Robust Estimation and Model Ensemble Selection

考虑一个包含 n 行（案例）和 d 列（变量）的数据表。经典的鲁棒性模型假设大多数情况都是无污染的。因此，可能需要识别和过滤少数受污染的病例。不幸的是，这种范式并不现实，并且在高维中会崩溃。如果一个单元格（数据表中的单个条目）被污染的概率较小且独立，那么一个案例（数据表中的一行）被污染的概率为 1-(1-p)^d，很快就会超过 0.5。例如，如果 p=0.01 且 d=100，则概率为 0.63397。 我以前的学生 Fatemah Alqallaf 的博士论文引起了人们对异常值传播问题的关注。 Alqallaf、Van Aelst、Yohai 和 Zamar（2009）继续进行这项研究，并表明传统的高击穿点估计不再能抵抗独立污染。因此，我以前的学生 Andy Leung 的博士论文构建了多元位置和散点估计，可以抵抗案例和单元格异常值。我的新研究生 Glenn McGuinness 和 Malvika Mitra 将考虑将这种方法扩展到其他多元模型，包括聚类分析。***廉价的数据收集/存储产生了变量丰富、案例贫乏的数据。我们经常发现这些数据集中的大量变量都是噪声变量，它们会损害而不是帮助推理任务。此外，剩余的有用变量可能是部分冗余的，并且这些变量的子集可能比整个集合更好地进行预测。那么最好基于不同的变量子集集成多个模型。这导致了非常广泛的研究工作——集成选择——模型选择的概括。我以前的学生 Jabed Tomal 的博士论文提出了一种称为方阵的临时程序。我的博士生 Anthony Christidis 将考虑通过优化新的损失函数来更结构化地选择最佳集成，该函数会惩罚为集成选择的模型的稀疏性和多样性。在几位研究生的帮助下，我希望研究线性和非线性回归和分类的扩展。 ***我对聚类分析也很感兴趣。一种流行的聚类方法是 K 均值。然而，这种方法不能抵抗异常值。 我的博士生 Juan D. Gonzalez（阿根廷布宜诺斯艾利斯大学）正在开发一种强大的替代方案，使用 Tau 尺度（Yohai 和 Zamar，1989）而不是从点到聚类中心的平均距离。将此过程应用于数字化图像处理也是他的研究项目的一部分。

Consider a data table with n rows - the cases - and d columns - the variables. The classical robustness model assumes that the majority of the cases are contamination free. Hence, a minority of contaminated cases may need to be identified and filtered. Unfortunately this paradigm is not realistic and collapses in high dimensions. If there is a small and independent probability p that a cell (individual entry in the data table) is contaminated then the probability that a case (a row in the data table) is contaminated is 1-(1-p)^d which can quickly go over 0.5. For example, if p=0.01 and d=100 the probability is 0.63397. The PhD thesis of my former student Fatemah Alqallaf, brings attention to this problem called propagation of outliers. Alqallaf, Van Aelst, Yohai and Zamar (2009) continue this research and shows that traditional high breakdown point estimates are no longer resistant against independent contamination. Hence, the PhD thesis of my former student Andy Leung constructed multivariate location and scatter estimates which are resistant to casewise and cellwise outliers. Extensions of this approach to other multivariate models including cluster analysis will be considered by my new graduate students Glenn McGuinness and Malvika Mitra.***Cheap collection/storage of data produced variable-rich-case-poor data. It is common to find that a large number of the variables in these datasets are noise variables which hurt rather than help inference tasks. Moreover, the remaining useful variables may be partially redundant and subsets of these variables may predict better than the full set. Then it may be better to ensemble several models based on diverse subsets of variables. This lead to a very broad research endeavor - ensemble selection - a generalization of model selection. The PhD thesis of my former student Jabed Tomal proposes an ad-hoc procedure called phalanxes. My PhD student Anthony Christidis will consider a more structured selection of optimal ensembles by optimizing a new loss function that penalizes lack of sparseness and lack of diversity of the models selected for the ensemble. With the help of several grad students I wish to study extensions to linear and nonlinear regression and classification. ***I am also interested in cluster analysis. A popular clustering method is K-means. However, this method is not resistant against outliers. My PhD student Juan D. Gonzalez (University of Buenos Aires, Argentina) is developing a robust alternative by using the Tau-scale (Yohai and Zamar, 1989) instead of the average of the distances from the points to their cluster centers. Application of this procedure to the processing of digitized images is also part of his research project.