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/ linear_model / _least_angle.py
"""
Least Angle Regression algorithm. See the documentation on the
Generalized Linear Model for a complete discussion.
"""
# 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
import numpy as np
from scipy import linalg, interpolate
from scipy.linalg.lapack import get_lapack_funcs
from joblib import Parallel, delayed
from ._base import LinearModel
from ..base import RegressorMixin, MultiOutputMixin
from ..utils import arrayfuncs, as_float_array, check_X_y
from ..model_selection import check_cv
from ..exceptions import ConvergenceWarning
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 : None or array-like of shape (n_samples, n_features)
Input data. Note that if X is None then the Gram matrix must be
specified, i.e., cannot be None or False.
.. deprecated:: 0.21
The use of ``X`` is ``None`` in combination with ``Gram`` is not
``None`` will be removed in v0.23. Use :func:`lars_path_gram`
instead.
y : None or array-like of shape (n_samples,)
Input targets.
Xy : array-like of shape (n_samples,) or (n_samples, n_targets), \
default=None
Xy = np.dot(X.T, y) that can be precomputed. It is useful
only when the Gram matrix is precomputed.
Gram : None, 'auto', array-like of shape (n_features, n_features), \
default=None
Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram
matrix is precomputed from the given X, if there are more samples
than features.
.. deprecated:: 0.21
The use of ``X`` is ``None`` in combination with ``Gram`` is not
None will be removed in v0.23. Use :func:`lars_path_gram` instead.
max_iter : int, default=500
Maximum number of iterations to perform, set to infinity for no limit.
alpha_min : float, default=0
Minimum correlation along the path. It corresponds to the
regularization parameter alpha parameter in the Lasso.
method : {'lar', 'lasso'}, default='lar'
Specifies the returned model. Select ``'lar'`` for Least Angle
Regression, ``'lasso'`` for the Lasso.
copy_X : bool, default=True
If ``False``, ``X`` is overwritten.
eps : float, optional
The machine-precision regularization in the computation of the
Cholesky diagonal factors. Increase this for very ill-conditioned
systems. By default, ``np.finfo(np.float).eps`` is used.
copy_Gram : bool, default=True
If ``False``, ``Gram`` is overwritten.
verbose : int, default=0
Controls output verbosity.
return_path : bool, default=True
If ``return_path==True`` returns the entire path, else returns only the
last point of the path.
return_n_iter : bool, default=False
Whether to return the number of iterations.
positive : bool, default=False
Restrict coefficients to be >= 0.
This option is only allowed with method 'lasso'. Note that the model
coefficients will not converge to the ordinary-least-squares solution
for small values of alpha. Only coefficients 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.
Returns
-------
alphas : array-like of 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-like of shape (n_alphas,)
Indices of active variables at the end of the path.
coefs : array-like of 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
--------
lars_path_gram
lasso_path
lasso_path_gram
LassoLars
Lars
LassoLarsCV
LarsCV
sklearn.decomposition.sparse_encode
References
----------
.. [1] "Least Angle Regression", Efron et al.
http://statweb.stanford.edu/~tibs/ftp/lars.pdf
.. [2] `Wikipedia entry on the Least-angle regression
<https://en.wikipedia.org/wiki/Least-angle_regression>`_
.. [3] `Wikipedia entry on the Lasso
<https://en.wikipedia.org/wiki/Lasso_(statistics)>`_
"""
if X is None and Gram is not None:
warnings.warn('Use lars_path_gram to avoid passing X and y. '
'The current option will be removed in v0.23.',
FutureWarning)
return _lars_path_solver(
X=X, y=y, Xy=Xy, Gram=Gram, n_samples=None, max_iter=max_iter,
alpha_min=alpha_min, method=method, copy_X=copy_X,
eps=eps, copy_Gram=copy_Gram, verbose=verbose, return_path=return_path,
return_n_iter=return_n_iter, positive=positive)
def lars_path_gram(Xy, Gram, n_samples, 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):
"""lars_path in the sufficient stats mode [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
----------
Xy : array-like of shape (n_samples,) or (n_samples, n_targets)
Xy = np.dot(X.T, y).
Gram : array-like of shape (n_features, n_features)
Gram = np.dot(X.T * X).
n_samples : int or float
Equivalent size of sample.
max_iter : int, default=500
Maximum number of iterations to perform, set to infinity for no limit.
alpha_min : float, default=0
Minimum correlation along the path. It corresponds to the
regularization parameter alpha parameter in the Lasso.
method : {'lar', 'lasso'}, default='lar'
Specifies the returned model. Select ``'lar'`` for Least Angle
Regression, ``'lasso'`` for the Lasso.
copy_X : bool, default=True
If ``False``, ``X`` is overwritten.
eps : float, optional
The machine-precision regularization in the computation of the
Cholesky diagonal factors. Increase this for very ill-conditioned
systems. By default, ``np.finfo(np.float).eps`` is used.
copy_Gram : bool, default=True
If ``False``, ``Gram`` is overwritten.
verbose : int, default=0
Controls output verbosity.
return_path : bool, default=True
If ``return_path==True`` returns the entire path, else returns only the
last point of the path.
return_n_iter : bool, default=False
Whether to return the number of iterations.
positive : bool, default=False
Restrict coefficients to be >= 0.
This option is only allowed with method 'lasso'. Note that the model
coefficients will not converge to the ordinary-least-squares solution
for small values of alpha. Only coefficients 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.
Returns
-------
alphas : array-like of 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-like of shape (n_alphas,)
Indices of active variables at the end of the path.
coefs : array-like of 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
--------
lars_path
lasso_path
lasso_path_gram
LassoLars
Lars
LassoLarsCV
LarsCV
sklearn.decomposition.sparse_encode
References
----------
.. [1] "Least Angle Regression", Efron et al.
http://statweb.stanford.edu/~tibs/ftp/lars.pdf
.. [2] `Wikipedia entry on the Least-angle regression
<https://en.wikipedia.org/wiki/Least-angle_regression>`_
.. [3] `Wikipedia entry on the Lasso
<https://en.wikipedia.org/wiki/Lasso_(statistics)>`_
"""
return _lars_path_solver(
X=None, y=None, Xy=Xy, Gram=Gram, n_samples=n_samples,
max_iter=max_iter, alpha_min=alpha_min, method=method,
copy_X=copy_X, eps=eps, copy_Gram=copy_Gram,
verbose=verbose, return_path=return_path,
return_n_iter=return_n_iter, positive=positive)
def _lars_path_solver(X, y, Xy=None, Gram=None, n_samples=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 : None or ndarray, of shape (n_samples, n_features)
Input data. Note that if X is None then Gram must be specified,
i.e., cannot be None or False.
y : None or ndarray, of shape (n_samples,)
Input targets.
Xy : array-like of shape (n_samples,) or (n_samples, n_targets), \
default=None
Xy = np.dot(X.T, y) that can be precomputed. It is useful
only when the Gram matrix is precomputed.
Gram : None, 'auto' or array-like of shape (n_features, n_features), \
default=None
Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram
matrix is precomputed from the given X, if there are more samples
than features.
n_samples : int or float, default=None
Equivalent size of sample.
max_iter : int, default=500
Maximum number of iterations to perform, set to infinity for no limit.
alpha_min : float, default=0
Minimum correlation along the path. It corresponds to the
regularization parameter alpha parameter in the Lasso.
method : {'lar', 'lasso'}, default='lar'
Specifies the returned model. Select ``'lar'`` for Least Angle
Regression, ``'lasso'`` for the Lasso.
copy_X : bool, default=True
If ``False``, ``X`` is overwritten.
eps : float, optional
The machine-precision regularization in the computation of the
Cholesky diagonal factors. Increase this for very ill-conditioned
systems. By default, ``np.finfo(np.float).eps`` is used
copy_Gram : bool, default=True
If ``False``, ``Gram`` is overwritten.
verbose : int, default=0
Controls output verbosity.
return_path : bool, default=True
If ``return_path==True`` returns the entire path, else returns only the
last point of the path.
return_n_iter : bool, default=False
Whether to return the number of iterations.
positive : bool, default=False
Restrict coefficients to be >= 0.
This option is only allowed with method 'lasso'. Note that the model
coefficients will not converge to the ordinary-least-squares solution
for small values of alpha. Only coefficients 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.
Returns
-------
alphas : array-like of 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-like of shape (n_alphas,)
Indices of active variables at the end of the path.
coefs : array-like of 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", Efron et al.
http://statweb.stanford.edu/~tibs/ftp/lars.pdf
.. [2] `Wikipedia entry on the Least-angle regression
<https://en.wikipedia.org/wiki/Least-angle_regression>`_
.. [3] `Wikipedia entry on the Lasso
<https://en.wikipedia.org/wiki/Lasso_(statistics)>`_
"""
if method == 'lar' and positive:
raise ValueError(
"Positive constraint not supported for 'lar' "
"coding method."
)
n_samples = n_samples if n_samples is not None else y.size
if Xy is None:
Cov = np.dot(X.T, y)
else:
Cov = Xy.copy()
if Gram is None or Gram is False:
Gram = None
if X is None:
raise ValueError('X and Gram cannot both be unspecified.')
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, str) and Gram == 'auto' or Gram is True:
if Gram is True or X.shape[0] > X.shape[1]:
Gram = np.dot(X.T, X)
else:
Gram = None
elif copy_Gram:
Gram = Gram.copy()
if Gram is None:
n_features = X.shape[1]
else:
n_features = Cov.shape[0]
if Gram.shape != (n_features, n_features):
raise ValueError('The shapes of the inputs Gram and Xy'
' do not match.')
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.
if Gram is None:
L = np.empty((max_features, max_features), dtype=X.dtype)
swap, nrm2 = linalg.get_blas_funcs(('swap', 'nrm2'), (X,))
else:
L = np.empty((max_features, max_features), dtype=Gram.dtype)
swap, nrm2 = linalg.get_blas_funcs(('swap', 'nrm2'), (Cov,))
solve_cholesky, = get_lapack_funcs(('potrs',), (L,))
if verbose:
if verbose > 1:
print("Step\t\tAdded\t\tDropped\t\tActive set size\t\tC")
else:
sys.stdout.write('.')
sys.stdout.flush()
tiny32 = np.finfo(np.float32).tiny # to avoid division by 0 warning
equality_tolerance = np.finfo(np.float32).eps
if Gram is not None:
Gram_copy = Gram.copy()
Cov_copy = Cov.copy()
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
# 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.'
' Reduce max_iter or increase eps parameters.'
% (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, _ = 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, _ = 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 + tiny32))
if positive:
gamma_ = min(g1, C / AA)
else:
g2 = arrayfuncs.min_pos((C + Cov) / (AA + corr_eq_dir + tiny32))
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))
coefs[-add_features:] = 0
alphas = np.resize(alphas, n_iter + add_features)
alphas[-add_features:] = 0
coef = coefs[n_iter]
prev_coef = coefs[n_iter - 1]
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
for ii in idx:
arrayfuncs.cholesky_delete(L[:n_active, :n_active], ii)
n_active -= 1
# 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
temp = Cov_copy[drop_idx] - np.dot(Gram_copy[drop_idx], coef)
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(MultiOutputMixin, RegressorMixin, LinearModel):
"""Least Angle Regression model a.k.a. LAR
Read more in the :ref:`User Guide <least_angle_regression>`.
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).
verbose : bool or int, default=False
Sets the verbosity amount
normalize : bool, default=True
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``.
precompute : bool, 'auto' or array-like , default='auto'
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.
n_nonzero_coefs : int, default=500
Target number of non-zero coefficients. Use ``np.inf`` for no limit.
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.
By default, ``np.finfo(np.float).eps`` is used.
copy_X : bool, default=True
If ``True``, X will be copied; else, it may be overwritten.
fit_path : bool, default=True
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-like of 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-like of 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-like of shape (n_features,) or (n_targets, n_features)
Parameter vector (w in the formulation formula).
intercept_ : float or array-like of 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
>>> reg = linear_model.Lars(n_nonzero_coefs=1)
>>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1.1111, 0, -1.1111])
Lars(n_nonzero_coefs=1)
>>> print(reg.coef_)
[ 0. -1.11...]
See also
--------
lars_path, LarsCV
sklearn.decomposition.sparse_encode
"""
method = 'lar'
positive = False
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):
self.fit_intercept = fit_intercept
self.verbose = verbose
self.normalize = normalize
self.precompute = precompute
self.n_nonzero_coefs = n_nonzero_coefs
self.eps = eps
self.copy_X = copy_X
self.fit_path = fit_path
@staticmethod
def _get_gram(precompute, X, y):
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)):
precompute = np.dot(X.T, X)
return precompute
def _fit(self, X, y, max_iter, alpha, fit_path, Xy=None):
"""Auxiliary method to fit the model using X, y as training data"""
n_features = X.shape[1]
X, y, X_offset, y_offset, X_scale = self._preprocess_data(
X, y, self.fit_intercept, self.normalize, self.copy_X)
if y.ndim == 1:
y = y[:, np.newaxis]
n_targets = y.shape[1]
Gram = self._get_gram(self.precompute, X, y)
self.alphas_ = []
self.n_iter_ = []
self.coef_ = np.empty((n_targets, n_features))
if fit_path:
self.active_ = []
self.coef_path_ = []
for k in range(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_[k] = 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:
for k in range(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_offset, y_offset, X_scale)
return self
def fit(self, X, y, Xy=None):
"""Fit the model using X, y as training data.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Training data.
y : array-like of shape (n_samples,) or (n_samples, n_targets)
Target values.
Xy : array-like of shape (n_samples,) or (n_samples, n_targets), \
default=None
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)
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
self._fit(X, y, max_iter=max_iter, alpha=alpha, fit_path=self.fit_path,
Xy=Xy)
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, default=1.0
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 : 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).
verbose : bool or int, default=False
Sets the verbosity amount
normalize : bool, default=True
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``.
precompute : bool, 'auto' or array-like, default='auto'
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 : int, default=500
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.
By default, ``np.finfo(np.float).eps`` is used.
copy_X : bool, default=True
If True, X will be copied; else, it may be overwritten.
fit_path : bool, default=True
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.
positive : bool, 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 coefficients 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.
Attributes
----------
alphas_ : array-like of 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-like of 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-like of shape (n_features,) or (n_targets, n_features)
Parameter vector (w in the formulation formula).
intercept_ : float or array-like of 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
>>> reg = linear_model.LassoLars(alpha=0.01)
>>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1, 0, -1])
LassoLars(alpha=0.01)
>>> print(reg.coef_)
[ 0. -0.963257...]
See also
--------
lars_path
lasso_path
Lasso
LassoCV
LassoLarsCV
LassoLarsIC
sklearn.decomposition.sparse_encode
"""
method = 'lasso'
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.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-like of shape (n_samples, n_features)
The data to fit the LARS on
y_train : array-like of shape (n_samples,)
The target variable to fit LARS on
X_test : array-like of shape (n_samples, n_features)
The data to compute the residues on
y_test : array-like of shape (n_samples,)
The target variable to compute the residues on
Gram : None, 'auto' or array-like of shape (n_features, n_features), \
default=None
Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram
matrix is precomputed from the given X, if there are more samples
than features
copy : bool, default=True
Whether X_train, X_test, y_train and y_test should be copied;
if False, they may be overwritten.
method : {'lar' , 'lasso'}, default='lar'
Specifies the returned model. Select ``'lar'`` for Least Angle
Regression, ``'lasso'`` for the Lasso.
verbose : bool or int, default=False
Sets the amount of verbosity
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).
positive : bool, 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 : bool, default=True
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``.
max_iter : int, default=500
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.
By default, ``np.finfo(np.float).eps`` is used
Returns
--------
alphas : array-like of 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-like of shape (n_features, n_alphas)
Coefficients along the path
residues : array-like of 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.
See glossary entry for :term:`cross-validation estimator`.
Read more in the :ref:`User Guide <least_angle_regression>`.
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).
verbose : bool or int, default=False
Sets the verbosity amount
max_iter : int, default=500
Maximum number of iterations to perform.
normalize : bool, default=True
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``.
precompute : bool, 'auto' or array-like , default='auto'
Whether to use a precomputed Gram matrix to speed up
calculations. If set to ``'auto'`` let us decide. The Gram matrix
cannot be passed as argument since we will use only subsets of X.
cv : int, cross-validation generator or an iterable, default=None
Determines the cross-validation splitting strategy.
Possible inputs for cv are:
- None, to use the default 5-fold cross-validation,
- integer, to specify the number of folds.
- :term:`CV splitter`,
- An iterable yielding (train, test) splits as arrays of indices.
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.
.. versionchanged:: 0.22
``cv`` default value if None changed from 3-fold to 5-fold.
max_n_alphas : int, default=1000
The maximum number of points on the path used to compute the
residuals in the cross-validation
n_jobs : int or None, default=None
Number of CPUs to use during the cross validation.
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
for more details.
eps : float, optional
The machine-precision regularization in the computation of the
Cholesky diagonal factors. Increase this for very ill-conditioned
systems. By default, ``np.finfo(np.float).eps`` is used.
copy_X : bool, default=True
If ``True``, X will be copied; else, it may be overwritten.
Attributes
----------
coef_ : array-like of shape (n_features,)
parameter vector (w in the formulation formula)
intercept_ : float
independent term in decision function
coef_path_ : array-like of shape (n_features, n_alphas)
the varying values of the coefficients along the path
alpha_ : float
the estimated regularization parameter alpha
alphas_ : array-like of shape (n_alphas,)
the different values of alpha along the path
cv_alphas_ : array-like of shape (n_cv_alphas,)
all the values of alpha along the path for the different folds
mse_path_ : array-like of 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.
Examples
--------
>>> from sklearn.linear_model import LarsCV
>>> from sklearn.datasets import make_regression
>>> X, y = make_regression(n_samples=200, noise=4.0, random_state=0)
>>> reg = LarsCV(cv=5).fit(X, y)
>>> reg.score(X, y)
0.9996...
>>> reg.alpha_
0.0254...
>>> reg.predict(X[:1,])
array([154.0842...])
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=None, eps=np.finfo(np.float).eps,
copy_X=True):
self.max_iter = max_iter
self.cv = cv
self.max_n_alphas = max_n_alphas
self.n_jobs = n_jobs
super().__init__(fit_intercept=fit_intercept,
verbose=verbose, normalize=normalize,
precompute=precompute,
n_nonzero_coefs=500,
eps=eps, copy_X=copy_X, fit_path=True)
def _more_tags(self):
return {'multioutput': False}
def fit(self, X, y):
"""Fit the model using X, y as training data.
Parameters
----------
X : array-like of shape (n_samples, n_features)
Training data.
y : array-like of shape (n_samples,)
Target values.
Returns
-------
self : object
returns an instance of self.
"""
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, classifier=False)
# As we use cross-validation, the Gram matrix is not precomputed here
Gram = self.precompute
if hasattr(Gram, '__array__'):
warnings.warn('Parameter "precompute" cannot be an array in '
'%s. Automatically switch to "auto" instead.'
% self.__class__.__name__)
Gram = 'auto'
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.split(X, y))
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, _, _, 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.mse_path_ = mse_path
# Now compute the full model
# it will call a lasso internally when self if LassoLarsCV
# as self.method == 'lasso'
self._fit(X, y, max_iter=self.max_iter, alpha=best_alpha,
Xy=None, fit_path=True)
return self
class LassoLarsCV(LarsCV):
"""Cross-validated Lasso, using the LARS algorithm.
See glossary entry for :term:`cross-validation estimator`.
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 : 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).
verbose : bool or int, default=False
Sets the verbosity amount
max_iter : int, default=500
Maximum number of iterations to perform.
normalize : bool, default=True
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``.
precompute : bool or 'auto' , default='auto'
Whether to use a precomputed Gram matrix to speed up
calculations. If set to ``'auto'`` let us decide. The Gram matrix
cannot be passed as argument since we will use only subsets of X.
cv : int, cross-validation generator or an iterable, default=None
Determines the cross-validation splitting strategy.
Possible inputs for cv are:
- None, to use the default 5-fold cross-validation,
- integer, to specify the number of folds.
- :term:`CV splitter`,
- An iterable yielding (train, test) splits as arrays of indices.
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.
.. versionchanged:: 0.22
``cv`` default value if None changed from 3-fold to 5-fold.
max_n_alphas : int, default=1000
The maximum number of points on the path used to compute the
residuals in the cross-validation
n_jobs : int or None, default=None
Number of CPUs to use during the cross validation.
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
for more details.
eps : float, optional
The machine-precision regularization in the computation of the
Cholesky diagonal factors. Increase this for very ill-conditioned
systems. By default, ``np.finfo(np.float).eps`` is used.
copy_X : bool, default=True
If True, X will be copied; else, it may be overwritten.
positive : bool, 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 coefficients 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.
Attributes
----------
coef_ : array-like of shape (n_features,)
parameter vector (w in the formulation formula)
intercept_ : float
independent term in decision function.
coef_path_ : array-like of shape (n_features, n_alphas)
the varying values of the coefficients along the path
alpha_ : float
the estimated regularization parameter alpha
alphas_ : array-like of shape (n_alphas,)
the different values of alpha along the path
cv_alphas_ : array-like of shape (n_cv_alphas,)
all the values of alpha along the path for the different folds
mse_path_ : array-like of 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.
Examples
--------
>>> from sklearn.linear_model import LassoLarsCV
>>> from sklearn.datasets import make_regression
>>> X, y = make_regression(noise=4.0, random_state=0)
>>> reg = LassoLarsCV(cv=5).fit(X, y)
>>> reg.score(X, y)
0.9992...
>>> reg.alpha_
0.0484...
>>> reg.predict(X[:1,])
array([-77.8723...])
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'
def __init__(self, fit_intercept=True, verbose=False, max_iter=500,
normalize=True, precompute='auto', cv=None,
max_n_alphas=1000, n_jobs=None, eps=np.finfo(np.float).eps,
copy_X=True, positive=False):
self.fit_intercept = fit_intercept
self.verbose = verbose
self.max_iter = max_iter
self.normalize = normalize
self.precompute = precompute
self.cv = cv
self.max_n_alphas = max_n_alphas
self.n_jobs = n_jobs
self.eps = eps
self.copy_X = copy_X
self.positive = positive
# XXX : we don't use super().__init__
# to avoid setting n_nonzero_coefs
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'}, default='aic'
The type of criterion to use.
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).
verbose : bool or int, default=False
Sets the verbosity amount
normalize : bool, default=True
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``.
precompute : bool, 'auto' or array-like, default='auto'
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 : int, default=500
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.
By default, ``np.finfo(np.float).eps`` is used
copy_X : bool, default=True
If True, X will be copied; else, it may be overwritten.
positive : bool, 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 coefficients 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.
Attributes
----------
coef_ : array-like of 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-like of shape (n_alphas,)
The value of the information criteria ('aic', 'bic') across all
alphas. The alpha which has the smallest information criterion is
chosen. This value is larger by a factor of ``n_samples`` compared to
Eqns. 2.15 and 2.16 in (Zou et al, 2007).
Examples
--------
>>> from sklearn import linear_model
>>> reg = linear_model.LassoLarsIC(criterion='bic')
>>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1.1111, 0, -1.1111])
LassoLarsIC(criterion='bic')
>>> print(reg.coef_)
[ 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.
https://en.wikipedia.org/wiki/Akaike_information_criterion
https://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
self.fit_path = True
def _more_tags(self):
return {'multioutput': False}
def fit(self, X, y, copy_X=None):
"""Fit the model using X, y as training data.
Parameters
----------
X : array-like of shape (n_samples, n_features)
training data.
y : array-like of shape (n_samples,)
target values. Will be cast to X's dtype if necessary
copy_X : bool, default=None
If provided, this parameter will override the choice
of copy_X made at instance creation.
If ``True``, X will be copied; else, it may be overwritten.
Returns
-------
self : object
returns an instance of self.
"""
if copy_X is None:
copy_X = self.copy_X
X, y = check_X_y(X, y, y_numeric=True)
X, y, Xmean, ymean, Xstd = LinearModel._preprocess_data(
X, y, self.fit_intercept, self.normalize, copy_X)
max_iter = self.max_iter
Gram = self.precompute
alphas_, _, 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)
sigma2 = np.var(y)
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_
eps64 = np.finfo('float64').eps
self.criterion_ = (n_samples * mean_squared_error / (sigma2 + eps64) +
K * df) # Eqns. 2.15--16 in (Zou et al, 2007)
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