Distance and Dissimilarity Information in Statistical Model Building

本研究的目的是极大地扩展统计工具的集合，这些统计工具利用两两距离和不相似信息在不同尺度的回归、分类和变量/模式选择的统计模型构建中。这包括使用涉及非度量的两两不相似信息的信息。在这项工作中，不同的信息可能是主观的、有噪声的、不完整的、局限于非线性流形的、可能来自多个来源的、可能不一致的。先前的结果已经表明如何将这些信息嵌入到欧几里得空间中，从而可以使用在欧几里得空间中操作的方法。最近有两个新颖且非常强大的工具，距离相关和距离分量，提供了对任意变量组之间的相关性和分布相等性的原则测试，仅基于成对欧几里得距离，基本上不需要分布假设。因此，结合将非度量信息嵌入欧几里得空间，然后使用距离相关和距离分量对欧几里得数据进行操作的方法，为使用“杂乱”成对数据提供了一种重要的新方法。此外，距离相关和距离分量正通过参数化、调整和测试技术扩展到某些回归、分类和变量/模式选择问题，在适当情况下先采用嵌入技术。该计划实施方面的一系列任务，为非传统信息的回归、分类和变量/模式选择等主要统计任务提供了原则性的进展。这项工作为统计学家/数据分析师和各种科学领域的建模者提供了一套实用工具的广泛扩展，用于从从小到大的数据集中提取信息以预测、分类和选择重要的变量/模式，其中包括来自各种结构的距离或不相似性信息，这些信息在实践中变得越来越可用和重要。拟议的工作为改进统计数据分析提供了一套新的重要和有用的工具，这些工具将得到广泛传播，并对社会产生影响，因为它们有助于研究人员从包含公共利益信息的生物、医学、环境和其他数据集中提取信息。该项目包括在一个重要的STEM领域培养一名博士生。

The objective of this research is to greatly expand the collection of statistical tools that exploit pairwise distance and dissimilarity information in statistical model building for regression, classification, and variable/pattern selection at different scales. This includes use of information that involves non-metric pairwise dissimilarity information. In this work dissimilarity information may be subjective, noisy, incomplete, confined within a nonlinear manifold, may come from multiple sources and may be inconsistent. Previous results have shown how this information may be embedded into a Euclidean space, so that methods that operate in a Euclidean space can be used. Two recent novel and very powerful tools, distance correlation and distance components, have provided for principled testing of correlations between arbitrary groups of variables and testing of equality of distributions, based only on pairwise Euclidean distances, and requiring essentially no distributional assumptions. Thus, combining methods that embed non-metric information into a Euclidean space followed by use of distance correlation and distance components that operate on Euclidean data provide an important new approach to using "messy" pairwise data. Furthermore distance correlation and distance components are being extended to certain regression, classification and variable/pattern selection problems via parametrization, tuning and testing techniques, preceded, when appropriate by embedding techniques. A series of tasks to implement aspects of this program provides advances in the major statistical tasks of regression, classification and variable/pattern selection for non-traditional information in a principled way.This work provides a vast extension of the set of practical tools available to the statistician/data analyst and to modelers in a wide variety of scientific fields to extract information to predict, classify, and select important variables/patterns from data sets from small to large, that include distance or dissimilarity information from a variety of structures that are becoming increasingly available and important in practice. The proposed work provides a new set of important and useful tools for improved statistical data analysis that will be widely disseminated, and impact society to the extent that they provide aid to researchers in the extraction of information in biological, medical, environmental and other data sets that contain information of public interest. The project includes high level training of a Ph.D. student in an important STEM area.