Robust Estimation and Model Ensemble Selection

点击翻译按钮获取中文摘要

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.