Robust Methods for Exploring Multivariate Data

摘要 PI：David Tyler (DMS-0305858) 标题：探索多元数据的稳健方法该研究项目的总体目标是开发计算上可行、概念上有吸引力且理论上可靠的稳健方法，用于探索多元数据集并进行推断。要研究的主要估计类别是研究者最近引入的具有辅助尺度的多元再降序 M 估计。此类估计基于将分散分量划分为令人讨厌的“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鲁棒多元估计问题的新颖解释，并使来自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元设置。特别是，它允许将回归 MM 估计推广到多元数据的 MM 估计。鉴于回归 MM 估计是 S-plus 中默认的稳健回归估计，多元 MM 估计的理论和计算发展预计将作为多元数据分析的标准方法产生广泛影响。除了 MM 估计之外，带有辅助尺度的降序 M 估计还包括多元 S 估计和多元约束 M 估计。对这些和其他具有辅助尺度的多元 M 估计的鲁棒性属性（包括影响函数、相对效率和最大偏差函数）进行一般统一研究。多元位置和分散稳健估计背后的方法和思想在概念上足够广泛，可以扩展到其他设置，例如多元线性模型和结构化协方差问题，并且此类扩展有待研究。多元位置和分散在许多经典统计过程中发挥着核心作用，例如主成分分析、判别分析和典型相关分析，这些过程通常应用于心理学、生物学、地质学和其他领域等不同学科。因此，进一步发展对多变量位置和分散的稳健估计可能会对这些科学领域的数据分析方法产生重大影响。除了多元位置和分散的鲁棒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的第一步。基于具有辅助尺度的降序 M 估计的多元数据探索性方法的开发是该研究项目的另一个主要目标。这种针对高维数据的探索方法与例如数据挖掘和图像数据等领域中出现的当代数据问题相关。对于此类数据问题，作为信号加噪声产生的数据的经典模型是不合适的，最好将数据视为作为嵌入在大量杂波中的信号加噪声产生的数据。稳健的多变量方法特别适合后一种数据观点。研究人员注意到该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.