Robust Estimation and Model Ensemble Selection

考虑一个数据表，它有n行(案例)和d列(变量)。经典的稳健性模型假定大多数情况下是无污染的。因此，可能需要确定和过滤少数受污染的病例。不幸的是，这种模式并不现实，而且在高维度崩溃。如果一个单元格(数据表中的单个条目)受到污染的概率很小且独立，则案例(数据表中的一行)受到污染的概率为1-(1-p)^d，可能很快超过0.5。例如，如果p=0.01d=100时，概率为0.63397。我以前的学生法特玛·阿尔卡拉夫的博士论文引起了人们对这个名为异常值传播的问题的关注。Alqallaf，Van Aelst，Yohai和Zamar(2009)继续这项研究，并表明传统的高崩溃点估计不再抵抗独立的污染。因此，我以前的学生梁安迪的博士论文构建了多变量位置和散布估计，它抵抗个案和单元型异常值。我的新研究生格伦·麦吉尼斯(Glenn McGuinness)和马尔维卡·米特拉(Malvika Mitra)将考虑将这种方法扩展到其他多变量模型，包括聚类分析。*廉价的数据收集/存储产生了变量丰富、病例贫乏的数据。人们经常发现，这些数据集中的大量变量都是噪声变量，这对推理任务不利而不是帮助。此外，剩余的有用变量可能是部分冗余的，这些变量的子集可能比全集预测得更好。那么，根据变量的不同子集整合几个模型可能会更好。这导致了一个非常广泛的研究努力--总体选择--模型选择的泛化。我以前的学生贾贝德·托马尔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 Christian dis将考虑通过优化一个新的损失函数来选择更有结构的最佳合奏，该函数惩罚了为合奏选择的模型的稀疏性和缺乏多样性。在几个研究生的帮助下，我希望学习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-Means。然而，这种方法对异常值没有抵抗力。我的博士生Juan D.Gonzalez(阿根廷布宜诺斯艾利斯大学)正在开发一种稳健的替代方法，使用Tau-Scale(Yohai和Zamar，1989)，而不是从点到集群中心的平均距离。将这一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.