""" Robust location and covariance estimators. Here are implemented estimators that are resistant to outliers. """ # Author: Virgile Fritsch # # License: BSD 3 clause import warnings from numbers import Integral, Real import numpy as np from scipy import linalg from scipy.stats import chi2 from . import empirical_covariance, EmpiricalCovariance from ..utils.extmath import fast_logdet from ..utils import check_random_state, check_array from ..utils._param_validation import Interval # Minimum Covariance Determinant # Implementing of an algorithm by Rousseeuw & Van Driessen described in # (A Fast Algorithm for the Minimum Covariance Determinant Estimator, # 1999, American Statistical Association and the American Society # for Quality, TECHNOMETRICS) # XXX Is this really a public function? It's not listed in the docs or # exported by sklearn.covariance. Deprecate? def c_step( X, n_support, remaining_iterations=30, initial_estimates=None, verbose=False, cov_computation_method=empirical_covariance, random_state=None, ): """C_step procedure described in [Rouseeuw1984]_ aiming at computing MCD. Parameters ---------- X : array-like of shape (n_samples, n_features) Data set in which we look for the n_support observations whose scatter matrix has minimum determinant. n_support : int Number of observations to compute the robust estimates of location and covariance from. This parameter must be greater than `n_samples / 2`. remaining_iterations : int, default=30 Number of iterations to perform. According to [Rouseeuw1999]_, two iterations are sufficient to get close to the minimum, and we never need more than 30 to reach convergence. initial_estimates : tuple of shape (2,), default=None Initial estimates of location and shape from which to run the c_step procedure: - initial_estimates[0]: an initial location estimate - initial_estimates[1]: an initial covariance estimate verbose : bool, default=False Verbose mode. cov_computation_method : callable, \ default=:func:`sklearn.covariance.empirical_covariance` The function which will be used to compute the covariance. Must return array of shape (n_features, n_features). random_state : int, RandomState instance or None, default=None Determines the pseudo random number generator for shuffling the data. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. Returns ------- location : ndarray of shape (n_features,) Robust location estimates. covariance : ndarray of shape (n_features, n_features) Robust covariance estimates. support : ndarray of shape (n_samples,) A mask for the `n_support` observations whose scatter matrix has minimum determinant. References ---------- .. [Rouseeuw1999] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS """ X = np.asarray(X) random_state = check_random_state(random_state) return _c_step( X, n_support, remaining_iterations=remaining_iterations, initial_estimates=initial_estimates, verbose=verbose, cov_computation_method=cov_computation_method, random_state=random_state, ) def _c_step( X, n_support, random_state, remaining_iterations=30, initial_estimates=None, verbose=False, cov_computation_method=empirical_covariance, ): n_samples, n_features = X.shape dist = np.inf # Initialisation support = np.zeros(n_samples, dtype=bool) if initial_estimates is None: # compute initial robust estimates from a random subset support[random_state.permutation(n_samples)[:n_support]] = True else: # get initial robust estimates from the function parameters location = initial_estimates[0] covariance = initial_estimates[1] # run a special iteration for that case (to get an initial support) precision = linalg.pinvh(covariance) X_centered = X - location dist = (np.dot(X_centered, precision) * X_centered).sum(1) # compute new estimates support[np.argsort(dist)[:n_support]] = True X_support = X[support] location = X_support.mean(0) covariance = cov_computation_method(X_support) # Iterative procedure for Minimum Covariance Determinant computation det = fast_logdet(covariance) # If the data already has singular covariance, calculate the precision, # as the loop below will not be entered. if np.isinf(det): precision = linalg.pinvh(covariance) previous_det = np.inf while det < previous_det and remaining_iterations > 0 and not np.isinf(det): # save old estimates values previous_location = location previous_covariance = covariance previous_det = det previous_support = support # compute a new support from the full data set mahalanobis distances precision = linalg.pinvh(covariance) X_centered = X - location dist = (np.dot(X_centered, precision) * X_centered).sum(axis=1) # compute new estimates support = np.zeros(n_samples, dtype=bool) support[np.argsort(dist)[:n_support]] = True X_support = X[support] location = X_support.mean(axis=0) covariance = cov_computation_method(X_support) det = fast_logdet(covariance) # update remaining iterations for early stopping remaining_iterations -= 1 previous_dist = dist dist = (np.dot(X - location, precision) * (X - location)).sum(axis=1) # Check if best fit already found (det => 0, logdet => -inf) if np.isinf(det): results = location, covariance, det, support, dist # Check convergence if np.allclose(det, previous_det): # c_step procedure converged if verbose: print( "Optimal couple (location, covariance) found before" " ending iterations (%d left)" % (remaining_iterations) ) results = location, covariance, det, support, dist elif det > previous_det: # determinant has increased (should not happen) warnings.warn( "Determinant has increased; this should not happen: " "log(det) > log(previous_det) (%.15f > %.15f). " "You may want to try with a higher value of " "support_fraction (current value: %.3f)." % (det, previous_det, n_support / n_samples), RuntimeWarning, ) results = ( previous_location, previous_covariance, previous_det, previous_support, previous_dist, ) # Check early stopping if remaining_iterations == 0: if verbose: print("Maximum number of iterations reached") results = location, covariance, det, support, dist return results def select_candidates( X, n_support, n_trials, select=1, n_iter=30, verbose=False, cov_computation_method=empirical_covariance, random_state=None, ): """Finds the best pure subset of observations to compute MCD from it. The purpose of this function is to find the best sets of n_support observations with respect to a minimization of their covariance matrix determinant. Equivalently, it removes n_samples-n_support observations to construct what we call a pure data set (i.e. not containing outliers). The list of the observations of the pure data set is referred to as the `support`. Starting from a random support, the pure data set is found by the c_step procedure introduced by Rousseeuw and Van Driessen in [RV]_. Parameters ---------- X : array-like of shape (n_samples, n_features) Data (sub)set in which we look for the n_support purest observations. n_support : int The number of samples the pure data set must contain. This parameter must be in the range `[(n + p + 1)/2] < n_support < n`. n_trials : int or tuple of shape (2,) Number of different initial sets of observations from which to run the algorithm. This parameter should be a strictly positive integer. Instead of giving a number of trials to perform, one can provide a list of initial estimates that will be used to iteratively run c_step procedures. In this case: - n_trials[0]: array-like, shape (n_trials, n_features) is the list of `n_trials` initial location estimates - n_trials[1]: array-like, shape (n_trials, n_features, n_features) is the list of `n_trials` initial covariances estimates select : int, default=1 Number of best candidates results to return. This parameter must be a strictly positive integer. n_iter : int, default=30 Maximum number of iterations for the c_step procedure. (2 is enough to be close to the final solution. "Never" exceeds 20). This parameter must be a strictly positive integer. verbose : bool, default=False Control the output verbosity. cov_computation_method : callable, \ default=:func:`sklearn.covariance.empirical_covariance` The function which will be used to compute the covariance. Must return an array of shape (n_features, n_features). random_state : int, RandomState instance or None, default=None Determines the pseudo random number generator for shuffling the data. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. See Also --------- c_step Returns ------- best_locations : ndarray of shape (select, n_features) The `select` location estimates computed from the `select` best supports found in the data set (`X`). best_covariances : ndarray of shape (select, n_features, n_features) The `select` covariance estimates computed from the `select` best supports found in the data set (`X`). best_supports : ndarray of shape (select, n_samples) The `select` best supports found in the data set (`X`). References ---------- .. [RV] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS """ random_state = check_random_state(random_state) if isinstance(n_trials, Integral): run_from_estimates = False elif isinstance(n_trials, tuple): run_from_estimates = True estimates_list = n_trials n_trials = estimates_list[0].shape[0] else: raise TypeError( "Invalid 'n_trials' parameter, expected tuple or integer, got %s (%s)" % (n_trials, type(n_trials)) ) # compute `n_trials` location and shape estimates candidates in the subset all_estimates = [] if not run_from_estimates: # perform `n_trials` computations from random initial supports for j in range(n_trials): all_estimates.append( _c_step( X, n_support, remaining_iterations=n_iter, verbose=verbose, cov_computation_method=cov_computation_method, random_state=random_state, ) ) else: # perform computations from every given initial estimates for j in range(n_trials): initial_estimates = (estimates_list[0][j], estimates_list[1][j]) all_estimates.append( _c_step( X, n_support, remaining_iterations=n_iter, initial_estimates=initial_estimates, verbose=verbose, cov_computation_method=cov_computation_method, random_state=random_state, ) ) all_locs_sub, all_covs_sub, all_dets_sub, all_supports_sub, all_ds_sub = zip( *all_estimates ) # find the `n_best` best results among the `n_trials` ones index_best = np.argsort(all_dets_sub)[:select] best_locations = np.asarray(all_locs_sub)[index_best] best_covariances = np.asarray(all_covs_sub)[index_best] best_supports = np.asarray(all_supports_sub)[index_best] best_ds = np.asarray(all_ds_sub)[index_best] return best_locations, best_covariances, best_supports, best_ds def fast_mcd( X, support_fraction=None, cov_computation_method=empirical_covariance, random_state=None, ): """Estimate the Minimum Covariance Determinant matrix. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples, n_features) The data matrix, with p features and n samples. support_fraction : float, default=None The proportion of points to be included in the support of the raw MCD estimate. Default is `None`, which implies that the minimum value of `support_fraction` will be used within the algorithm: `(n_sample + n_features + 1) / 2`. This parameter must be in the range (0, 1). cov_computation_method : callable, \ default=:func:`sklearn.covariance.empirical_covariance` The function which will be used to compute the covariance. Must return an array of shape (n_features, n_features). random_state : int, RandomState instance or None, default=None Determines the pseudo random number generator for shuffling the data. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. Returns ------- location : ndarray of shape (n_features,) Robust location of the data. covariance : ndarray of shape (n_features, n_features) Robust covariance of the features. support : ndarray of shape (n_samples,), dtype=bool A mask of the observations that have been used to compute the robust location and covariance estimates of the data set. Notes ----- The FastMCD algorithm has been introduced by Rousseuw and Van Driessen in "A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS". The principle is to compute robust estimates and random subsets before pooling them into a larger subsets, and finally into the full data set. Depending on the size of the initial sample, we have one, two or three such computation levels. Note that only raw estimates are returned. If one is interested in the correction and reweighting steps described in [RouseeuwVan]_, see the MinCovDet object. References ---------- .. [RouseeuwVan] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS .. [Butler1993] R. W. Butler, P. L. Davies and M. Jhun, Asymptotics For The Minimum Covariance Determinant Estimator, The Annals of Statistics, 1993, Vol. 21, No. 3, 1385-1400 """ random_state = check_random_state(random_state) X = check_array(X, ensure_min_samples=2, estimator="fast_mcd") n_samples, n_features = X.shape # minimum breakdown value if support_fraction is None: n_support = int(np.ceil(0.5 * (n_samples + n_features + 1))) else: n_support = int(support_fraction * n_samples) # 1-dimensional case quick computation # (Rousseeuw, P. J. and Leroy, A. M. (2005) References, in Robust # Regression and Outlier Detection, John Wiley & Sons, chapter 4) if n_features == 1: if n_support < n_samples: # find the sample shortest halves X_sorted = np.sort(np.ravel(X)) diff = X_sorted[n_support:] - X_sorted[: (n_samples - n_support)] halves_start = np.where(diff == np.min(diff))[0] # take the middle points' mean to get the robust location estimate location = ( 0.5 * (X_sorted[n_support + halves_start] + X_sorted[halves_start]).mean() ) support = np.zeros(n_samples, dtype=bool) X_centered = X - location support[np.argsort(np.abs(X_centered), 0)[:n_support]] = True covariance = np.asarray([[np.var(X[support])]]) location = np.array([location]) # get precision matrix in an optimized way precision = linalg.pinvh(covariance) dist = (np.dot(X_centered, precision) * (X_centered)).sum(axis=1) else: support = np.ones(n_samples, dtype=bool) covariance = np.asarray([[np.var(X)]]) location = np.asarray([np.mean(X)]) X_centered = X - location # get precision matrix in an optimized way precision = linalg.pinvh(covariance) dist = (np.dot(X_centered, precision) * (X_centered)).sum(axis=1) # Starting FastMCD algorithm for p-dimensional case if (n_samples > 500) and (n_features > 1): # 1. Find candidate supports on subsets # a. split the set in subsets of size ~ 300 n_subsets = n_samples // 300 n_samples_subsets = n_samples // n_subsets samples_shuffle = random_state.permutation(n_samples) h_subset = int(np.ceil(n_samples_subsets * (n_support / float(n_samples)))) # b. perform a total of 500 trials n_trials_tot = 500 # c. select 10 best (location, covariance) for each subset n_best_sub = 10 n_trials = max(10, n_trials_tot // n_subsets) n_best_tot = n_subsets * n_best_sub all_best_locations = np.zeros((n_best_tot, n_features)) try: all_best_covariances = np.zeros((n_best_tot, n_features, n_features)) except MemoryError: # The above is too big. Let's try with something much small # (and less optimal) n_best_tot = 10 all_best_covariances = np.zeros((n_best_tot, n_features, n_features)) n_best_sub = 2 for i in range(n_subsets): low_bound = i * n_samples_subsets high_bound = low_bound + n_samples_subsets current_subset = X[samples_shuffle[low_bound:high_bound]] best_locations_sub, best_covariances_sub, _, _ = select_candidates( current_subset, h_subset, n_trials, select=n_best_sub, n_iter=2, cov_computation_method=cov_computation_method, random_state=random_state, ) subset_slice = np.arange(i * n_best_sub, (i + 1) * n_best_sub) all_best_locations[subset_slice] = best_locations_sub all_best_covariances[subset_slice] = best_covariances_sub # 2. Pool the candidate supports into a merged set # (possibly the full dataset) n_samples_merged = min(1500, n_samples) h_merged = int(np.ceil(n_samples_merged * (n_support / float(n_samples)))) if n_samples > 1500: n_best_merged = 10 else: n_best_merged = 1 # find the best couples (location, covariance) on the merged set selection = random_state.permutation(n_samples)[:n_samples_merged] locations_merged, covariances_merged, supports_merged, d = select_candidates( X[selection], h_merged, n_trials=(all_best_locations, all_best_covariances), select=n_best_merged, cov_computation_method=cov_computation_method, random_state=random_state, ) # 3. Finally get the overall best (locations, covariance) couple if n_samples < 1500: # directly get the best couple (location, covariance) location = locations_merged[0] covariance = covariances_merged[0] support = np.zeros(n_samples, dtype=bool) dist = np.zeros(n_samples) support[selection] = supports_merged[0] dist[selection] = d[0] else: # select the best couple on the full dataset locations_full, covariances_full, supports_full, d = select_candidates( X, n_support, n_trials=(locations_merged, covariances_merged), select=1, cov_computation_method=cov_computation_method, random_state=random_state, ) location = locations_full[0] covariance = covariances_full[0] support = supports_full[0] dist = d[0] elif n_features > 1: # 1. Find the 10 best couples (location, covariance) # considering two iterations n_trials = 30 n_best = 10 locations_best, covariances_best, _, _ = select_candidates( X, n_support, n_trials=n_trials, select=n_best, n_iter=2, cov_computation_method=cov_computation_method, random_state=random_state, ) # 2. Select the best couple on the full dataset amongst the 10 locations_full, covariances_full, supports_full, d = select_candidates( X, n_support, n_trials=(locations_best, covariances_best), select=1, cov_computation_method=cov_computation_method, random_state=random_state, ) location = locations_full[0] covariance = covariances_full[0] support = supports_full[0] dist = d[0] return location, covariance, support, dist class MinCovDet(EmpiricalCovariance): """Minimum Covariance Determinant (MCD): robust estimator of covariance. The Minimum Covariance Determinant covariance estimator is to be applied on Gaussian-distributed data, but could still be relevant on data drawn from a unimodal, symmetric distribution. It is not meant to be used with multi-modal data (the algorithm used to fit a MinCovDet object is likely to fail in such a case). One should consider projection pursuit methods to deal with multi-modal datasets. Read more in the :ref:`User Guide `. Parameters ---------- store_precision : bool, default=True Specify if the estimated precision is stored. assume_centered : bool, default=False If True, the support of the robust location and the covariance estimates is computed, and a covariance estimate is recomputed from it, without centering the data. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, the robust location and covariance are directly computed with the FastMCD algorithm without additional treatment. support_fraction : float, default=None The proportion of points to be included in the support of the raw MCD estimate. Default is None, which implies that the minimum value of support_fraction will be used within the algorithm: `(n_sample + n_features + 1) / 2`. The parameter must be in the range (0, 1]. random_state : int, RandomState instance or None, default=None Determines the pseudo random number generator for shuffling the data. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. Attributes ---------- raw_location_ : ndarray of shape (n_features,) The raw robust estimated location before correction and re-weighting. raw_covariance_ : ndarray of shape (n_features, n_features) The raw robust estimated covariance before correction and re-weighting. raw_support_ : ndarray of shape (n_samples,) A mask of the observations that have been used to compute the raw robust estimates of location and shape, before correction and re-weighting. location_ : ndarray of shape (n_features,) Estimated robust location. covariance_ : ndarray of shape (n_features, n_features) Estimated robust covariance matrix. precision_ : ndarray of shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True) support_ : ndarray of shape (n_samples,) A mask of the observations that have been used to compute the robust estimates of location and shape. dist_ : ndarray of shape (n_samples,) Mahalanobis distances of the training set (on which :meth:`fit` is called) observations. 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 -------- EllipticEnvelope : An object for detecting outliers in a Gaussian distributed dataset. EmpiricalCovariance : Maximum likelihood covariance estimator. GraphicalLasso : Sparse inverse covariance estimation with an l1-penalized estimator. GraphicalLassoCV : Sparse inverse covariance with cross-validated choice of the l1 penalty. LedoitWolf : LedoitWolf Estimator. OAS : Oracle Approximating Shrinkage Estimator. ShrunkCovariance : Covariance estimator with shrinkage. References ---------- .. [Rouseeuw1984] P. J. Rousseeuw. Least median of squares regression. J. Am Stat Ass, 79:871, 1984. .. [Rousseeuw] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS .. [ButlerDavies] R. W. Butler, P. L. Davies and M. Jhun, Asymptotics For The Minimum Covariance Determinant Estimator, The Annals of Statistics, 1993, Vol. 21, No. 3, 1385-1400 Examples -------- >>> import numpy as np >>> from sklearn.covariance import MinCovDet >>> from sklearn.datasets import make_gaussian_quantiles >>> real_cov = np.array([[.8, .3], ... [.3, .4]]) >>> rng = np.random.RandomState(0) >>> X = rng.multivariate_normal(mean=[0, 0], ... cov=real_cov, ... size=500) >>> cov = MinCovDet(random_state=0).fit(X) >>> cov.covariance_ array([[0.7411..., 0.2535...], [0.2535..., 0.3053...]]) >>> cov.location_ array([0.0813... , 0.0427...]) """ _parameter_constraints: dict = { **EmpiricalCovariance._parameter_constraints, "support_fraction": [Interval(Real, 0, 1, closed="right"), None], "random_state": ["random_state"], } _nonrobust_covariance = staticmethod(empirical_covariance) def __init__( self, *, store_precision=True, assume_centered=False, support_fraction=None, random_state=None, ): self.store_precision = store_precision self.assume_centered = assume_centered self.support_fraction = support_fraction self.random_state = random_state def fit(self, X, y=None): """Fit a Minimum Covariance Determinant with the FastMCD algorithm. Parameters ---------- X : array-like of shape (n_samples, n_features) Training data, where `n_samples` is the number of samples and `n_features` is the number of features. y : Ignored Not used, present for API consistency by convention. Returns ------- self : object Returns the instance itself. """ self._validate_params() X = self._validate_data(X, ensure_min_samples=2, estimator="MinCovDet") random_state = check_random_state(self.random_state) n_samples, n_features = X.shape # check that the empirical covariance is full rank if (linalg.svdvals(np.dot(X.T, X)) > 1e-8).sum() != n_features: warnings.warn( "The covariance matrix associated to your dataset is not full rank" ) # compute and store raw estimates raw_location, raw_covariance, raw_support, raw_dist = fast_mcd( X, support_fraction=self.support_fraction, cov_computation_method=self._nonrobust_covariance, random_state=random_state, ) if self.assume_centered: raw_location = np.zeros(n_features) raw_covariance = self._nonrobust_covariance( X[raw_support], assume_centered=True ) # get precision matrix in an optimized way precision = linalg.pinvh(raw_covariance) raw_dist = np.sum(np.dot(X, precision) * X, 1) self.raw_location_ = raw_location self.raw_covariance_ = raw_covariance self.raw_support_ = raw_support self.location_ = raw_location self.support_ = raw_support self.dist_ = raw_dist # obtain consistency at normal models self.correct_covariance(X) # re-weight estimator self.reweight_covariance(X) return self def correct_covariance(self, data): """Apply a correction to raw Minimum Covariance Determinant estimates. Correction using the empirical correction factor suggested by Rousseeuw and Van Driessen in [RVD]_. Parameters ---------- data : array-like of shape (n_samples, n_features) The data matrix, with p features and n samples. The data set must be the one which was used to compute the raw estimates. Returns ------- covariance_corrected : ndarray of shape (n_features, n_features) Corrected robust covariance estimate. References ---------- .. [RVD] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS """ # Check that the covariance of the support data is not equal to 0. # Otherwise self.dist_ = 0 and thus correction = 0. n_samples = len(self.dist_) n_support = np.sum(self.support_) if n_support < n_samples and np.allclose(self.raw_covariance_, 0): raise ValueError( "The covariance matrix of the support data " "is equal to 0, try to increase support_fraction" ) correction = np.median(self.dist_) / chi2(data.shape[1]).isf(0.5) covariance_corrected = self.raw_covariance_ * correction self.dist_ /= correction return covariance_corrected def reweight_covariance(self, data): """Re-weight raw Minimum Covariance Determinant estimates. Re-weight observations using Rousseeuw's method (equivalent to deleting outlying observations from the data set before computing location and covariance estimates) described in [RVDriessen]_. Parameters ---------- data : array-like of shape (n_samples, n_features) The data matrix, with p features and n samples. The data set must be the one which was used to compute the raw estimates. Returns ------- location_reweighted : ndarray of shape (n_features,) Re-weighted robust location estimate. covariance_reweighted : ndarray of shape (n_features, n_features) Re-weighted robust covariance estimate. support_reweighted : ndarray of shape (n_samples,), dtype=bool A mask of the observations that have been used to compute the re-weighted robust location and covariance estimates. References ---------- .. [RVDriessen] A Fast Algorithm for the Minimum Covariance Determinant Estimator, 1999, American Statistical Association and the American Society for Quality, TECHNOMETRICS """ n_samples, n_features = data.shape mask = self.dist_ < chi2(n_features).isf(0.025) if self.assume_centered: location_reweighted = np.zeros(n_features) else: location_reweighted = data[mask].mean(0) covariance_reweighted = self._nonrobust_covariance( data[mask], assume_centered=self.assume_centered ) support_reweighted = np.zeros(n_samples, dtype=bool) support_reweighted[mask] = True self._set_covariance(covariance_reweighted) self.location_ = location_reweighted self.support_ = support_reweighted X_centered = data - self.location_ self.dist_ = np.sum(np.dot(X_centered, self.get_precision()) * X_centered, 1) return location_reweighted, covariance_reweighted, support_reweighted