1911 lines
73 KiB
Python
1911 lines
73 KiB
Python
"""
|
|
Ridge regression
|
|
"""
|
|
|
|
# Author: Mathieu Blondel <mathieu@mblondel.org>
|
|
# Reuben Fletcher-Costin <reuben.fletchercostin@gmail.com>
|
|
# Fabian Pedregosa <fabian@fseoane.net>
|
|
# Michael Eickenberg <michael.eickenberg@nsup.org>
|
|
# License: BSD 3 clause
|
|
|
|
|
|
from abc import ABCMeta, abstractmethod
|
|
import warnings
|
|
|
|
import numpy as np
|
|
from scipy import linalg
|
|
from scipy import sparse
|
|
from scipy.sparse import linalg as sp_linalg
|
|
|
|
from ._base import LinearClassifierMixin, LinearModel, _rescale_data
|
|
from ._sag import sag_solver
|
|
from ..base import RegressorMixin, MultiOutputMixin, is_classifier
|
|
from ..utils.extmath import safe_sparse_dot
|
|
from ..utils.extmath import row_norms
|
|
from ..utils import check_array
|
|
from ..utils import check_consistent_length
|
|
from ..utils import compute_sample_weight
|
|
from ..utils import column_or_1d
|
|
from ..utils.validation import _check_sample_weight
|
|
from ..utils.validation import _deprecate_positional_args
|
|
from ..preprocessing import LabelBinarizer
|
|
from ..model_selection import GridSearchCV
|
|
from ..metrics import check_scoring
|
|
from ..exceptions import ConvergenceWarning
|
|
from ..utils.sparsefuncs import mean_variance_axis
|
|
|
|
|
|
def _solve_sparse_cg(X, y, alpha, max_iter=None, tol=1e-3, verbose=0,
|
|
X_offset=None, X_scale=None):
|
|
|
|
def _get_rescaled_operator(X):
|
|
|
|
X_offset_scale = X_offset / X_scale
|
|
|
|
def matvec(b):
|
|
return X.dot(b) - b.dot(X_offset_scale)
|
|
|
|
def rmatvec(b):
|
|
return X.T.dot(b) - X_offset_scale * np.sum(b)
|
|
|
|
X1 = sparse.linalg.LinearOperator(shape=X.shape,
|
|
matvec=matvec,
|
|
rmatvec=rmatvec)
|
|
return X1
|
|
|
|
n_samples, n_features = X.shape
|
|
|
|
if X_offset is None or X_scale is None:
|
|
X1 = sp_linalg.aslinearoperator(X)
|
|
else:
|
|
X1 = _get_rescaled_operator(X)
|
|
|
|
coefs = np.empty((y.shape[1], n_features), dtype=X.dtype)
|
|
|
|
if n_features > n_samples:
|
|
def create_mv(curr_alpha):
|
|
def _mv(x):
|
|
return X1.matvec(X1.rmatvec(x)) + curr_alpha * x
|
|
return _mv
|
|
else:
|
|
def create_mv(curr_alpha):
|
|
def _mv(x):
|
|
return X1.rmatvec(X1.matvec(x)) + curr_alpha * x
|
|
return _mv
|
|
|
|
for i in range(y.shape[1]):
|
|
y_column = y[:, i]
|
|
|
|
mv = create_mv(alpha[i])
|
|
if n_features > n_samples:
|
|
# kernel ridge
|
|
# w = X.T * inv(X X^t + alpha*Id) y
|
|
C = sp_linalg.LinearOperator(
|
|
(n_samples, n_samples), matvec=mv, dtype=X.dtype)
|
|
# FIXME atol
|
|
try:
|
|
coef, info = sp_linalg.cg(C, y_column, tol=tol, atol='legacy')
|
|
except TypeError:
|
|
# old scipy
|
|
coef, info = sp_linalg.cg(C, y_column, tol=tol)
|
|
coefs[i] = X1.rmatvec(coef)
|
|
else:
|
|
# linear ridge
|
|
# w = inv(X^t X + alpha*Id) * X.T y
|
|
y_column = X1.rmatvec(y_column)
|
|
C = sp_linalg.LinearOperator(
|
|
(n_features, n_features), matvec=mv, dtype=X.dtype)
|
|
# FIXME atol
|
|
try:
|
|
coefs[i], info = sp_linalg.cg(C, y_column, maxiter=max_iter,
|
|
tol=tol, atol='legacy')
|
|
except TypeError:
|
|
# old scipy
|
|
coefs[i], info = sp_linalg.cg(C, y_column, maxiter=max_iter,
|
|
tol=tol)
|
|
|
|
if info < 0:
|
|
raise ValueError("Failed with error code %d" % info)
|
|
|
|
if max_iter is None and info > 0 and verbose:
|
|
warnings.warn("sparse_cg did not converge after %d iterations." %
|
|
info, ConvergenceWarning)
|
|
|
|
return coefs
|
|
|
|
|
|
def _solve_lsqr(X, y, alpha, max_iter=None, tol=1e-3):
|
|
n_samples, n_features = X.shape
|
|
coefs = np.empty((y.shape[1], n_features), dtype=X.dtype)
|
|
n_iter = np.empty(y.shape[1], dtype=np.int32)
|
|
|
|
# According to the lsqr documentation, alpha = damp^2.
|
|
sqrt_alpha = np.sqrt(alpha)
|
|
|
|
for i in range(y.shape[1]):
|
|
y_column = y[:, i]
|
|
info = sp_linalg.lsqr(X, y_column, damp=sqrt_alpha[i],
|
|
atol=tol, btol=tol, iter_lim=max_iter)
|
|
coefs[i] = info[0]
|
|
n_iter[i] = info[2]
|
|
|
|
return coefs, n_iter
|
|
|
|
|
|
def _solve_cholesky(X, y, alpha):
|
|
# w = inv(X^t X + alpha*Id) * X.T y
|
|
n_features = X.shape[1]
|
|
n_targets = y.shape[1]
|
|
|
|
A = safe_sparse_dot(X.T, X, dense_output=True)
|
|
Xy = safe_sparse_dot(X.T, y, dense_output=True)
|
|
|
|
one_alpha = np.array_equal(alpha, len(alpha) * [alpha[0]])
|
|
|
|
if one_alpha:
|
|
A.flat[::n_features + 1] += alpha[0]
|
|
return linalg.solve(A, Xy, sym_pos=True,
|
|
overwrite_a=True).T
|
|
else:
|
|
coefs = np.empty([n_targets, n_features], dtype=X.dtype)
|
|
for coef, target, current_alpha in zip(coefs, Xy.T, alpha):
|
|
A.flat[::n_features + 1] += current_alpha
|
|
coef[:] = linalg.solve(A, target, sym_pos=True,
|
|
overwrite_a=False).ravel()
|
|
A.flat[::n_features + 1] -= current_alpha
|
|
return coefs
|
|
|
|
|
|
def _solve_cholesky_kernel(K, y, alpha, sample_weight=None, copy=False):
|
|
# dual_coef = inv(X X^t + alpha*Id) y
|
|
n_samples = K.shape[0]
|
|
n_targets = y.shape[1]
|
|
|
|
if copy:
|
|
K = K.copy()
|
|
|
|
alpha = np.atleast_1d(alpha)
|
|
one_alpha = (alpha == alpha[0]).all()
|
|
has_sw = isinstance(sample_weight, np.ndarray) \
|
|
or sample_weight not in [1.0, None]
|
|
|
|
if has_sw:
|
|
# Unlike other solvers, we need to support sample_weight directly
|
|
# because K might be a pre-computed kernel.
|
|
sw = np.sqrt(np.atleast_1d(sample_weight))
|
|
y = y * sw[:, np.newaxis]
|
|
K *= np.outer(sw, sw)
|
|
|
|
if one_alpha:
|
|
# Only one penalty, we can solve multi-target problems in one time.
|
|
K.flat[::n_samples + 1] += alpha[0]
|
|
|
|
try:
|
|
# Note: we must use overwrite_a=False in order to be able to
|
|
# use the fall-back solution below in case a LinAlgError
|
|
# is raised
|
|
dual_coef = linalg.solve(K, y, sym_pos=True,
|
|
overwrite_a=False)
|
|
except np.linalg.LinAlgError:
|
|
warnings.warn("Singular matrix in solving dual problem. Using "
|
|
"least-squares solution instead.")
|
|
dual_coef = linalg.lstsq(K, y)[0]
|
|
|
|
# K is expensive to compute and store in memory so change it back in
|
|
# case it was user-given.
|
|
K.flat[::n_samples + 1] -= alpha[0]
|
|
|
|
if has_sw:
|
|
dual_coef *= sw[:, np.newaxis]
|
|
|
|
return dual_coef
|
|
else:
|
|
# One penalty per target. We need to solve each target separately.
|
|
dual_coefs = np.empty([n_targets, n_samples], K.dtype)
|
|
|
|
for dual_coef, target, current_alpha in zip(dual_coefs, y.T, alpha):
|
|
K.flat[::n_samples + 1] += current_alpha
|
|
|
|
dual_coef[:] = linalg.solve(K, target, sym_pos=True,
|
|
overwrite_a=False).ravel()
|
|
|
|
K.flat[::n_samples + 1] -= current_alpha
|
|
|
|
if has_sw:
|
|
dual_coefs *= sw[np.newaxis, :]
|
|
|
|
return dual_coefs.T
|
|
|
|
|
|
def _solve_svd(X, y, alpha):
|
|
U, s, Vt = linalg.svd(X, full_matrices=False)
|
|
idx = s > 1e-15 # same default value as scipy.linalg.pinv
|
|
s_nnz = s[idx][:, np.newaxis]
|
|
UTy = np.dot(U.T, y)
|
|
d = np.zeros((s.size, alpha.size), dtype=X.dtype)
|
|
d[idx] = s_nnz / (s_nnz ** 2 + alpha)
|
|
d_UT_y = d * UTy
|
|
return np.dot(Vt.T, d_UT_y).T
|
|
|
|
|
|
def _get_valid_accept_sparse(is_X_sparse, solver):
|
|
if is_X_sparse and solver in ['auto', 'sag', 'saga']:
|
|
return 'csr'
|
|
else:
|
|
return ['csr', 'csc', 'coo']
|
|
|
|
|
|
@_deprecate_positional_args
|
|
def ridge_regression(X, y, alpha, *, sample_weight=None, solver='auto',
|
|
max_iter=None, tol=1e-3, verbose=0, random_state=None,
|
|
return_n_iter=False, return_intercept=False,
|
|
check_input=True):
|
|
"""Solve the ridge equation by the method of normal equations.
|
|
|
|
Read more in the :ref:`User Guide <ridge_regression>`.
|
|
|
|
Parameters
|
|
----------
|
|
X : {ndarray, sparse matrix, LinearOperator} of shape \
|
|
(n_samples, n_features)
|
|
Training data
|
|
|
|
y : ndarray of shape (n_samples,) or (n_samples, n_targets)
|
|
Target values
|
|
|
|
alpha : float or array-like of shape (n_targets,)
|
|
Regularization strength; must be a positive float. Regularization
|
|
improves the conditioning of the problem and reduces the variance of
|
|
the estimates. Larger values specify stronger regularization.
|
|
Alpha corresponds to ``1 / (2C)`` in other linear models such as
|
|
:class:`~sklearn.linear_model.LogisticRegression` or
|
|
:class:`sklearn.svm.LinearSVC`. If an array is passed, penalties are
|
|
assumed to be specific to the targets. Hence they must correspond in
|
|
number.
|
|
|
|
sample_weight : float or array-like of shape (n_samples,), default=None
|
|
Individual weights for each sample. If given a float, every sample
|
|
will have the same weight. If sample_weight is not None and
|
|
solver='auto', the solver will be set to 'cholesky'.
|
|
|
|
.. versionadded:: 0.17
|
|
|
|
solver : {'auto', 'svd', 'cholesky', 'lsqr', 'sparse_cg', 'sag', 'saga'}, \
|
|
default='auto'
|
|
Solver to use in the computational routines:
|
|
|
|
- 'auto' chooses the solver automatically based on the type of data.
|
|
|
|
- 'svd' uses a Singular Value Decomposition of X to compute the Ridge
|
|
coefficients. More stable for singular matrices than 'cholesky'.
|
|
|
|
- 'cholesky' uses the standard scipy.linalg.solve function to
|
|
obtain a closed-form solution via a Cholesky decomposition of
|
|
dot(X.T, X)
|
|
|
|
- 'sparse_cg' uses the conjugate gradient solver as found in
|
|
scipy.sparse.linalg.cg. As an iterative algorithm, this solver is
|
|
more appropriate than 'cholesky' for large-scale data
|
|
(possibility to set `tol` and `max_iter`).
|
|
|
|
- 'lsqr' uses the dedicated regularized least-squares routine
|
|
scipy.sparse.linalg.lsqr. It is the fastest and uses an iterative
|
|
procedure.
|
|
|
|
- 'sag' uses a Stochastic Average Gradient descent, and 'saga' uses
|
|
its improved, unbiased version named SAGA. Both methods also use an
|
|
iterative procedure, and are often faster than other solvers when
|
|
both n_samples and n_features are large. Note that 'sag' and
|
|
'saga' fast convergence is only guaranteed on features with
|
|
approximately the same scale. You can preprocess the data with a
|
|
scaler from sklearn.preprocessing.
|
|
|
|
|
|
All last five solvers support both dense and sparse data. However, only
|
|
'sag' and 'sparse_cg' supports sparse input when `fit_intercept` is
|
|
True.
|
|
|
|
.. versionadded:: 0.17
|
|
Stochastic Average Gradient descent solver.
|
|
.. versionadded:: 0.19
|
|
SAGA solver.
|
|
|
|
max_iter : int, default=None
|
|
Maximum number of iterations for conjugate gradient solver.
|
|
For the 'sparse_cg' and 'lsqr' solvers, the default value is determined
|
|
by scipy.sparse.linalg. For 'sag' and saga solver, the default value is
|
|
1000.
|
|
|
|
tol : float, default=1e-3
|
|
Precision of the solution.
|
|
|
|
verbose : int, default=0
|
|
Verbosity level. Setting verbose > 0 will display additional
|
|
information depending on the solver used.
|
|
|
|
random_state : int, RandomState instance, default=None
|
|
Used when ``solver`` == 'sag' or 'saga' to shuffle the data.
|
|
See :term:`Glossary <random_state>` for details.
|
|
|
|
return_n_iter : bool, default=False
|
|
If True, the method also returns `n_iter`, the actual number of
|
|
iteration performed by the solver.
|
|
|
|
.. versionadded:: 0.17
|
|
|
|
return_intercept : bool, default=False
|
|
If True and if X is sparse, the method also returns the intercept,
|
|
and the solver is automatically changed to 'sag'. This is only a
|
|
temporary fix for fitting the intercept with sparse data. For dense
|
|
data, use sklearn.linear_model._preprocess_data before your regression.
|
|
|
|
.. versionadded:: 0.17
|
|
|
|
check_input : bool, default=True
|
|
If False, the input arrays X and y will not be checked.
|
|
|
|
.. versionadded:: 0.21
|
|
|
|
Returns
|
|
-------
|
|
coef : ndarray of shape (n_features,) or (n_targets, n_features)
|
|
Weight vector(s).
|
|
|
|
n_iter : int, optional
|
|
The actual number of iteration performed by the solver.
|
|
Only returned if `return_n_iter` is True.
|
|
|
|
intercept : float or ndarray of shape (n_targets,)
|
|
The intercept of the model. Only returned if `return_intercept`
|
|
is True and if X is a scipy sparse array.
|
|
|
|
Notes
|
|
-----
|
|
This function won't compute the intercept.
|
|
"""
|
|
return _ridge_regression(X, y, alpha,
|
|
sample_weight=sample_weight,
|
|
solver=solver,
|
|
max_iter=max_iter,
|
|
tol=tol,
|
|
verbose=verbose,
|
|
random_state=random_state,
|
|
return_n_iter=return_n_iter,
|
|
return_intercept=return_intercept,
|
|
X_scale=None,
|
|
X_offset=None,
|
|
check_input=check_input)
|
|
|
|
|
|
def _ridge_regression(X, y, alpha, sample_weight=None, solver='auto',
|
|
max_iter=None, tol=1e-3, verbose=0, random_state=None,
|
|
return_n_iter=False, return_intercept=False,
|
|
X_scale=None, X_offset=None, check_input=True):
|
|
|
|
has_sw = sample_weight is not None
|
|
|
|
if solver == 'auto':
|
|
if return_intercept:
|
|
# only sag supports fitting intercept directly
|
|
solver = "sag"
|
|
elif not sparse.issparse(X):
|
|
solver = "cholesky"
|
|
else:
|
|
solver = "sparse_cg"
|
|
|
|
if solver not in ('sparse_cg', 'cholesky', 'svd', 'lsqr', 'sag', 'saga'):
|
|
raise ValueError("Known solvers are 'sparse_cg', 'cholesky', 'svd'"
|
|
" 'lsqr', 'sag' or 'saga'. Got %s." % solver)
|
|
|
|
if return_intercept and solver != 'sag':
|
|
raise ValueError("In Ridge, only 'sag' solver can directly fit the "
|
|
"intercept. Please change solver to 'sag' or set "
|
|
"return_intercept=False.")
|
|
|
|
if check_input:
|
|
_dtype = [np.float64, np.float32]
|
|
_accept_sparse = _get_valid_accept_sparse(sparse.issparse(X), solver)
|
|
X = check_array(X, accept_sparse=_accept_sparse, dtype=_dtype,
|
|
order="C")
|
|
y = check_array(y, dtype=X.dtype, ensure_2d=False, order=None)
|
|
check_consistent_length(X, y)
|
|
|
|
n_samples, n_features = X.shape
|
|
|
|
if y.ndim > 2:
|
|
raise ValueError("Target y has the wrong shape %s" % str(y.shape))
|
|
|
|
ravel = False
|
|
if y.ndim == 1:
|
|
y = y.reshape(-1, 1)
|
|
ravel = True
|
|
|
|
n_samples_, n_targets = y.shape
|
|
|
|
if n_samples != n_samples_:
|
|
raise ValueError("Number of samples in X and y does not correspond:"
|
|
" %d != %d" % (n_samples, n_samples_))
|
|
|
|
if has_sw:
|
|
sample_weight = _check_sample_weight(sample_weight, X, dtype=X.dtype)
|
|
|
|
if solver not in ['sag', 'saga']:
|
|
# SAG supports sample_weight directly. For other solvers,
|
|
# we implement sample_weight via a simple rescaling.
|
|
X, y = _rescale_data(X, y, sample_weight)
|
|
|
|
# There should be either 1 or n_targets penalties
|
|
alpha = np.asarray(alpha, dtype=X.dtype).ravel()
|
|
if alpha.size not in [1, n_targets]:
|
|
raise ValueError("Number of targets and number of penalties "
|
|
"do not correspond: %d != %d"
|
|
% (alpha.size, n_targets))
|
|
|
|
if alpha.size == 1 and n_targets > 1:
|
|
alpha = np.repeat(alpha, n_targets)
|
|
|
|
n_iter = None
|
|
if solver == 'sparse_cg':
|
|
coef = _solve_sparse_cg(X, y, alpha,
|
|
max_iter=max_iter,
|
|
tol=tol,
|
|
verbose=verbose,
|
|
X_offset=X_offset,
|
|
X_scale=X_scale)
|
|
|
|
elif solver == 'lsqr':
|
|
coef, n_iter = _solve_lsqr(X, y, alpha, max_iter, tol)
|
|
|
|
elif solver == 'cholesky':
|
|
if n_features > n_samples:
|
|
K = safe_sparse_dot(X, X.T, dense_output=True)
|
|
try:
|
|
dual_coef = _solve_cholesky_kernel(K, y, alpha)
|
|
|
|
coef = safe_sparse_dot(X.T, dual_coef, dense_output=True).T
|
|
except linalg.LinAlgError:
|
|
# use SVD solver if matrix is singular
|
|
solver = 'svd'
|
|
else:
|
|
try:
|
|
coef = _solve_cholesky(X, y, alpha)
|
|
except linalg.LinAlgError:
|
|
# use SVD solver if matrix is singular
|
|
solver = 'svd'
|
|
|
|
elif solver in ['sag', 'saga']:
|
|
# precompute max_squared_sum for all targets
|
|
max_squared_sum = row_norms(X, squared=True).max()
|
|
|
|
coef = np.empty((y.shape[1], n_features), dtype=X.dtype)
|
|
n_iter = np.empty(y.shape[1], dtype=np.int32)
|
|
intercept = np.zeros((y.shape[1], ), dtype=X.dtype)
|
|
for i, (alpha_i, target) in enumerate(zip(alpha, y.T)):
|
|
init = {'coef': np.zeros((n_features + int(return_intercept), 1),
|
|
dtype=X.dtype)}
|
|
coef_, n_iter_, _ = sag_solver(
|
|
X, target.ravel(), sample_weight, 'squared', alpha_i, 0,
|
|
max_iter, tol, verbose, random_state, False, max_squared_sum,
|
|
init, is_saga=solver == 'saga')
|
|
if return_intercept:
|
|
coef[i] = coef_[:-1]
|
|
intercept[i] = coef_[-1]
|
|
else:
|
|
coef[i] = coef_
|
|
n_iter[i] = n_iter_
|
|
|
|
if intercept.shape[0] == 1:
|
|
intercept = intercept[0]
|
|
coef = np.asarray(coef)
|
|
|
|
if solver == 'svd':
|
|
if sparse.issparse(X):
|
|
raise TypeError('SVD solver does not support sparse'
|
|
' inputs currently')
|
|
coef = _solve_svd(X, y, alpha)
|
|
|
|
if ravel:
|
|
# When y was passed as a 1d-array, we flatten the coefficients.
|
|
coef = coef.ravel()
|
|
|
|
if return_n_iter and return_intercept:
|
|
return coef, n_iter, intercept
|
|
elif return_intercept:
|
|
return coef, intercept
|
|
elif return_n_iter:
|
|
return coef, n_iter
|
|
else:
|
|
return coef
|
|
|
|
|
|
class _BaseRidge(LinearModel, metaclass=ABCMeta):
|
|
@abstractmethod
|
|
@_deprecate_positional_args
|
|
def __init__(self, alpha=1.0, *, fit_intercept=True, normalize=False,
|
|
copy_X=True, max_iter=None, tol=1e-3, solver="auto",
|
|
random_state=None):
|
|
self.alpha = alpha
|
|
self.fit_intercept = fit_intercept
|
|
self.normalize = normalize
|
|
self.copy_X = copy_X
|
|
self.max_iter = max_iter
|
|
self.tol = tol
|
|
self.solver = solver
|
|
self.random_state = random_state
|
|
|
|
def fit(self, X, y, sample_weight=None):
|
|
|
|
# all other solvers work at both float precision levels
|
|
_dtype = [np.float64, np.float32]
|
|
_accept_sparse = _get_valid_accept_sparse(sparse.issparse(X),
|
|
self.solver)
|
|
X, y = self._validate_data(X, y,
|
|
accept_sparse=_accept_sparse,
|
|
dtype=_dtype,
|
|
multi_output=True, y_numeric=True)
|
|
if sparse.issparse(X) and self.fit_intercept:
|
|
if self.solver not in ['auto', 'sparse_cg', 'sag']:
|
|
raise ValueError(
|
|
"solver='{}' does not support fitting the intercept "
|
|
"on sparse data. Please set the solver to 'auto' or "
|
|
"'sparse_cg', 'sag', or set `fit_intercept=False`"
|
|
.format(self.solver))
|
|
if (self.solver == 'sag' and self.max_iter is None and
|
|
self.tol > 1e-4):
|
|
warnings.warn(
|
|
'"sag" solver requires many iterations to fit '
|
|
'an intercept with sparse inputs. Either set the '
|
|
'solver to "auto" or "sparse_cg", or set a low '
|
|
'"tol" and a high "max_iter" (especially if inputs are '
|
|
'not standardized).')
|
|
solver = 'sag'
|
|
else:
|
|
solver = 'sparse_cg'
|
|
else:
|
|
solver = self.solver
|
|
|
|
if sample_weight is not None:
|
|
sample_weight = _check_sample_weight(sample_weight, X,
|
|
dtype=X.dtype)
|
|
|
|
# when X is sparse we only remove offset from y
|
|
X, y, X_offset, y_offset, X_scale = self._preprocess_data(
|
|
X, y, self.fit_intercept, self.normalize, self.copy_X,
|
|
sample_weight=sample_weight, return_mean=True)
|
|
|
|
if solver == 'sag' and sparse.issparse(X) and self.fit_intercept:
|
|
self.coef_, self.n_iter_, self.intercept_ = _ridge_regression(
|
|
X, y, alpha=self.alpha, sample_weight=sample_weight,
|
|
max_iter=self.max_iter, tol=self.tol, solver='sag',
|
|
random_state=self.random_state, return_n_iter=True,
|
|
return_intercept=True, check_input=False)
|
|
# add the offset which was subtracted by _preprocess_data
|
|
self.intercept_ += y_offset
|
|
|
|
else:
|
|
if sparse.issparse(X) and self.fit_intercept:
|
|
# required to fit intercept with sparse_cg solver
|
|
params = {'X_offset': X_offset, 'X_scale': X_scale}
|
|
else:
|
|
# for dense matrices or when intercept is set to 0
|
|
params = {}
|
|
|
|
self.coef_, self.n_iter_ = _ridge_regression(
|
|
X, y, alpha=self.alpha, sample_weight=sample_weight,
|
|
max_iter=self.max_iter, tol=self.tol, solver=solver,
|
|
random_state=self.random_state, return_n_iter=True,
|
|
return_intercept=False, check_input=False, **params)
|
|
self._set_intercept(X_offset, y_offset, X_scale)
|
|
|
|
return self
|
|
|
|
|
|
class Ridge(MultiOutputMixin, RegressorMixin, _BaseRidge):
|
|
"""Linear least squares with l2 regularization.
|
|
|
|
Minimizes the objective function::
|
|
|
|
||y - Xw||^2_2 + alpha * ||w||^2_2
|
|
|
|
This model solves a regression model where the loss function is
|
|
the linear least squares function and regularization is given by
|
|
the l2-norm. Also known as Ridge Regression or Tikhonov regularization.
|
|
This estimator has built-in support for multi-variate regression
|
|
(i.e., when y is a 2d-array of shape (n_samples, n_targets)).
|
|
|
|
Read more in the :ref:`User Guide <ridge_regression>`.
|
|
|
|
Parameters
|
|
----------
|
|
alpha : {float, ndarray of shape (n_targets,)}, default=1.0
|
|
Regularization strength; must be a positive float. Regularization
|
|
improves the conditioning of the problem and reduces the variance of
|
|
the estimates. Larger values specify stronger regularization.
|
|
Alpha corresponds to ``1 / (2C)`` in other linear models such as
|
|
:class:`~sklearn.linear_model.LogisticRegression` or
|
|
:class:`sklearn.svm.LinearSVC`. If an array is passed, penalties are
|
|
assumed to be specific to the targets. Hence they must correspond in
|
|
number.
|
|
|
|
fit_intercept : bool, default=True
|
|
Whether to fit the intercept for this model. If set
|
|
to false, no intercept will be used in calculations
|
|
(i.e. ``X`` and ``y`` are expected to be centered).
|
|
|
|
normalize : bool, default=False
|
|
This parameter is ignored when ``fit_intercept`` is set to False.
|
|
If True, the regressors X will be normalized before regression by
|
|
subtracting the mean and dividing by the l2-norm.
|
|
If you wish to standardize, please use
|
|
:class:`sklearn.preprocessing.StandardScaler` before calling ``fit``
|
|
on an estimator with ``normalize=False``.
|
|
|
|
copy_X : bool, default=True
|
|
If True, X will be copied; else, it may be overwritten.
|
|
|
|
max_iter : int, default=None
|
|
Maximum number of iterations for conjugate gradient solver.
|
|
For 'sparse_cg' and 'lsqr' solvers, the default value is determined
|
|
by scipy.sparse.linalg. For 'sag' solver, the default value is 1000.
|
|
|
|
tol : float, default=1e-3
|
|
Precision of the solution.
|
|
|
|
solver : {'auto', 'svd', 'cholesky', 'lsqr', 'sparse_cg', 'sag', 'saga'}, \
|
|
default='auto'
|
|
Solver to use in the computational routines:
|
|
|
|
- 'auto' chooses the solver automatically based on the type of data.
|
|
|
|
- 'svd' uses a Singular Value Decomposition of X to compute the Ridge
|
|
coefficients. More stable for singular matrices than 'cholesky'.
|
|
|
|
- 'cholesky' uses the standard scipy.linalg.solve function to
|
|
obtain a closed-form solution.
|
|
|
|
- 'sparse_cg' uses the conjugate gradient solver as found in
|
|
scipy.sparse.linalg.cg. As an iterative algorithm, this solver is
|
|
more appropriate than 'cholesky' for large-scale data
|
|
(possibility to set `tol` and `max_iter`).
|
|
|
|
- 'lsqr' uses the dedicated regularized least-squares routine
|
|
scipy.sparse.linalg.lsqr. It is the fastest and uses an iterative
|
|
procedure.
|
|
|
|
- 'sag' uses a Stochastic Average Gradient descent, and 'saga' uses
|
|
its improved, unbiased version named SAGA. Both methods also use an
|
|
iterative procedure, and are often faster than other solvers when
|
|
both n_samples and n_features are large. Note that 'sag' and
|
|
'saga' fast convergence is only guaranteed on features with
|
|
approximately the same scale. You can preprocess the data with a
|
|
scaler from sklearn.preprocessing.
|
|
|
|
All last five solvers support both dense and sparse data. However, only
|
|
'sag' and 'sparse_cg' supports sparse input when `fit_intercept` is
|
|
True.
|
|
|
|
.. versionadded:: 0.17
|
|
Stochastic Average Gradient descent solver.
|
|
.. versionadded:: 0.19
|
|
SAGA solver.
|
|
|
|
random_state : int, RandomState instance, default=None
|
|
Used when ``solver`` == 'sag' or 'saga' to shuffle the data.
|
|
See :term:`Glossary <random_state>` for details.
|
|
|
|
.. versionadded:: 0.17
|
|
`random_state` to support Stochastic Average Gradient.
|
|
|
|
Attributes
|
|
----------
|
|
coef_ : ndarray of shape (n_features,) or (n_targets, n_features)
|
|
Weight vector(s).
|
|
|
|
intercept_ : float or ndarray of shape (n_targets,)
|
|
Independent term in decision function. Set to 0.0 if
|
|
``fit_intercept = False``.
|
|
|
|
n_iter_ : None or ndarray of shape (n_targets,)
|
|
Actual number of iterations for each target. Available only for
|
|
sag and lsqr solvers. Other solvers will return None.
|
|
|
|
.. versionadded:: 0.17
|
|
|
|
See also
|
|
--------
|
|
RidgeClassifier : Ridge classifier
|
|
RidgeCV : Ridge regression with built-in cross validation
|
|
:class:`sklearn.kernel_ridge.KernelRidge` : Kernel ridge regression
|
|
combines ridge regression with the kernel trick
|
|
|
|
Examples
|
|
--------
|
|
>>> from sklearn.linear_model import Ridge
|
|
>>> import numpy as np
|
|
>>> n_samples, n_features = 10, 5
|
|
>>> rng = np.random.RandomState(0)
|
|
>>> y = rng.randn(n_samples)
|
|
>>> X = rng.randn(n_samples, n_features)
|
|
>>> clf = Ridge(alpha=1.0)
|
|
>>> clf.fit(X, y)
|
|
Ridge()
|
|
"""
|
|
@_deprecate_positional_args
|
|
def __init__(self, alpha=1.0, *, fit_intercept=True, normalize=False,
|
|
copy_X=True, max_iter=None, tol=1e-3, solver="auto",
|
|
random_state=None):
|
|
super().__init__(
|
|
alpha=alpha, fit_intercept=fit_intercept,
|
|
normalize=normalize, copy_X=copy_X,
|
|
max_iter=max_iter, tol=tol, solver=solver,
|
|
random_state=random_state)
|
|
|
|
def fit(self, X, y, sample_weight=None):
|
|
"""Fit Ridge regression model.
|
|
|
|
Parameters
|
|
----------
|
|
X : {ndarray, sparse matrix} of shape (n_samples, n_features)
|
|
Training data
|
|
|
|
y : ndarray of shape (n_samples,) or (n_samples, n_targets)
|
|
Target values
|
|
|
|
sample_weight : float or ndarray of shape (n_samples,), default=None
|
|
Individual weights for each sample. If given a float, every sample
|
|
will have the same weight.
|
|
|
|
Returns
|
|
-------
|
|
self : returns an instance of self.
|
|
"""
|
|
return super().fit(X, y, sample_weight=sample_weight)
|
|
|
|
|
|
class RidgeClassifier(LinearClassifierMixin, _BaseRidge):
|
|
"""Classifier using Ridge regression.
|
|
|
|
This classifier first converts the target values into ``{-1, 1}`` and
|
|
then treats the problem as a regression task (multi-output regression in
|
|
the multiclass case).
|
|
|
|
Read more in the :ref:`User Guide <ridge_regression>`.
|
|
|
|
Parameters
|
|
----------
|
|
alpha : float, default=1.0
|
|
Regularization strength; must be a positive float. Regularization
|
|
improves the conditioning of the problem and reduces the variance of
|
|
the estimates. Larger values specify stronger regularization.
|
|
Alpha corresponds to ``1 / (2C)`` in other linear models such as
|
|
:class:`~sklearn.linear_model.LogisticRegression` or
|
|
:class:`sklearn.svm.LinearSVC`.
|
|
|
|
fit_intercept : bool, default=True
|
|
Whether to calculate the intercept for this model. If set to false, no
|
|
intercept will be used in calculations (e.g. data is expected to be
|
|
already centered).
|
|
|
|
normalize : bool, default=False
|
|
This parameter is ignored when ``fit_intercept`` is set to False.
|
|
If True, the regressors X will be normalized before regression by
|
|
subtracting the mean and dividing by the l2-norm.
|
|
If you wish to standardize, please use
|
|
:class:`sklearn.preprocessing.StandardScaler` before calling ``fit``
|
|
on an estimator with ``normalize=False``.
|
|
|
|
copy_X : bool, default=True
|
|
If True, X will be copied; else, it may be overwritten.
|
|
|
|
max_iter : int, default=None
|
|
Maximum number of iterations for conjugate gradient solver.
|
|
The default value is determined by scipy.sparse.linalg.
|
|
|
|
tol : float, default=1e-3
|
|
Precision of the solution.
|
|
|
|
class_weight : dict or 'balanced', default=None
|
|
Weights associated with classes in the form ``{class_label: weight}``.
|
|
If not given, all classes are supposed to have weight one.
|
|
|
|
The "balanced" mode uses the values of y to automatically adjust
|
|
weights inversely proportional to class frequencies in the input data
|
|
as ``n_samples / (n_classes * np.bincount(y))``.
|
|
|
|
solver : {'auto', 'svd', 'cholesky', 'lsqr', 'sparse_cg', 'sag', 'saga'}, \
|
|
default='auto'
|
|
Solver to use in the computational routines:
|
|
|
|
- 'auto' chooses the solver automatically based on the type of data.
|
|
|
|
- 'svd' uses a Singular Value Decomposition of X to compute the Ridge
|
|
coefficients. More stable for singular matrices than 'cholesky'.
|
|
|
|
- 'cholesky' uses the standard scipy.linalg.solve function to
|
|
obtain a closed-form solution.
|
|
|
|
- 'sparse_cg' uses the conjugate gradient solver as found in
|
|
scipy.sparse.linalg.cg. As an iterative algorithm, this solver is
|
|
more appropriate than 'cholesky' for large-scale data
|
|
(possibility to set `tol` and `max_iter`).
|
|
|
|
- 'lsqr' uses the dedicated regularized least-squares routine
|
|
scipy.sparse.linalg.lsqr. It is the fastest and uses an iterative
|
|
procedure.
|
|
|
|
- 'sag' uses a Stochastic Average Gradient descent, and 'saga' uses
|
|
its unbiased and more flexible version named SAGA. Both methods
|
|
use an iterative procedure, and are often faster than other solvers
|
|
when both n_samples and n_features are large. Note that 'sag' and
|
|
'saga' fast convergence is only guaranteed on features with
|
|
approximately the same scale. You can preprocess the data with a
|
|
scaler from sklearn.preprocessing.
|
|
|
|
.. versionadded:: 0.17
|
|
Stochastic Average Gradient descent solver.
|
|
.. versionadded:: 0.19
|
|
SAGA solver.
|
|
|
|
random_state : int, RandomState instance, default=None
|
|
Used when ``solver`` == 'sag' or 'saga' to shuffle the data.
|
|
See :term:`Glossary <random_state>` for details.
|
|
|
|
Attributes
|
|
----------
|
|
coef_ : ndarray of shape (1, n_features) or (n_classes, n_features)
|
|
Coefficient of the features in the decision function.
|
|
|
|
``coef_`` is of shape (1, n_features) when the given problem is binary.
|
|
|
|
intercept_ : float or ndarray of shape (n_targets,)
|
|
Independent term in decision function. Set to 0.0 if
|
|
``fit_intercept = False``.
|
|
|
|
n_iter_ : None or ndarray of shape (n_targets,)
|
|
Actual number of iterations for each target. Available only for
|
|
sag and lsqr solvers. Other solvers will return None.
|
|
|
|
classes_ : ndarray of shape (n_classes,)
|
|
The classes labels.
|
|
|
|
See Also
|
|
--------
|
|
Ridge : Ridge regression.
|
|
RidgeClassifierCV : Ridge classifier with built-in cross validation.
|
|
|
|
Notes
|
|
-----
|
|
For multi-class classification, n_class classifiers are trained in
|
|
a one-versus-all approach. Concretely, this is implemented by taking
|
|
advantage of the multi-variate response support in Ridge.
|
|
|
|
Examples
|
|
--------
|
|
>>> from sklearn.datasets import load_breast_cancer
|
|
>>> from sklearn.linear_model import RidgeClassifier
|
|
>>> X, y = load_breast_cancer(return_X_y=True)
|
|
>>> clf = RidgeClassifier().fit(X, y)
|
|
>>> clf.score(X, y)
|
|
0.9595...
|
|
"""
|
|
@_deprecate_positional_args
|
|
def __init__(self, alpha=1.0, *, fit_intercept=True, normalize=False,
|
|
copy_X=True, max_iter=None, tol=1e-3, class_weight=None,
|
|
solver="auto", random_state=None):
|
|
super().__init__(
|
|
alpha=alpha, fit_intercept=fit_intercept, normalize=normalize,
|
|
copy_X=copy_X, max_iter=max_iter, tol=tol, solver=solver,
|
|
random_state=random_state)
|
|
self.class_weight = class_weight
|
|
|
|
def fit(self, X, y, sample_weight=None):
|
|
"""Fit Ridge classifier model.
|
|
|
|
Parameters
|
|
----------
|
|
X : {ndarray, sparse matrix} of shape (n_samples, n_features)
|
|
Training data.
|
|
|
|
y : ndarray of shape (n_samples,)
|
|
Target values.
|
|
|
|
sample_weight : float or ndarray of shape (n_samples,), default=None
|
|
Individual weights for each sample. If given a float, every sample
|
|
will have the same weight.
|
|
|
|
.. versionadded:: 0.17
|
|
*sample_weight* support to Classifier.
|
|
|
|
Returns
|
|
-------
|
|
self : object
|
|
Instance of the estimator.
|
|
"""
|
|
_accept_sparse = _get_valid_accept_sparse(sparse.issparse(X),
|
|
self.solver)
|
|
X, y = self._validate_data(X, y, accept_sparse=_accept_sparse,
|
|
multi_output=True, y_numeric=False)
|
|
sample_weight = _check_sample_weight(sample_weight, X, dtype=X.dtype)
|
|
|
|
self._label_binarizer = LabelBinarizer(pos_label=1, neg_label=-1)
|
|
Y = self._label_binarizer.fit_transform(y)
|
|
if not self._label_binarizer.y_type_.startswith('multilabel'):
|
|
y = column_or_1d(y, warn=True)
|
|
else:
|
|
# we don't (yet) support multi-label classification in Ridge
|
|
raise ValueError(
|
|
"%s doesn't support multi-label classification" % (
|
|
self.__class__.__name__))
|
|
|
|
if self.class_weight:
|
|
# modify the sample weights with the corresponding class weight
|
|
sample_weight = (sample_weight *
|
|
compute_sample_weight(self.class_weight, y))
|
|
|
|
super().fit(X, Y, sample_weight=sample_weight)
|
|
return self
|
|
|
|
@property
|
|
def classes_(self):
|
|
return self._label_binarizer.classes_
|
|
|
|
|
|
def _check_gcv_mode(X, gcv_mode):
|
|
possible_gcv_modes = [None, 'auto', 'svd', 'eigen']
|
|
if gcv_mode not in possible_gcv_modes:
|
|
raise ValueError(
|
|
"Unknown value for 'gcv_mode'. "
|
|
"Got {} instead of one of {}" .format(
|
|
gcv_mode, possible_gcv_modes))
|
|
if gcv_mode in ['eigen', 'svd']:
|
|
return gcv_mode
|
|
# if X has more rows than columns, use decomposition of X^T.X,
|
|
# otherwise X.X^T
|
|
if X.shape[0] > X.shape[1]:
|
|
return 'svd'
|
|
return 'eigen'
|
|
|
|
|
|
def _find_smallest_angle(query, vectors):
|
|
"""Find the column of vectors that is most aligned with the query.
|
|
|
|
Both query and the columns of vectors must have their l2 norm equal to 1.
|
|
|
|
Parameters
|
|
----------
|
|
query : ndarray of shape (n_samples,)
|
|
Normalized query vector.
|
|
|
|
vectors : ndarray of shape (n_samples, n_features)
|
|
Vectors to which we compare query, as columns. Must be normalized.
|
|
"""
|
|
abs_cosine = np.abs(query.dot(vectors))
|
|
index = np.argmax(abs_cosine)
|
|
return index
|
|
|
|
|
|
class _X_CenterStackOp(sparse.linalg.LinearOperator):
|
|
"""Behaves as centered and scaled X with an added intercept column.
|
|
|
|
This operator behaves as
|
|
np.hstack([X - sqrt_sw[:, None] * X_mean, sqrt_sw[:, None]])
|
|
"""
|
|
|
|
def __init__(self, X, X_mean, sqrt_sw):
|
|
n_samples, n_features = X.shape
|
|
super().__init__(X.dtype, (n_samples, n_features + 1))
|
|
self.X = X
|
|
self.X_mean = X_mean
|
|
self.sqrt_sw = sqrt_sw
|
|
|
|
def _matvec(self, v):
|
|
v = v.ravel()
|
|
return safe_sparse_dot(
|
|
self.X, v[:-1], dense_output=True
|
|
) - self.sqrt_sw * self.X_mean.dot(v[:-1]) + v[-1] * self.sqrt_sw
|
|
|
|
def _matmat(self, v):
|
|
return (
|
|
safe_sparse_dot(self.X, v[:-1], dense_output=True) -
|
|
self.sqrt_sw[:, None] * self.X_mean.dot(v[:-1]) + v[-1] *
|
|
self.sqrt_sw[:, None])
|
|
|
|
def _transpose(self):
|
|
return _XT_CenterStackOp(self.X, self.X_mean, self.sqrt_sw)
|
|
|
|
|
|
class _XT_CenterStackOp(sparse.linalg.LinearOperator):
|
|
"""Behaves as transposed centered and scaled X with an intercept column.
|
|
|
|
This operator behaves as
|
|
np.hstack([X - sqrt_sw[:, None] * X_mean, sqrt_sw[:, None]]).T
|
|
"""
|
|
|
|
def __init__(self, X, X_mean, sqrt_sw):
|
|
n_samples, n_features = X.shape
|
|
super().__init__(X.dtype, (n_features + 1, n_samples))
|
|
self.X = X
|
|
self.X_mean = X_mean
|
|
self.sqrt_sw = sqrt_sw
|
|
|
|
def _matvec(self, v):
|
|
v = v.ravel()
|
|
n_features = self.shape[0]
|
|
res = np.empty(n_features, dtype=self.X.dtype)
|
|
res[:-1] = (
|
|
safe_sparse_dot(self.X.T, v, dense_output=True) -
|
|
(self.X_mean * self.sqrt_sw.dot(v))
|
|
)
|
|
res[-1] = np.dot(v, self.sqrt_sw)
|
|
return res
|
|
|
|
def _matmat(self, v):
|
|
n_features = self.shape[0]
|
|
res = np.empty((n_features, v.shape[1]), dtype=self.X.dtype)
|
|
res[:-1] = (
|
|
safe_sparse_dot(self.X.T, v, dense_output=True) -
|
|
self.X_mean[:, None] * self.sqrt_sw.dot(v)
|
|
)
|
|
res[-1] = np.dot(self.sqrt_sw, v)
|
|
return res
|
|
|
|
|
|
class _IdentityRegressor:
|
|
"""Fake regressor which will directly output the prediction."""
|
|
|
|
def decision_function(self, y_predict):
|
|
return y_predict
|
|
|
|
def predict(self, y_predict):
|
|
return y_predict
|
|
|
|
|
|
class _IdentityClassifier(LinearClassifierMixin):
|
|
"""Fake classifier which will directly output the prediction.
|
|
|
|
We inherit from LinearClassifierMixin to get the proper shape for the
|
|
output `y`.
|
|
"""
|
|
def __init__(self, classes):
|
|
self.classes_ = classes
|
|
|
|
def decision_function(self, y_predict):
|
|
return y_predict
|
|
|
|
|
|
class _RidgeGCV(LinearModel):
|
|
"""Ridge regression with built-in Generalized Cross-Validation.
|
|
|
|
It allows efficient Leave-One-Out cross-validation.
|
|
|
|
This class is not intended to be used directly. Use RidgeCV instead.
|
|
|
|
Notes
|
|
-----
|
|
|
|
We want to solve (K + alpha*Id)c = y,
|
|
where K = X X^T is the kernel matrix.
|
|
|
|
Let G = (K + alpha*Id).
|
|
|
|
Dual solution: c = G^-1y
|
|
Primal solution: w = X^T c
|
|
|
|
Compute eigendecomposition K = Q V Q^T.
|
|
Then G^-1 = Q (V + alpha*Id)^-1 Q^T,
|
|
where (V + alpha*Id) is diagonal.
|
|
It is thus inexpensive to inverse for many alphas.
|
|
|
|
Let loov be the vector of prediction values for each example
|
|
when the model was fitted with all examples but this example.
|
|
|
|
loov = (KG^-1Y - diag(KG^-1)Y) / diag(I-KG^-1)
|
|
|
|
Let looe be the vector of prediction errors for each example
|
|
when the model was fitted with all examples but this example.
|
|
|
|
looe = y - loov = c / diag(G^-1)
|
|
|
|
The best score (negative mean squared error or user-provided scoring) is
|
|
stored in the `best_score_` attribute, and the selected hyperparameter in
|
|
`alpha_`.
|
|
|
|
References
|
|
----------
|
|
http://cbcl.mit.edu/publications/ps/MIT-CSAIL-TR-2007-025.pdf
|
|
https://www.mit.edu/~9.520/spring07/Classes/rlsslides.pdf
|
|
"""
|
|
@_deprecate_positional_args
|
|
def __init__(self, alphas=(0.1, 1.0, 10.0), *,
|
|
fit_intercept=True, normalize=False,
|
|
scoring=None, copy_X=True,
|
|
gcv_mode=None, store_cv_values=False,
|
|
is_clf=False):
|
|
self.alphas = np.asarray(alphas)
|
|
self.fit_intercept = fit_intercept
|
|
self.normalize = normalize
|
|
self.scoring = scoring
|
|
self.copy_X = copy_X
|
|
self.gcv_mode = gcv_mode
|
|
self.store_cv_values = store_cv_values
|
|
self.is_clf = is_clf
|
|
|
|
@staticmethod
|
|
def _decomp_diag(v_prime, Q):
|
|
# compute diagonal of the matrix: dot(Q, dot(diag(v_prime), Q^T))
|
|
return (v_prime * Q ** 2).sum(axis=-1)
|
|
|
|
@staticmethod
|
|
def _diag_dot(D, B):
|
|
# compute dot(diag(D), B)
|
|
if len(B.shape) > 1:
|
|
# handle case where B is > 1-d
|
|
D = D[(slice(None), ) + (np.newaxis, ) * (len(B.shape) - 1)]
|
|
return D * B
|
|
|
|
def _compute_gram(self, X, sqrt_sw):
|
|
"""Computes the Gram matrix XX^T with possible centering.
|
|
|
|
Parameters
|
|
----------
|
|
X : {ndarray, sparse matrix} of shape (n_samples, n_features)
|
|
The preprocessed design matrix.
|
|
|
|
sqrt_sw : ndarray of shape (n_samples,)
|
|
square roots of sample weights
|
|
|
|
Returns
|
|
-------
|
|
gram : ndarray of shape (n_samples, n_samples)
|
|
The Gram matrix.
|
|
X_mean : ndarray of shape (n_feature,)
|
|
The weighted mean of ``X`` for each feature.
|
|
|
|
Notes
|
|
-----
|
|
When X is dense the centering has been done in preprocessing
|
|
so the mean is 0 and we just compute XX^T.
|
|
|
|
When X is sparse it has not been centered in preprocessing, but it has
|
|
been scaled by sqrt(sample weights).
|
|
|
|
When self.fit_intercept is False no centering is done.
|
|
|
|
The centered X is never actually computed because centering would break
|
|
the sparsity of X.
|
|
"""
|
|
center = self.fit_intercept and sparse.issparse(X)
|
|
if not center:
|
|
# in this case centering has been done in preprocessing
|
|
# or we are not fitting an intercept.
|
|
X_mean = np.zeros(X.shape[1], dtype=X.dtype)
|
|
return safe_sparse_dot(X, X.T, dense_output=True), X_mean
|
|
# X is sparse
|
|
n_samples = X.shape[0]
|
|
sample_weight_matrix = sparse.dia_matrix(
|
|
(sqrt_sw, 0), shape=(n_samples, n_samples))
|
|
X_weighted = sample_weight_matrix.dot(X)
|
|
X_mean, _ = mean_variance_axis(X_weighted, axis=0)
|
|
X_mean *= n_samples / sqrt_sw.dot(sqrt_sw)
|
|
X_mX = sqrt_sw[:, None] * safe_sparse_dot(
|
|
X_mean, X.T, dense_output=True)
|
|
X_mX_m = np.outer(sqrt_sw, sqrt_sw) * np.dot(X_mean, X_mean)
|
|
return (safe_sparse_dot(X, X.T, dense_output=True) + X_mX_m
|
|
- X_mX - X_mX.T, X_mean)
|
|
|
|
def _compute_covariance(self, X, sqrt_sw):
|
|
"""Computes covariance matrix X^TX with possible centering.
|
|
|
|
Parameters
|
|
----------
|
|
X : sparse matrix of shape (n_samples, n_features)
|
|
The preprocessed design matrix.
|
|
|
|
sqrt_sw : ndarray of shape (n_samples,)
|
|
square roots of sample weights
|
|
|
|
Returns
|
|
-------
|
|
covariance : ndarray of shape (n_features, n_features)
|
|
The covariance matrix.
|
|
X_mean : ndarray of shape (n_feature,)
|
|
The weighted mean of ``X`` for each feature.
|
|
|
|
Notes
|
|
-----
|
|
Since X is sparse it has not been centered in preprocessing, but it has
|
|
been scaled by sqrt(sample weights).
|
|
|
|
When self.fit_intercept is False no centering is done.
|
|
|
|
The centered X is never actually computed because centering would break
|
|
the sparsity of X.
|
|
"""
|
|
if not self.fit_intercept:
|
|
# in this case centering has been done in preprocessing
|
|
# or we are not fitting an intercept.
|
|
X_mean = np.zeros(X.shape[1], dtype=X.dtype)
|
|
return safe_sparse_dot(X.T, X, dense_output=True), X_mean
|
|
# this function only gets called for sparse X
|
|
n_samples = X.shape[0]
|
|
sample_weight_matrix = sparse.dia_matrix(
|
|
(sqrt_sw, 0), shape=(n_samples, n_samples))
|
|
X_weighted = sample_weight_matrix.dot(X)
|
|
X_mean, _ = mean_variance_axis(X_weighted, axis=0)
|
|
X_mean = X_mean * n_samples / sqrt_sw.dot(sqrt_sw)
|
|
weight_sum = sqrt_sw.dot(sqrt_sw)
|
|
return (safe_sparse_dot(X.T, X, dense_output=True) -
|
|
weight_sum * np.outer(X_mean, X_mean),
|
|
X_mean)
|
|
|
|
def _sparse_multidot_diag(self, X, A, X_mean, sqrt_sw):
|
|
"""Compute the diagonal of (X - X_mean).dot(A).dot((X - X_mean).T)
|
|
without explicitely centering X nor computing X.dot(A)
|
|
when X is sparse.
|
|
|
|
Parameters
|
|
----------
|
|
X : sparse matrix of shape (n_samples, n_features)
|
|
|
|
A : ndarray of shape (n_features, n_features)
|
|
|
|
X_mean : ndarray of shape (n_features,)
|
|
|
|
sqrt_sw : ndarray of shape (n_features,)
|
|
square roots of sample weights
|
|
|
|
Returns
|
|
-------
|
|
diag : np.ndarray, shape (n_samples,)
|
|
The computed diagonal.
|
|
"""
|
|
intercept_col = scale = sqrt_sw
|
|
batch_size = X.shape[1]
|
|
diag = np.empty(X.shape[0], dtype=X.dtype)
|
|
for start in range(0, X.shape[0], batch_size):
|
|
batch = slice(start, min(X.shape[0], start + batch_size), 1)
|
|
X_batch = np.empty(
|
|
(X[batch].shape[0], X.shape[1] + self.fit_intercept),
|
|
dtype=X.dtype
|
|
)
|
|
if self.fit_intercept:
|
|
X_batch[:, :-1] = X[batch].A - X_mean * scale[batch][:, None]
|
|
X_batch[:, -1] = intercept_col[batch]
|
|
else:
|
|
X_batch = X[batch].A
|
|
diag[batch] = (X_batch.dot(A) * X_batch).sum(axis=1)
|
|
return diag
|
|
|
|
def _eigen_decompose_gram(self, X, y, sqrt_sw):
|
|
"""Eigendecomposition of X.X^T, used when n_samples <= n_features."""
|
|
# if X is dense it has already been centered in preprocessing
|
|
K, X_mean = self._compute_gram(X, sqrt_sw)
|
|
if self.fit_intercept:
|
|
# to emulate centering X with sample weights,
|
|
# ie removing the weighted average, we add a column
|
|
# containing the square roots of the sample weights.
|
|
# by centering, it is orthogonal to the other columns
|
|
K += np.outer(sqrt_sw, sqrt_sw)
|
|
eigvals, Q = linalg.eigh(K)
|
|
QT_y = np.dot(Q.T, y)
|
|
return X_mean, eigvals, Q, QT_y
|
|
|
|
def _solve_eigen_gram(self, alpha, y, sqrt_sw, X_mean, eigvals, Q, QT_y):
|
|
"""Compute dual coefficients and diagonal of G^-1.
|
|
|
|
Used when we have a decomposition of X.X^T (n_samples <= n_features).
|
|
"""
|
|
w = 1. / (eigvals + alpha)
|
|
if self.fit_intercept:
|
|
# the vector containing the square roots of the sample weights (1
|
|
# when no sample weights) is the eigenvector of XX^T which
|
|
# corresponds to the intercept; we cancel the regularization on
|
|
# this dimension. the corresponding eigenvalue is
|
|
# sum(sample_weight).
|
|
normalized_sw = sqrt_sw / np.linalg.norm(sqrt_sw)
|
|
intercept_dim = _find_smallest_angle(normalized_sw, Q)
|
|
w[intercept_dim] = 0 # cancel regularization for the intercept
|
|
|
|
c = np.dot(Q, self._diag_dot(w, QT_y))
|
|
G_inverse_diag = self._decomp_diag(w, Q)
|
|
# handle case where y is 2-d
|
|
if len(y.shape) != 1:
|
|
G_inverse_diag = G_inverse_diag[:, np.newaxis]
|
|
return G_inverse_diag, c
|
|
|
|
def _eigen_decompose_covariance(self, X, y, sqrt_sw):
|
|
"""Eigendecomposition of X^T.X, used when n_samples > n_features
|
|
and X is sparse.
|
|
"""
|
|
n_samples, n_features = X.shape
|
|
cov = np.empty((n_features + 1, n_features + 1), dtype=X.dtype)
|
|
cov[:-1, :-1], X_mean = self._compute_covariance(X, sqrt_sw)
|
|
if not self.fit_intercept:
|
|
cov = cov[:-1, :-1]
|
|
# to emulate centering X with sample weights,
|
|
# ie removing the weighted average, we add a column
|
|
# containing the square roots of the sample weights.
|
|
# by centering, it is orthogonal to the other columns
|
|
# when all samples have the same weight we add a column of 1
|
|
else:
|
|
cov[-1] = 0
|
|
cov[:, -1] = 0
|
|
cov[-1, -1] = sqrt_sw.dot(sqrt_sw)
|
|
nullspace_dim = max(0, n_features - n_samples)
|
|
eigvals, V = linalg.eigh(cov)
|
|
# remove eigenvalues and vectors in the null space of X^T.X
|
|
eigvals = eigvals[nullspace_dim:]
|
|
V = V[:, nullspace_dim:]
|
|
return X_mean, eigvals, V, X
|
|
|
|
def _solve_eigen_covariance_no_intercept(
|
|
self, alpha, y, sqrt_sw, X_mean, eigvals, V, X):
|
|
"""Compute dual coefficients and diagonal of G^-1.
|
|
|
|
Used when we have a decomposition of X^T.X
|
|
(n_samples > n_features and X is sparse), and not fitting an intercept.
|
|
"""
|
|
w = 1 / (eigvals + alpha)
|
|
A = (V * w).dot(V.T)
|
|
AXy = A.dot(safe_sparse_dot(X.T, y, dense_output=True))
|
|
y_hat = safe_sparse_dot(X, AXy, dense_output=True)
|
|
hat_diag = self._sparse_multidot_diag(X, A, X_mean, sqrt_sw)
|
|
if len(y.shape) != 1:
|
|
# handle case where y is 2-d
|
|
hat_diag = hat_diag[:, np.newaxis]
|
|
return (1 - hat_diag) / alpha, (y - y_hat) / alpha
|
|
|
|
def _solve_eigen_covariance_intercept(
|
|
self, alpha, y, sqrt_sw, X_mean, eigvals, V, X):
|
|
"""Compute dual coefficients and diagonal of G^-1.
|
|
|
|
Used when we have a decomposition of X^T.X
|
|
(n_samples > n_features and X is sparse),
|
|
and we are fitting an intercept.
|
|
"""
|
|
# the vector [0, 0, ..., 0, 1]
|
|
# is the eigenvector of X^TX which
|
|
# corresponds to the intercept; we cancel the regularization on
|
|
# this dimension. the corresponding eigenvalue is
|
|
# sum(sample_weight), e.g. n when uniform sample weights.
|
|
intercept_sv = np.zeros(V.shape[0])
|
|
intercept_sv[-1] = 1
|
|
intercept_dim = _find_smallest_angle(intercept_sv, V)
|
|
w = 1 / (eigvals + alpha)
|
|
w[intercept_dim] = 1 / eigvals[intercept_dim]
|
|
A = (V * w).dot(V.T)
|
|
# add a column to X containing the square roots of sample weights
|
|
X_op = _X_CenterStackOp(X, X_mean, sqrt_sw)
|
|
AXy = A.dot(X_op.T.dot(y))
|
|
y_hat = X_op.dot(AXy)
|
|
hat_diag = self._sparse_multidot_diag(X, A, X_mean, sqrt_sw)
|
|
# return (1 - hat_diag), (y - y_hat)
|
|
if len(y.shape) != 1:
|
|
# handle case where y is 2-d
|
|
hat_diag = hat_diag[:, np.newaxis]
|
|
return (1 - hat_diag) / alpha, (y - y_hat) / alpha
|
|
|
|
def _solve_eigen_covariance(
|
|
self, alpha, y, sqrt_sw, X_mean, eigvals, V, X):
|
|
"""Compute dual coefficients and diagonal of G^-1.
|
|
|
|
Used when we have a decomposition of X^T.X
|
|
(n_samples > n_features and X is sparse).
|
|
"""
|
|
if self.fit_intercept:
|
|
return self._solve_eigen_covariance_intercept(
|
|
alpha, y, sqrt_sw, X_mean, eigvals, V, X)
|
|
return self._solve_eigen_covariance_no_intercept(
|
|
alpha, y, sqrt_sw, X_mean, eigvals, V, X)
|
|
|
|
def _svd_decompose_design_matrix(self, X, y, sqrt_sw):
|
|
# X already centered
|
|
X_mean = np.zeros(X.shape[1], dtype=X.dtype)
|
|
if self.fit_intercept:
|
|
# to emulate fit_intercept=True situation, add a column
|
|
# containing the square roots of the sample weights
|
|
# by centering, the other columns are orthogonal to that one
|
|
intercept_column = sqrt_sw[:, None]
|
|
X = np.hstack((X, intercept_column))
|
|
U, singvals, _ = linalg.svd(X, full_matrices=0)
|
|
singvals_sq = singvals ** 2
|
|
UT_y = np.dot(U.T, y)
|
|
return X_mean, singvals_sq, U, UT_y
|
|
|
|
def _solve_svd_design_matrix(
|
|
self, alpha, y, sqrt_sw, X_mean, singvals_sq, U, UT_y):
|
|
"""Compute dual coefficients and diagonal of G^-1.
|
|
|
|
Used when we have an SVD decomposition of X
|
|
(n_samples > n_features and X is dense).
|
|
"""
|
|
w = ((singvals_sq + alpha) ** -1) - (alpha ** -1)
|
|
if self.fit_intercept:
|
|
# detect intercept column
|
|
normalized_sw = sqrt_sw / np.linalg.norm(sqrt_sw)
|
|
intercept_dim = _find_smallest_angle(normalized_sw, U)
|
|
# cancel the regularization for the intercept
|
|
w[intercept_dim] = - (alpha ** -1)
|
|
c = np.dot(U, self._diag_dot(w, UT_y)) + (alpha ** -1) * y
|
|
G_inverse_diag = self._decomp_diag(w, U) + (alpha ** -1)
|
|
if len(y.shape) != 1:
|
|
# handle case where y is 2-d
|
|
G_inverse_diag = G_inverse_diag[:, np.newaxis]
|
|
return G_inverse_diag, c
|
|
|
|
def fit(self, X, y, sample_weight=None):
|
|
"""Fit Ridge regression model with gcv.
|
|
|
|
Parameters
|
|
----------
|
|
X : {ndarray, sparse matrix} of shape (n_samples, n_features)
|
|
Training data. Will be cast to float64 if necessary.
|
|
|
|
y : ndarray of shape (n_samples,) or (n_samples, n_targets)
|
|
Target values. Will be cast to float64 if necessary.
|
|
|
|
sample_weight : float or ndarray of shape (n_samples,), default=None
|
|
Individual weights for each sample. If given a float, every sample
|
|
will have the same weight.
|
|
|
|
Returns
|
|
-------
|
|
self : object
|
|
"""
|
|
X, y = self._validate_data(X, y, accept_sparse=['csr', 'csc', 'coo'],
|
|
dtype=[np.float64],
|
|
multi_output=True, y_numeric=True)
|
|
|
|
if sample_weight is not None:
|
|
sample_weight = _check_sample_weight(sample_weight, X,
|
|
dtype=X.dtype)
|
|
|
|
if np.any(self.alphas <= 0):
|
|
raise ValueError(
|
|
"alphas must be positive. Got {} containing some "
|
|
"negative or null value instead.".format(self.alphas))
|
|
|
|
X, y, X_offset, y_offset, X_scale = LinearModel._preprocess_data(
|
|
X, y, self.fit_intercept, self.normalize, self.copy_X,
|
|
sample_weight=sample_weight)
|
|
|
|
gcv_mode = _check_gcv_mode(X, self.gcv_mode)
|
|
|
|
if gcv_mode == 'eigen':
|
|
decompose = self._eigen_decompose_gram
|
|
solve = self._solve_eigen_gram
|
|
elif gcv_mode == 'svd':
|
|
if sparse.issparse(X):
|
|
decompose = self._eigen_decompose_covariance
|
|
solve = self._solve_eigen_covariance
|
|
else:
|
|
decompose = self._svd_decompose_design_matrix
|
|
solve = self._solve_svd_design_matrix
|
|
|
|
n_samples = X.shape[0]
|
|
|
|
if sample_weight is not None:
|
|
X, y = _rescale_data(X, y, sample_weight)
|
|
sqrt_sw = np.sqrt(sample_weight)
|
|
else:
|
|
sqrt_sw = np.ones(n_samples, dtype=X.dtype)
|
|
|
|
X_mean, *decomposition = decompose(X, y, sqrt_sw)
|
|
|
|
scorer = check_scoring(self, scoring=self.scoring, allow_none=True)
|
|
error = scorer is None
|
|
|
|
n_y = 1 if len(y.shape) == 1 else y.shape[1]
|
|
|
|
if self.store_cv_values:
|
|
self.cv_values_ = np.empty(
|
|
(n_samples * n_y, len(self.alphas)), dtype=X.dtype)
|
|
|
|
best_coef, best_score, best_alpha = None, None, None
|
|
|
|
for i, alpha in enumerate(self.alphas):
|
|
G_inverse_diag, c = solve(
|
|
float(alpha), y, sqrt_sw, X_mean, *decomposition)
|
|
if error:
|
|
squared_errors = (c / G_inverse_diag) ** 2
|
|
alpha_score = -squared_errors.mean()
|
|
if self.store_cv_values:
|
|
self.cv_values_[:, i] = squared_errors.ravel()
|
|
else:
|
|
predictions = y - (c / G_inverse_diag)
|
|
if self.store_cv_values:
|
|
self.cv_values_[:, i] = predictions.ravel()
|
|
|
|
if self.is_clf:
|
|
identity_estimator = _IdentityClassifier(
|
|
classes=np.arange(n_y)
|
|
)
|
|
predictions_, y_ = predictions, y.argmax(axis=1)
|
|
else:
|
|
identity_estimator = _IdentityRegressor()
|
|
predictions_, y_ = predictions.ravel(), y.ravel()
|
|
|
|
alpha_score = scorer(identity_estimator, predictions_, y_)
|
|
|
|
if (best_score is None) or (alpha_score > best_score):
|
|
best_coef, best_score, best_alpha = c, alpha_score, alpha
|
|
|
|
self.alpha_ = best_alpha
|
|
self.best_score_ = best_score
|
|
self.dual_coef_ = best_coef
|
|
self.coef_ = safe_sparse_dot(self.dual_coef_.T, X)
|
|
|
|
X_offset += X_mean * X_scale
|
|
self._set_intercept(X_offset, y_offset, X_scale)
|
|
|
|
if self.store_cv_values:
|
|
if len(y.shape) == 1:
|
|
cv_values_shape = n_samples, len(self.alphas)
|
|
else:
|
|
cv_values_shape = n_samples, n_y, len(self.alphas)
|
|
self.cv_values_ = self.cv_values_.reshape(cv_values_shape)
|
|
|
|
return self
|
|
|
|
|
|
class _BaseRidgeCV(LinearModel):
|
|
@_deprecate_positional_args
|
|
def __init__(self, alphas=(0.1, 1.0, 10.0), *,
|
|
fit_intercept=True, normalize=False, scoring=None,
|
|
cv=None, gcv_mode=None,
|
|
store_cv_values=False):
|
|
self.alphas = np.asarray(alphas)
|
|
self.fit_intercept = fit_intercept
|
|
self.normalize = normalize
|
|
self.scoring = scoring
|
|
self.cv = cv
|
|
self.gcv_mode = gcv_mode
|
|
self.store_cv_values = store_cv_values
|
|
|
|
def fit(self, X, y, sample_weight=None):
|
|
"""Fit Ridge regression model with cv.
|
|
|
|
Parameters
|
|
----------
|
|
X : ndarray of shape (n_samples, n_features)
|
|
Training data. If using GCV, will be cast to float64
|
|
if necessary.
|
|
|
|
y : ndarray of shape (n_samples,) or (n_samples, n_targets)
|
|
Target values. Will be cast to X's dtype if necessary.
|
|
|
|
sample_weight : float or ndarray of shape (n_samples,), default=None
|
|
Individual weights for each sample. If given a float, every sample
|
|
will have the same weight.
|
|
|
|
Returns
|
|
-------
|
|
self : object
|
|
|
|
Notes
|
|
-----
|
|
When sample_weight is provided, the selected hyperparameter may depend
|
|
on whether we use generalized cross-validation (cv=None or cv='auto')
|
|
or another form of cross-validation, because only generalized
|
|
cross-validation takes the sample weights into account when computing
|
|
the validation score.
|
|
"""
|
|
cv = self.cv
|
|
if cv is None:
|
|
estimator = _RidgeGCV(self.alphas,
|
|
fit_intercept=self.fit_intercept,
|
|
normalize=self.normalize,
|
|
scoring=self.scoring,
|
|
gcv_mode=self.gcv_mode,
|
|
store_cv_values=self.store_cv_values,
|
|
is_clf=is_classifier(self))
|
|
estimator.fit(X, y, sample_weight=sample_weight)
|
|
self.alpha_ = estimator.alpha_
|
|
self.best_score_ = estimator.best_score_
|
|
if self.store_cv_values:
|
|
self.cv_values_ = estimator.cv_values_
|
|
else:
|
|
if self.store_cv_values:
|
|
raise ValueError("cv!=None and store_cv_values=True "
|
|
" are incompatible")
|
|
parameters = {'alpha': self.alphas}
|
|
solver = 'sparse_cg' if sparse.issparse(X) else 'auto'
|
|
model = RidgeClassifier if is_classifier(self) else Ridge
|
|
gs = GridSearchCV(model(fit_intercept=self.fit_intercept,
|
|
normalize=self.normalize,
|
|
solver=solver),
|
|
parameters, cv=cv, scoring=self.scoring)
|
|
gs.fit(X, y, sample_weight=sample_weight)
|
|
estimator = gs.best_estimator_
|
|
self.alpha_ = gs.best_estimator_.alpha
|
|
self.best_score_ = gs.best_score_
|
|
|
|
self.coef_ = estimator.coef_
|
|
self.intercept_ = estimator.intercept_
|
|
self.n_features_in_ = estimator.n_features_in_
|
|
|
|
return self
|
|
|
|
|
|
class RidgeCV(MultiOutputMixin, RegressorMixin, _BaseRidgeCV):
|
|
"""Ridge regression with built-in cross-validation.
|
|
|
|
See glossary entry for :term:`cross-validation estimator`.
|
|
|
|
By default, it performs Generalized Cross-Validation, which is a form of
|
|
efficient Leave-One-Out cross-validation.
|
|
|
|
Read more in the :ref:`User Guide <ridge_regression>`.
|
|
|
|
Parameters
|
|
----------
|
|
alphas : ndarray of shape (n_alphas,), default=(0.1, 1.0, 10.0)
|
|
Array of alpha values to try.
|
|
Regularization strength; must be a positive float. Regularization
|
|
improves the conditioning of the problem and reduces the variance of
|
|
the estimates. Larger values specify stronger regularization.
|
|
Alpha corresponds to ``1 / (2C)`` in other linear models such as
|
|
:class:`~sklearn.linear_model.LogisticRegression` or
|
|
:class:`sklearn.svm.LinearSVC`.
|
|
If using generalized cross-validation, alphas must be positive.
|
|
|
|
fit_intercept : bool, default=True
|
|
Whether to calculate the intercept for this model. If set
|
|
to false, no intercept will be used in calculations
|
|
(i.e. data is expected to be centered).
|
|
|
|
normalize : bool, default=False
|
|
This parameter is ignored when ``fit_intercept`` is set to False.
|
|
If True, the regressors X will be normalized before regression by
|
|
subtracting the mean and dividing by the l2-norm.
|
|
If you wish to standardize, please use
|
|
:class:`sklearn.preprocessing.StandardScaler` before calling ``fit``
|
|
on an estimator with ``normalize=False``.
|
|
|
|
scoring : string, callable, default=None
|
|
A string (see model evaluation documentation) or
|
|
a scorer callable object / function with signature
|
|
``scorer(estimator, X, y)``.
|
|
If None, the negative mean squared error if cv is 'auto' or None
|
|
(i.e. when using generalized cross-validation), and r2 score otherwise.
|
|
|
|
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 efficient Leave-One-Out cross-validation
|
|
(also known as Generalized 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, if ``y`` is binary or multiclass,
|
|
:class:`sklearn.model_selection.StratifiedKFold` is used, else,
|
|
:class:`sklearn.model_selection.KFold` is used.
|
|
|
|
Refer :ref:`User Guide <cross_validation>` for the various
|
|
cross-validation strategies that can be used here.
|
|
|
|
gcv_mode : {'auto', 'svd', eigen'}, default='auto'
|
|
Flag indicating which strategy to use when performing
|
|
Generalized Cross-Validation. Options are::
|
|
|
|
'auto' : use 'svd' if n_samples > n_features, otherwise use 'eigen'
|
|
'svd' : force use of singular value decomposition of X when X is
|
|
dense, eigenvalue decomposition of X^T.X when X is sparse.
|
|
'eigen' : force computation via eigendecomposition of X.X^T
|
|
|
|
The 'auto' mode is the default and is intended to pick the cheaper
|
|
option of the two depending on the shape of the training data.
|
|
|
|
store_cv_values : bool, default=False
|
|
Flag indicating if the cross-validation values corresponding to
|
|
each alpha should be stored in the ``cv_values_`` attribute (see
|
|
below). This flag is only compatible with ``cv=None`` (i.e. using
|
|
Generalized Cross-Validation).
|
|
|
|
Attributes
|
|
----------
|
|
cv_values_ : ndarray of shape (n_samples, n_alphas) or \
|
|
shape (n_samples, n_targets, n_alphas), optional
|
|
Cross-validation values for each alpha (only available if \
|
|
``store_cv_values=True`` and ``cv=None``). After ``fit()`` has been \
|
|
called, this attribute will contain the mean squared errors \
|
|
(by default) or the values of the ``{loss,score}_func`` function \
|
|
(if provided in the constructor).
|
|
|
|
coef_ : ndarray of shape (n_features) or (n_targets, n_features)
|
|
Weight vector(s).
|
|
|
|
intercept_ : float or ndarray of shape (n_targets,)
|
|
Independent term in decision function. Set to 0.0 if
|
|
``fit_intercept = False``.
|
|
|
|
alpha_ : float
|
|
Estimated regularization parameter.
|
|
|
|
best_score_ : float
|
|
Score of base estimator with best alpha.
|
|
|
|
Examples
|
|
--------
|
|
>>> from sklearn.datasets import load_diabetes
|
|
>>> from sklearn.linear_model import RidgeCV
|
|
>>> X, y = load_diabetes(return_X_y=True)
|
|
>>> clf = RidgeCV(alphas=[1e-3, 1e-2, 1e-1, 1]).fit(X, y)
|
|
>>> clf.score(X, y)
|
|
0.5166...
|
|
|
|
See also
|
|
--------
|
|
Ridge : Ridge regression
|
|
RidgeClassifier : Ridge classifier
|
|
RidgeClassifierCV : Ridge classifier with built-in cross validation
|
|
"""
|
|
|
|
|
|
class RidgeClassifierCV(LinearClassifierMixin, _BaseRidgeCV):
|
|
"""Ridge classifier with built-in cross-validation.
|
|
|
|
See glossary entry for :term:`cross-validation estimator`.
|
|
|
|
By default, it performs Generalized Cross-Validation, which is a form of
|
|
efficient Leave-One-Out cross-validation. Currently, only the n_features >
|
|
n_samples case is handled efficiently.
|
|
|
|
Read more in the :ref:`User Guide <ridge_regression>`.
|
|
|
|
Parameters
|
|
----------
|
|
alphas : ndarray of shape (n_alphas,), default=(0.1, 1.0, 10.0)
|
|
Array of alpha values to try.
|
|
Regularization strength; must be a positive float. Regularization
|
|
improves the conditioning of the problem and reduces the variance of
|
|
the estimates. Larger values specify stronger regularization.
|
|
Alpha corresponds to ``1 / (2C)`` in other linear models such as
|
|
:class:`~sklearn.linear_model.LogisticRegression` or
|
|
:class:`sklearn.svm.LinearSVC`.
|
|
|
|
fit_intercept : bool, default=True
|
|
Whether to calculate the intercept for this model. If set
|
|
to false, no intercept will be used in calculations
|
|
(i.e. data is expected to be centered).
|
|
|
|
normalize : bool, default=False
|
|
This parameter is ignored when ``fit_intercept`` is set to False.
|
|
If True, the regressors X will be normalized before regression by
|
|
subtracting the mean and dividing by the l2-norm.
|
|
If you wish to standardize, please use
|
|
:class:`sklearn.preprocessing.StandardScaler` before calling ``fit``
|
|
on an estimator with ``normalize=False``.
|
|
|
|
scoring : string, callable, default=None
|
|
A string (see model evaluation documentation) or
|
|
a scorer callable object / function with signature
|
|
``scorer(estimator, X, y)``.
|
|
|
|
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 efficient Leave-One-Out cross-validation
|
|
- integer, to specify the number of folds.
|
|
- :term:`CV splitter`,
|
|
- An iterable yielding (train, test) splits as arrays of indices.
|
|
|
|
Refer :ref:`User Guide <cross_validation>` for the various
|
|
cross-validation strategies that can be used here.
|
|
|
|
class_weight : dict or 'balanced', default=None
|
|
Weights associated with classes in the form ``{class_label: weight}``.
|
|
If not given, all classes are supposed to have weight one.
|
|
|
|
The "balanced" mode uses the values of y to automatically adjust
|
|
weights inversely proportional to class frequencies in the input data
|
|
as ``n_samples / (n_classes * np.bincount(y))``
|
|
|
|
store_cv_values : bool, default=False
|
|
Flag indicating if the cross-validation values corresponding to
|
|
each alpha should be stored in the ``cv_values_`` attribute (see
|
|
below). This flag is only compatible with ``cv=None`` (i.e. using
|
|
Generalized Cross-Validation).
|
|
|
|
Attributes
|
|
----------
|
|
cv_values_ : ndarray of shape (n_samples, n_targets, n_alphas), optional
|
|
Cross-validation values for each alpha (if ``store_cv_values=True`` and
|
|
``cv=None``). After ``fit()`` has been called, this attribute will
|
|
contain the mean squared errors (by default) or the values of the
|
|
``{loss,score}_func`` function (if provided in the constructor). This
|
|
attribute exists only when ``store_cv_values`` is True.
|
|
|
|
coef_ : ndarray of shape (1, n_features) or (n_targets, n_features)
|
|
Coefficient of the features in the decision function.
|
|
|
|
``coef_`` is of shape (1, n_features) when the given problem is binary.
|
|
|
|
intercept_ : float or ndarray of shape (n_targets,)
|
|
Independent term in decision function. Set to 0.0 if
|
|
``fit_intercept = False``.
|
|
|
|
alpha_ : float
|
|
Estimated regularization parameter.
|
|
|
|
best_score_ : float
|
|
Score of base estimator with best alpha.
|
|
|
|
classes_ : ndarray of shape (n_classes,)
|
|
The classes labels.
|
|
|
|
Examples
|
|
--------
|
|
>>> from sklearn.datasets import load_breast_cancer
|
|
>>> from sklearn.linear_model import RidgeClassifierCV
|
|
>>> X, y = load_breast_cancer(return_X_y=True)
|
|
>>> clf = RidgeClassifierCV(alphas=[1e-3, 1e-2, 1e-1, 1]).fit(X, y)
|
|
>>> clf.score(X, y)
|
|
0.9630...
|
|
|
|
See also
|
|
--------
|
|
Ridge : Ridge regression
|
|
RidgeClassifier : Ridge classifier
|
|
RidgeCV : Ridge regression with built-in cross validation
|
|
|
|
Notes
|
|
-----
|
|
For multi-class classification, n_class classifiers are trained in
|
|
a one-versus-all approach. Concretely, this is implemented by taking
|
|
advantage of the multi-variate response support in Ridge.
|
|
"""
|
|
@_deprecate_positional_args
|
|
def __init__(self, alphas=(0.1, 1.0, 10.0), *, fit_intercept=True,
|
|
normalize=False, scoring=None, cv=None, class_weight=None,
|
|
store_cv_values=False):
|
|
super().__init__(
|
|
alphas=alphas, fit_intercept=fit_intercept, normalize=normalize,
|
|
scoring=scoring, cv=cv, store_cv_values=store_cv_values)
|
|
self.class_weight = class_weight
|
|
|
|
def fit(self, X, y, sample_weight=None):
|
|
"""Fit Ridge classifier with cv.
|
|
|
|
Parameters
|
|
----------
|
|
X : ndarray of shape (n_samples, n_features)
|
|
Training vectors, where n_samples is the number of samples
|
|
and n_features is the number of features. When using GCV,
|
|
will be cast to float64 if necessary.
|
|
|
|
y : ndarray of shape (n_samples,)
|
|
Target values. Will be cast to X's dtype if necessary.
|
|
|
|
sample_weight : float or ndarray of shape (n_samples,), default=None
|
|
Individual weights for each sample. If given a float, every sample
|
|
will have the same weight.
|
|
|
|
Returns
|
|
-------
|
|
self : object
|
|
"""
|
|
X, y = self._validate_data(X, y, accept_sparse=['csr', 'csc', 'coo'],
|
|
multi_output=True, y_numeric=False)
|
|
sample_weight = _check_sample_weight(sample_weight, X, dtype=X.dtype)
|
|
|
|
self._label_binarizer = LabelBinarizer(pos_label=1, neg_label=-1)
|
|
Y = self._label_binarizer.fit_transform(y)
|
|
if not self._label_binarizer.y_type_.startswith('multilabel'):
|
|
y = column_or_1d(y, warn=True)
|
|
|
|
if self.class_weight:
|
|
# modify the sample weights with the corresponding class weight
|
|
sample_weight = (sample_weight *
|
|
compute_sample_weight(self.class_weight, y))
|
|
|
|
target = Y if self.cv is None else y
|
|
_BaseRidgeCV.fit(self, X, target, sample_weight=sample_weight)
|
|
return self
|
|
|
|
@property
|
|
def classes_(self):
|
|
return self._label_binarizer.classes_
|