763 lines
32 KiB
Python
763 lines
32 KiB
Python
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"""
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Robust location and covariance estimators.
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Here are implemented estimators that are resistant to outliers.
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"""
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# Author: Virgile Fritsch <virgile.fritsch@inria.fr>
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#
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# License: BSD 3 clause
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import warnings
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import numbers
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import numpy as np
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from scipy import linalg
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from scipy.stats import chi2
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from . import empirical_covariance, EmpiricalCovariance
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from ..utils.extmath import fast_logdet
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from ..utils import check_random_state, check_array
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from ..utils.validation import _deprecate_positional_args
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# Minimum Covariance Determinant
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# Implementing of an algorithm by Rousseeuw & Van Driessen described in
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# (A Fast Algorithm for the Minimum Covariance Determinant Estimator,
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# 1999, American Statistical Association and the American Society
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# for Quality, TECHNOMETRICS)
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# XXX Is this really a public function? It's not listed in the docs or
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# exported by sklearn.covariance. Deprecate?
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def c_step(X, n_support, remaining_iterations=30, initial_estimates=None,
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verbose=False, cov_computation_method=empirical_covariance,
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random_state=None):
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"""C_step procedure described in [Rouseeuw1984]_ aiming at computing MCD.
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Parameters
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----------
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X : array-like of shape (n_samples, n_features)
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Data set in which we look for the n_support observations whose
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scatter matrix has minimum determinant.
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n_support : int,
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Number of observations to compute the robust estimates of location
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and covariance from. This parameter must be greater than
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`n_samples / 2`.
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remaining_iterations : int, default=30
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Number of iterations to perform.
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According to [Rouseeuw1999]_, two iterations are sufficient to get
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close to the minimum, and we never need more than 30 to reach
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convergence.
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initial_estimates : tuple of shape (2,), default=None
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Initial estimates of location and shape from which to run the c_step
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procedure:
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- initial_estimates[0]: an initial location estimate
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- initial_estimates[1]: an initial covariance estimate
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verbose : bool, defaut=False
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Verbose mode.
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cov_computation_method : callable, \
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default=:func:`sklearn.covariance.empirical_covariance`
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The function which will be used to compute the covariance.
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Must return array of shape (n_features, n_features).
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random_state : int or RandomState instance, default=None
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Determines the pseudo random number generator for shuffling the data.
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Pass an int for reproducible results across multiple function calls.
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See :term: `Glossary <random_state>`.
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Returns
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-------
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location : ndarray of shape (n_features,)
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Robust location estimates.
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covariance : ndarray of shape (n_features, n_features)
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Robust covariance estimates.
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support : ndarray of shape (n_samples,)
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A mask for the `n_support` observations whose scatter matrix has
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minimum determinant.
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References
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----------
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.. [Rouseeuw1999] A Fast Algorithm for the Minimum Covariance Determinant
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Estimator, 1999, American Statistical Association and the American
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Society for Quality, TECHNOMETRICS
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"""
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X = np.asarray(X)
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random_state = check_random_state(random_state)
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return _c_step(X, n_support, remaining_iterations=remaining_iterations,
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initial_estimates=initial_estimates, verbose=verbose,
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cov_computation_method=cov_computation_method,
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random_state=random_state)
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def _c_step(X, n_support, random_state, remaining_iterations=30,
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initial_estimates=None, verbose=False,
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cov_computation_method=empirical_covariance):
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n_samples, n_features = X.shape
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dist = np.inf
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# Initialisation
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support = np.zeros(n_samples, dtype=bool)
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if initial_estimates is None:
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# compute initial robust estimates from a random subset
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support[random_state.permutation(n_samples)[:n_support]] = True
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else:
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# get initial robust estimates from the function parameters
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location = initial_estimates[0]
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covariance = initial_estimates[1]
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# run a special iteration for that case (to get an initial support)
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precision = linalg.pinvh(covariance)
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X_centered = X - location
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dist = (np.dot(X_centered, precision) * X_centered).sum(1)
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# compute new estimates
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support[np.argsort(dist)[:n_support]] = True
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X_support = X[support]
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location = X_support.mean(0)
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covariance = cov_computation_method(X_support)
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# Iterative procedure for Minimum Covariance Determinant computation
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det = fast_logdet(covariance)
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# If the data already has singular covariance, calculate the precision,
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# as the loop below will not be entered.
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if np.isinf(det):
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precision = linalg.pinvh(covariance)
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previous_det = np.inf
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while (det < previous_det and remaining_iterations > 0
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and not np.isinf(det)):
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# save old estimates values
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previous_location = location
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previous_covariance = covariance
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previous_det = det
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previous_support = support
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# compute a new support from the full data set mahalanobis distances
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precision = linalg.pinvh(covariance)
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X_centered = X - location
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dist = (np.dot(X_centered, precision) * X_centered).sum(axis=1)
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# compute new estimates
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support = np.zeros(n_samples, dtype=bool)
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support[np.argsort(dist)[:n_support]] = True
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X_support = X[support]
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location = X_support.mean(axis=0)
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covariance = cov_computation_method(X_support)
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det = fast_logdet(covariance)
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# update remaining iterations for early stopping
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remaining_iterations -= 1
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previous_dist = dist
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dist = (np.dot(X - location, precision) * (X - location)).sum(axis=1)
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# Check if best fit already found (det => 0, logdet => -inf)
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if np.isinf(det):
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results = location, covariance, det, support, dist
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# Check convergence
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if np.allclose(det, previous_det):
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# c_step procedure converged
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if verbose:
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print("Optimal couple (location, covariance) found before"
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" ending iterations (%d left)" % (remaining_iterations))
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results = location, covariance, det, support, dist
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elif det > previous_det:
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# determinant has increased (should not happen)
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warnings.warn("Determinant has increased; this should not happen: "
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"log(det) > log(previous_det) (%.15f > %.15f). "
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"You may want to try with a higher value of "
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"support_fraction (current value: %.3f)."
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% (det, previous_det, n_support / n_samples),
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RuntimeWarning)
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results = previous_location, previous_covariance, \
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previous_det, previous_support, previous_dist
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# Check early stopping
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if remaining_iterations == 0:
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if verbose:
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print('Maximum number of iterations reached')
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results = location, covariance, det, support, dist
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return results
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def select_candidates(X, n_support, n_trials, select=1, n_iter=30,
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verbose=False,
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cov_computation_method=empirical_covariance,
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random_state=None):
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"""Finds the best pure subset of observations to compute MCD from it.
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The purpose of this function is to find the best sets of n_support
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observations with respect to a minimization of their covariance
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matrix determinant. Equivalently, it removes n_samples-n_support
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observations to construct what we call a pure data set (i.e. not
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containing outliers). The list of the observations of the pure
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data set is referred to as the `support`.
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Starting from a random support, the pure data set is found by the
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c_step procedure introduced by Rousseeuw and Van Driessen in
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[RV]_.
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Parameters
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----------
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X : array-like of shape (n_samples, n_features)
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Data (sub)set in which we look for the n_support purest observations.
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n_support : int
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The number of samples the pure data set must contain.
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This parameter must be in the range `[(n + p + 1)/2] < n_support < n`.
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n_trials : int or tuple of shape (2,)
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Number of different initial sets of observations from which to
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run the algorithm. This parameter should be a strictly positive
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integer.
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Instead of giving a number of trials to perform, one can provide a
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list of initial estimates that will be used to iteratively run
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c_step procedures. In this case:
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- n_trials[0]: array-like, shape (n_trials, n_features)
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is the list of `n_trials` initial location estimates
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- n_trials[1]: array-like, shape (n_trials, n_features, n_features)
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is the list of `n_trials` initial covariances estimates
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select : int, default=1
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Number of best candidates results to return. This parameter must be
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a strictly positive integer.
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n_iter : int, default=30
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Maximum number of iterations for the c_step procedure.
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(2 is enough to be close to the final solution. "Never" exceeds 20).
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This parameter must be a strictly positive integer.
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verbose : bool, default False
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Control the output verbosity.
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cov_computation_method : callable, \
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default=:func:`sklearn.covariance.empirical_covariance`
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The function which will be used to compute the covariance.
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Must return an array of shape (n_features, n_features).
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random_state : int or RandomState instance, default=None
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Determines the pseudo random number generator for shuffling the data.
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Pass an int for reproducible results across multiple function calls.
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See :term: `Glossary <random_state>`.
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See Also
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---------
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c_step
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Returns
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-------
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best_locations : ndarray of shape (select, n_features)
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The `select` location estimates computed from the `select` best
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supports found in the data set (`X`).
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best_covariances : ndarray of shape (select, n_features, n_features)
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The `select` covariance estimates computed from the `select`
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best supports found in the data set (`X`).
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best_supports : ndarray of shape (select, n_samples)
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The `select` best supports found in the data set (`X`).
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References
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----------
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.. [RV] A Fast Algorithm for the Minimum Covariance Determinant
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Estimator, 1999, American Statistical Association and the American
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Society for Quality, TECHNOMETRICS
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"""
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random_state = check_random_state(random_state)
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if isinstance(n_trials, numbers.Integral):
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run_from_estimates = False
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elif isinstance(n_trials, tuple):
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run_from_estimates = True
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estimates_list = n_trials
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n_trials = estimates_list[0].shape[0]
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else:
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raise TypeError("Invalid 'n_trials' parameter, expected tuple or "
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" integer, got %s (%s)" % (n_trials, type(n_trials)))
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# compute `n_trials` location and shape estimates candidates in the subset
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all_estimates = []
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if not run_from_estimates:
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# perform `n_trials` computations from random initial supports
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for j in range(n_trials):
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all_estimates.append(
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_c_step(
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X, n_support, remaining_iterations=n_iter, verbose=verbose,
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cov_computation_method=cov_computation_method,
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random_state=random_state))
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else:
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# perform computations from every given initial estimates
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for j in range(n_trials):
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initial_estimates = (estimates_list[0][j], estimates_list[1][j])
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all_estimates.append(_c_step(
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X, n_support, remaining_iterations=n_iter,
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initial_estimates=initial_estimates, verbose=verbose,
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cov_computation_method=cov_computation_method,
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random_state=random_state))
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all_locs_sub, all_covs_sub, all_dets_sub, all_supports_sub, all_ds_sub = \
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zip(*all_estimates)
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# find the `n_best` best results among the `n_trials` ones
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index_best = np.argsort(all_dets_sub)[:select]
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best_locations = np.asarray(all_locs_sub)[index_best]
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best_covariances = np.asarray(all_covs_sub)[index_best]
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best_supports = np.asarray(all_supports_sub)[index_best]
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best_ds = np.asarray(all_ds_sub)[index_best]
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return best_locations, best_covariances, best_supports, best_ds
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def fast_mcd(X, support_fraction=None,
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cov_computation_method=empirical_covariance,
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random_state=None):
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"""Estimates the Minimum Covariance Determinant matrix.
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Read more in the :ref:`User Guide <robust_covariance>`.
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Parameters
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----------
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X : array-like of shape (n_samples, n_features)
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The data matrix, with p features and n samples.
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support_fraction : float, default=None
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The proportion of points to be included in the support of the raw
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MCD estimate. Default is `None`, which implies that the minimum
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value of `support_fraction` will be used within the algorithm:
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`(n_sample + n_features + 1) / 2`. This parameter must be in the
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range (0, 1).
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cov_computation_method : callable, \
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default=:func:`sklearn.covariance.empirical_covariance`
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The function which will be used to compute the covariance.
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Must return an array of shape (n_features, n_features).
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random_state : int or RandomState instance, default=None
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Determines the pseudo random number generator for shuffling the data.
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Pass an int for reproducible results across multiple function calls.
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See :term: `Glossary <random_state>`.
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Returns
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-------
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location : ndarray of shape (n_features,)
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Robust location of the data.
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covariance : ndarray of shape (n_features, n_features)
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Robust covariance of the features.
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support : ndarray of shape (n_samples,), dtype=bool
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A mask of the observations that have been used to compute
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the robust location and covariance estimates of the data set.
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Notes
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-----
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The FastMCD algorithm has been introduced by Rousseuw and Van Driessen
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in "A Fast Algorithm for the Minimum Covariance Determinant Estimator,
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1999, American Statistical Association and the American Society
|
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for Quality, TECHNOMETRICS".
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|
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 <robust_covariance>`.
|
||
|
|
|
||
|
|
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 or RandomState instance, 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 <random_state>`.
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
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...])
|
||
|
|
|
||
|
|
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
|
||
|
|
"""
|
||
|
|
_nonrobust_covariance = staticmethod(empirical_covariance)
|
||
|
|
|
||
|
|
@_deprecate_positional_args
|
||
|
|
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):
|
||
|
|
"""Fits 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 consistence purpose.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
self : object
|
||
|
|
"""
|
||
|
|
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.
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||
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|
n_samples = len(self.dist_)
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||
|
|
n_support = np.sum(self.support_)
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||
|
|
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
|