Distance and Dissimilarity Information in Statistical Model Building

这项研究的目的是极大地扩大统计工具的集合，这些工具在统计模型建立中利用成对距离和不同信息，用于不同尺度上的回归、分类和变量/模式选择。这包括使用涉及非度量成对相异信息的信息。在这项工作中，相异信息可能是主观的、有噪声的、不完整的，被限制在一个非线性流形中，可能来自多个来源，并且可能是不一致的。以前的结果已经显示了如何将这些信息嵌入到欧几里德空间中，以便可以使用在欧几里德空间中操作的方法。最近的两个新的和非常强大的工具，距离相关和距离分量，提供了对任意变量组之间的相关性的原则性检验和对分布相等性的检验，仅基于成对的欧几里得距离，并且基本上不需要分布假设。因此，将非度量信息嵌入到欧氏空间，然后使用距离相关性和对欧氏数据进行操作的距离分量相结合的方法，提供了一种重要的新方法来使用“杂乱的”成对数据。此外，距离相关和距离分量正在通过参数化、调整和测试技术被扩展到某些回归、分类和变量/模式选择问题，在适当的情况下通过嵌入技术在此之前。实施本计划各方面的一系列任务以原则性的方式提供了非传统信息的回归、分类和变量/模式选择等主要统计任务的进展。这项工作为统计学家/数据分析师和各种科学领域的建模人员提供了大量实用工具集，以从数据集中提取信息以从数据集中从小到大进行预测、分类和选择重要变量/模式，这些变量/模式包括来自各种结构的距离或不同信息，这些结构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，这些工具将被广泛传播，并在一定程度上帮助研究人员从包含公共利益的信息的生物、医学、环境和其他数据集中提取信息，从而对社会产生影响。该项目包括在一个重要的STEM领域对一名博士生进行高级培训。

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.