"""GraphLasso: sparse inverse covariance estimation with an l1-penalized
estimator.
"""
# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
# License: BSD 3 clause
# Copyright: INRIA
import warnings
import operator
import sys
import time
import numpy as np
from scipy import linalg
from .empirical_covariance_ import (empirical_covariance, EmpiricalCovariance,
log_likelihood)
from ..exceptions import ConvergenceWarning
from ..utils.validation import check_random_state, check_array
from ..utils import deprecated
from ..linear_model import lars_path
from ..linear_model import cd_fast
from ..model_selection import check_cv, cross_val_score
from ..externals.joblib import Parallel, delayed
import collections
# Helper functions to compute the objective and dual objective functions
# of the l1-penalized estimator
def _objective(mle, precision_, alpha):
"""Evaluation of the graph-lasso objective function
the objective function is made of a shifted scaled version of the
normalized log-likelihood (i.e. its empirical mean over the samples) and a
penalisation term to promote sparsity
"""
p = precision_.shape[0]
cost = - 2. * log_likelihood(mle, precision_) + p * np.log(2 * np.pi)
cost += alpha * (np.abs(precision_).sum()
- np.abs(np.diag(precision_)).sum())
return cost
def _dual_gap(emp_cov, precision_, alpha):
"""Expression of the dual gap convergence criterion
The specific definition is given in Duchi "Projected Subgradient Methods
for Learning Sparse Gaussians".
"""
gap = np.sum(emp_cov * precision_)
gap -= precision_.shape[0]
gap += alpha * (np.abs(precision_).sum()
- np.abs(np.diag(precision_)).sum())
return gap
def alpha_max(emp_cov):
"""Find the maximum alpha for which there are some non-zeros off-diagonal.
Parameters
----------
emp_cov : 2D array, (n_features, n_features)
The sample covariance matrix
Notes
-----
This results from the bound for the all the Lasso that are solved
in GraphLasso: each time, the row of cov corresponds to Xy. As the
bound for alpha is given by `max(abs(Xy))`, the result follows.
"""
A = np.copy(emp_cov)
A.flat[::A.shape[0] + 1] = 0
return np.max(np.abs(A))
# The g-lasso algorithm
def graph_lasso(emp_cov, alpha, cov_init=None, mode='cd', tol=1e-4,
enet_tol=1e-4, max_iter=100, verbose=False,
return_costs=False, eps=np.finfo(np.float64).eps,
return_n_iter=False):
"""l1-penalized covariance estimator
Read more in the :ref:`User Guide <sparse_inverse_covariance>`.
Parameters
----------
emp_cov : 2D ndarray, shape (n_features, n_features)
Empirical covariance from which to compute the covariance estimate.
alpha : positive float
The regularization parameter: the higher alpha, the more
regularization, the sparser the inverse covariance.
cov_init : 2D array (n_features, n_features), optional
The initial guess for the covariance.
mode : {'cd', 'lars'}
The Lasso solver to use: coordinate descent or LARS. Use LARS for
very sparse underlying graphs, where p > n. Elsewhere prefer cd
which is more numerically stable.
tol : positive float, optional
The tolerance to declare convergence: if the dual gap goes below
this value, iterations are stopped.
enet_tol : positive float, optional
The tolerance for the elastic net solver used to calculate the descent
direction. This parameter controls the accuracy of the search direction
for a given column update, not of the overall parameter estimate. Only
used for mode='cd'.
max_iter : integer, optional
The maximum number of iterations.
verbose : boolean, optional
If verbose is True, the objective function and dual gap are
printed at each iteration.
return_costs : boolean, optional
If return_costs is True, the objective function and dual gap
at each iteration are returned.
eps : float, optional
The machine-precision regularization in the computation of the
Cholesky diagonal factors. Increase this for very ill-conditioned
systems.
return_n_iter : bool, optional
Whether or not to return the number of iterations.
Returns
-------
covariance : 2D ndarray, shape (n_features, n_features)
The estimated covariance matrix.
precision : 2D ndarray, shape (n_features, n_features)
The estimated (sparse) precision matrix.
costs : list of (objective, dual_gap) pairs
The list of values of the objective function and the dual gap at
each iteration. Returned only if return_costs is True.
n_iter : int
Number of iterations. Returned only if `return_n_iter` is set to True.
See Also
--------
GraphLasso, GraphLassoCV
Notes
-----
The algorithm employed to solve this problem is the GLasso algorithm,
from the Friedman 2008 Biostatistics paper. It is the same algorithm
as in the R `glasso` package.
One possible difference with the `glasso` R package is that the
diagonal coefficients are not penalized.
"""
_, n_features = emp_cov.shape
if alpha == 0:
if return_costs:
precision_ = linalg.inv(emp_cov)
cost = - 2. * log_likelihood(emp_cov, precision_)
cost += n_features * np.log(2 * np.pi)
d_gap = np.sum(emp_cov * precision_) - n_features
if return_n_iter:
return emp_cov, precision_, (cost, d_gap), 0
else:
return emp_cov, precision_, (cost, d_gap)
else:
if return_n_iter:
return emp_cov, linalg.inv(emp_cov), 0
else:
return emp_cov, linalg.inv(emp_cov)
if cov_init is None:
covariance_ = emp_cov.copy()
else:
covariance_ = cov_init.copy()
# As a trivial regularization (Tikhonov like), we scale down the
# off-diagonal coefficients of our starting point: This is needed, as
# in the cross-validation the cov_init can easily be
# ill-conditioned, and the CV loop blows. Beside, this takes
# conservative stand-point on the initial conditions, and it tends to
# make the convergence go faster.
covariance_ *= 0.95
diagonal = emp_cov.flat[::n_features + 1]
covariance_.flat[::n_features + 1] = diagonal
precision_ = linalg.pinvh(covariance_)
indices = np.arange(n_features)
costs = list()
# The different l1 regression solver have different numerical errors
if mode == 'cd':
errors = dict(over='raise', invalid='ignore')
else:
errors = dict(invalid='raise')
try:
# be robust to the max_iter=0 edge case, see:
# https://github.com/scikit-learn/scikit-learn/issues/4134
d_gap = np.inf
for i in range(max_iter):
for idx in range(n_features):
sub_covariance = np.ascontiguousarray(
covariance_[indices != idx].T[indices != idx])
row = emp_cov[idx, indices != idx]
with np.errstate(**errors):
if mode == 'cd':
# Use coordinate descent
coefs = -(precision_[indices != idx, idx]
/ (precision_[idx, idx] + 1000 * eps))
coefs, _, _, _ = cd_fast.enet_coordinate_descent_gram(
coefs, alpha, 0, sub_covariance, row, row,
max_iter, enet_tol, check_random_state(None), False)
else:
# Use LARS
_, _, coefs = lars_path(
sub_covariance, row, Xy=row, Gram=sub_covariance,
alpha_min=alpha / (n_features - 1), copy_Gram=True,
eps=eps, method='lars', return_path=False)
# Update the precision matrix
precision_[idx, idx] = (
1. / (covariance_[idx, idx]
- np.dot(covariance_[indices != idx, idx], coefs)))
precision_[indices != idx, idx] = (- precision_[idx, idx]
* coefs)
precision_[idx, indices != idx] = (- precision_[idx, idx]
* coefs)
coefs = np.dot(sub_covariance, coefs)
covariance_[idx, indices != idx] = coefs
covariance_[indices != idx, idx] = coefs
d_gap = _dual_gap(emp_cov, precision_, alpha)
cost = _objective(emp_cov, precision_, alpha)
if verbose:
print(
'[graph_lasso] Iteration % 3i, cost % 3.2e, dual gap %.3e'
% (i, cost, d_gap))
if return_costs:
costs.append((cost, d_gap))
if np.abs(d_gap) < tol:
break
if not np.isfinite(cost) and i > 0:
raise FloatingPointError('Non SPD result: the system is '
'too ill-conditioned for this solver')
else:
warnings.warn('graph_lasso: did not converge after %i iteration:'
' dual gap: %.3e' % (max_iter, d_gap),
ConvergenceWarning)
except FloatingPointError as e:
e.args = (e.args[0]
+ '. The system is too ill-conditioned for this solver',)
raise e
if return_costs:
if return_n_iter:
return covariance_, precision_, costs, i + 1
else:
return covariance_, precision_, costs
else:
if return_n_iter:
return covariance_, precision_, i + 1
else:
return covariance_, precision_
class GraphLasso(EmpiricalCovariance):
"""Sparse inverse covariance estimation with an l1-penalized estimator.
Read more in the :ref:`User Guide <sparse_inverse_covariance>`.
Parameters
----------
alpha : positive float, default 0.01
The regularization parameter: the higher alpha, the more
regularization, the sparser the inverse covariance.
mode : {'cd', 'lars'}, default 'cd'
The Lasso solver to use: coordinate descent or LARS. Use LARS for
very sparse underlying graphs, where p > n. Elsewhere prefer cd
which is more numerically stable.
tol : positive float, default 1e-4
The tolerance to declare convergence: if the dual gap goes below
this value, iterations are stopped.
enet_tol : positive float, optional
The tolerance for the elastic net solver used to calculate the descent
direction. This parameter controls the accuracy of the search direction
for a given column update, not of the overall parameter estimate. Only
used for mode='cd'.
max_iter : integer, default 100
The maximum number of iterations.
verbose : boolean, default False
If verbose is True, the objective function and dual gap are
plotted at each iteration.
assume_centered : boolean, default False
If True, data are not centered before computation.
Useful when working with data whose mean is almost, but not exactly
zero.
If False, data are centered before computation.
Attributes
----------
covariance_ : array-like, shape (n_features, n_features)
Estimated covariance matrix
precision_ : array-like, shape (n_features, n_features)
Estimated pseudo inverse matrix.
n_iter_ : int
Number of iterations run.
See Also
--------
graph_lasso, GraphLassoCV
"""
def __init__(self, alpha=.01, mode='cd', tol=1e-4, enet_tol=1e-4,
max_iter=100, verbose=False, assume_centered=False):
super(GraphLasso, self).__init__(assume_centered=assume_centered)
self.alpha = alpha
self.mode = mode
self.tol = tol
self.enet_tol = enet_tol
self.max_iter = max_iter
self.verbose = verbose
[docs] def fit(self, X, y=None):
"""Fits the GraphLasso model to X.
Parameters
----------
X : ndarray, shape (n_samples, n_features)
Data from which to compute the covariance estimate
y : (ignored)
"""
# Covariance does not make sense for a single feature
X = check_array(X, ensure_min_features=2, ensure_min_samples=2,
estimator=self)
if self.assume_centered:
self.location_ = np.zeros(X.shape[1])
else:
self.location_ = X.mean(0)
emp_cov = empirical_covariance(
X, assume_centered=self.assume_centered)
self.covariance_, self.precision_, self.n_iter_ = graph_lasso(
emp_cov, alpha=self.alpha, mode=self.mode, tol=self.tol,
enet_tol=self.enet_tol, max_iter=self.max_iter,
verbose=self.verbose, return_n_iter=True)
return self
# Cross-validation with GraphLasso
def graph_lasso_path(X, alphas, cov_init=None, X_test=None, mode='cd',
tol=1e-4, enet_tol=1e-4, max_iter=100, verbose=False):
"""l1-penalized covariance estimator along a path of decreasing alphas
Read more in the :ref:`User Guide <sparse_inverse_covariance>`.
Parameters
----------
X : 2D ndarray, shape (n_samples, n_features)
Data from which to compute the covariance estimate.
alphas : list of positive floats
The list of regularization parameters, decreasing order.
X_test : 2D array, shape (n_test_samples, n_features), optional
Optional test matrix to measure generalisation error.
mode : {'cd', 'lars'}
The Lasso solver to use: coordinate descent or LARS. Use LARS for
very sparse underlying graphs, where p > n. Elsewhere prefer cd
which is more numerically stable.
tol : positive float, optional
The tolerance to declare convergence: if the dual gap goes below
this value, iterations are stopped.
enet_tol : positive float, optional
The tolerance for the elastic net solver used to calculate the descent
direction. This parameter controls the accuracy of the search direction
for a given column update, not of the overall parameter estimate. Only
used for mode='cd'.
max_iter : integer, optional
The maximum number of iterations.
verbose : integer, optional
The higher the verbosity flag, the more information is printed
during the fitting.
Returns
-------
covariances_ : List of 2D ndarray, shape (n_features, n_features)
The estimated covariance matrices.
precisions_ : List of 2D ndarray, shape (n_features, n_features)
The estimated (sparse) precision matrices.
scores_ : List of float
The generalisation error (log-likelihood) on the test data.
Returned only if test data is passed.
"""
inner_verbose = max(0, verbose - 1)
emp_cov = empirical_covariance(X)
if cov_init is None:
covariance_ = emp_cov.copy()
else:
covariance_ = cov_init
covariances_ = list()
precisions_ = list()
scores_ = list()
if X_test is not None:
test_emp_cov = empirical_covariance(X_test)
for alpha in alphas:
try:
# Capture the errors, and move on
covariance_, precision_ = graph_lasso(
emp_cov, alpha=alpha, cov_init=covariance_, mode=mode, tol=tol,
enet_tol=enet_tol, max_iter=max_iter, verbose=inner_verbose)
covariances_.append(covariance_)
precisions_.append(precision_)
if X_test is not None:
this_score = log_likelihood(test_emp_cov, precision_)
except FloatingPointError:
this_score = -np.inf
covariances_.append(np.nan)
precisions_.append(np.nan)
if X_test is not None:
if not np.isfinite(this_score):
this_score = -np.inf
scores_.append(this_score)
if verbose == 1:
sys.stderr.write('.')
elif verbose > 1:
if X_test is not None:
print('[graph_lasso_path] alpha: %.2e, score: %.2e'
% (alpha, this_score))
else:
print('[graph_lasso_path] alpha: %.2e' % alpha)
if X_test is not None:
return covariances_, precisions_, scores_
return covariances_, precisions_
class GraphLassoCV(GraphLasso):
"""Sparse inverse covariance w/ cross-validated choice of the l1 penalty
Read more in the :ref:`User Guide <sparse_inverse_covariance>`.
Parameters
----------
alphas : integer, or list positive float, optional
If an integer is given, it fixes the number of points on the
grids of alpha to be used. If a list is given, it gives the
grid to be used. See the notes in the class docstring for
more details.
n_refinements : strictly positive integer
The number of times the grid is refined. Not used if explicit
values of alphas are passed.
cv : int, cross-validation generator or an iterable, optional
Determines the cross-validation splitting strategy.
Possible inputs for cv are:
- None, to use the default 3-fold cross-validation,
- integer, to specify the number of folds.
- An object to be used as a cross-validation generator.
- An iterable yielding train/test splits.
For integer/None inputs :class:`KFold` is used.
Refer :ref:`User Guide <cross_validation>` for the various
cross-validation strategies that can be used here.
tol : positive float, optional
The tolerance to declare convergence: if the dual gap goes below
this value, iterations are stopped.
enet_tol : positive float, optional
The tolerance for the elastic net solver used to calculate the descent
direction. This parameter controls the accuracy of the search direction
for a given column update, not of the overall parameter estimate. Only
used for mode='cd'.
max_iter : integer, optional
Maximum number of iterations.
mode : {'cd', 'lars'}
The Lasso solver to use: coordinate descent or LARS. Use LARS for
very sparse underlying graphs, where number of features is greater
than number of samples. Elsewhere prefer cd which is more numerically
stable.
n_jobs : int, optional
number of jobs to run in parallel (default 1).
verbose : boolean, optional
If verbose is True, the objective function and duality gap are
printed at each iteration.
assume_centered : Boolean
If True, data are not centered before computation.
Useful when working with data whose mean is almost, but not exactly
zero.
If False, data are centered before computation.
Attributes
----------
covariance_ : numpy.ndarray, shape (n_features, n_features)
Estimated covariance matrix.
precision_ : numpy.ndarray, shape (n_features, n_features)
Estimated precision matrix (inverse covariance).
alpha_ : float
Penalization parameter selected.
cv_alphas_ : list of float
All penalization parameters explored.
grid_scores_ : 2D numpy.ndarray (n_alphas, n_folds)
Log-likelihood score on left-out data across folds.
n_iter_ : int
Number of iterations run for the optimal alpha.
See Also
--------
graph_lasso, GraphLasso
Notes
-----
The search for the optimal penalization parameter (alpha) is done on an
iteratively refined grid: first the cross-validated scores on a grid are
computed, then a new refined grid is centered around the maximum, and so
on.
One of the challenges which is faced here is that the solvers can
fail to converge to a well-conditioned estimate. The corresponding
values of alpha then come out as missing values, but the optimum may
be close to these missing values.
"""
def __init__(self, alphas=4, n_refinements=4, cv=None, tol=1e-4,
enet_tol=1e-4, max_iter=100, mode='cd', n_jobs=1,
verbose=False, assume_centered=False):
super(GraphLassoCV, self).__init__(
mode=mode, tol=tol, verbose=verbose, enet_tol=enet_tol,
max_iter=max_iter, assume_centered=assume_centered)
self.alphas = alphas
self.n_refinements = n_refinements
self.cv = cv
self.n_jobs = n_jobs
@property
@deprecated("Attribute grid_scores was deprecated in version 0.19 and "
"will be removed in 0.21. Use ``grid_scores_`` instead")
def grid_scores(self):
return self.grid_scores_
[docs] def fit(self, X, y=None):
"""Fits the GraphLasso covariance model to X.
Parameters
----------
X : ndarray, shape (n_samples, n_features)
Data from which to compute the covariance estimate
y : (ignored)
"""
# Covariance does not make sense for a single feature
X = check_array(X, ensure_min_features=2, estimator=self)
if self.assume_centered:
self.location_ = np.zeros(X.shape[1])
else:
self.location_ = X.mean(0)
emp_cov = empirical_covariance(
X, assume_centered=self.assume_centered)
cv = check_cv(self.cv, y, classifier=False)
# List of (alpha, scores, covs)
path = list()
n_alphas = self.alphas
inner_verbose = max(0, self.verbose - 1)
if isinstance(n_alphas, collections.Sequence):
alphas = self.alphas
n_refinements = 1
else:
n_refinements = self.n_refinements
alpha_1 = alpha_max(emp_cov)
alpha_0 = 1e-2 * alpha_1
alphas = np.logspace(np.log10(alpha_0), np.log10(alpha_1),
n_alphas)[::-1]
t0 = time.time()
for i in range(n_refinements):
with warnings.catch_warnings():
# No need to see the convergence warnings on this grid:
# they will always be points that will not converge
# during the cross-validation
warnings.simplefilter('ignore', ConvergenceWarning)
# Compute the cross-validated loss on the current grid
# NOTE: Warm-restarting graph_lasso_path has been tried, and
# this did not allow to gain anything (same execution time with
# or without).
this_path = Parallel(
n_jobs=self.n_jobs,
verbose=self.verbose
)(delayed(graph_lasso_path)(X[train], alphas=alphas,
X_test=X[test], mode=self.mode,
tol=self.tol,
enet_tol=self.enet_tol,
max_iter=int(.1 * self.max_iter),
verbose=inner_verbose)
for train, test in cv.split(X, y))
# Little danse to transform the list in what we need
covs, _, scores = zip(*this_path)
covs = zip(*covs)
scores = zip(*scores)
path.extend(zip(alphas, scores, covs))
path = sorted(path, key=operator.itemgetter(0), reverse=True)
# Find the maximum (avoid using built in 'max' function to
# have a fully-reproducible selection of the smallest alpha
# in case of equality)
best_score = -np.inf
last_finite_idx = 0
for index, (alpha, scores, _) in enumerate(path):
this_score = np.mean(scores)
if this_score >= .1 / np.finfo(np.float64).eps:
this_score = np.nan
if np.isfinite(this_score):
last_finite_idx = index
if this_score >= best_score:
best_score = this_score
best_index = index
# Refine the grid
if best_index == 0:
# We do not need to go back: we have chosen
# the highest value of alpha for which there are
# non-zero coefficients
alpha_1 = path[0][0]
alpha_0 = path[1][0]
elif (best_index == last_finite_idx
and not best_index == len(path) - 1):
# We have non-converged models on the upper bound of the
# grid, we need to refine the grid there
alpha_1 = path[best_index][0]
alpha_0 = path[best_index + 1][0]
elif best_index == len(path) - 1:
alpha_1 = path[best_index][0]
alpha_0 = 0.01 * path[best_index][0]
else:
alpha_1 = path[best_index - 1][0]
alpha_0 = path[best_index + 1][0]
if not isinstance(n_alphas, collections.Sequence):
alphas = np.logspace(np.log10(alpha_1), np.log10(alpha_0),
n_alphas + 2)
alphas = alphas[1:-1]
if self.verbose and n_refinements > 1:
print('[GraphLassoCV] Done refinement % 2i out of %i: % 3is'
% (i + 1, n_refinements, time.time() - t0))
path = list(zip(*path))
grid_scores = list(path[1])
alphas = list(path[0])
# Finally, compute the score with alpha = 0
alphas.append(0)
grid_scores.append(cross_val_score(EmpiricalCovariance(), X,
cv=cv, n_jobs=self.n_jobs,
verbose=inner_verbose))
self.grid_scores_ = np.array(grid_scores)
best_alpha = alphas[best_index]
self.alpha_ = best_alpha
self.cv_alphas_ = alphas
# Finally fit the model with the selected alpha
self.covariance_, self.precision_, self.n_iter_ = graph_lasso(
emp_cov, alpha=best_alpha, mode=self.mode, tol=self.tol,
enet_tol=self.enet_tol, max_iter=self.max_iter,
verbose=inner_verbose, return_n_iter=True)
return self
