Robust Methods for Exploring Multivariate Data

摘要PI：大卫泰勒（DMS-0305858）标题：稳健的方法，探索多元数据本研究项目的总体目标是开发计算上可行的，概念上有吸引力的和理论上可辩护的稳健的方法，探索和推断的多元数据集。主要研究的估计类是最近研究者引入的带有辅助尺度的多元降阶M-估计。这类估计是基于将散射分量划分为滋扰"尺度"分量和结构"形状"分量的关键思想。这种划分方法产生了一个新的解释稳健的多元估计问题，并使概念从单变量稳健统计和稳健回归容易扩展到多变量设置。特别是，它允许推广的回归MM-估计MM-估计的多元数据。鉴于回归MM估计是S+中的默认稳健回归估计，多元MM估计的理论和计算发展预计将作为多元数据分析中的标准方法产生广泛影响。除了MM-估计外，带辅助尺度的降阶M-估计还包括多元S-估计和多元约束M-估计。一般统一的鲁棒性的研究，包括影响函数，相对效率，和最大偏差函数，这些和其他多变量M-估计与辅助尺度进行。多元位置和散布的稳健估计的方法和思想在概念上足够广泛，可以扩展到其他环境，例如多元线性模型和结构协方差问题，并且这种扩展将被研究。多元位置和散布在许多经典的统计过程中起着核心作用，例如主成分分析，判别分析和典型相关分析，其通常应用于诸如心理学、生物学、地质学和其他领域的不同学科。因此，多变量位置和分散的鲁棒估计的进一步发展可能会对这些科学领域的数据分析方法产生重大影响。除了多变量位置和散布的鲁棒估计的内在重要性之外，这样的估计也是对高维数据集进行更深入分析的重要的第一步。本研究的另一个主要目标是发展基于辅助尺度的降阶M-估计的多元数据探索性方法。这种用于高维数据的探索方法与例如在诸如数据挖掘和图像数据的领域中出现的当代数据问题有关。对于这样的数据问题，作为信号加噪声产生的数据的经典模型是不合适的，数据最好被看作是作为嵌入在大量杂波中的信号加噪声产生的。稳健的多变量方法特别适用于后一种数据视图。研究人员注意到这种方法与聚类分析和计算机视觉等其他领域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.