136 lines
5.7 KiB
Python
136 lines
5.7 KiB
Python
"""
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=======================================
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Robust vs Empirical covariance estimate
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=======================================
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The usual covariance maximum likelihood estimate is very sensitive to
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the presence of outliers in the data set. In such a case, one would
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have better to use a robust estimator of covariance to garanty that
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the estimation is resistant to "errorneous" observations in the data
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set.
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The Minimum Covariance Determinant estimator is a robust,
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high-breakdown point (i.e. it can be used to estimate the covariance
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matrix of highly contaminated datasets, up to
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:math:`\frac{n_samples-n_features-1}{2}` outliers) estimator of
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covariance. The idea is to find :math:`\frac{n_samples+n_features+1}{2}`
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observations whose empirical covariance has the smallest determinant,
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yielding a "pure" subset of observations from which to compute
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standards estimates of location and covariance. After a correction
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step aiming at compensating the fact the the estimates were learnt
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from only a portion of the initial data, we end up with robust
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estimates of the data set location and covariance.
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The Minimum Covariance Determinant estimator (MCD) has been introduced
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by P.J.Rousseuw in [1].
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In this example, we compare the estimation errors that are made when
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using three types of location and covariance estimates on contaminated
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gaussian distributed data sets:
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- The mean and the empirical covariance of the full dataset, which break
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down as soon as there are outliers in the data set
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- The robust MCD, that has a low error provided n_samples > 5 * n_features
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- The mean and the empirical covariance of the observations that are known
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to be good ones. This can be considered as a "perfect" MCD estimation,
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so one can trust our implementation by comparing to this case.
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[1] P. J. Rousseeuw. Least median of squares regression. J. Am
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Stat Ass, 79:871, 1984.
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[2] Johanna Hardin, David M Rocke. Journal of Computational and
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Graphical Statistics. December 1, 2005, 14(4): 928-946.
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"""
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print __doc__
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import numpy as np
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import pylab as pl
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import matplotlib.font_manager
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from sklearn.covariance import EmpiricalCovariance, MinCovDet
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# example settings
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n_samples = 80
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n_features = 5
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repeat = 10
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range_n_outliers = np.concatenate(
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(np.linspace(0, n_samples / 8, 5),
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np.linspace(n_samples / 8, n_samples / 2, 5)[1:-1]))
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# definition of arrays to store results
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err_loc_mcd = np.zeros((range_n_outliers.size, repeat))
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err_cov_mcd = np.zeros((range_n_outliers.size, repeat))
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err_loc_emp_full = np.zeros((range_n_outliers.size, repeat))
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err_cov_emp_full = np.zeros((range_n_outliers.size, repeat))
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err_loc_emp_pure = np.zeros((range_n_outliers.size, repeat))
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err_cov_emp_pure = np.zeros((range_n_outliers.size, repeat))
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# computation
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for i, n_outliers in enumerate(range_n_outliers):
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for j in range(repeat):
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# generate data
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X = np.random.randn(n_samples, n_features)
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# add some outliers
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outliers_index = np.random.permutation(n_samples)[:n_outliers]
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outliers_offset = 10. * \
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(np.random.randint(2, size=(n_outliers, n_features)) - 0.5)
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X[outliers_index] += outliers_offset
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inliers_mask = np.ones(n_samples).astype(bool)
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inliers_mask[outliers_index] = False
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# fit a Minimum Covariance Determinant (MCD) robust estimator to data
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S = MinCovDet().fit(X)
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# compare raw robust estimates with the true location and covariance
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err_loc_mcd[i, j] = np.sum(S.location_ ** 2)
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err_cov_mcd[i, j] = S.error_norm(np.eye(n_features))
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# compare estimators learnt from the full data set with true parameters
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err_loc_emp_full[i, j] = np.sum(X.mean(0) ** 2)
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err_cov_emp_full[i, j] = EmpiricalCovariance().fit(X).error_norm(
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np.eye(n_features))
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# compare with an empirical covariance learnt from a pure data set
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# (i.e. "perfect" MCD)
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pure_X = X[inliers_mask]
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pure_location = pure_X.mean(0)
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pure_emp_cov = EmpiricalCovariance().fit(pure_X)
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err_loc_emp_pure[i, j] = np.sum(pure_location ** 2)
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err_cov_emp_pure[i, j] = pure_emp_cov.error_norm(np.eye(n_features))
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# Display results
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font_prop = matplotlib.font_manager.FontProperties(size=11)
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pl.subplot(2, 1, 1)
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pl.errorbar(range_n_outliers, err_loc_mcd.mean(1),
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yerr=err_loc_mcd.std(1) / np.sqrt(repeat),
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label="Robust location", color='m')
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pl.errorbar(range_n_outliers, err_loc_emp_full.mean(1),
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yerr=err_loc_emp_full.std(1) / np.sqrt(repeat),
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label="Full data set mean", color='green')
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pl.errorbar(range_n_outliers, err_loc_emp_pure.mean(1),
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yerr=err_loc_emp_pure.std(1) / np.sqrt(repeat),
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label="Pure data set mean", color='black')
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pl.title("Influence of outliers on the location estimation")
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pl.ylabel(r"Error ($||\mu - \hat{\mu}||_2^2$)")
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pl.legend(loc="upper left", prop=font_prop)
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pl.subplot(2, 1, 2)
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x_size = range_n_outliers.size
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pl.errorbar(range_n_outliers, err_cov_mcd.mean(1),
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yerr=err_cov_mcd.std(1),
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label="Robust covariance (MCD)", color='m')
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pl.errorbar(range_n_outliers[:(x_size / 5 + 1)],
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err_cov_emp_full.mean(1)[:(x_size / 5 + 1)],
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yerr=err_cov_emp_full.std(1)[:(x_size / 5 + 1)],
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label="Full data set empirical covariance", color='green')
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pl.plot(range_n_outliers[(x_size / 5):(x_size / 2 - 1)],
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err_cov_emp_full.mean(1)[(x_size / 5):(x_size / 2 - 1)],
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color='green', ls='--')
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pl.errorbar(range_n_outliers, err_cov_emp_pure.mean(1),
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yerr=err_cov_emp_pure.std(1),
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label="Pure data set empirical covariance", color='black')
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pl.title("Influence of outliers on the covariance estimation")
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pl.xlabel("Amount of contamination (%)")
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pl.ylabel("RMSE")
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pl.legend(loc="upper center", prop=font_prop)
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pl.show()
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