Robust Methods for Exploring Multivariate Data

摘要：David Tyler （DMS-0305858）题目：探索多元数据的鲁棒方法本研究项目的总体目标是开发计算上可行、概念上吸引人、理论上站得住的用于探索和推断多元数据集的鲁棒方法。研究的主要估计类型是最近研究者引入的带辅助尺度的多元重降m估计。这类估计基于将分散组件划分为讨厌的“规模”组件和结构“形状”组件的关键思想。这种划分方法产生了对鲁棒多变量估计问题的一种新的解释，并使单变量鲁棒统计和鲁棒回归的概念能够很容易地扩展到多变量设置。特别是，它允许将回归mm -估计推广到多变量数据的mm -估计。鉴于回归mm估计是S-plus中默认的稳健回归估计，多变量mm估计的理论和计算发展有望作为多变量数据分析的标准方法产生广泛影响。除mm估计外，辅助尺度的重降m估计还包括多变量s估计和多变量约束m估计。对这些和其他带有辅助尺度的多变量m估计的鲁棒性特性，包括影响函数、相对效率和最大偏差函数，进行一般的统一研究。基于多变量位置和分散的稳健估计的方法和思想在概念上足够广泛，可以扩展到其他设置，例如多变量线性模型和结构化协方差问题，并且将对此类扩展进行研究。多元定位和离散在许多经典的统计过程中发挥着核心作用，如主成分分析、区别分析和典型相关分析，这些分析通常应用于心理学、生物学、地质学等不同学科。因此，进一步发展多元位置和散射的稳健估计可以对这些科学领域的数据分析方法产生重大影响。除了多元位置和分散的稳健估计的内在重要性之外，这种估计也是对高维数据集进行更深入分析的重要第一步。基于辅助尺度的重降m估计的多元数据的探索性方法的发展是本研究项目的另一个主要目标。这种对高维数据的探索方法与当前出现的数据问题有关，例如，在数据挖掘和图像数据等领域。对于这类数据问题，以信号加噪声的形式产生数据的经典模型是不合适的，最好将数据视为嵌入在大量杂波中的信号加噪声。稳健的多变量方法特别适用于后一种数据视图。研究者注意到这种方法与其他领域如聚类分析和计算机视觉开发的方法之间的重要联系。将对这些联系进行更深入的调查。

AbstractPI: David Tyler (DMS-0305858)Title: Robust Methods for Exploring Multivariate DataThe overall goal of this research project is to develop computationally feasible, conceptually appealing and theoretically defensible robust methods for exploring and making inferences about a multivariate data set. The main class of estimates to be studied is the multivariate redescending M-estimates with auxiliary scale recently introduced by the investigator. This class of estimates is based upon the key idea of partitioning the scatter component into a nuisance "scale" component and a structural "shape" component. This partitioning method produces a novel interpretation of robust multivariate estimation problems, and enables concepts from univariate robust statistics and from robust regression to be readily extended to the multivariate setting. In particular, it allows for the generalization of the regression MM-estimates to MM-estimates for multivariate data. Given that the regression MM-estimates are the default robust regression estimates in S-plus, theoretical and computational developments for the multivariate MM-estimates are expected to have wide impact as a standard method in the analysis of multivariate data. Aside from the MM-estimates, the redescending M-estimates with auxiliary scale also include the multivariate S-estimates and the multivariate constrained M-estimates. A general unifying study of the robustness properties, including influence functions, relative efficiencies, and maximum bias functions, of these and other multivariate M-estimates with auxiliary scale is to be undertaken. The methods and ideas underlying the robust estimates of multivariate location and scatter are conceptually broad enough to be extended to other settings, such as to multivariate linear models and to structured covariance problems, and such extensions are to be investigated.Multivariate location and scatter play a central role in many classical statistical procedures, such as principal component analysis, discriminate analysis, and canonical correlation analysis, which are routinely applied in such diverse disciplines as psychology, biology, geology, and other fields. Hence, the further development of robust estimates for multivariate location and scatter can have a substantial impact on data analysis methods in these scientific areas. Aside from the intrinsic importance of robust estimates of multivariate location and scatter, such estimates also serve as an important first step to a deeper analysis of a high dimensional data set. The development of exploratory methods for multivariate data based on the redescending M-estimates with auxiliary scale is another primary goal of this research project. Such exploratory methods for high dimensional data are pertinent to contemporary data problems arising, for example, in areas such as data mining and in image data. For such data problems, the classical model of data arising as signal plus noise is inappropriate and the data is better viewed as arising as signal plus noise embedded within a mass of clutter. Robust multivariate methods are particularly apt for this latter view of data. The investigator has noted important links between this methodology and methodologies developed in other areas such as cluster analysis and computer vision. A deeper investigation into these links will be undertaken.