Robust Estimation and Model Ensemble Selection

考虑一个数据表，它有n行（cases）和d列（variables）。经典的鲁棒性模型假设大多数情况下是无污染的。因此，可能需要识别和过滤少数受污染的病例。不幸的是，这种范式是不现实的，并在高维度中崩溃。如果一个单元格（数据表中的单个条目）被污染的概率p很小且独立，那么一个案例（数据表中的一行）被污染的概率是1-（1-p）^d，它可以很快超过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这个称为离群值传播的问题的关注。Alqallaf、货车Aelst、Yohai和Zelbow（2009年）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变量位置和散布估计，这些估计对casewise和cellwise离群值具有抵抗力。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/存储产生了可变的富病例贫数据。通常会发现，这些数据集中的大量变量是噪声变量，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全集预测得更好。然后，基于不同的变量子集集成几个模型可能会更好。这导致了一个非常广泛的研究奋进-系综选择-模型选择的推广。我以前的学生Jabed Tomal的博士论文提出了一种称为方阵的特别程序。我的博士生Anthony Christidis将通过优化一个新的损失函数来考虑更结构化的最佳集合选择，该损失函数会惩罚为集合选择的模型的稀疏性和多样性的缺乏。在几个格拉德生的帮助下，我希望学习线性和非线性回归和分类的扩展。**一种流行的聚类方法是K-means。然而，这种方法对离群值没有抵抗力。 我的博士生胡安·D. Gonzalez（University of Buenos Aires，Argentina）正在开发一种稳健的替代方法，它使用Tau尺度（Yohai and Zelvis，1989）代替从点到其聚类中心的平均距离。将此程序应用于数字化图像处理也是他研究项目的一部分。

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.