621 lines
22 KiB
Python
621 lines
22 KiB
Python
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"""
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Generalized Linear Models.
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"""
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# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
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# Fabian Pedregosa <fabian.pedregosa@inria.fr>
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# Olivier Grisel <olivier.grisel@ensta.org>
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# Vincent Michel <vincent.michel@inria.fr>
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# Peter Prettenhofer <peter.prettenhofer@gmail.com>
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# Mathieu Blondel <mathieu@mblondel.org>
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# Lars Buitinck
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# Maryan Morel <maryan.morel@polytechnique.edu>
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# Giorgio Patrini <giorgio.patrini@anu.edu.au>
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# License: BSD 3 clause
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from abc import ABCMeta, abstractmethod
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import numbers
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import warnings
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import numpy as np
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import scipy.sparse as sp
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from scipy import linalg
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from scipy import sparse
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from scipy.special import expit
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from joblib import Parallel, delayed
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from ..base import (BaseEstimator, ClassifierMixin, RegressorMixin,
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MultiOutputMixin)
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from ..utils import check_array
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from ..utils.validation import FLOAT_DTYPES
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from ..utils.validation import _deprecate_positional_args
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from ..utils import check_random_state
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from ..utils.extmath import safe_sparse_dot
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from ..utils.sparsefuncs import mean_variance_axis, inplace_column_scale
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from ..utils.fixes import sparse_lsqr
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from ..utils._seq_dataset import ArrayDataset32, CSRDataset32
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from ..utils._seq_dataset import ArrayDataset64, CSRDataset64
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from ..utils.validation import check_is_fitted, _check_sample_weight
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from ..preprocessing import normalize as f_normalize
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# TODO: bayesian_ridge_regression and bayesian_regression_ard
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# should be squashed into its respective objects.
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SPARSE_INTERCEPT_DECAY = 0.01
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# For sparse data intercept updates are scaled by this decay factor to avoid
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# intercept oscillation.
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def make_dataset(X, y, sample_weight, random_state=None):
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"""Create ``Dataset`` abstraction for sparse and dense inputs.
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This also returns the ``intercept_decay`` which is different
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for sparse datasets.
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Parameters
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----------
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X : array_like, shape (n_samples, n_features)
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Training data
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y : array_like, shape (n_samples, )
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Target values.
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sample_weight : numpy array of shape (n_samples,)
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The weight of each sample
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random_state : int, RandomState instance or None (default)
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Determines random number generation for dataset shuffling and noise.
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Pass an int for reproducible output across multiple function calls.
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See :term:`Glossary <random_state>`.
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Returns
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-------
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dataset
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The ``Dataset`` abstraction
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intercept_decay
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The intercept decay
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"""
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rng = check_random_state(random_state)
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# seed should never be 0 in SequentialDataset64
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seed = rng.randint(1, np.iinfo(np.int32).max)
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if X.dtype == np.float32:
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CSRData = CSRDataset32
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ArrayData = ArrayDataset32
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else:
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CSRData = CSRDataset64
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ArrayData = ArrayDataset64
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if sp.issparse(X):
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dataset = CSRData(X.data, X.indptr, X.indices, y, sample_weight,
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seed=seed)
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intercept_decay = SPARSE_INTERCEPT_DECAY
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else:
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X = np.ascontiguousarray(X)
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dataset = ArrayData(X, y, sample_weight, seed=seed)
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intercept_decay = 1.0
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return dataset, intercept_decay
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def _preprocess_data(X, y, fit_intercept, normalize=False, copy=True,
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sample_weight=None, return_mean=False, check_input=True):
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"""Center and scale data.
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Centers data to have mean zero along axis 0. If fit_intercept=False or if
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the X is a sparse matrix, no centering is done, but normalization can still
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be applied. The function returns the statistics necessary to reconstruct
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the input data, which are X_offset, y_offset, X_scale, such that the output
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X = (X - X_offset) / X_scale
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X_scale is the L2 norm of X - X_offset. If sample_weight is not None,
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then the weighted mean of X and y is zero, and not the mean itself. If
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return_mean=True, the mean, eventually weighted, is returned, independently
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of whether X was centered (option used for optimization with sparse data in
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coordinate_descend).
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This is here because nearly all linear models will want their data to be
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centered. This function also systematically makes y consistent with X.dtype
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"""
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if isinstance(sample_weight, numbers.Number):
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sample_weight = None
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if sample_weight is not None:
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sample_weight = np.asarray(sample_weight)
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if check_input:
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X = check_array(X, copy=copy, accept_sparse=['csr', 'csc'],
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dtype=FLOAT_DTYPES)
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elif copy:
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if sp.issparse(X):
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X = X.copy()
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else:
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X = X.copy(order='K')
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y = np.asarray(y, dtype=X.dtype)
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if fit_intercept:
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if sp.issparse(X):
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X_offset, X_var = mean_variance_axis(X, axis=0)
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if not return_mean:
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X_offset[:] = X.dtype.type(0)
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if normalize:
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# TODO: f_normalize could be used here as well but the function
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# inplace_csr_row_normalize_l2 must be changed such that it
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# can return also the norms computed internally
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# transform variance to norm in-place
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X_var *= X.shape[0]
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X_scale = np.sqrt(X_var, X_var)
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del X_var
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X_scale[X_scale == 0] = 1
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inplace_column_scale(X, 1. / X_scale)
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else:
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X_scale = np.ones(X.shape[1], dtype=X.dtype)
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else:
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X_offset = np.average(X, axis=0, weights=sample_weight)
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X -= X_offset
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if normalize:
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X, X_scale = f_normalize(X, axis=0, copy=False,
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return_norm=True)
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else:
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X_scale = np.ones(X.shape[1], dtype=X.dtype)
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y_offset = np.average(y, axis=0, weights=sample_weight)
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y = y - y_offset
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else:
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X_offset = np.zeros(X.shape[1], dtype=X.dtype)
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X_scale = np.ones(X.shape[1], dtype=X.dtype)
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if y.ndim == 1:
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y_offset = X.dtype.type(0)
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else:
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y_offset = np.zeros(y.shape[1], dtype=X.dtype)
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return X, y, X_offset, y_offset, X_scale
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# TODO: _rescale_data should be factored into _preprocess_data.
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# Currently, the fact that sag implements its own way to deal with
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# sample_weight makes the refactoring tricky.
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def _rescale_data(X, y, sample_weight):
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"""Rescale data sample-wise by square root of sample_weight.
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For many linear models, this enables easy support for sample_weight.
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Returns
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-------
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X_rescaled : {array-like, sparse matrix}
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y_rescaled : {array-like, sparse matrix}
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"""
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n_samples = X.shape[0]
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sample_weight = np.asarray(sample_weight)
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if sample_weight.ndim == 0:
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sample_weight = np.full(n_samples, sample_weight,
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dtype=sample_weight.dtype)
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sample_weight = np.sqrt(sample_weight)
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sw_matrix = sparse.dia_matrix((sample_weight, 0),
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shape=(n_samples, n_samples))
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X = safe_sparse_dot(sw_matrix, X)
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y = safe_sparse_dot(sw_matrix, y)
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return X, y
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class LinearModel(BaseEstimator, metaclass=ABCMeta):
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"""Base class for Linear Models"""
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@abstractmethod
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def fit(self, X, y):
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"""Fit model."""
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def _decision_function(self, X):
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check_is_fitted(self)
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X = check_array(X, accept_sparse=['csr', 'csc', 'coo'])
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return safe_sparse_dot(X, self.coef_.T,
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dense_output=True) + self.intercept_
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def predict(self, X):
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"""
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Predict using the linear model.
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Parameters
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----------
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X : array_like or sparse matrix, shape (n_samples, n_features)
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Samples.
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Returns
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-------
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C : array, shape (n_samples,)
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Returns predicted values.
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"""
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return self._decision_function(X)
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_preprocess_data = staticmethod(_preprocess_data)
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def _set_intercept(self, X_offset, y_offset, X_scale):
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"""Set the intercept_
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"""
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if self.fit_intercept:
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self.coef_ = self.coef_ / X_scale
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self.intercept_ = y_offset - np.dot(X_offset, self.coef_.T)
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else:
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self.intercept_ = 0.
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def _more_tags(self):
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return {'requires_y': True}
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# XXX Should this derive from LinearModel? It should be a mixin, not an ABC.
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# Maybe the n_features checking can be moved to LinearModel.
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class LinearClassifierMixin(ClassifierMixin):
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"""Mixin for linear classifiers.
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Handles prediction for sparse and dense X.
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"""
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def decision_function(self, X):
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"""
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Predict confidence scores for samples.
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The confidence score for a sample is the signed distance of that
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sample to the hyperplane.
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Parameters
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----------
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X : array_like or sparse matrix, shape (n_samples, n_features)
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Samples.
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Returns
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-------
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array, shape=(n_samples,) if n_classes == 2 else (n_samples, n_classes)
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Confidence scores per (sample, class) combination. In the binary
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case, confidence score for self.classes_[1] where >0 means this
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class would be predicted.
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"""
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check_is_fitted(self)
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X = check_array(X, accept_sparse='csr')
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n_features = self.coef_.shape[1]
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if X.shape[1] != n_features:
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raise ValueError("X has %d features per sample; expecting %d"
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% (X.shape[1], n_features))
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scores = safe_sparse_dot(X, self.coef_.T,
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dense_output=True) + self.intercept_
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return scores.ravel() if scores.shape[1] == 1 else scores
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def predict(self, X):
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"""
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Predict class labels for samples in X.
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Parameters
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----------
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X : array_like or sparse matrix, shape (n_samples, n_features)
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Samples.
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Returns
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-------
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C : array, shape [n_samples]
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Predicted class label per sample.
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"""
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scores = self.decision_function(X)
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if len(scores.shape) == 1:
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indices = (scores > 0).astype(np.int)
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else:
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indices = scores.argmax(axis=1)
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return self.classes_[indices]
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def _predict_proba_lr(self, X):
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"""Probability estimation for OvR logistic regression.
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Positive class probabilities are computed as
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1. / (1. + np.exp(-self.decision_function(X)));
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multiclass is handled by normalizing that over all classes.
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"""
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prob = self.decision_function(X)
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expit(prob, out=prob)
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if prob.ndim == 1:
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return np.vstack([1 - prob, prob]).T
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else:
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# OvR normalization, like LibLinear's predict_probability
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prob /= prob.sum(axis=1).reshape((prob.shape[0], -1))
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return prob
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class SparseCoefMixin:
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"""Mixin for converting coef_ to and from CSR format.
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L1-regularizing estimators should inherit this.
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"""
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def densify(self):
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"""
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Convert coefficient matrix to dense array format.
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Converts the ``coef_`` member (back) to a numpy.ndarray. This is the
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default format of ``coef_`` and is required for fitting, so calling
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this method is only required on models that have previously been
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sparsified; otherwise, it is a no-op.
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Returns
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-------
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self
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Fitted estimator.
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"""
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msg = "Estimator, %(name)s, must be fitted before densifying."
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check_is_fitted(self, msg=msg)
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if sp.issparse(self.coef_):
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self.coef_ = self.coef_.toarray()
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return self
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def sparsify(self):
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"""
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Convert coefficient matrix to sparse format.
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Converts the ``coef_`` member to a scipy.sparse matrix, which for
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L1-regularized models can be much more memory- and storage-efficient
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than the usual numpy.ndarray representation.
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The ``intercept_`` member is not converted.
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Returns
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-------
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self
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Fitted estimator.
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Notes
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-----
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For non-sparse models, i.e. when there are not many zeros in ``coef_``,
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this may actually *increase* memory usage, so use this method with
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care. A rule of thumb is that the number of zero elements, which can
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be computed with ``(coef_ == 0).sum()``, must be more than 50% for this
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|
to provide significant benefits.
|
||
|
|
|
||
|
|
After calling this method, further fitting with the partial_fit
|
||
|
|
method (if any) will not work until you call densify.
|
||
|
|
"""
|
||
|
|
msg = "Estimator, %(name)s, must be fitted before sparsifying."
|
||
|
|
check_is_fitted(self, msg=msg)
|
||
|
|
self.coef_ = sp.csr_matrix(self.coef_)
|
||
|
|
return self
|
||
|
|
|
||
|
|
|
||
|
|
class LinearRegression(MultiOutputMixin, RegressorMixin, LinearModel):
|
||
|
|
"""
|
||
|
|
Ordinary least squares Linear Regression.
|
||
|
|
|
||
|
|
LinearRegression fits a linear model with coefficients w = (w1, ..., wp)
|
||
|
|
to minimize the residual sum of squares between the observed targets in
|
||
|
|
the dataset, and the targets predicted by the linear approximation.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
fit_intercept : bool, 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).
|
||
|
|
|
||
|
|
normalize : bool, 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
|
||
|
|
:class:`sklearn.preprocessing.StandardScaler` before calling ``fit`` on
|
||
|
|
an estimator with ``normalize=False``.
|
||
|
|
|
||
|
|
copy_X : bool, default=True
|
||
|
|
If True, X will be copied; else, it may be overwritten.
|
||
|
|
|
||
|
|
n_jobs : int, default=None
|
||
|
|
The number of jobs to use for the computation. This will only provide
|
||
|
|
speedup for n_targets > 1 and sufficient large problems.
|
||
|
|
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
|
||
|
|
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
|
||
|
|
for more details.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
coef_ : array of shape (n_features, ) or (n_targets, n_features)
|
||
|
|
Estimated coefficients for the linear regression problem.
|
||
|
|
If multiple targets are passed during the fit (y 2D), this
|
||
|
|
is a 2D array of shape (n_targets, n_features), while if only
|
||
|
|
one target is passed, this is a 1D array of length n_features.
|
||
|
|
|
||
|
|
rank_ : int
|
||
|
|
Rank of matrix `X`. Only available when `X` is dense.
|
||
|
|
|
||
|
|
singular_ : array of shape (min(X, y),)
|
||
|
|
Singular values of `X`. Only available when `X` is dense.
|
||
|
|
|
||
|
|
intercept_ : float or array of shape (n_targets,)
|
||
|
|
Independent term in the linear model. Set to 0.0 if
|
||
|
|
`fit_intercept = False`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
sklearn.linear_model.Ridge : Ridge regression addresses some of the
|
||
|
|
problems of Ordinary Least Squares by imposing a penalty on the
|
||
|
|
size of the coefficients with l2 regularization.
|
||
|
|
sklearn.linear_model.Lasso : The Lasso is a linear model that estimates
|
||
|
|
sparse coefficients with l1 regularization.
|
||
|
|
sklearn.linear_model.ElasticNet : Elastic-Net is a linear regression
|
||
|
|
model trained with both l1 and l2 -norm regularization of the
|
||
|
|
coefficients.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
From the implementation point of view, this is just plain Ordinary
|
||
|
|
Least Squares (scipy.linalg.lstsq) wrapped as a predictor object.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from sklearn.linear_model import LinearRegression
|
||
|
|
>>> X = np.array([[1, 1], [1, 2], [2, 2], [2, 3]])
|
||
|
|
>>> # y = 1 * x_0 + 2 * x_1 + 3
|
||
|
|
>>> y = np.dot(X, np.array([1, 2])) + 3
|
||
|
|
>>> reg = LinearRegression().fit(X, y)
|
||
|
|
>>> reg.score(X, y)
|
||
|
|
1.0
|
||
|
|
>>> reg.coef_
|
||
|
|
array([1., 2.])
|
||
|
|
>>> reg.intercept_
|
||
|
|
3.0000...
|
||
|
|
>>> reg.predict(np.array([[3, 5]]))
|
||
|
|
array([16.])
|
||
|
|
"""
|
||
|
|
@_deprecate_positional_args
|
||
|
|
def __init__(self, *, fit_intercept=True, normalize=False, copy_X=True,
|
||
|
|
n_jobs=None):
|
||
|
|
self.fit_intercept = fit_intercept
|
||
|
|
self.normalize = normalize
|
||
|
|
self.copy_X = copy_X
|
||
|
|
self.n_jobs = n_jobs
|
||
|
|
|
||
|
|
def fit(self, X, y, sample_weight=None):
|
||
|
|
"""
|
||
|
|
Fit linear model.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : {array-like, sparse matrix} of shape (n_samples, n_features)
|
||
|
|
Training data
|
||
|
|
|
||
|
|
y : array-like of shape (n_samples,) or (n_samples, n_targets)
|
||
|
|
Target values. Will be cast to X's dtype if necessary
|
||
|
|
|
||
|
|
sample_weight : array-like of shape (n_samples,), default=None
|
||
|
|
Individual weights for each sample
|
||
|
|
|
||
|
|
.. versionadded:: 0.17
|
||
|
|
parameter *sample_weight* support to LinearRegression.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
self : returns an instance of self.
|
||
|
|
"""
|
||
|
|
|
||
|
|
n_jobs_ = self.n_jobs
|
||
|
|
X, y = self._validate_data(X, y, accept_sparse=['csr', 'csc', 'coo'],
|
||
|
|
y_numeric=True, multi_output=True)
|
||
|
|
|
||
|
|
if sample_weight is not None:
|
||
|
|
sample_weight = _check_sample_weight(sample_weight, X,
|
||
|
|
dtype=X.dtype)
|
||
|
|
|
||
|
|
X, y, X_offset, y_offset, X_scale = self._preprocess_data(
|
||
|
|
X, y, fit_intercept=self.fit_intercept, normalize=self.normalize,
|
||
|
|
copy=self.copy_X, sample_weight=sample_weight,
|
||
|
|
return_mean=True)
|
||
|
|
|
||
|
|
if sample_weight is not None:
|
||
|
|
# Sample weight can be implemented via a simple rescaling.
|
||
|
|
X, y = _rescale_data(X, y, sample_weight)
|
||
|
|
|
||
|
|
if sp.issparse(X):
|
||
|
|
X_offset_scale = X_offset / X_scale
|
||
|
|
|
||
|
|
def matvec(b):
|
||
|
|
return X.dot(b) - b.dot(X_offset_scale)
|
||
|
|
|
||
|
|
def rmatvec(b):
|
||
|
|
return X.T.dot(b) - X_offset_scale * np.sum(b)
|
||
|
|
|
||
|
|
X_centered = sparse.linalg.LinearOperator(shape=X.shape,
|
||
|
|
matvec=matvec,
|
||
|
|
rmatvec=rmatvec)
|
||
|
|
|
||
|
|
if y.ndim < 2:
|
||
|
|
out = sparse_lsqr(X_centered, y)
|
||
|
|
self.coef_ = out[0]
|
||
|
|
self._residues = out[3]
|
||
|
|
else:
|
||
|
|
# sparse_lstsq cannot handle y with shape (M, K)
|
||
|
|
outs = Parallel(n_jobs=n_jobs_)(
|
||
|
|
delayed(sparse_lsqr)(X_centered, y[:, j].ravel())
|
||
|
|
for j in range(y.shape[1]))
|
||
|
|
self.coef_ = np.vstack([out[0] for out in outs])
|
||
|
|
self._residues = np.vstack([out[3] for out in outs])
|
||
|
|
else:
|
||
|
|
self.coef_, self._residues, self.rank_, self.singular_ = \
|
||
|
|
linalg.lstsq(X, y)
|
||
|
|
self.coef_ = self.coef_.T
|
||
|
|
|
||
|
|
if y.ndim == 1:
|
||
|
|
self.coef_ = np.ravel(self.coef_)
|
||
|
|
self._set_intercept(X_offset, y_offset, X_scale)
|
||
|
|
return self
|
||
|
|
|
||
|
|
|
||
|
|
def _pre_fit(X, y, Xy, precompute, normalize, fit_intercept, copy,
|
||
|
|
check_input=True, sample_weight=None):
|
||
|
|
"""Aux function used at beginning of fit in linear models
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
order : 'F', 'C' or None, default=None
|
||
|
|
Whether X and y will be forced to be fortran or c-style. Only relevant
|
||
|
|
if sample_weight is not None.
|
||
|
|
"""
|
||
|
|
n_samples, n_features = X.shape
|
||
|
|
|
||
|
|
if sparse.isspmatrix(X):
|
||
|
|
# copy is not needed here as X is not modified inplace when X is sparse
|
||
|
|
precompute = False
|
||
|
|
X, y, X_offset, y_offset, X_scale = _preprocess_data(
|
||
|
|
X, y, fit_intercept=fit_intercept, normalize=normalize,
|
||
|
|
copy=False, return_mean=True, check_input=check_input)
|
||
|
|
else:
|
||
|
|
# copy was done in fit if necessary
|
||
|
|
X, y, X_offset, y_offset, X_scale = _preprocess_data(
|
||
|
|
X, y, fit_intercept=fit_intercept, normalize=normalize, copy=copy,
|
||
|
|
check_input=check_input, sample_weight=sample_weight)
|
||
|
|
if sample_weight is not None:
|
||
|
|
X, y = _rescale_data(X, y, sample_weight=sample_weight)
|
||
|
|
if hasattr(precompute, '__array__') and (
|
||
|
|
fit_intercept and not np.allclose(X_offset, np.zeros(n_features)) or
|
||
|
|
normalize and not np.allclose(X_scale, np.ones(n_features))):
|
||
|
|
warnings.warn("Gram matrix was provided but X was centered"
|
||
|
|
" to fit intercept, "
|
||
|
|
"or X was normalized : recomputing Gram matrix.",
|
||
|
|
UserWarning)
|
||
|
|
# recompute Gram
|
||
|
|
precompute = 'auto'
|
||
|
|
Xy = None
|
||
|
|
|
||
|
|
# precompute if n_samples > n_features
|
||
|
|
if isinstance(precompute, str) and precompute == 'auto':
|
||
|
|
precompute = (n_samples > n_features)
|
||
|
|
|
||
|
|
if precompute is True:
|
||
|
|
# make sure that the 'precompute' array is contiguous.
|
||
|
|
precompute = np.empty(shape=(n_features, n_features), dtype=X.dtype,
|
||
|
|
order='C')
|
||
|
|
np.dot(X.T, X, out=precompute)
|
||
|
|
|
||
|
|
if not hasattr(precompute, '__array__'):
|
||
|
|
Xy = None # cannot use Xy if precompute is not Gram
|
||
|
|
|
||
|
|
if hasattr(precompute, '__array__') and Xy is None:
|
||
|
|
common_dtype = np.find_common_type([X.dtype, y.dtype], [])
|
||
|
|
if y.ndim == 1:
|
||
|
|
# Xy is 1d, make sure it is contiguous.
|
||
|
|
Xy = np.empty(shape=n_features, dtype=common_dtype, order='C')
|
||
|
|
np.dot(X.T, y, out=Xy)
|
||
|
|
else:
|
||
|
|
# Make sure that Xy is always F contiguous even if X or y are not
|
||
|
|
# contiguous: the goal is to make it fast to extract the data for a
|
||
|
|
# specific target.
|
||
|
|
n_targets = y.shape[1]
|
||
|
|
Xy = np.empty(shape=(n_features, n_targets), dtype=common_dtype,
|
||
|
|
order='F')
|
||
|
|
np.dot(y.T, X, out=Xy.T)
|
||
|
|
|
||
|
|
return X, y, X_offset, y_offset, X_scale, precompute, Xy
|