least_angle.py

"""
Least Angle Regression algorithm. See the documentation on the
Generalized Linear Model for a complete discussion.
"""
from __future__ import print_function

# Author: Fabian Pedregosa <fabian.pedregosa@inria.fr>
#         Alexandre Gramfort <alexandre.gramfort@inria.fr>
#         Gael Varoquaux
#
# License: BSD 3 clause

from math import log
import sys
import warnings
from distutils.version import LooseVersion

import numpy as np
from scipy import linalg, interpolate
from scipy.linalg.lapack import get_lapack_funcs

from .base import LinearModel
from ..base import RegressorMixin
from ..utils import arrayfuncs, as_float_array, check_X_y
from ..cross_validation import check_cv
from ..utils import ConvergenceWarning
from ..externals.joblib import Parallel, delayed
from ..externals.six.moves import xrange
from ..externals.six import string_types

import scipy
solve_triangular_args = {}
if LooseVersion(scipy.__version__) >= LooseVersion('0.12'):
    solve_triangular_args = {'check_finite': False}


def lars_path(X, y, Xy=None, Gram=None, max_iter=500,
              alpha_min=0, method='lar', copy_X=True,
              eps=np.finfo(np.float).eps,
              copy_Gram=True, verbose=0, return_path=True,
              return_n_iter=False, positive=False):
    """Compute Least Angle Regression or Lasso path using LARS algorithm [1]

    The optimization objective for the case method='lasso' is::

    (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1

    in the case of method='lars', the objective function is only known in
    the form of an implicit equation (see discussion in [1])

    Read more in the :ref:`User Guide <least_angle_regression>`.

    Parameters
    -----------
    X : array, shape: (n_samples, n_features)
        Input data.

    y : array, shape: (n_samples)
        Input targets.

    positive : boolean (default=False)
        Restrict coefficients to be >= 0.
        When using this option together with method 'lasso' the model
        coefficients will not converge to the ordinary-least-squares solution
        for small values of alpha (neither will they when using method 'lar'
        ..). Only coeffiencts up to the smallest alpha value (``alphas_[alphas_ >
        0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso
        algorithm are typically in congruence with the solution of the
        coordinate descent lasso_path function.

    max_iter : integer, optional (default=500)
        Maximum number of iterations to perform, set to infinity for no limit.

    Gram : None, 'auto', array, shape: (n_features, n_features), optional
        Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram
        matrix is precomputed from the given X, if there are more samples
        than features.

    alpha_min : float, optional (default=0)
        Minimum correlation along the path. It corresponds to the
        regularization parameter alpha parameter in the Lasso.

    method : {'lar', 'lasso'}, optional (default='lar')
        Specifies the returned model. Select ``'lar'`` for Least Angle
        Regression, ``'lasso'`` for the Lasso.

    eps : float, optional (default=``np.finfo(np.float).eps``)
        The machine-precision regularization in the computation of the
        Cholesky diagonal factors. Increase this for very ill-conditioned
        systems.

    copy_X : bool, optional (default=True)
        If ``False``, ``X`` is overwritten.

    copy_Gram : bool, optional (default=True)
        If ``False``, ``Gram`` is overwritten.

    verbose : int (default=0)
        Controls output verbosity.

    return_path : bool, optional (default=True)
        If ``return_path==True`` returns the entire path, else returns only the
        last point of the path.

    return_n_iter : bool, optional (default=False)
        Whether to return the number of iterations.

    Returns
    --------
    alphas : array, shape: [n_alphas + 1]
        Maximum of covariances (in absolute value) at each iteration.
        ``n_alphas`` is either ``max_iter``, ``n_features`` or the
        number of nodes in the path with ``alpha >= alpha_min``, whichever
        is smaller.

    active : array, shape [n_alphas]
        Indices of active variables at the end of the path.

    coefs : array, shape (n_features, n_alphas + 1)
        Coefficients along the path

    n_iter : int
        Number of iterations run. Returned only if return_n_iter is set
        to True.

    See also
    --------
    lasso_path
    LassoLars
    Lars
    LassoLarsCV
    LarsCV
    sklearn.decomposition.sparse_encode

    References
    ----------
    .. [1] "Least Angle Regression", Effron et al.
           http://www-stat.stanford.edu/~tibs/ftp/lars.pdf

    .. [2] `Wikipedia entry on the Least-angle regression
           <http://en.wikipedia.org/wiki/Least-angle_regression>`_

    .. [3] `Wikipedia entry on the Lasso
           <http://en.wikipedia.org/wiki/Lasso_(statistics)#Lasso_method>`_

    """

    n_features = X.shape[1]
    n_samples = y.size
    max_features = min(max_iter, n_features)

    if return_path:
        coefs = np.zeros((max_features + 1, n_features))
        alphas = np.zeros(max_features + 1)
    else:
        coef, prev_coef = np.zeros(n_features), np.zeros(n_features)
        alpha, prev_alpha = np.array([0.]), np.array([0.])  # better ideas?

    n_iter, n_active = 0, 0
    active, indices = list(), np.arange(n_features)
    # holds the sign of covariance
    sign_active = np.empty(max_features, dtype=np.int8)
    drop = False

    # will hold the cholesky factorization. Only lower part is
    # referenced.
    # We are initializing this to "zeros" and not empty, because
    # it is passed to scipy linalg functions and thus if it has NaNs,
    # even if they are in the upper part that it not used, we
    # get errors raised.
    # Once we support only scipy > 0.12 we can use check_finite=False and
    # go back to "empty"
    L = np.zeros((max_features, max_features), dtype=X.dtype)
    swap, nrm2 = linalg.get_blas_funcs(('swap', 'nrm2'), (X,))
    solve_cholesky, = get_lapack_funcs(('potrs',), (X,))

    if Gram is None:
        if copy_X:
            # force copy. setting the array to be fortran-ordered
            # speeds up the calculation of the (partial) Gram matrix
            # and allows to easily swap columns
            X = X.copy('F')
    elif isinstance(Gram, string_types) and Gram == 'auto':
        Gram = None
        if X.shape[0] > X.shape[1]:
            Gram = np.dot(X.T, X)
    elif copy_Gram:
        Gram = Gram.copy()

    if Xy is None:
        Cov = np.dot(X.T, y)
    else:
        Cov = Xy.copy()

    if verbose:
        if verbose > 1:
            print("Step\t\tAdded\t\tDropped\t\tActive set size\t\tC")
        else:
            sys.stdout.write('.')
            sys.stdout.flush()

    tiny = np.finfo(np.float).tiny  # to avoid division by 0 warning
    tiny32 = np.finfo(np.float32).tiny  # to avoid division by 0 warning
    equality_tolerance = np.finfo(np.float32).eps

    while True:
        if Cov.size:
            if positive:
                C_idx = np.argmax(Cov)
            else:
                C_idx = np.argmax(np.abs(Cov))

            C_ = Cov[C_idx]

            if positive:
                C = C_
            else:
                C = np.fabs(C_)
        else:
            C = 0.

        if return_path:
            alpha = alphas[n_iter, np.newaxis]
            coef = coefs[n_iter]
            prev_alpha = alphas[n_iter - 1, np.newaxis]
            prev_coef = coefs[n_iter - 1]

        alpha[0] = C / n_samples
        if alpha[0] <= alpha_min + equality_tolerance:  # early stopping
            if abs(alpha[0] - alpha_min) > equality_tolerance:
                # interpolation factor 0 <= ss < 1
                if n_iter > 0:
                    # In the first iteration, all alphas are zero, the formula
                    # below would make ss a NaN
                    ss = ((prev_alpha[0] - alpha_min) /
                          (prev_alpha[0] - alpha[0]))
                    coef[:] = prev_coef + ss * (coef - prev_coef)
                alpha[0] = alpha_min
            if return_path:
                coefs[n_iter] = coef
            break

        if n_iter >= max_iter or n_active >= n_features:
            break

        if not drop:

            ##########################################################
            # Append x_j to the Cholesky factorization of (Xa * Xa') #
            #                                                        #
            #            ( L   0 )                                   #
            #     L  ->  (       )  , where L * w = Xa' x_j          #
            #            ( w   z )    and z = ||x_j||                #
            #                                                        #
            ##########################################################

            if positive:
                sign_active[n_active] = np.ones_like(C_)
            else:
                sign_active[n_active] = np.sign(C_)
            m, n = n_active, C_idx + n_active

            Cov[C_idx], Cov[0] = swap(Cov[C_idx], Cov[0])
            indices[n], indices[m] = indices[m], indices[n]
            Cov_not_shortened = Cov
            Cov = Cov[1:]  # remove Cov[0]

            if Gram is None:
                X.T[n], X.T[m] = swap(X.T[n], X.T[m])
                c = nrm2(X.T[n_active]) ** 2
                L[n_active, :n_active] = \
                    np.dot(X.T[n_active], X.T[:n_active].T)
            else:
                # swap does only work inplace if matrix is fortran
                # contiguous ...
                Gram[m], Gram[n] = swap(Gram[m], Gram[n])
                Gram[:, m], Gram[:, n] = swap(Gram[:, m], Gram[:, n])
                c = Gram[n_active, n_active]
                L[n_active, :n_active] = Gram[n_active, :n_active]

            # Update the cholesky decomposition for the Gram matrix
            if n_active:
                linalg.solve_triangular(L[:n_active, :n_active],
                                        L[n_active, :n_active],
                                        trans=0, lower=1,
                                        overwrite_b=True,
                                        **solve_triangular_args)

            v = np.dot(L[n_active, :n_active], L[n_active, :n_active])
            diag = max(np.sqrt(np.abs(c - v)), eps)
            L[n_active, n_active] = diag

            if diag < 1e-7:
                # The system is becoming too ill-conditioned.
                # We have degenerate vectors in our active set.
                # We'll 'drop for good' the last regressor added.

                # Note: this case is very rare. It is no longer triggered by the
                # test suite. The `equality_tolerance` margin added in 0.16.0 to
                # get early stopping to work consistently on all versions of
                # Python including 32 bit Python under Windows seems to make it
                # very difficult to trigger the 'drop for good' strategy.
                warnings.warn('Regressors in active set degenerate. '
                              'Dropping a regressor, after %i iterations, '
                              'i.e. alpha=%.3e, '
                              'with an active set of %i regressors, and '
                              'the smallest cholesky pivot element being %.3e'
                              % (n_iter, alpha, n_active, diag),
                              ConvergenceWarning)
                # XXX: need to figure a 'drop for good' way
                Cov = Cov_not_shortened
                Cov[0] = 0
                Cov[C_idx], Cov[0] = swap(Cov[C_idx], Cov[0])
                continue

            active.append(indices[n_active])
            n_active += 1

            if verbose > 1:
                print("%s\t\t%s\t\t%s\t\t%s\t\t%s" % (n_iter, active[-1], '',
                                                      n_active, C))

        if method == 'lasso' and n_iter > 0 and prev_alpha[0] < alpha[0]:
            # alpha is increasing. This is because the updates of Cov are
            # bringing in too much numerical error that is greater than
            # than the remaining correlation with the
            # regressors. Time to bail out
            warnings.warn('Early stopping the lars path, as the residues '
                          'are small and the current value of alpha is no '
                          'longer well controlled. %i iterations, alpha=%.3e, '
                          'previous alpha=%.3e, with an active set of %i '
                          'regressors.'
                          % (n_iter, alpha, prev_alpha, n_active),
                          ConvergenceWarning)
            break

        # least squares solution
        least_squares, info = solve_cholesky(L[:n_active, :n_active],
                                             sign_active[:n_active],
                                             lower=True)

        if least_squares.size == 1 and least_squares == 0:
            # This happens because sign_active[:n_active] = 0
            least_squares[...] = 1
            AA = 1.
        else:
            # is this really needed ?
            AA = 1. / np.sqrt(np.sum(least_squares * sign_active[:n_active]))

            if not np.isfinite(AA):
                # L is too ill-conditioned
                i = 0
                L_ = L[:n_active, :n_active].copy()
                while not np.isfinite(AA):
                    L_.flat[::n_active + 1] += (2 ** i) * eps
                    least_squares, info = solve_cholesky(
                        L_, sign_active[:n_active], lower=True)
                    tmp = max(np.sum(least_squares * sign_active[:n_active]),
                              eps)
                    AA = 1. / np.sqrt(tmp)
                    i += 1
            least_squares *= AA

        if Gram is None:
            # equiangular direction of variables in the active set
            eq_dir = np.dot(X.T[:n_active].T, least_squares)
            # correlation between each unactive variables and
            # eqiangular vector
            corr_eq_dir = np.dot(X.T[n_active:], eq_dir)
        else:
            # if huge number of features, this takes 50% of time, I
            # think could be avoided if we just update it using an
            # orthogonal (QR) decomposition of X
            corr_eq_dir = np.dot(Gram[:n_active, n_active:].T,
                                 least_squares)

        g1 = arrayfuncs.min_pos((C - Cov) / (AA - corr_eq_dir + tiny))
        if positive:
            gamma_ = min(g1, C / AA)
        else:
            g2 = arrayfuncs.min_pos((C + Cov) / (AA + corr_eq_dir + tiny))
            gamma_ = min(g1, g2, C / AA)

        # TODO: better names for these variables: z
        drop = False
        z = -coef[active] / (least_squares + tiny32)
        z_pos = arrayfuncs.min_pos(z)
        if z_pos < gamma_:
            # some coefficients have changed sign
            idx = np.where(z == z_pos)[0][::-1]

            # update the sign, important for LAR
            sign_active[idx] = -sign_active[idx]

            if method == 'lasso':
                gamma_ = z_pos
            drop = True

        n_iter += 1

        if return_path:
            if n_iter >= coefs.shape[0]:
                del coef, alpha, prev_alpha, prev_coef
                # resize the coefs and alphas array
                add_features = 2 * max(1, (max_features - n_active))
                coefs = np.resize(coefs, (n_iter + add_features, n_features))
                alphas = np.resize(alphas, n_iter + add_features)
            coef = coefs[n_iter]
            prev_coef = coefs[n_iter - 1]
            alpha = alphas[n_iter, np.newaxis]
            prev_alpha = alphas[n_iter - 1, np.newaxis]
        else:
            # mimic the effect of incrementing n_iter on the array references
            prev_coef = coef
            prev_alpha[0] = alpha[0]
            coef = np.zeros_like(coef)

        coef[active] = prev_coef[active] + gamma_ * least_squares

        # update correlations
        Cov -= gamma_ * corr_eq_dir

        # See if any coefficient has changed sign
        if drop and method == 'lasso':

            # handle the case when idx is not length of 1
            [arrayfuncs.cholesky_delete(L[:n_active, :n_active], ii) for ii in
                idx]

            n_active -= 1
            m, n = idx, n_active
            # handle the case when idx is not length of 1
            drop_idx = [active.pop(ii) for ii in idx]

            if Gram is None:
                # propagate dropped variable
                for ii in idx:
                    for i in range(ii, n_active):
                        X.T[i], X.T[i + 1] = swap(X.T[i], X.T[i + 1])
                        # yeah this is stupid
                        indices[i], indices[i + 1] = indices[i + 1], indices[i]

                # TODO: this could be updated
                residual = y - np.dot(X[:, :n_active], coef[active])
                temp = np.dot(X.T[n_active], residual)

                Cov = np.r_[temp, Cov]
            else:
                for ii in idx:
                    for i in range(ii, n_active):
                        indices[i], indices[i + 1] = indices[i + 1], indices[i]
                        Gram[i], Gram[i + 1] = swap(Gram[i], Gram[i + 1])
                        Gram[:, i], Gram[:, i + 1] = swap(Gram[:, i],
                                                          Gram[:, i + 1])

                # Cov_n = Cov_j + x_j * X + increment(betas) TODO:
                # will this still work with multiple drops ?

                # recompute covariance. Probably could be done better
                # wrong as Xy is not swapped with the rest of variables

                # TODO: this could be updated
                residual = y - np.dot(X, coef)
                temp = np.dot(X.T[drop_idx], residual)
                Cov = np.r_[temp, Cov]

            sign_active = np.delete(sign_active, idx)
            sign_active = np.append(sign_active, 0.)  # just to maintain size
            if verbose > 1:
                print("%s\t\t%s\t\t%s\t\t%s\t\t%s" % (n_iter, '', drop_idx,
                                                      n_active, abs(temp)))

    if return_path:
        # resize coefs in case of early stop
        alphas = alphas[:n_iter + 1]
        coefs = coefs[:n_iter + 1]

        if return_n_iter:
            return alphas, active, coefs.T, n_iter
        else:
            return alphas, active, coefs.T
    else:
        if return_n_iter:
            return alpha, active, coef, n_iter
        else:
            return alpha, active, coef


###############################################################################
# Estimator classes

class Lars(LinearModel, RegressorMixin):
    """Least Angle Regression model a.k.a. LAR

    Read more in the :ref:`User Guide <least_angle_regression>`.

    Parameters
    ----------
    n_nonzero_coefs : int, optional
        Target number of non-zero coefficients. Use ``np.inf`` for no limit.

    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).

    positive : boolean (default=False)
        Restrict coefficients to be >= 0. Be aware that you might want to
        remove fit_intercept which is set True by default.

    verbose : boolean or integer, optional
        Sets the verbosity amount

    normalize : boolean, optional, default False
        If ``True``, the regressors X will be normalized before regression.

    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.

    copy_X : boolean, optional, default True
        If ``True``, X will be copied; else, it may be overwritten.

    eps : float, optional
        The machine-precision regularization in the computation of the
        Cholesky diagonal factors. Increase this for very ill-conditioned
        systems. Unlike the ``tol`` parameter in some iterative
        optimization-based algorithms, this parameter does not control
        the tolerance of the optimization.

    fit_path : boolean
        If True the full path is stored in the ``coef_path_`` attribute.
        If you compute the solution for a large problem or many targets,
        setting ``fit_path`` to ``False`` will lead to a speedup, especially
        with a small alpha.

    Attributes
    ----------
    alphas_ : array, shape (n_alphas + 1,) | list of n_targets such arrays
        Maximum of covariances (in absolute value) at each iteration. \
        ``n_alphas`` is either ``n_nonzero_coefs`` or ``n_features``, \
        whichever is smaller.

    active_ : list, length = n_alphas | list of n_targets such lists
        Indices of active variables at the end of the path.

    coef_path_ : array, shape (n_features, n_alphas + 1) \
        | list of n_targets such arrays
        The varying values of the coefficients along the path. It is not
        present if the ``fit_path`` parameter is ``False``.

    coef_ : array, shape (n_features,) or (n_targets, n_features)
        Parameter vector (w in the formulation formula).

    intercept_ : float | array, shape (n_targets,)
        Independent term in decision function.

    n_iter_ : array-like or int
        The number of iterations taken by lars_path to find the
        grid of alphas for each target.

    Examples
    --------
    >>> from sklearn import linear_model
    >>> clf = linear_model.Lars(n_nonzero_coefs=1)
    >>> clf.fit([[-1, 1], [0, 0], [1, 1]], [-1.1111, 0, -1.1111])
    ... # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
    Lars(copy_X=True, eps=..., fit_intercept=True, fit_path=True,
       n_nonzero_coefs=1, normalize=True, positive=False, precompute='auto',
       verbose=False)
    >>> print(clf.coef_) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
    [ 0. -1.11...]

    See also
    --------
    lars_path, LarsCV
    sklearn.decomposition.sparse_encode

    """
    def __init__(self, fit_intercept=True, verbose=False, normalize=True,
                 precompute='auto', n_nonzero_coefs=500,
                 eps=np.finfo(np.float).eps, copy_X=True, fit_path=True,
                 positive=False):
        self.fit_intercept = fit_intercept
        self.verbose = verbose
        self.normalize = normalize
        self.method = 'lar'
        self.precompute = precompute
        self.n_nonzero_coefs = n_nonzero_coefs
        self.positive = positive
        self.eps = eps
        self.copy_X = copy_X
        self.fit_path = fit_path

    def _get_gram(self):
        # precompute if n_samples > n_features
        precompute = self.precompute
        if hasattr(precompute, '__array__'):
            Gram = precompute
        elif precompute == 'auto':
            Gram = 'auto'
        else:
            Gram = None
        return Gram

    def fit(self, X, y, Xy=None):
        """Fit the model using X, y as training data.

        parameters
        ----------
        X : array-like, shape (n_samples, n_features)
            Training data.

        y : array-like, shape (n_samples,) or (n_samples, n_targets)
            Target values.

        Xy : array-like, shape (n_samples,) or (n_samples, n_targets), \
                optional
            Xy = np.dot(X.T, y) that can be precomputed. It is useful
            only when the Gram matrix is precomputed.

        returns
        -------
        self : object
            returns an instance of self.
        """
        X, y = check_X_y(X, y, y_numeric=True, multi_output=True)
        n_features = X.shape[1]

        X, y, X_mean, y_mean, X_std = self._center_data(X, y,
                                                        self.fit_intercept,
                                                        self.normalize,
                                                        self.copy_X)

        if y.ndim == 1:
            y = y[:, np.newaxis]

        n_targets = y.shape[1]

        alpha = getattr(self, 'alpha', 0.)
        if hasattr(self, 'n_nonzero_coefs'):
            alpha = 0.  # n_nonzero_coefs parametrization takes priority
            max_iter = self.n_nonzero_coefs
        else:
            max_iter = self.max_iter

        precompute = self.precompute
        if not hasattr(precompute, '__array__') and (
                precompute is True or
                (precompute == 'auto' and X.shape[0] > X.shape[1]) or
                (precompute == 'auto' and y.shape[1] > 1)):
            Gram = np.dot(X.T, X)
        else:
            Gram = self._get_gram()

        self.alphas_ = []
        self.n_iter_ = []

        if self.fit_path:
            self.coef_ = []
            self.active_ = []
            self.coef_path_ = []
            for k in xrange(n_targets):
                this_Xy = None if Xy is None else Xy[:, k]
                alphas, active, coef_path, n_iter_ = lars_path(
                    X, y[:, k], Gram=Gram, Xy=this_Xy, copy_X=self.copy_X,
                    copy_Gram=True, alpha_min=alpha, method=self.method,
                    verbose=max(0, self.verbose - 1), max_iter=max_iter,
                    eps=self.eps, return_path=True,
                    return_n_iter=True, positive=self.positive)
                self.alphas_.append(alphas)
                self.active_.append(active)
                self.n_iter_.append(n_iter_)
                self.coef_path_.append(coef_path)
                self.coef_.append(coef_path[:, -1])

            if n_targets == 1:
                self.alphas_, self.active_, self.coef_path_, self.coef_ = [
                    a[0] for a in (self.alphas_, self.active_, self.coef_path_,
                                   self.coef_)]
                self.n_iter_ = self.n_iter_[0]
        else:
            self.coef_ = np.empty((n_targets, n_features))
            for k in xrange(n_targets):
                this_Xy = None if Xy is None else Xy[:, k]
                alphas, _, self.coef_[k], n_iter_ = lars_path(
                    X, y[:, k], Gram=Gram, Xy=this_Xy, copy_X=self.copy_X,
                    copy_Gram=True, alpha_min=alpha, method=self.method,
                    verbose=max(0, self.verbose - 1), max_iter=max_iter,
                    eps=self.eps, return_path=False, return_n_iter=True,
                    positive=self.positive)
                self.alphas_.append(alphas)
                self.n_iter_.append(n_iter_)
            if n_targets == 1:
                self.alphas_ = self.alphas_[0]
                self.n_iter_ = self.n_iter_[0]
        self._set_intercept(X_mean, y_mean, X_std)
        return self


class LassoLars(Lars):
    """Lasso model fit with Least Angle Regression a.k.a. Lars

    It is a Linear Model trained with an L1 prior as regularizer.

    The optimization objective for Lasso is::

    (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1

    Read more in the :ref:`User Guide <least_angle_regression>`.

    Parameters
    ----------
    alpha : float
        Constant that multiplies the penalty term. Defaults to 1.0.
        ``alpha = 0`` is equivalent to an ordinary least square, solved
        by :class:`LinearRegression`. For numerical reasons, using
        ``alpha = 0`` with the LassoLars object is not advised and you
        should prefer the LinearRegression object.

    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).

    positive : boolean (default=False)
        Restrict coefficients to be >= 0. Be aware that you might want to
        remove fit_intercept which is set True by default.
        Under the positive restriction the model coefficients will not converge
        to the ordinary-least-squares solution for small values of alpha.
        Only coeffiencts up to the smallest alpha value (``alphas_[alphas_ >
        0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso
        algorithm are typically in congruence with the solution of the
        coordinate descent Lasso estimator.

    verbose : boolean or integer, optional
        Sets the verbosity amount

    normalize : boolean, optional, default False
        If True, the regressors X will be normalized before regression.

    copy_X : boolean, optional, default True
        If True, X will be copied; else, it may be overwritten.

    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.

    max_iter : integer, optional
        Maximum number of iterations to perform.

    eps : float, optional
        The machine-precision regularization in the computation of the
        Cholesky diagonal factors. Increase this for very ill-conditioned
        systems. Unlike the ``tol`` parameter in some iterative
        optimization-based algorithms, this parameter does not control
        the tolerance of the optimization.

    fit_path : boolean
        If ``True`` the full path is stored in the ``coef_path_`` attribute.
        If you compute the solution for a large problem or many targets,
        setting ``fit_path`` to ``False`` will lead to a speedup, especially
        with a small alpha.

    Attributes
    ----------
    alphas_ : array, shape (n_alphas + 1,) | list of n_targets such arrays
        Maximum of covariances (in absolute value) at each iteration. \
        ``n_alphas`` is either ``max_iter``, ``n_features``, or the number of \
        nodes in the path with correlation greater than ``alpha``, whichever \
        is smaller.

    active_ : list, length = n_alphas | list of n_targets such lists
        Indices of active variables at the end of the path.

    coef_path_ : array, shape (n_features, n_alphas + 1) or list
        If a list is passed it's expected to be one of n_targets such arrays.
        The varying values of the coefficients along the path. It is not
        present if the ``fit_path`` parameter is ``False``.

    coef_ : array, shape (n_features,) or (n_targets, n_features)
        Parameter vector (w in the formulation formula).

    intercept_ : float | array, shape (n_targets,)
        Independent term in decision function.

    n_iter_ : array-like or int.
        The number of iterations taken by lars_path to find the
        grid of alphas for each target.

    Examples
    --------
    >>> from sklearn import linear_model
    >>> clf = linear_model.LassoLars(alpha=0.01)
    >>> clf.fit([[-1, 1], [0, 0], [1, 1]], [-1, 0, -1])
    ... # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
    LassoLars(alpha=0.01, copy_X=True, eps=..., fit_intercept=True,
         fit_path=True, max_iter=500, normalize=True, positive=False,
         precompute='auto', verbose=False)
    >>> print(clf.coef_) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
    [ 0.         -0.963257...]

    See also
    --------
    lars_path
    lasso_path
    Lasso
    LassoCV
    LassoLarsCV
    sklearn.decomposition.sparse_encode

    """

    def __init__(self, alpha=1.0, fit_intercept=True, verbose=False,
                 normalize=True, precompute='auto', max_iter=500,
                 eps=np.finfo(np.float).eps, copy_X=True, fit_path=True,
                 positive=False):
        self.alpha = alpha
        self.fit_intercept = fit_intercept
        self.max_iter = max_iter
        self.verbose = verbose
        self.normalize = normalize
        self.method = 'lasso'
        self.positive = positive
        self.precompute = precompute
        self.copy_X = copy_X
        self.eps = eps
        self.fit_path = fit_path


###############################################################################
# Cross-validated estimator classes

def _check_copy_and_writeable(array, copy=False):
    if copy or not array.flags.writeable:
        return array.copy()
    return array


def _lars_path_residues(X_train, y_train, X_test, y_test, Gram=None,
                        copy=True, method='lars', verbose=False,
                        fit_intercept=True, normalize=True, max_iter=500,
                        eps=np.finfo(np.float).eps, positive=False):
    """Compute the residues on left-out data for a full LARS path

    Parameters
    -----------
    X_train : array, shape (n_samples, n_features)
        The data to fit the LARS on

    y_train : array, shape (n_samples)
        The target variable to fit LARS on

    X_test : array, shape (n_samples, n_features)
        The data to compute the residues on

    y_test : array, shape (n_samples)
        The target variable to compute the residues on

    Gram : None, 'auto', array, shape: (n_features, n_features), optional
        Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram
        matrix is precomputed from the given X, if there are more samples
        than features

    copy : boolean, optional
        Whether X_train, X_test, y_train and y_test should be copied;
        if False, they may be overwritten.

    method : 'lar' | 'lasso'
        Specifies the returned model. Select ``'lar'`` for Least Angle
        Regression, ``'lasso'`` for the Lasso.

    verbose : integer, optional
        Sets the amount of verbosity

    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).

    positive : boolean (default=False)
        Restrict coefficients to be >= 0. Be aware that you might want to
        remove fit_intercept which is set True by default.
        See reservations for using this option in combination with method
        'lasso' for expected small values of alpha in the doc of LassoLarsCV
        and LassoLarsIC.

    normalize : boolean, optional, default False
        If True, the regressors X will be normalized before regression.

    max_iter : integer, optional
        Maximum number of iterations to perform.

    eps : float, optional
        The machine-precision regularization in the computation of the
        Cholesky diagonal factors. Increase this for very ill-conditioned
        systems. Unlike the ``tol`` parameter in some iterative
        optimization-based algorithms, this parameter does not control
        the tolerance of the optimization.


    Returns
    --------
    alphas : array, shape (n_alphas,)
        Maximum of covariances (in absolute value) at each iteration.
        ``n_alphas`` is either ``max_iter`` or ``n_features``, whichever
        is smaller.

    active : list
        Indices of active variables at the end of the path.

    coefs : array, shape (n_features, n_alphas)
        Coefficients along the path

    residues : array, shape (n_alphas, n_samples)
        Residues of the prediction on the test data
    """
    X_train = _check_copy_and_writeable(X_train, copy)
    y_train = _check_copy_and_writeable(y_train, copy)
    X_test = _check_copy_and_writeable(X_test, copy)
    y_test = _check_copy_and_writeable(y_test, copy)

    if fit_intercept:
        X_mean = X_train.mean(axis=0)
        X_train -= X_mean
        X_test -= X_mean
        y_mean = y_train.mean(axis=0)
        y_train = as_float_array(y_train, copy=False)
        y_train -= y_mean
        y_test = as_float_array(y_test, copy=False)
        y_test -= y_mean

    if normalize:
        norms = np.sqrt(np.sum(X_train ** 2, axis=0))
        nonzeros = np.flatnonzero(norms)
        X_train[:, nonzeros] /= norms[nonzeros]

    alphas, active, coefs = lars_path(
        X_train, y_train, Gram=Gram, copy_X=False, copy_Gram=False,
        method=method, verbose=max(0, verbose - 1), max_iter=max_iter, eps=eps,
        positive=positive)
    if normalize:
        coefs[nonzeros] /= norms[nonzeros][:, np.newaxis]
    residues = np.dot(X_test, coefs) - y_test[:, np.newaxis]
    return alphas, active, coefs, residues.T


class LarsCV(Lars):
    """Cross-validated Least Angle Regression model

    Read more in the :ref:`User Guide <least_angle_regression>`.

    Parameters
    ----------
    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).

    positive : boolean (default=False)
        Restrict coefficients to be >= 0. Be aware that you might want to
        remove fit_intercept which is set True by default.

    verbose : boolean or integer, optional
        Sets the verbosity amount

    normalize : boolean, optional, default False
        If True, the regressors X will be normalized before regression.

    copy_X : boolean, optional, default True
        If ``True``, X will be copied; else, it may be overwritten.

    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.

    max_iter: integer, optional
        Maximum number of iterations to perform.

    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, :class:`KFold` is used.

        Refer :ref:`User Guide <cross_validation>` for the various
        cross-validation strategies that can be used here.

    max_n_alphas : integer, optional
        The maximum number of points on the path used to compute the
        residuals in the cross-validation

    n_jobs : integer, optional
        Number of CPUs to use during the cross validation. If ``-1``, use
        all the CPUs

    eps : float, optional
        The machine-precision regularization in the computation of the
        Cholesky diagonal factors. Increase this for very ill-conditioned
        systems.


    Attributes
    ----------
    coef_ : array, shape (n_features,)
        parameter vector (w in the formulation formula)

    intercept_ : float
        independent term in decision function

    coef_path_ : array, shape (n_features, n_alphas)
        the varying values of the coefficients along the path

    alpha_ : float
        the estimated regularization parameter alpha

    alphas_ : array, shape (n_alphas,)
        the different values of alpha along the path

    cv_alphas_ : array, shape (n_cv_alphas,)
        all the values of alpha along the path for the different folds

    cv_mse_path_ : array, shape (n_folds, n_cv_alphas)
        the mean square error on left-out for each fold along the path
        (alpha values given by ``cv_alphas``)

    n_iter_ : array-like or int
        the number of iterations run by Lars with the optimal alpha.

    See also
    --------
    lars_path, LassoLars, LassoLarsCV
    """

    method = 'lar'

    def __init__(self, fit_intercept=True, verbose=False, max_iter=500,
                 normalize=True, precompute='auto', cv=None,
                 max_n_alphas=1000, n_jobs=1, eps=np.finfo(np.float).eps,
                 copy_X=True, positive=False):
        self.fit_intercept = fit_intercept
        self.positive = positive
        self.max_iter = max_iter
        self.verbose = verbose
        self.normalize = normalize
        self.precompute = precompute
        self.copy_X = copy_X
        self.cv = cv
        self.max_n_alphas = max_n_alphas
        self.n_jobs = n_jobs
        self.eps = eps

    def fit(self, X, y):
        """Fit the model using X, y as training data.

        Parameters
        ----------
        X : array-like, shape (n_samples, n_features)
            Training data.

        y : array-like, shape (n_samples,)
            Target values.

        Returns
        -------
        self : object
            returns an instance of self.
        """
        self.fit_path = True
        X, y = check_X_y(X, y, y_numeric=True)
        X = as_float_array(X, copy=self.copy_X)
        y = as_float_array(y, copy=self.copy_X)

        # init cross-validation generator
        cv = check_cv(self.cv, X, y, classifier=False)

        Gram = 'auto' if self.precompute else None

        cv_paths = Parallel(n_jobs=self.n_jobs, verbose=self.verbose)(
            delayed(_lars_path_residues)(
                X[train], y[train], X[test], y[test], Gram=Gram, copy=False,
                method=self.method, verbose=max(0, self.verbose - 1),
                normalize=self.normalize, fit_intercept=self.fit_intercept,
                max_iter=self.max_iter, eps=self.eps, positive=self.positive)
            for train, test in cv)
        all_alphas = np.concatenate(list(zip(*cv_paths))[0])
        # Unique also sorts
        all_alphas = np.unique(all_alphas)
        # Take at most max_n_alphas values
        stride = int(max(1, int(len(all_alphas) / float(self.max_n_alphas))))
        all_alphas = all_alphas[::stride]

        mse_path = np.empty((len(all_alphas), len(cv_paths)))
        for index, (alphas, active, coefs, residues) in enumerate(cv_paths):
            alphas = alphas[::-1]
            residues = residues[::-1]
            if alphas[0] != 0:
                alphas = np.r_[0, alphas]
                residues = np.r_[residues[0, np.newaxis], residues]
            if alphas[-1] != all_alphas[-1]:
                alphas = np.r_[alphas, all_alphas[-1]]
                residues = np.r_[residues, residues[-1, np.newaxis]]
            this_residues = interpolate.interp1d(alphas,
                                                 residues,
                                                 axis=0)(all_alphas)
            this_residues **= 2
            mse_path[:, index] = np.mean(this_residues, axis=-1)

        mask = np.all(np.isfinite(mse_path), axis=-1)
        all_alphas = all_alphas[mask]
        mse_path = mse_path[mask]
        # Select the alpha that minimizes left-out error
        i_best_alpha = np.argmin(mse_path.mean(axis=-1))
        best_alpha = all_alphas[i_best_alpha]

        # Store our parameters
        self.alpha_ = best_alpha
        self.cv_alphas_ = all_alphas
        self.cv_mse_path_ = mse_path

        # Now compute the full model
        # it will call a lasso internally when self if LassoLarsCV
        # as self.method == 'lasso'
        Lars.fit(self, X, y)
        return self

    @property
    def alpha(self):
        # impedance matching for the above Lars.fit (should not be documented)
        return self.alpha_


class LassoLarsCV(LarsCV):
    """Cross-validated Lasso, using the LARS algorithm

    The optimization objective for Lasso is::

    (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1

    Read more in the :ref:`User Guide <least_angle_regression>`.

    Parameters
    ----------
    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).

    positive : boolean (default=False)
        Restrict coefficients to be >= 0. Be aware that you might want to
        remove fit_intercept which is set True by default.
        Under the positive restriction the model coefficients do not converge
        to the ordinary-least-squares solution for small values of alpha.
        Only coeffiencts up to the smallest alpha value (``alphas_[alphas_ >
        0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso
        algorithm are typically in congruence with the solution of the
        coordinate descent Lasso estimator.
        As a consequence using LassoLarsCV only makes sense for problems where
        a sparse solution is expected and/or reached.

    verbose : boolean or integer, optional
        Sets the verbosity amount

    normalize : boolean, optional, default False
        If True, the regressors X will be normalized before regression.

    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.

    max_iter : integer, optional
        Maximum number of iterations to perform.

    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, :class:`KFold` is used.

        Refer :ref:`User Guide <cross_validation>` for the various
        cross-validation strategies that can be used here.

    max_n_alphas : integer, optional
        The maximum number of points on the path used to compute the
        residuals in the cross-validation

    n_jobs : integer, optional
        Number of CPUs to use during the cross validation. If ``-1``, use
        all the CPUs

    eps : float, optional
        The machine-precision regularization in the computation of the
        Cholesky diagonal factors. Increase this for very ill-conditioned
        systems.

    copy_X : boolean, optional, default True
        If True, X will be copied; else, it may be overwritten.

    Attributes
    ----------
    coef_ : array, shape (n_features,)
        parameter vector (w in the formulation formula)

    intercept_ : float
        independent term in decision function.

    coef_path_ : array, shape (n_features, n_alphas)
        the varying values of the coefficients along the path

    alpha_ : float
        the estimated regularization parameter alpha

    alphas_ : array, shape (n_alphas,)
        the different values of alpha along the path

    cv_alphas_ : array, shape (n_cv_alphas,)
        all the values of alpha along the path for the different folds

    cv_mse_path_ : array, shape (n_folds, n_cv_alphas)
        the mean square error on left-out for each fold along the path
        (alpha values given by ``cv_alphas``)

    n_iter_ : array-like or int
        the number of iterations run by Lars with the optimal alpha.

    Notes
    -----

    The object solves the same problem as the LassoCV object. However,
    unlike the LassoCV, it find the relevant alphas values by itself.
    In general, because of this property, it will be more stable.
    However, it is more fragile to heavily multicollinear datasets.

    It is more efficient than the LassoCV if only a small number of
    features are selected compared to the total number, for instance if
    there are very few samples compared to the number of features.

    See also
    --------
    lars_path, LassoLars, LarsCV, LassoCV
    """

    method = 'lasso'


class LassoLarsIC(LassoLars):
    """Lasso model fit with Lars using BIC or AIC for model selection

    The optimization objective for Lasso is::

    (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1

    AIC is the Akaike information criterion and BIC is the Bayes
    Information criterion. Such criteria are useful to select the value
    of the regularization parameter by making a trade-off between the
    goodness of fit and the complexity of the model. A good model should
    explain well the data while being simple.

    Read more in the :ref:`User Guide <least_angle_regression>`.

    Parameters
    ----------
    criterion : 'bic' | 'aic'
        The type of criterion to use.

    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).

    positive : boolean (default=False)
        Restrict coefficients to be >= 0. Be aware that you might want to
        remove fit_intercept which is set True by default.
        Under the positive restriction the model coefficients do not converge
        to the ordinary-least-squares solution for small values of alpha.
        Only coeffiencts up to the smallest alpha value (``alphas_[alphas_ >
        0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso
        algorithm are typically in congruence with the solution of the
        coordinate descent Lasso estimator.
        As a consequence using LassoLarsIC only makes sense for problems where
        a sparse solution is expected and/or reached.

    verbose : boolean or integer, optional
        Sets the verbosity amount

    normalize : boolean, optional, default False
        If True, the regressors X will be normalized before regression.

    copy_X : boolean, optional, default True
        If True, X will be copied; else, it may be overwritten.

    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.

    max_iter : integer, optional
        Maximum number of iterations to perform. Can be used for
        early stopping.

    eps : float, optional
        The machine-precision regularization in the computation of the
        Cholesky diagonal factors. Increase this for very ill-conditioned
        systems. Unlike the ``tol`` parameter in some iterative
        optimization-based algorithms, this parameter does not control
        the tolerance of the optimization.


    Attributes
    ----------
    coef_ : array, shape (n_features,)
        parameter vector (w in the formulation formula)

    intercept_ : float
        independent term in decision function.

    alpha_ : float
        the alpha parameter chosen by the information criterion

    n_iter_ : int
        number of iterations run by lars_path to find the grid of
        alphas.

    criterion_ : array, shape (n_alphas,)
        The value of the information criteria ('aic', 'bic') across all
        alphas. The alpha which has the smallest information criteria
        is chosen.

    Examples
    --------
    >>> from sklearn import linear_model
    >>> clf = linear_model.LassoLarsIC(criterion='bic')
    >>> clf.fit([[-1, 1], [0, 0], [1, 1]], [-1.1111, 0, -1.1111])
    ... # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
    LassoLarsIC(copy_X=True, criterion='bic', eps=..., fit_intercept=True,
          max_iter=500, normalize=True, positive=False, precompute='auto',
          verbose=False)
    >>> print(clf.coef_) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
    [ 0.  -1.11...]

    Notes
    -----
    The estimation of the number of degrees of freedom is given by:

    "On the degrees of freedom of the lasso"
    Hui Zou, Trevor Hastie, and Robert Tibshirani
    Ann. Statist. Volume 35, Number 5 (2007), 2173-2192.

    http://en.wikipedia.org/wiki/Akaike_information_criterion
    http://en.wikipedia.org/wiki/Bayesian_information_criterion

    See also
    --------
    lars_path, LassoLars, LassoLarsCV
    """
    def __init__(self, criterion='aic', fit_intercept=True, verbose=False,
                 normalize=True, precompute='auto', max_iter=500,
                 eps=np.finfo(np.float).eps, copy_X=True, positive=False):
        self.criterion = criterion
        self.fit_intercept = fit_intercept
        self.positive = positive
        self.max_iter = max_iter
        self.verbose = verbose
        self.normalize = normalize
        self.copy_X = copy_X
        self.precompute = precompute
        self.eps = eps

    def fit(self, X, y, copy_X=True):
        """Fit the model using X, y as training data.

        Parameters
        ----------
        X : array-like, shape (n_samples, n_features)
            training data.

        y : array-like, shape (n_samples,)
            target values.

        copy_X : boolean, optional, default True
            If ``True``, X will be copied; else, it may be overwritten.

        Returns
        -------
        self : object
            returns an instance of self.
        """
        self.fit_path = True
        X, y = check_X_y(X, y, y_numeric=True)

        X, y, Xmean, ymean, Xstd = LinearModel._center_data(
            X, y, self.fit_intercept, self.normalize, self.copy_X)
        max_iter = self.max_iter

        Gram = self._get_gram()

        alphas_, active_, coef_path_, self.n_iter_ = lars_path(
            X, y, Gram=Gram, copy_X=copy_X, copy_Gram=True, alpha_min=0.0,
            method='lasso', verbose=self.verbose, max_iter=max_iter,
            eps=self.eps, return_n_iter=True, positive=self.positive)

        n_samples = X.shape[0]

        if self.criterion == 'aic':
            K = 2  # AIC
        elif self.criterion == 'bic':
            K = log(n_samples)  # BIC
        else:
            raise ValueError('criterion should be either bic or aic')

        R = y[:, np.newaxis] - np.dot(X, coef_path_)  # residuals
        mean_squared_error = np.mean(R ** 2, axis=0)

        df = np.zeros(coef_path_.shape[1], dtype=np.int)  # Degrees of freedom
        for k, coef in enumerate(coef_path_.T):
            mask = np.abs(coef) > np.finfo(coef.dtype).eps
            if not np.any(mask):
                continue
            # get the number of degrees of freedom equal to:
            # Xc = X[:, mask]
            # Trace(Xc * inv(Xc.T, Xc) * Xc.T) ie the number of non-zero coefs
            df[k] = np.sum(mask)

        self.alphas_ = alphas_
        with np.errstate(divide='ignore'):
            self.criterion_ = n_samples * np.log(mean_squared_error) + K * df
        n_best = np.argmin(self.criterion_)

        self.alpha_ = alphas_[n_best]
        self.coef_ = coef_path_[:, n_best]
        self._set_intercept(Xmean, ymean, Xstd)
        return self