"""
Generalized Linear models.
"""
# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
# Fabian Pedregosa <fabian.pedregosa@inria.fr>
# Olivier Grisel <olivier.grisel@ensta.org>
# Vincent Michel <vincent.michel@inria.fr>
# Peter Prettenhofer <peter.prettenhofer@gmail.com>
# Mathieu Blondel <mathieu@mblondel.org>
# Lars Buitinck
# Maryan Morel <maryan.morel@polytechnique.edu>
# Giorgio Patrini <giorgio.patrini@anu.edu.au>
# License: BSD 3 clause
from __future__ import division
from abc import ABCMeta, abstractmethod
import numbers
import warnings
import numpy as np
import scipy.sparse as sp
from scipy import linalg
from scipy import sparse
from ..externals import six
from ..externals.joblib import Parallel, delayed
from ..base import BaseEstimator, ClassifierMixin, RegressorMixin
from ..utils import check_array, check_X_y, deprecated, as_float_array
from ..utils.validation import FLOAT_DTYPES
from ..utils import check_random_state
from ..utils.extmath import safe_sparse_dot
from ..utils.sparsefuncs import mean_variance_axis, inplace_column_scale
from ..utils.fixes import sparse_lsqr
from ..utils.seq_dataset import ArrayDataset, CSRDataset
from ..utils.validation import check_is_fitted
from ..exceptions import NotFittedError
from ..preprocessing.data import normalize as f_normalize
# TODO: bayesian_ridge_regression and bayesian_regression_ard
# should be squashed into its respective objects.
SPARSE_INTERCEPT_DECAY = 0.01
# For sparse data intercept updates are scaled by this decay factor to avoid
# intercept oscillation.
def make_dataset(X, y, sample_weight, random_state=None):
"""Create ``Dataset`` abstraction for sparse and dense inputs.
This also returns the ``intercept_decay`` which is different
for sparse datasets.
"""
rng = check_random_state(random_state)
# seed should never be 0 in SequentialDataset
seed = rng.randint(1, np.iinfo(np.int32).max)
if sp.issparse(X):
dataset = CSRDataset(X.data, X.indptr, X.indices, y, sample_weight,
seed=seed)
intercept_decay = SPARSE_INTERCEPT_DECAY
else:
dataset = ArrayDataset(X, y, sample_weight, seed=seed)
intercept_decay = 1.0
return dataset, intercept_decay
@deprecated("sparse_center_data was deprecated in version 0.18 and will be "
"removed in 0.20. Use utilities in preprocessing.data instead")
def sparse_center_data(X, y, fit_intercept, normalize=False):
"""
Compute information needed to center data to have mean zero along
axis 0. Be aware that X will not be centered since it would break
the sparsity, but will be normalized if asked so.
"""
if fit_intercept:
# we might require not to change the csr matrix sometimes
# store a copy if normalize is True.
# Change dtype to float64 since mean_variance_axis accepts
# it that way.
if sp.isspmatrix(X) and X.getformat() == 'csr':
X = sp.csr_matrix(X, copy=normalize, dtype=np.float64)
else:
X = sp.csc_matrix(X, copy=normalize, dtype=np.float64)
X_offset, X_var = mean_variance_axis(X, axis=0)
if normalize:
# transform variance to std in-place
X_var *= X.shape[0]
X_std = np.sqrt(X_var, X_var)
del X_var
X_std[X_std == 0] = 1
inplace_column_scale(X, 1. / X_std)
else:
X_std = np.ones(X.shape[1])
y_offset = y.mean(axis=0)
y = y - y_offset
else:
X_offset = np.zeros(X.shape[1])
X_std = np.ones(X.shape[1])
y_offset = 0. if y.ndim == 1 else np.zeros(y.shape[1], dtype=X.dtype)
return X, y, X_offset, y_offset, X_std
@deprecated("center_data was deprecated in version 0.18 and will be removed "
"in 0.20. Use utilities in preprocessing.data instead")
def center_data(X, y, fit_intercept, normalize=False, copy=True,
sample_weight=None):
"""
Centers data to have mean zero along axis 0. This is here because
nearly all linear models will want their data to be centered.
If sample_weight is not None, then the weighted mean of X and y
is zero, and not the mean itself
"""
X = as_float_array(X, copy)
if fit_intercept:
if isinstance(sample_weight, numbers.Number):
sample_weight = None
if sp.issparse(X):
X_offset = np.zeros(X.shape[1])
X_std = np.ones(X.shape[1])
else:
X_offset = np.average(X, axis=0, weights=sample_weight)
X -= X_offset
# XXX: currently scaled to variance=n_samples
if normalize:
X_std = np.sqrt(np.sum(X ** 2, axis=0))
X_std[X_std == 0] = 1
X /= X_std
else:
X_std = np.ones(X.shape[1])
y_offset = np.average(y, axis=0, weights=sample_weight)
y = y - y_offset
else:
X_offset = np.zeros(X.shape[1])
X_std = np.ones(X.shape[1])
y_offset = 0. if y.ndim == 1 else np.zeros(y.shape[1], dtype=X.dtype)
return X, y, X_offset, y_offset, X_std
def _preprocess_data(X, y, fit_intercept, normalize=False, copy=True,
sample_weight=None, return_mean=False):
"""
Centers data to have mean zero along axis 0. If fit_intercept=False or if
the X is a sparse matrix, no centering is done, but normalization can still
be applied. The function returns the statistics necessary to reconstruct
the input data, which are X_offset, y_offset, X_scale, such that the output
X = (X - X_offset) / X_scale
X_scale is the L2 norm of X - X_offset. If sample_weight is not None,
then the weighted mean of X and y is zero, and not the mean itself. If
return_mean=True, the mean, eventually weighted, is returned, independently
of whether X was centered (option used for optimization with sparse data in
coordinate_descend).
This is here because nearly all linear models will want their data to be
centered. This function also systematically makes y consistent with X.dtype
"""
if isinstance(sample_weight, numbers.Number):
sample_weight = None
X = check_array(X, copy=copy, accept_sparse=['csr', 'csc'],
dtype=FLOAT_DTYPES)
y = np.asarray(y, dtype=X.dtype)
if fit_intercept:
if sp.issparse(X):
X_offset, X_var = mean_variance_axis(X, axis=0)
if not return_mean:
X_offset[:] = X.dtype.type(0)
if normalize:
# TODO: f_normalize could be used here as well but the function
# inplace_csr_row_normalize_l2 must be changed such that it
# can return also the norms computed internally
# transform variance to norm in-place
X_var *= X.shape[0]
X_scale = np.sqrt(X_var, X_var)
del X_var
X_scale[X_scale == 0] = 1
inplace_column_scale(X, 1. / X_scale)
else:
X_scale = np.ones(X.shape[1], dtype=X.dtype)
else:
X_offset = np.average(X, axis=0, weights=sample_weight)
X -= X_offset
if normalize:
X, X_scale = f_normalize(X, axis=0, copy=False,
return_norm=True)
else:
X_scale = np.ones(X.shape[1], dtype=X.dtype)
y_offset = np.average(y, axis=0, weights=sample_weight)
y = y - y_offset
else:
X_offset = np.zeros(X.shape[1], dtype=X.dtype)
X_scale = np.ones(X.shape[1], dtype=X.dtype)
if y.ndim == 1:
y_offset = X.dtype.type(0)
else:
y_offset = np.zeros(y.shape[1], dtype=X.dtype)
return X, y, X_offset, y_offset, X_scale
# TODO: _rescale_data should be factored into _preprocess_data.
# Currently, the fact that sag implements its own way to deal with
# sample_weight makes the refactoring tricky.
def _rescale_data(X, y, sample_weight):
"""Rescale data so as to support sample_weight"""
n_samples = X.shape[0]
sample_weight = sample_weight * np.ones(n_samples)
sample_weight = np.sqrt(sample_weight)
sw_matrix = sparse.dia_matrix((sample_weight, 0),
shape=(n_samples, n_samples))
X = safe_sparse_dot(sw_matrix, X)
y = safe_sparse_dot(sw_matrix, y)
return X, y
class LinearModel(six.with_metaclass(ABCMeta, BaseEstimator)):
"""Base class for Linear Models"""
@abstractmethod
def fit(self, X, y):
"""Fit model."""
def _decision_function(self, X):
check_is_fitted(self, "coef_")
X = check_array(X, accept_sparse=['csr', 'csc', 'coo'])
return safe_sparse_dot(X, self.coef_.T,
dense_output=True) + self.intercept_
def predict(self, X):
"""Predict using the linear model
Parameters
----------
X : {array-like, sparse matrix}, shape = (n_samples, n_features)
Samples.
Returns
-------
C : array, shape = (n_samples,)
Returns predicted values.
"""
return self._decision_function(X)
_preprocess_data = staticmethod(_preprocess_data)
def _set_intercept(self, X_offset, y_offset, X_scale):
"""Set the intercept_
"""
if self.fit_intercept:
self.coef_ = self.coef_ / X_scale
self.intercept_ = y_offset - np.dot(X_offset, self.coef_.T)
else:
self.intercept_ = 0.
# XXX Should this derive from LinearModel? It should be a mixin, not an ABC.
# Maybe the n_features checking can be moved to LinearModel.
class LinearClassifierMixin(ClassifierMixin):
"""Mixin for linear classifiers.
Handles prediction for sparse and dense X.
"""
def decision_function(self, X):
"""Predict confidence scores for samples.
The confidence score for a sample is the signed distance of that
sample to the hyperplane.
Parameters
----------
X : {array-like, sparse matrix}, shape = (n_samples, n_features)
Samples.
Returns
-------
array, shape=(n_samples,) if n_classes == 2 else (n_samples, n_classes)
Confidence scores per (sample, class) combination. In the binary
case, confidence score for self.classes_[1] where >0 means this
class would be predicted.
"""
if not hasattr(self, 'coef_') or self.coef_ is None:
raise NotFittedError("This %(name)s instance is not fitted "
"yet" % {'name': type(self).__name__})
X = check_array(X, accept_sparse='csr')
n_features = self.coef_.shape[1]
if X.shape[1] != n_features:
raise ValueError("X has %d features per sample; expecting %d"
% (X.shape[1], n_features))
scores = safe_sparse_dot(X, self.coef_.T,
dense_output=True) + self.intercept_
return scores.ravel() if scores.shape[1] == 1 else scores
def predict(self, X):
"""Predict class labels for samples in X.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
Samples.
Returns
-------
C : array, shape = [n_samples]
Predicted class label per sample.
"""
scores = self.decision_function(X)
if len(scores.shape) == 1:
indices = (scores > 0).astype(np.int)
else:
indices = scores.argmax(axis=1)
return self.classes_[indices]
def _predict_proba_lr(self, X):
"""Probability estimation for OvR logistic regression.
Positive class probabilities are computed as
1. / (1. + np.exp(-self.decision_function(X)));
multiclass is handled by normalizing that over all classes.
"""
prob = self.decision_function(X)
prob *= -1
np.exp(prob, prob)
prob += 1
np.reciprocal(prob, prob)
if prob.ndim == 1:
return np.vstack([1 - prob, prob]).T
else:
# OvR normalization, like LibLinear's predict_probability
prob /= prob.sum(axis=1).reshape((prob.shape[0], -1))
return prob
class SparseCoefMixin(object):
"""Mixin for converting coef_ to and from CSR format.
L1-regularizing estimators should inherit this.
"""
def densify(self):
"""Convert coefficient matrix to dense array format.
Converts the ``coef_`` member (back) to a numpy.ndarray. This is the
default format of ``coef_`` and is required for fitting, so calling
this method is only required on models that have previously been
sparsified; otherwise, it is a no-op.
Returns
-------
self : estimator
"""
msg = "Estimator, %(name)s, must be fitted before densifying."
check_is_fitted(self, "coef_", msg=msg)
if sp.issparse(self.coef_):
self.coef_ = self.coef_.toarray()
return self
def sparsify(self):
"""Convert coefficient matrix to sparse format.
Converts the ``coef_`` member to a scipy.sparse matrix, which for
L1-regularized models can be much more memory- and storage-efficient
than the usual numpy.ndarray representation.
The ``intercept_`` member is not converted.
Notes
-----
For non-sparse models, i.e. when there are not many zeros in ``coef_``,
this may actually *increase* memory usage, so use this method with
care. A rule of thumb is that the number of zero elements, which can
be computed with ``(coef_ == 0).sum()``, must be more than 50% for this
to provide significant benefits.
After calling this method, further fitting with the partial_fit
method (if any) will not work until you call densify.
Returns
-------
self : estimator
"""
msg = "Estimator, %(name)s, must be fitted before sparsifying."
check_is_fitted(self, "coef_", msg=msg)
self.coef_ = sp.csr_matrix(self.coef_)
return self
class LinearRegression(LinearModel, RegressorMixin):
"""
Ordinary least squares Linear Regression.
Parameters
----------
fit_intercept : boolean, optional, default True
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
:class:`sklearn.preprocessing.StandardScaler` before calling ``fit`` on
an estimator with ``normalize=False``.
copy_X : boolean, optional, default True
If True, X will be copied; else, it may be overwritten.
n_jobs : int, optional, default 1
The number of jobs to use for the computation.
If -1 all CPUs are used. This will only provide speedup for
n_targets > 1 and sufficient large problems.
Attributes
----------
coef_ : array, 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.
intercept_ : array
Independent term in the linear model.
Notes
-----
From the implementation point of view, this is just plain Ordinary
Least Squares (scipy.linalg.lstsq) wrapped as a predictor object.
"""
def __init__(self, fit_intercept=True, normalize=False, copy_X=True,
n_jobs=1):
self.fit_intercept = fit_intercept
self.normalize = normalize
self.copy_X = copy_X
self.n_jobs = n_jobs
[docs] def fit(self, X, y, sample_weight=None):
"""
Fit linear model.
Parameters
----------
X : numpy array or sparse matrix of shape [n_samples,n_features]
Training data
y : numpy array of shape [n_samples, n_targets]
Target values. Will be cast to X's dtype if necessary
sample_weight : numpy array of shape [n_samples]
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 = check_X_y(X, y, accept_sparse=['csr', 'csc', 'coo'],
y_numeric=True, multi_output=True)
if sample_weight is not None and np.atleast_1d(sample_weight).ndim > 1:
raise ValueError("Sample weights must be 1D array or scalar")
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)
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):
if y.ndim < 2:
out = sparse_lsqr(X, 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, 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):
"""Aux function used at beginning of fit in linear models"""
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)
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)
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, six.string_types) 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
