Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness
Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness
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DOI：
10.1007/s00190-005-0477-7
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发表时间：
2005-06
影响因子：
4.4
通讯作者：
Peiliang Xu
中科院分区：
文献类型：
--
作者：
Peiliang Xu
本文的主要研究成果如下：（1）提出了一种符号约束鲁棒估计方法，该方法可以容忍50%的数据污染，同时具有较高的最小二乘可比效率。由于目标函数与最小二乘法相同，该方法也可称为符号约束鲁棒最小二乘法。该方法的迭代版本已被实施，并显示出能够抵抗超过50%的污染。作为副产品，也可以获得尺度参数的稳健估计。与最小二乘中位数法和重复中位数法使用尽可能少的数据来求解不同，符号约束鲁棒最小二乘方法试图使用尽可能多的好数据来获得鲁棒解，因此不会受到部分数据之间的部分多重共线性或某些数据聚集在一起的影响;（2）虽然M-估计的崩溃点为1/（t+1），但我们证明了观测值的权很容易使结果恶化，并使Huber型M-估计的崩溃点为零。（3）通过假设异常值的符号服从先验分布，我们提出了主观崩溃点的概念，它可以被认为是Donoho和Huber的随机崩溃的扩展，但在解释地球科学和图像重建中的实际问题时可能是重要的;（4）证明了即使不存在高度集中的好数据或高度集中的异常值，最小二乘中值法仍能被单个异常值破坏。
The findings of this paper are summarized as follows: (1) We propose a sign-constrained robust estimation method, which can tolerate 50% of data contamination and meanwhile achieve high, least-squares-comparable efficiency. Since the objective function is identical with least squares, the method may also be called sign-constrained robust least squares. An iterative version of the method has been implemented and shown to be capable of resisting against more than 50% of contamination. As a by-product, a robust estimate of scale parameter can also be obtained. Unlike the least median of squares method and repeated medians, which use a least possible number of data to derive the solution, the sign-constrained robust least squares method attempts to employ a maximum possible number of good data to derive the robust solution, and thus will not be affected by partial near multi-collinearity among part of the data or if some of the data are clustered together; (2) although M-estimates have been reported to have a breakdown point of 1/(t+1), we have shown that the weights of observations can readily deteriorate such results and bring the breakdown point of M-estimates of Huber’s type to zero. The same zero breakdown point of theL1-norm method is also derived, again due to the weights of observations; (3) by assuming a prior distribution for the signs of outliers, we have developed the concept of subjective breakdown point, which may be thought of as an extension of stochastic breakdown by Donoho and Huber but can be important in explaining real-life problems in Earth Sciences and image reconstruction; and finally, (4) We have shown that the least median of squares method can still break down with a single outlier, even if no highly concentrated good data nor highly concentrated outliers exist.