带惩罚截尾平方的稳健回归的快速算法
A Fast Algorithm for Robust Regression with Penalised Trimmed Squares
摘要：
由于掩蔽效应，包含高杠杆离群值的组的存在使得线性回归成为一个困难的问题。现有的基于最小二乘的高崩溃估计往往不能成功地检测有限样本中的掩蔽高杠杆离群值。作者在\cite{ZiouAv:05,ZiAvPi:07}中引入了一种替代LTS估计量的方法，称为惩罚修剪平方（PTS）估计量，它似乎对掩蔽问题不太敏感。该估计量定义为二次混合线性规划（QMIP）问题，在目标函数中包含每个观测值的惩罚成本，作为任意可行回归线的残差上限。由于PTS不需要预先设定从数据集中删除的离群值的数量，因此相对于其他估计量，它具有更好的效率。然而，由于所得QMIP问题的计算复杂性很高，中等规模回归问题的精确解是不可行的。在本文中，我们进一步建立的PTS估计的理论性质，如高故障和效率，并提出了一个近似算法称为快速PTS计算PTS估计大数据集有效。在具有不同程度的离群点污染的基准实例集上进行的大量计算实验表明，该算法在合理的计算时间内识别高杠杆离群点组方面表现良好。
英文摘要：
The presence of groups containing high leverage outliers makes linear regression a difficult problem due to the masking effect. The available high breakdown estimators based on Least Trimmed Squares often do not succeed in detecting masked high leverage outliers in finite samples. An alternative to the LTS estimator, called Penalised Trimmed Squares (PTS) estimator, was introduced by the authors in \cite{ZiouAv:05,ZiAvPi:07} and it appears to be less sensitive to the masking problem. This estimator is defined by a Quadratic Mixed Integer Programming (QMIP) problem, where in the objective function a penalty cost for each observation is included which serves as an upper bound on the residual error for any feasible regression line. Since the PTS does not require presetting the number of outliers to delete from the data set, it has better efficiency with respect to other estimators. However, due to the high computational complexity of the resulting QMIP problem, exact solutions for moderately large regression problems is infeasible. In this paper we further establish the theoretical properties of the PTS estimator, such as high breakdown and efficiency, and propose an approximate algorithm called Fast-PTS to compute the PTS estimator for large data sets efficiently. Extensive computational experiments on sets of benchmark instances with varying degrees of outlier contamination, indicate that the proposed algorithm performs well in identifying groups of high leverage outliers in reasonable computational time.
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