Robust Methods for Exploring Multivariate Data

本研究项目的总体目标是开发在计算上可行的、在概念上有吸引力的、在理论上可防御的、用于探索和推断多变量数据集的稳健方法。被研究的主要估计类是研究者最近引入的带辅助尺度的多元重降M-估计。这类估计是基于将散布分量划分为讨厌的“比例”分量和结构的“形状”分量的关键思想。这种划分方法对稳健的多变量估计问题产生了一种新的解释，并使来自单变量稳健统计和稳健回归的概念能够容易地扩展到多变量环境。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模型中默认的稳健回归估计，多元MM-估计作为多元数据分析的一种标准方法，其理论和计算的发展可望产生广泛的影响。除MM估计外，带辅助尺度的降序M估计还包括多元S估计和多元约束M估计。对这些估计和其他带辅助尺度的多元M估计的稳健性，包括影响函数、相对效率和最大偏差函数，进行了一般的统一研究。多元位置和离散度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，如多元线性模型和结构协方差问题，这种扩展还有待研究。多变量位置和离散度在许多经典统计过程中发挥着核心作用，如主成分分析、判别分析和典型相关分析，这些程序通常应用于心理学、生物学、地质学和其他领域。因此，对多变量位置和散布的稳健估计的进一步发展可以对这些科学领域的数据分析方法产生重大影响。除了多变量位置和散布的稳健估计的内在重要性外，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。发展基于辅助尺度重降M估计的多变量数据探索性方法是本研究项目的另一个主要目标。这种用于高维数据的探索方法与例如在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，数据更好地被看作是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观点。调查员注意到这一方法与集群分析和计算机视觉等其他领域制定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.