio are given. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors. See Glossary for more details. -
random_state : int, RandomState instance or None, optional, default None -
The seed of the pseudo random number generator that selects a random feature to update. If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random. Used when selection == ‘random’. -
selection : str, default ‘cyclic’ -
If set to ‘random’, a random coefficient is updated every iteration rather than looping over features sequentially by default. This (setting to ‘random’) often leads to significantly faster convergence especially when tol is higher than 1e-4. |
| Attributes: |
-
intercept_ : array, shape (n_tasks,) -
Independent term in decision function. -
coef_ : array, shape (n_tasks, n_features) -
Parameter vector (W in the cost function formula). Note that coef_ stores the transpose of W, W.T. -
alpha_ : float -
The amount of penalization chosen by cross validation -
mse_path_ : array, shape (n_alphas, n_folds) or (n_l1_ratio, n_alphas, n_folds) -
mean square error for the test set on each fold, varying alpha -
alphas_ : numpy array, shape (n_alphas,) or (n_l1_ratio, n_alphas) -
The grid of alphas used for fitting, for each l1_ratio -
l1_ratio_ : float -
best l1_ratio obtained by cross-validation. -
n_iter_ : int -
number of iterations run by the coordinate descent solver to reach the specified tolerance for the optimal alpha. |
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.MultiTaskElasticNetCV(cv=3)
>>> clf.fit([[0,0], [1, 1], [2, 2]],
... [[0, 0], [1, 1], [2, 2]])
...
MultiTaskElasticNetCV(alphas=None, copy_X=True, cv=3, eps=0.001,
fit_intercept=True, l1_ratio=0.5, max_iter=1000, n_alphas=100,
n_jobs=None, normalize=False, random_state=None, selection='cyclic',
tol=0.0001, verbose=0)
>>> print(clf.coef_)
[[0.52875032 0.46958558]
[0.52875032 0.46958558]]
>>> print(clf.intercept_)
[0.00166409 0.00166409]
Methods
fit(X, y) | Fit linear model with coordinate descent |
get_params([deep]) | Get parameters for this estimator. |
path(X, y[, l1_ratio, eps, n_alphas, …]) | Compute elastic net path with coordinate descent |
predict(X) | Predict using the linear model |
score(X, y[, sample_weight]) | Returns the coefficient of determination R^2 of the prediction. |
set_params(**params) | Set the parameters of this estimator. |
-
__init__(l1_ratio=0.5, eps=0.001, n_alphas=100, alphas=None, fit_intercept=True, normalize=False, max_iter=1000, tol=0.0001, cv=’warn’, copy_X=True, verbose=0, n_jobs=None, random_state=None, selection=’cyclic’) [source]
-
fit(X, y) [source]
-
Fit linear model with coordinate descent
Fit is on grid of alphas and best alpha estimated by cross-validation.
| Parameters: |
-
X : {array-like}, shape (n_samples, n_features) -
Training data. Pass directly as Fortran-contiguous data to avoid unnecessary memory duplication. If y is mono-output, X can be sparse. -
y : array-like, shape (n_samples,) or (n_samples, n_targets) -
Target values |
-
get_params(deep=True) [source]
-
Get parameters for this estimator.
| Parameters: |
-
deep : boolean, optional -
If True, will return the parameters for this estimator and contained subobjects that are estimators. |
| Returns: |
-
params : mapping of string to any -
Parameter names mapped to their values. |
-
static path(X, y, l1_ratio=0.5, eps=0.001, n_alphas=100, alphas=None, precompute=’auto’, Xy=None, copy_X=True, coef_init=None, verbose=False, return_n_iter=False, positive=False, check_input=True, **params) [source]
-
Compute elastic net path with coordinate descent
The elastic net optimization function varies for mono and multi-outputs.
For mono-output tasks it is:
1 / (2 * n_samples) * ||y - Xw||^2_2
+ alpha * l1_ratio * ||w||_1
+ 0.5 * alpha * (1 - l1_ratio) * ||w||^2_2
For multi-output tasks it is:
(1 / (2 * n_samples)) * ||Y - XW||^Fro_2
+ alpha * l1_ratio * ||W||_21
+ 0.5 * alpha * (1 - l1_ratio) * ||W||_Fro^2
Where:
||W||_21 = \sum_i \sqrt{\sum_j w_{ij}^2}
i.e. the sum of norm of each row.
Read more in the User Guide.
| Parameters: |
-
X : {array-like}, shape (n_samples, n_features) -
Training data. Pass directly as Fortran-contiguous data to avoid unnecessary memory duplication. If y is mono-output then X can be sparse. -
y : ndarray, shape (n_samples,) or (n_samples, n_outputs) -
Target values -
l1_ratio : float, optional -
float between 0 and 1 passed to elastic net (scaling between l1 and l2 penalties). l1_ratio=1 corresponds to the Lasso -
eps : float -
Length of the path. eps=1e-3 means that alpha_min / alpha_max = 1e-3 -
n_alphas : int, optional -
Number of alphas along the regularization path -
alphas : ndarray, optional -
List of alphas where to compute the models. If None alphas are set automatically -
precompute : True | False | ‘auto’ | array-like -
Whether to use a precomputed Gram matrix to speed up calculations. If set to 'auto' let us decide. The Gram matrix can also be passed as argument. -
Xy : array-like, optional -
Xy = np.dot(X.T, y) that can be precomputed. It is useful only when the Gram matrix is precomputed. -
copy_X : boolean, optional, default True -
If True, X will be copied; else, it may be overwritten. -
coef_init : array, shape (n_features, ) | None -
The initial values of the coefficients. -
verbose : bool or integer -
Amount of verbosity. -
return_n_iter : bool -
whether to return the number of iterations or not. -
positive : bool, default False -
If set to True, forces coefficients to be positive. (Only allowed when y.ndim == 1). -
check_input : bool, default True -
Skip input validation checks, including the Gram matrix when provided assuming there are handled by the caller when check_input=False. -
**params : kwargs -
keyword arguments passed to the coordinate descent solver. |
| Returns: |
-
alphas : array, shape (n_alphas,) -
The alphas along the path where models are computed. -
coefs : array, shape (n_features, n_alphas) or (n_outputs, n_features, n_alphas) -
Coefficients along the path. -
dual_gaps : array, shape (n_alphas,) -
The dual gaps at the end of the optimization for each alpha. -
n_iters : array-like, shape (n_alphas,) -
The number of iterations taken by the coordinate descent optimizer to reach the specified tolerance for each alpha. (Is returned when return_n_iter is set to True). |
Notes
For an example, see examples/linear_model/plot_lasso_coordinate_descent_path.py.
-
predict(X) [source]
-
Predict using the linear model
| Parameters: |
-
X : array_like or sparse matrix, shape (n_samples, n_features) -
Samples. |
| Returns: |
-
C : array, shape (n_samples,) -
Returns predicted values. |
-
score(X, y, sample_weight=None) [source]
-
Returns the coefficient of determination R^2 of the prediction.
The coefficient R^2 is defined as (1 - u/v), where u is the 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: |
-
X : array-like, shape = (n_samples, n_features) -
Test samples. For some estimators this may be a precomputed kernel matrix instead, shape = (n_samples, n_samples_fitted], where n_samples_fitted is the number of samples used in the fitting for the estimator. -
y : array-like, shape = (n_samples) or (n_samples, n_outputs) -
True values for X. -
sample_weight : array-like, shape = [n_samples], optional -
Sample weights. |
| Returns: |
-
score : float -
R^2 of self.predict(X) wrt. y. |
-
set_params(**params) [source]
-
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.