_base.py

"""
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 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 scipy.special import expit
from joblib import Parallel, delayed

from ..base import (BaseEstimator, ClassifierMixin, RegressorMixin,
                    MultiOutputMixin)
from ..utils import check_array, check_X_y
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 ArrayDataset32, CSRDataset32
from ..utils._seq_dataset import ArrayDataset64, CSRDataset64
from ..utils.validation import check_is_fitted, _check_sample_weight
from ..preprocessing 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.

    Parameters
    ----------
    X : array_like, shape (n_samples, n_features)
        Training data

    y : array_like, shape (n_samples, )
        Target values.

    sample_weight : numpy array of shape (n_samples,)
        The weight of each sample

    random_state : int, RandomState instance or None (default)
        Determines random number generation for dataset shuffling and noise.
        Pass an int for reproducible output across multiple function calls.
        See :term:`Glossary <random_state>`.

    Returns
    -------
    dataset
        The ``Dataset`` abstraction
    intercept_decay
        The intercept decay
    """

    rng = check_random_state(random_state)
    # seed should never be 0 in SequentialDataset64
    seed = rng.randint(1, np.iinfo(np.int32).max)

    if X.dtype == np.float32:
        CSRData = CSRDataset32
        ArrayData = ArrayDataset32
    else:
        CSRData = CSRDataset64
        ArrayData = ArrayDataset64

    if sp.issparse(X):
        dataset = CSRData(X.data, X.indptr, X.indices, y, sample_weight,
                          seed=seed)
        intercept_decay = SPARSE_INTERCEPT_DECAY
    else:
        X = np.ascontiguousarray(X)
        dataset = ArrayData(X, y, sample_weight, seed=seed)
        intercept_decay = 1.0

    return dataset, intercept_decay


def _preprocess_data(X, y, fit_intercept, normalize=False, copy=True,
                     sample_weight=None, return_mean=False, check_input=True):
    """Center and scale data.

    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
    if sample_weight is not None:
        sample_weight = np.asarray(sample_weight)

    if check_input:
        X = check_array(X, copy=copy, accept_sparse=['csr', 'csc'],
                        dtype=FLOAT_DTYPES)
    elif copy:
        if sp.issparse(X):
            X = X.copy()
        else:
            X = X.copy(order='K')

    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 sample-wise by square root of sample_weight.

    For many linear models, this enables easy support for sample_weight.

    Returns
    -------
    X_rescaled : {array-like, sparse matrix}

    y_rescaled : {array-like, sparse matrix}
    """
    n_samples = X.shape[0]
    sample_weight = np.asarray(sample_weight)
    if sample_weight.ndim == 0:
        sample_weight = np.full(n_samples, sample_weight,
                                dtype=sample_weight.dtype)
    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(BaseEstimator, metaclass=ABCMeta):
    """Base class for Linear Models"""

    @abstractmethod
    def fit(self, X, y):
        """Fit model."""

    def _decision_function(self, X):
        check_is_fitted(self)

        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 or 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 or 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.
        """
        check_is_fitted(self)

        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 or 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)
        expit(prob, out=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:
    """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
            Fitted estimator.
        """
        msg = "Estimator, %(name)s, must be fitted before densifying."
        check_is_fitted(self, 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.

        Returns
        -------
        self
            Fitted estimator.

        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.
        """
        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.])
    """

    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 = check_X_y(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