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通常的协方差最大似然估计对数据集里的异常点(outliers
)是非常敏感的。在这样的情况下,使用稳健的协方差估计,保证对即使数据集存在错误的观测,估计量也是一致的。
极小极大协方差行列式估计
极小极大协方差行列式估计量(Minimum Covariance Determinant Estimator, MCDE
), 通常被用来估计高度污染(contaminated
)的数据集的协方差。数据集的受污染程度,可以达到
个离群点。它的基本思想是,找到
个观测,它们的经验协方差有最小的行列式。这样就产生了一个“纯”的观测子集,根据它计算标准的协方差估计量。为了弥补估计量仅从一部分原始数据得到的偏差,我们需要经过一个校正步,并与稳健的协方差估计作比较。
评价
在这个例子里,我们在受污染的正态总体的数据集上对比不同估计量的估计误差:
-
完整数据集的均值和经验协方差,它受离群点的影响。
-
稳健的MCDE误差较小,但需满足“样本数大于特征数”的条件。
-
观测的均值和协方差是已知的,这里作为
perfect
估计,方便估计量的比较。
实例代码
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.font_manager
from sklearn.covariance import EmpiricalCovariance, MinCovDet
# example settings
n_samples = 80
n_features = 5
repeat = 10
range_n_outliers = np.concatenate(
(np.linspace(0, n_samples / 8, 5),
np.linspace(n_samples / 8, n_samples / 2, 5)[1:-1])).astype(np.int)
# definition of arrays to store results
err_loc_mcd = np.zeros((range_n_outliers.size, repeat))
err_cov_mcd = np.zeros((range_n_outliers.size, repeat))
err_loc_emp_full = np.zeros((range_n_outliers.size, repeat))
err_cov_emp_full = np.zeros((range_n_outliers.size, repeat))
err_loc_emp_pure = np.zeros((range_n_outliers.size, repeat))
err_cov_emp_pure = np.zeros((range_n_outliers.size, repeat))
# computation
for i, n_outliers in enumerate(range_n_outliers):
for j in range(repeat):
rng = np.random.RandomState(i * j)
# generate data
X = rng.randn(n_samples, n_features)
# add some outliers
outliers_index = rng.permutation(n_samples)[:n_outliers]
outliers_offset = 10. * \
(np.random.randint(2, size=(n_outliers, n_features)) - 0.5)
X[outliers_index] += outliers_offset
inliers_mask = np.ones(n_samples).astype(bool)
inliers_mask[outliers_index] = False
# fit a Minimum Covariance Determinant (MCD) robust estimator to data
mcd = MinCovDet().fit(X)
# compare raw robust estimates with the true location and covariance
err_loc_mcd[i, j] = np.sum(mcd.location_ ** 2)
err_cov_mcd[i, j] = mcd.error_norm(np.eye(n_features))
# compare estimators learned from the full data set with true
# parameters
err_loc_emp_full[i, j] = np.sum(X.mean(0) ** 2)
err_cov_emp_full[i, j] = EmpiricalCovariance().fit(X).error_norm(
np.eye(n_features))
# compare with an empirical covariance learned from a pure data set
# (i.e. "perfect" mcd)
pure_X = X[inliers_mask]
pure_location = pure_X.mean(0)
pure_emp_cov = EmpiricalCovariance().fit(pure_X)
err_loc_emp_pure[i, j] = np.sum(pure_location ** 2)
err_cov_emp_pure[i, j] = pure_emp_cov.error_norm(np.eye(n_features))
# Display results
font_prop = matplotlib.font_manager.FontProperties(size=11)
plt.subplot(2, 1, 1)
lw = 2
plt.errorbar(range_n_outliers, err_loc_mcd.mean(1),
yerr=err_loc_mcd.std(1) / np.sqrt(repeat),
label="Robust location", lw=lw, color='m')
plt.errorbar(range_n_outliers, err_loc_emp_full.mean(1),
yerr=err_loc_emp_full.std(1) / np.sqrt(repeat),
label="Full data set mean", lw=lw, color='green')
plt.errorbar(range_n_outliers, err_loc_emp_pure.mean(1),
yerr=err_loc_emp_pure.std(1) / np.sqrt(repeat),
label="Pure data set mean", lw=lw, color='black')
plt.title("Influence of outliers on the location estimation")
plt.ylabel(r"Error ($||\mu - \hat{\mu}||_2^2$)")
plt.legend(loc="upper left", prop=font_prop)
plt.subplot(2, 1, 2)
x_size = range_n_outliers.size
plt.errorbar(range_n_outliers, err_cov_mcd.mean(1),
yerr=err_cov_mcd.std(1),
label="Robust covariance (mcd)", color='m')
plt.errorbar(range_n_outliers[:(x_size // 5 + 1)],
err_cov_emp_full.mean(1)[:(x_size // 5 + 1)],
yerr=err_cov_emp_full.std(1)[:(x_size // 5 + 1)],
label="Full data set empirical covariance", color='green')
plt.plot(range_n_outliers[(x_size // 5):(x_size // 2 - 1)],
err_cov_emp_full.mean(1)[(x_size // 5):(x_size // 2 - 1)],
color='green', ls='--')
plt.errorbar(range_n_outliers, err_cov_emp_pure.mean(1),
yerr=err_cov_emp_pure.std(1),
label="Pure data set empirical covariance", color='black')
plt.title("Influence of outliers on the covariance estimation")
plt.xlabel("Amount of contamination (%)")
plt.ylabel("RMSE")
plt.legend(loc="upper center", prop=font_prop)
plt.show()
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