997 lines
34 KiB
Python
997 lines
34 KiB
Python
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"""GraphicalLasso: sparse inverse covariance estimation with an l1-penalized
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estimator.
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"""
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# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
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# License: BSD 3 clause
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# Copyright: INRIA
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import warnings
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import operator
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import sys
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import time
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from numbers import Integral, Real
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import numpy as np
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from scipy import linalg
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from . import empirical_covariance, EmpiricalCovariance, log_likelihood
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from ..exceptions import ConvergenceWarning
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from ..utils.validation import (
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_is_arraylike_not_scalar,
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check_random_state,
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check_scalar,
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)
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from ..utils.parallel import delayed, Parallel
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from ..utils._param_validation import Interval, StrOptions
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# mypy error: Module 'sklearn.linear_model' has no attribute '_cd_fast'
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from ..linear_model import _cd_fast as cd_fast # type: ignore
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from ..linear_model import lars_path_gram
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from ..model_selection import check_cv, cross_val_score
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# Helper functions to compute the objective and dual objective functions
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# of the l1-penalized estimator
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def _objective(mle, precision_, alpha):
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"""Evaluation of the graphical-lasso objective function
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the objective function is made of a shifted scaled version of the
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normalized log-likelihood (i.e. its empirical mean over the samples) and a
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penalisation term to promote sparsity
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"""
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p = precision_.shape[0]
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cost = -2.0 * log_likelihood(mle, precision_) + p * np.log(2 * np.pi)
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cost += alpha * (np.abs(precision_).sum() - np.abs(np.diag(precision_)).sum())
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return cost
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def _dual_gap(emp_cov, precision_, alpha):
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"""Expression of the dual gap convergence criterion
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The specific definition is given in Duchi "Projected Subgradient Methods
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for Learning Sparse Gaussians".
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"""
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gap = np.sum(emp_cov * precision_)
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gap -= precision_.shape[0]
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gap += alpha * (np.abs(precision_).sum() - np.abs(np.diag(precision_)).sum())
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return gap
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def alpha_max(emp_cov):
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"""Find the maximum alpha for which there are some non-zeros off-diagonal.
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Parameters
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----------
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emp_cov : ndarray of shape (n_features, n_features)
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The sample covariance matrix.
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Notes
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-----
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This results from the bound for the all the Lasso that are solved
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in GraphicalLasso: each time, the row of cov corresponds to Xy. As the
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bound for alpha is given by `max(abs(Xy))`, the result follows.
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"""
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A = np.copy(emp_cov)
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A.flat[:: A.shape[0] + 1] = 0
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return np.max(np.abs(A))
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# The g-lasso algorithm
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def graphical_lasso(
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emp_cov,
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alpha,
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*,
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cov_init=None,
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mode="cd",
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tol=1e-4,
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enet_tol=1e-4,
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max_iter=100,
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verbose=False,
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return_costs=False,
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eps=np.finfo(np.float64).eps,
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return_n_iter=False,
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):
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"""L1-penalized covariance estimator.
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Read more in the :ref:`User Guide <sparse_inverse_covariance>`.
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.. versionchanged:: v0.20
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graph_lasso has been renamed to graphical_lasso
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Parameters
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----------
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emp_cov : ndarray of shape (n_features, n_features)
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Empirical covariance from which to compute the covariance estimate.
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alpha : float
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The regularization parameter: the higher alpha, the more
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regularization, the sparser the inverse covariance.
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Range is (0, inf].
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cov_init : array of shape (n_features, n_features), default=None
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The initial guess for the covariance. If None, then the empirical
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covariance is used.
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mode : {'cd', 'lars'}, default='cd'
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The Lasso solver to use: coordinate descent or LARS. Use LARS for
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very sparse underlying graphs, where p > n. Elsewhere prefer cd
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which is more numerically stable.
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tol : float, default=1e-4
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The tolerance to declare convergence: if the dual gap goes below
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this value, iterations are stopped. Range is (0, inf].
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enet_tol : float, default=1e-4
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The tolerance for the elastic net solver used to calculate the descent
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direction. This parameter controls the accuracy of the search direction
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for a given column update, not of the overall parameter estimate. Only
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used for mode='cd'. Range is (0, inf].
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max_iter : int, default=100
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The maximum number of iterations.
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verbose : bool, default=False
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If verbose is True, the objective function and dual gap are
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printed at each iteration.
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return_costs : bool, default=False
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If return_costs is True, the objective function and dual gap
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at each iteration are returned.
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eps : float, default=eps
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The machine-precision regularization in the computation of the
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Cholesky diagonal factors. Increase this for very ill-conditioned
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systems. Default is `np.finfo(np.float64).eps`.
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return_n_iter : bool, default=False
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Whether or not to return the number of iterations.
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Returns
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-------
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covariance : ndarray of shape (n_features, n_features)
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The estimated covariance matrix.
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precision : ndarray of shape (n_features, n_features)
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The estimated (sparse) precision matrix.
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costs : list of (objective, dual_gap) pairs
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The list of values of the objective function and the dual gap at
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each iteration. Returned only if return_costs is True.
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n_iter : int
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Number of iterations. Returned only if `return_n_iter` is set to True.
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See Also
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--------
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GraphicalLasso : Sparse inverse covariance estimation
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with an l1-penalized estimator.
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GraphicalLassoCV : Sparse inverse covariance with
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cross-validated choice of the l1 penalty.
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Notes
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-----
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The algorithm employed to solve this problem is the GLasso algorithm,
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from the Friedman 2008 Biostatistics paper. It is the same algorithm
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as in the R `glasso` package.
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One possible difference with the `glasso` R package is that the
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diagonal coefficients are not penalized.
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"""
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_, n_features = emp_cov.shape
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if alpha == 0:
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if return_costs:
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precision_ = linalg.inv(emp_cov)
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cost = -2.0 * log_likelihood(emp_cov, precision_)
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cost += n_features * np.log(2 * np.pi)
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d_gap = np.sum(emp_cov * precision_) - n_features
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if return_n_iter:
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return emp_cov, precision_, (cost, d_gap), 0
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else:
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return emp_cov, precision_, (cost, d_gap)
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else:
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if return_n_iter:
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return emp_cov, linalg.inv(emp_cov), 0
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else:
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return emp_cov, linalg.inv(emp_cov)
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if cov_init is None:
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covariance_ = emp_cov.copy()
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else:
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covariance_ = cov_init.copy()
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# As a trivial regularization (Tikhonov like), we scale down the
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# off-diagonal coefficients of our starting point: This is needed, as
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# in the cross-validation the cov_init can easily be
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# ill-conditioned, and the CV loop blows. Beside, this takes
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# conservative stand-point on the initial conditions, and it tends to
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# make the convergence go faster.
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covariance_ *= 0.95
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diagonal = emp_cov.flat[:: n_features + 1]
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covariance_.flat[:: n_features + 1] = diagonal
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precision_ = linalg.pinvh(covariance_)
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indices = np.arange(n_features)
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costs = list()
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# The different l1 regression solver have different numerical errors
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if mode == "cd":
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errors = dict(over="raise", invalid="ignore")
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else:
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errors = dict(invalid="raise")
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try:
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# be robust to the max_iter=0 edge case, see:
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# https://github.com/scikit-learn/scikit-learn/issues/4134
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d_gap = np.inf
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# set a sub_covariance buffer
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sub_covariance = np.copy(covariance_[1:, 1:], order="C")
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for i in range(max_iter):
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for idx in range(n_features):
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# To keep the contiguous matrix `sub_covariance` equal to
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# covariance_[indices != idx].T[indices != idx]
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# we only need to update 1 column and 1 line when idx changes
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if idx > 0:
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di = idx - 1
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sub_covariance[di] = covariance_[di][indices != idx]
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sub_covariance[:, di] = covariance_[:, di][indices != idx]
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else:
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sub_covariance[:] = covariance_[1:, 1:]
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row = emp_cov[idx, indices != idx]
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with np.errstate(**errors):
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if mode == "cd":
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# Use coordinate descent
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coefs = -(
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precision_[indices != idx, idx]
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/ (precision_[idx, idx] + 1000 * eps)
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)
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coefs, _, _, _ = cd_fast.enet_coordinate_descent_gram(
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coefs,
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alpha,
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0,
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sub_covariance,
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row,
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row,
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max_iter,
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enet_tol,
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check_random_state(None),
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False,
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)
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else: # mode == "lars"
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_, _, coefs = lars_path_gram(
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Xy=row,
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Gram=sub_covariance,
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n_samples=row.size,
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alpha_min=alpha / (n_features - 1),
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copy_Gram=True,
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eps=eps,
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method="lars",
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return_path=False,
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)
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# Update the precision matrix
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precision_[idx, idx] = 1.0 / (
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covariance_[idx, idx]
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- np.dot(covariance_[indices != idx, idx], coefs)
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)
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precision_[indices != idx, idx] = -precision_[idx, idx] * coefs
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precision_[idx, indices != idx] = -precision_[idx, idx] * coefs
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coefs = np.dot(sub_covariance, coefs)
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covariance_[idx, indices != idx] = coefs
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covariance_[indices != idx, idx] = coefs
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if not np.isfinite(precision_.sum()):
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raise FloatingPointError(
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"The system is too ill-conditioned for this solver"
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)
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d_gap = _dual_gap(emp_cov, precision_, alpha)
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cost = _objective(emp_cov, precision_, alpha)
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if verbose:
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print(
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"[graphical_lasso] Iteration % 3i, cost % 3.2e, dual gap %.3e"
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% (i, cost, d_gap)
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)
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if return_costs:
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costs.append((cost, d_gap))
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if np.abs(d_gap) < tol:
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break
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if not np.isfinite(cost) and i > 0:
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raise FloatingPointError(
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"Non SPD result: the system is too ill-conditioned for this solver"
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)
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else:
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warnings.warn(
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"graphical_lasso: did not converge after %i iteration: dual gap: %.3e"
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% (max_iter, d_gap),
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ConvergenceWarning,
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)
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except FloatingPointError as e:
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e.args = (e.args[0] + ". The system is too ill-conditioned for this solver",)
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raise e
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if return_costs:
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if return_n_iter:
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return covariance_, precision_, costs, i + 1
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else:
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return covariance_, precision_, costs
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else:
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if return_n_iter:
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return covariance_, precision_, i + 1
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else:
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return covariance_, precision_
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class BaseGraphicalLasso(EmpiricalCovariance):
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_parameter_constraints: dict = {
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**EmpiricalCovariance._parameter_constraints,
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"tol": [Interval(Real, 0, None, closed="right")],
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"enet_tol": [Interval(Real, 0, None, closed="right")],
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"max_iter": [Interval(Integral, 0, None, closed="left")],
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"mode": [StrOptions({"cd", "lars"})],
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"verbose": ["verbose"],
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}
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_parameter_constraints.pop("store_precision")
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def __init__(
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self,
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tol=1e-4,
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enet_tol=1e-4,
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max_iter=100,
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mode="cd",
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verbose=False,
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assume_centered=False,
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):
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super().__init__(assume_centered=assume_centered)
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self.tol = tol
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self.enet_tol = enet_tol
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self.max_iter = max_iter
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self.mode = mode
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self.verbose = verbose
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class GraphicalLasso(BaseGraphicalLasso):
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"""Sparse inverse covariance estimation with an l1-penalized estimator.
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Read more in the :ref:`User Guide <sparse_inverse_covariance>`.
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.. versionchanged:: v0.20
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||
|
|
GraphLasso has been renamed to GraphicalLasso
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Parameters
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----------
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alpha : float, default=0.01
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The regularization parameter: the higher alpha, the more
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regularization, the sparser the inverse covariance.
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Range is (0, inf].
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mode : {'cd', 'lars'}, default='cd'
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The Lasso solver to use: coordinate descent or LARS. Use LARS for
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very sparse underlying graphs, where p > n. Elsewhere prefer cd
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which is more numerically stable.
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tol : float, default=1e-4
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The tolerance to declare convergence: if the dual gap goes below
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this value, iterations are stopped. Range is (0, inf].
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enet_tol : float, default=1e-4
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The tolerance for the elastic net solver used to calculate the descent
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direction. This parameter controls the accuracy of the search direction
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for a given column update, not of the overall parameter estimate. Only
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used for mode='cd'. Range is (0, inf].
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|
|
|
||
|
|
max_iter : int, default=100
|
||
|
|
The maximum number of iterations.
|
||
|
|
|
||
|
|
verbose : bool, default=False
|
||
|
|
If verbose is True, the objective function and dual gap are
|
||
|
|
plotted at each iteration.
|
||
|
|
|
||
|
|
assume_centered : bool, 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
|
||
|
|
----------
|
||
|
|
location_ : ndarray of shape (n_features,)
|
||
|
|
Estimated location, i.e. the estimated mean.
|
||
|
|
|
||
|
|
covariance_ : ndarray of shape (n_features, n_features)
|
||
|
|
Estimated covariance matrix
|
||
|
|
|
||
|
|
precision_ : ndarray of shape (n_features, n_features)
|
||
|
|
Estimated pseudo inverse matrix.
|
||
|
|
|
||
|
|
n_iter_ : int
|
||
|
|
Number of iterations run.
|
||
|
|
|
||
|
|
n_features_in_ : int
|
||
|
|
Number of features seen during :term:`fit`.
|
||
|
|
|
||
|
|
.. versionadded:: 0.24
|
||
|
|
|
||
|
|
feature_names_in_ : ndarray of shape (`n_features_in_`,)
|
||
|
|
Names of features seen during :term:`fit`. Defined only when `X`
|
||
|
|
has feature names that are all strings.
|
||
|
|
|
||
|
|
.. versionadded:: 1.0
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
graphical_lasso : L1-penalized covariance estimator.
|
||
|
|
GraphicalLassoCV : Sparse inverse covariance with
|
||
|
|
cross-validated choice of the l1 penalty.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from sklearn.covariance import GraphicalLasso
|
||
|
|
>>> true_cov = np.array([[0.8, 0.0, 0.2, 0.0],
|
||
|
|
... [0.0, 0.4, 0.0, 0.0],
|
||
|
|
... [0.2, 0.0, 0.3, 0.1],
|
||
|
|
... [0.0, 0.0, 0.1, 0.7]])
|
||
|
|
>>> np.random.seed(0)
|
||
|
|
>>> X = np.random.multivariate_normal(mean=[0, 0, 0, 0],
|
||
|
|
... cov=true_cov,
|
||
|
|
... size=200)
|
||
|
|
>>> cov = GraphicalLasso().fit(X)
|
||
|
|
>>> np.around(cov.covariance_, decimals=3)
|
||
|
|
array([[0.816, 0.049, 0.218, 0.019],
|
||
|
|
[0.049, 0.364, 0.017, 0.034],
|
||
|
|
[0.218, 0.017, 0.322, 0.093],
|
||
|
|
[0.019, 0.034, 0.093, 0.69 ]])
|
||
|
|
>>> np.around(cov.location_, decimals=3)
|
||
|
|
array([0.073, 0.04 , 0.038, 0.143])
|
||
|
|
"""
|
||
|
|
|
||
|
|
_parameter_constraints: dict = {
|
||
|
|
**BaseGraphicalLasso._parameter_constraints,
|
||
|
|
"alpha": [Interval(Real, 0, None, closed="right")],
|
||
|
|
}
|
||
|
|
|
||
|
|
def __init__(
|
||
|
|
self,
|
||
|
|
alpha=0.01,
|
||
|
|
*,
|
||
|
|
mode="cd",
|
||
|
|
tol=1e-4,
|
||
|
|
enet_tol=1e-4,
|
||
|
|
max_iter=100,
|
||
|
|
verbose=False,
|
||
|
|
assume_centered=False,
|
||
|
|
):
|
||
|
|
super().__init__(
|
||
|
|
tol=tol,
|
||
|
|
enet_tol=enet_tol,
|
||
|
|
max_iter=max_iter,
|
||
|
|
mode=mode,
|
||
|
|
verbose=verbose,
|
||
|
|
assume_centered=assume_centered,
|
||
|
|
)
|
||
|
|
self.alpha = alpha
|
||
|
|
|
||
|
|
def fit(self, X, y=None):
|
||
|
|
"""Fit the GraphicalLasso model to X.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like of shape (n_samples, n_features)
|
||
|
|
Data from which to compute the covariance estimate.
|
||
|
|
|
||
|
|
y : Ignored
|
||
|
|
Not used, present for API consistency by convention.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
self : object
|
||
|
|
Returns the instance itself.
|
||
|
|
"""
|
||
|
|
self._validate_params()
|
||
|
|
# Covariance does not make sense for a single feature
|
||
|
|
X = self._validate_data(X, ensure_min_features=2, ensure_min_samples=2)
|
||
|
|
|
||
|
|
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_ = graphical_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 GraphicalLasso
|
||
|
|
def graphical_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 : ndarray of shape (n_samples, n_features)
|
||
|
|
Data from which to compute the covariance estimate.
|
||
|
|
|
||
|
|
alphas : array-like of shape (n_alphas,)
|
||
|
|
The list of regularization parameters, decreasing order.
|
||
|
|
|
||
|
|
cov_init : array of shape (n_features, n_features), default=None
|
||
|
|
The initial guess for the covariance.
|
||
|
|
|
||
|
|
X_test : array of shape (n_test_samples, n_features), default=None
|
||
|
|
Optional test matrix to measure generalisation error.
|
||
|
|
|
||
|
|
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 : float, default=1e-4
|
||
|
|
The tolerance to declare convergence: if the dual gap goes below
|
||
|
|
this value, iterations are stopped. The tolerance must be a positive
|
||
|
|
number.
|
||
|
|
|
||
|
|
enet_tol : float, default=1e-4
|
||
|
|
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'. The tolerance must be a positive number.
|
||
|
|
|
||
|
|
max_iter : int, default=100
|
||
|
|
The maximum number of iterations. This parameter should be a strictly
|
||
|
|
positive integer.
|
||
|
|
|
||
|
|
verbose : int or bool, default=False
|
||
|
|
The higher the verbosity flag, the more information is printed
|
||
|
|
during the fitting.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
covariances_ : list of shape (n_alphas,) of ndarray of shape \
|
||
|
|
(n_features, n_features)
|
||
|
|
The estimated covariance matrices.
|
||
|
|
|
||
|
|
precisions_ : list of shape (n_alphas,) of ndarray of shape \
|
||
|
|
(n_features, n_features)
|
||
|
|
The estimated (sparse) precision matrices.
|
||
|
|
|
||
|
|
scores_ : list of shape (n_alphas,), dtype=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_ = graphical_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(
|
||
|
|
"[graphical_lasso_path] alpha: %.2e, score: %.2e"
|
||
|
|
% (alpha, this_score)
|
||
|
|
)
|
||
|
|
else:
|
||
|
|
print("[graphical_lasso_path] alpha: %.2e" % alpha)
|
||
|
|
if X_test is not None:
|
||
|
|
return covariances_, precisions_, scores_
|
||
|
|
return covariances_, precisions_
|
||
|
|
|
||
|
|
|
||
|
|
class GraphicalLassoCV(BaseGraphicalLasso):
|
||
|
|
"""Sparse inverse covariance w/ cross-validated choice of the l1 penalty.
|
||
|
|
|
||
|
|
See glossary entry for :term:`cross-validation estimator`.
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <sparse_inverse_covariance>`.
|
||
|
|
|
||
|
|
.. versionchanged:: v0.20
|
||
|
|
GraphLassoCV has been renamed to GraphicalLassoCV
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
alphas : int or array-like of shape (n_alphas,), dtype=float, default=4
|
||
|
|
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. Range is [1, inf) for an integer.
|
||
|
|
Range is (0, inf] for an array-like of floats.
|
||
|
|
|
||
|
|
n_refinements : int, default=4
|
||
|
|
The number of times the grid is refined. Not used if explicit
|
||
|
|
values of alphas are passed. Range is [1, inf).
|
||
|
|
|
||
|
|
cv : int, cross-validation generator or 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:: 0.20
|
||
|
|
``cv`` default value if None changed from 3-fold to 5-fold.
|
||
|
|
|
||
|
|
tol : float, default=1e-4
|
||
|
|
The tolerance to declare convergence: if the dual gap goes below
|
||
|
|
this value, iterations are stopped. Range is (0, inf].
|
||
|
|
|
||
|
|
enet_tol : float, default=1e-4
|
||
|
|
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'. Range is (0, inf].
|
||
|
|
|
||
|
|
max_iter : int, default=100
|
||
|
|
Maximum number of iterations.
|
||
|
|
|
||
|
|
mode : {'cd', 'lars'}, default='cd'
|
||
|
|
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, default=None
|
||
|
|
Number of jobs to run in parallel.
|
||
|
|
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
|
||
|
|
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
|
||
|
|
for more details.
|
||
|
|
|
||
|
|
.. versionchanged:: v0.20
|
||
|
|
`n_jobs` default changed from 1 to None
|
||
|
|
|
||
|
|
verbose : bool, default=False
|
||
|
|
If verbose is True, the objective function and duality gap are
|
||
|
|
printed at each iteration.
|
||
|
|
|
||
|
|
assume_centered : bool, 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
|
||
|
|
----------
|
||
|
|
location_ : ndarray of shape (n_features,)
|
||
|
|
Estimated location, i.e. the estimated mean.
|
||
|
|
|
||
|
|
covariance_ : ndarray of shape (n_features, n_features)
|
||
|
|
Estimated covariance matrix.
|
||
|
|
|
||
|
|
precision_ : ndarray of shape (n_features, n_features)
|
||
|
|
Estimated precision matrix (inverse covariance).
|
||
|
|
|
||
|
|
alpha_ : float
|
||
|
|
Penalization parameter selected.
|
||
|
|
|
||
|
|
cv_results_ : dict of ndarrays
|
||
|
|
A dict with keys:
|
||
|
|
|
||
|
|
alphas : ndarray of shape (n_alphas,)
|
||
|
|
All penalization parameters explored.
|
||
|
|
|
||
|
|
split(k)_test_score : ndarray of shape (n_alphas,)
|
||
|
|
Log-likelihood score on left-out data across (k)th fold.
|
||
|
|
|
||
|
|
.. versionadded:: 1.0
|
||
|
|
|
||
|
|
mean_test_score : ndarray of shape (n_alphas,)
|
||
|
|
Mean of scores over the folds.
|
||
|
|
|
||
|
|
.. versionadded:: 1.0
|
||
|
|
|
||
|
|
std_test_score : ndarray of shape (n_alphas,)
|
||
|
|
Standard deviation of scores over the folds.
|
||
|
|
|
||
|
|
.. versionadded:: 1.0
|
||
|
|
|
||
|
|
n_iter_ : int
|
||
|
|
Number of iterations run for the optimal alpha.
|
||
|
|
|
||
|
|
n_features_in_ : int
|
||
|
|
Number of features seen during :term:`fit`.
|
||
|
|
|
||
|
|
.. versionadded:: 0.24
|
||
|
|
|
||
|
|
feature_names_in_ : ndarray of shape (`n_features_in_`,)
|
||
|
|
Names of features seen during :term:`fit`. Defined only when `X`
|
||
|
|
has feature names that are all strings.
|
||
|
|
|
||
|
|
.. versionadded:: 1.0
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
graphical_lasso : L1-penalized covariance estimator.
|
||
|
|
GraphicalLasso : Sparse inverse covariance estimation
|
||
|
|
with an l1-penalized estimator.
|
||
|
|
|
||
|
|
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
|
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|
|
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.
|
||
|
|
|
||
|
|
In `fit`, once the best parameter `alpha` is found through
|
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|
|
cross-validation, the model is fit again using the entire training set.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import numpy as np
|
||
|
|
>>> from sklearn.covariance import GraphicalLassoCV
|
||
|
|
>>> true_cov = np.array([[0.8, 0.0, 0.2, 0.0],
|
||
|
|
... [0.0, 0.4, 0.0, 0.0],
|
||
|
|
... [0.2, 0.0, 0.3, 0.1],
|
||
|
|
... [0.0, 0.0, 0.1, 0.7]])
|
||
|
|
>>> np.random.seed(0)
|
||
|
|
>>> X = np.random.multivariate_normal(mean=[0, 0, 0, 0],
|
||
|
|
... cov=true_cov,
|
||
|
|
... size=200)
|
||
|
|
>>> cov = GraphicalLassoCV().fit(X)
|
||
|
|
>>> np.around(cov.covariance_, decimals=3)
|
||
|
|
array([[0.816, 0.051, 0.22 , 0.017],
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|
|
[0.051, 0.364, 0.018, 0.036],
|
||
|
|
[0.22 , 0.018, 0.322, 0.094],
|
||
|
|
[0.017, 0.036, 0.094, 0.69 ]])
|
||
|
|
>>> np.around(cov.location_, decimals=3)
|
||
|
|
array([0.073, 0.04 , 0.038, 0.143])
|
||
|
|
"""
|
||
|
|
|
||
|
|
_parameter_constraints: dict = {
|
||
|
|
**BaseGraphicalLasso._parameter_constraints,
|
||
|
|
"alphas": [Interval(Integral, 1, None, closed="left"), "array-like"],
|
||
|
|
"n_refinements": [Interval(Integral, 1, None, closed="left")],
|
||
|
|
"cv": ["cv_object"],
|
||
|
|
"n_jobs": [Integral, None],
|
||
|
|
}
|
||
|
|
|
||
|
|
def __init__(
|
||
|
|
self,
|
||
|
|
*,
|
||
|
|
alphas=4,
|
||
|
|
n_refinements=4,
|
||
|
|
cv=None,
|
||
|
|
tol=1e-4,
|
||
|
|
enet_tol=1e-4,
|
||
|
|
max_iter=100,
|
||
|
|
mode="cd",
|
||
|
|
n_jobs=None,
|
||
|
|
verbose=False,
|
||
|
|
assume_centered=False,
|
||
|
|
):
|
||
|
|
super().__init__(
|
||
|
|
tol=tol,
|
||
|
|
enet_tol=enet_tol,
|
||
|
|
max_iter=max_iter,
|
||
|
|
mode=mode,
|
||
|
|
verbose=verbose,
|
||
|
|
assume_centered=assume_centered,
|
||
|
|
)
|
||
|
|
self.alphas = alphas
|
||
|
|
self.n_refinements = n_refinements
|
||
|
|
self.cv = cv
|
||
|
|
self.n_jobs = n_jobs
|
||
|
|
|
||
|
|
def fit(self, X, y=None):
|
||
|
|
"""Fit the GraphicalLasso covariance model to X.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
X : array-like of shape (n_samples, n_features)
|
||
|
|
Data from which to compute the covariance estimate.
|
||
|
|
|
||
|
|
y : Ignored
|
||
|
|
Not used, present for API consistency by convention.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
self : object
|
||
|
|
Returns the instance itself.
|
||
|
|
"""
|
||
|
|
self._validate_params()
|
||
|
|
# Covariance does not make sense for a single feature
|
||
|
|
X = self._validate_data(X, ensure_min_features=2)
|
||
|
|
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 _is_arraylike_not_scalar(n_alphas):
|
||
|
|
for alpha in self.alphas:
|
||
|
|
check_scalar(
|
||
|
|
alpha,
|
||
|
|
"alpha",
|
||
|
|
Real,
|
||
|
|
min_val=0,
|
||
|
|
max_val=np.inf,
|
||
|
|
include_boundaries="right",
|
||
|
|
)
|
||
|
|
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 graphical_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(graphical_lasso_path)(
|
||
|
|
X[train],
|
||
|
|
alphas=alphas,
|
||
|
|
X_test=X[test],
|
||
|
|
mode=self.mode,
|
||
|
|
tol=self.tol,
|
||
|
|
enet_tol=self.enet_tol,
|
||
|
|
max_iter=int(0.1 * 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 >= 0.1 / 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 _is_arraylike_not_scalar(n_alphas):
|
||
|
|
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(
|
||
|
|
"[GraphicalLassoCV] 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,
|
||
|
|
)
|
||
|
|
)
|
||
|
|
grid_scores = np.array(grid_scores)
|
||
|
|
|
||
|
|
self.cv_results_ = {"alphas": np.array(alphas)}
|
||
|
|
|
||
|
|
for i in range(grid_scores.shape[1]):
|
||
|
|
self.cv_results_[f"split{i}_test_score"] = grid_scores[:, i]
|
||
|
|
|
||
|
|
self.cv_results_["mean_test_score"] = np.mean(grid_scores, axis=1)
|
||
|
|
self.cv_results_["std_test_score"] = np.std(grid_scores, axis=1)
|
||
|
|
|
||
|
|
best_alpha = alphas[best_index]
|
||
|
|
self.alpha_ = best_alpha
|
||
|
|
|
||
|
|
# Finally fit the model with the selected alpha
|
||
|
|
self.covariance_, self.precision_, self.n_iter_ = graphical_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
|