"""GraphicalLasso: sparse inverse covariance estimation with an l1-penalized estimator. """ # Author: Gael Varoquaux # License: BSD 3 clause # Copyright: INRIA from collections.abc import Sequence import warnings import operator import sys import time import numpy as np from scipy import linalg from joblib import Parallel, delayed from . import empirical_covariance, EmpiricalCovariance, log_likelihood from ..exceptions import ConvergenceWarning from ..utils.validation import check_random_state from ..utils.validation import _deprecate_positional_args # mypy error: Module 'sklearn.linear_model' has no attribute '_cd_fast' from ..linear_model import _cd_fast as cd_fast # type: ignore from ..linear_model import lars_path_gram from ..model_selection import check_cv, cross_val_score # Helper functions to compute the objective and dual objective functions # of the l1-penalized estimator def _objective(mle, precision_, alpha): """Evaluation of the graphical-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 : ndarray of shape (n_features, n_features) The sample covariance matrix. Notes ----- This results from the bound for the all the Lasso that are solved in GraphicalLasso: 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 @_deprecate_positional_args def graphical_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 `. .. versionchanged:: v0.20 graph_lasso has been renamed to graphical_lasso Parameters ---------- emp_cov : ndarray of shape (n_features, n_features) Empirical covariance from which to compute the covariance estimate. alpha : float The regularization parameter: the higher alpha, the more regularization, the sparser the inverse covariance. Range is (0, inf]. cov_init : array of shape (n_features, n_features), default=None The initial guess for the 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 : 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 The maximum number of iterations. verbose : bool, default=False If verbose is True, the objective function and dual gap are printed at each iteration. return_costs : bool, default=Flase If return_costs is True, the objective function and dual gap at each iteration are returned. eps : float, default=eps The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Default is `np.finfo(np.float64).eps`. return_n_iter : bool, default=False Whether or not to return the number of iterations. Returns ------- covariance : ndarray of shape (n_features, n_features) The estimated covariance matrix. precision : ndarray of 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 -------- GraphicalLasso, GraphicalLassoCV 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 # set a sub_covariance buffer sub_covariance = np.copy(covariance_[1:, 1:], order='C') for i in range(max_iter): for idx in range(n_features): # To keep the contiguous matrix `sub_covariance` equal to # covariance_[indices != idx].T[indices != idx] # we only need to update 1 column and 1 line when idx changes if idx > 0: di = idx - 1 sub_covariance[di] = covariance_[di][indices != idx] sub_covariance[:, di] = covariance_[:, di][indices != idx] else: sub_covariance[:] = covariance_[1:, 1:] 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_gram( Xy=row, Gram=sub_covariance, n_samples=row.size, 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 if not np.isfinite(precision_.sum()): raise FloatingPointError('The system is too ill-conditioned ' 'for this solver') d_gap = _dual_gap(emp_cov, precision_, alpha) cost = _objective(emp_cov, precision_, alpha) if verbose: print('[graphical_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('graphical_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 GraphicalLasso(EmpiricalCovariance): """Sparse inverse covariance estimation with an l1-penalized estimator. Read more in the :ref:`User Guide `. .. versionchanged:: v0.20 GraphLasso has been renamed to GraphicalLasso Parameters ---------- alpha : float, default=0.01 The regularization parameter: the higher alpha, the more regularization, the sparser the inverse covariance. Range is (0, inf]. 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. 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 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. 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]) See Also -------- graphical_lasso, GraphicalLassoCV """ @_deprecate_positional_args def __init__(self, alpha=.01, *, mode='cd', tol=1e-4, enet_tol=1e-4, max_iter=100, verbose=False, assume_centered=False): super().__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 def fit(self, X, y=None): """Fits 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 consistence purpose. Returns ------- self : object """ # Covariance does not make sense for a single feature X = self._validate_data(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_ = 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 `. 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(GraphicalLasso): """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 `. .. 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 (0, inf] when floats given. 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 ` 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 ` 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_alphas_ : list of shape (n_alphas,), dtype=float All penalization parameters explored. grid_scores_ : ndarray of shape (n_alphas, n_folds) Log-likelihood score on left-out data across folds. n_iter_ : int Number of iterations run for the optimal alpha. 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], [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]) See Also -------- graphical_lasso, GraphicalLasso 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. """ @_deprecate_positional_args 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__( 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 def fit(self, X, y=None): """Fits 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 consistence purpose. Returns ------- self : object """ # Covariance does not make sense for a single feature X = self._validate_data(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, 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 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(.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, 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('[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)) 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_ = 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