# LinearRegression#

Warning

In the old version of abess (before 0.4.0), this model is named abess.linear.abessLm. Please note that it will be deprecated in version 0.6.0.

class abess.linear.LinearRegression[source]#

Adaptive Best-Subset Selection(ABESS) algorithm for linear regression.

Parameters
• path_type ({"seq", "gs"}, optional, default="seq") --

The method to be used to select the optimal support size.

• For path_type = "seq", we solve the best subset selection problem for each size in support_size.

• For path_type = "gs", we solve the best subset selection problem with support size ranged in (s_min, s_max), where the specific support size to be considered is determined by golden section.

• support_size (array-like, optional) -- default=range(min(n, int(n/(log(log(n))log(p))))). An integer vector representing the alternative support sizes. Only used when path_type = "seq".

• s_min (int, optional, default=0) -- The lower bound of golden-section-search for sparsity searching.

• s_max (int, optional, default=min(n, int(n/(log(log(n))log(p)))).) -- The higher bound of golden-section-search for sparsity searching.

• group (int, optional, default=np.ones(p)) -- The group index for each variable.

• alpha (float, optional, default=0) --

Constant that multiples the L2 term in loss function, controlling regularization strength. It should be non-negative.

• If alpha = 0, it indicates ordinary least square.

• fit_intercept (bool, optional, default=True) -- Whether to consider intercept in the model. We assume that the data has been centered if fit_intercept=False.

• ic_type ({'aic', 'bic', 'gic', 'ebic', 'loss'}, optional, default='ebic') -- The type of criterion for choosing the support size if cv=1.

• ic_coef (float, optional, default=1.0) -- Constant that controls the regularization strength on chosen information criterion.

• cv (int, optional, default=1) --

The folds number when use the cross-validation method.

• If cv=1, cross-validation would not be used.

• If cv>1, support size will be chosen by CV's test loss, instead of IC.

• cv_score ({'test_loss', ...}, optional, default='test_loss') --

The score used on test data for CV.

• All methods support {'test_loss'}.

• LogisticRegression also supports {'roc_auc'}.

• MultinomialRegression also supports {'roc_auc_ovo', 'roc_auc_ovr'}, which indicate "One vs One/Rest" algorithm, respectively.

• thread (int, optional, default=1) --

• If thread = 0, the maximum number of threads supported by the device will be used.

• A_init (array-like, optional, default=None) -- Initial active set before the first splicing.

• always_select (array-like, optional, default=None) -- An array contains the indexes of variables we want to consider in the model. For group selection, it should be the indexes of groups (start from 0).

• max_iter (int, optional, default=20) -- Maximum number of iterations taken for the splicing algorithm to converge. Due to the limitation of loss reduction, the splicing algorithm must be able to converge. The number of iterations is only to simplify the implementation.

• is_warm_start (bool, optional, default=True) -- When tuning the optimal parameter combination, whether to use the last solution as a warm start to accelerate the iterative convergence of the splicing algorithm.

• screening_size (int, optional, default=-1) --

The number of variables remaining after screening. It should be a non-negative number smaller than p, but larger than any value in support_size.

• If screening_size=-1, screening will not be used.

• If screening_size=0, screening_size will be set as $$\\min(p, int(n / (\\log(\\log(n))\\log(p))))$$.

• primary_model_fit_max_iter (int, optional, default=10) -- The maximal number of iteration for primary_model_fit.

• primary_model_fit_epsilon (float, optional, default=1e-08) -- The epsilon (threshold) of iteration for primary_model_fit.

• splicing_type ({0, 1}, optional, default=0) -- The type of splicing: "0" for decreasing by half, "1" for decresing by one.

• important_search (int, optional, default=128) -- The size of inactive set during updating active set when splicing. It should be a non-positive integer and if important_search=0, it would be set as the size of whole inactive set.

Examples

Results may differ with different version of numpy.

>>> ### Sparsity known
>>>
>>> from abess.linear import LinearRegression
>>> from abess.datasets import make_glm_data
>>> import numpy as np
>>> np.random.seed(12345)
>>> data = make_glm_data(n = 100, p = 50, k = 10, family = 'gaussian')
>>> model = LinearRegression(support_size = 10)
>>> model.fit(data.x, data.y)
LinearRegression(support_size=10)
>>> model.predict(data.x)[:4]
array([ -91.02169383,  100.7302593 , -226.99517096,    9.47389912])

>>> ### Sparsity unknown
>>>
>>> # path_type="seq"
>>> model = LinearRegression(path_type = "seq")
>>> model.fit(data.x, data.y)
LinearRegression()
>>> model.predict(data.x)[:4]
array([ -91.02169383,  100.7302593 , -226.99517096,    9.47389912])
>>>
>>> # path_type="gs"
>>> model = LinearRegression(path_type="gs")
>>> model.fit(data.x, data.y)
LinearRegression(path_type='gs')
>>> model.predict(data.x)[:4]
array([ -91.02169383,  100.7302593 , -226.99517096,    9.47389912])

coef_#

Estimated coefficients for the best subset selection problem.

Type

array-like, shape(p_features, ) or (p_features, M_responses)

intercept_#

The intercept in the model when fit_intercept=True.

Type

float or array-like, shape(M_responses,)

train_loss_#

The loss on training data.

Type

float

eval_loss_#
• If cv=1, it stores the score under chosen information criterion.

• If cv>1, it stores the test loss under cross-validation.

Type

float

References

• Junxian Zhu, Canhong Wen, Jin Zhu, Heping Zhang, and Xueqin Wang. A polynomial algorithm for best-subset selection problem. Proceedings of the National Academy of Sciences, 117(52):33117-33123, 2020.

__init__(path_type='seq', support_size=None, s_min=None, s_max=None, group=None, alpha=None, fit_intercept=True, ic_type='ebic', ic_coef=1.0, cv=1, cv_score='test_loss', thread=1, A_init=None, always_select=None, max_iter=20, exchange_num=5, is_warm_start=True, splicing_type=0, important_search=128, screening_size=-1, covariance_update=False)[source]#
predict(X)[source]#

Predict on given data.

Parameters

X (array-like, shape(n_samples, p_features)) -- Sample matrix to be predicted.

Returns

y -- Prediction of the mean on given X.

Return type

array-like, shape(n_samples,)

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

Give data, and it returns the coefficient of determination.

Parameters
• X (array-like, shape(n_samples, p_features)) -- Sample matrix.

• y (array-like, shape(n_samples, p_features)) -- Real response for given X.

• sample_weight (array-like, shape(n_samples,), default=None) -- Sample weights.

Returns

score -- $$R^2$$ score.

Return type

float

fit(X=None, y=None, is_normal=True, sample_weight=None, cv_fold_id=None, sparse_matrix=False, beta_low=None, beta_high=None)#

The fit function is used to transfer the information of data and return the fit result.

Parameters
• X (array-like of shape(n_samples, p_features)) -- Training data matrix. It should be a numpy array.

• y (array-like of shape(n_samples,) or (n_samples, M_responses)) --

Training response values. It should be a numpy array.

• For regression problem, the element of y should be float.

• For classification problem, the element of y should be either 0 or 1. In multinomial regression, the p features are actually dummy variables.

• For survival data, y should be a $$n \times 2$$ array, where the columns indicates "censoring" and "time", respectively.

• is_normal (bool, optional, default=True) -- whether normalize the variables array before fitting the algorithm.

• sample_weight (array-like, shape (n_samples,), optional) -- Individual weights for each sample. Only used for is_weight=True. Default=np.ones(n).

• cv_fold_id (array-like, shape (n_samples,), optional, default=None) -- An array indicates different folds in CV. Samples in the same fold should be given the same number.

• sparse_matrix (bool, optional, default=False) -- Set as True to treat X as sparse matrix during fitting. It would be automatically set as True when X has the sparse matrix type defined in scipy.sparse.

get_params(deep=True)#

Get parameters for this estimator.

Parameters

deep (bool, default=True) -- If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns

params -- Parameter names mapped to their values.

Return type

dict

set_params(**params)#

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

**params (dict) -- Estimator parameters.

Returns

self -- Estimator instance.

Return type

estimator instance

__new__(*args, **kwargs)#