Measure of Heterogeneity for Complex Data Objects

技术进步导致了大型复杂数据对象(如矩阵、张量、函数和流形)的常规收集。从庞大而复杂的数据来源中理解和得出结论，并以合理的科学理论为基础，是具有挑战性的。困难程度通常随着数据的大小和复杂程度而增加。大多数现有的统计工具都是将数字作为信息单位来处理的，它们变得不够充分，因为例如，一个人的脸可以数字化成许多数字，但必须作为一个整体进行分析，以保留必要的信息。该项目的目标是为这一需要开发适当的统计方法和软件。即将开发的概念和方法将为复杂数据的分析提供重要工具，这些数据在科学和工程中都有广泛的应用。该项目不仅将推进统计方法和传播计算机软件，而且还将为重大和具有挑战性的科学和公共卫生问题提供解决办法。在理解和诊断阿尔茨海默病和乳腺癌方面的应用是具有重大公共卫生意义的两个例子。此外，通过吸引博士后和博士后学生、初级教员和暑期实习生，PI将利用这一项目培训新一代统计学家和数据科学家。为了考虑复杂的数据结构，并在统计分析过程中保留必要的信息，重要的是将某些结构作为观察点，例如在给定时间收集的人的功能磁共振成像(MRI)。这样的高维点被称为张量，它可能适合也可能不适合传统定义的欧几里德空间。该项目旨在开发统计方法，将张量作为数据对象进行分析，并在可能的非欧几里德空间中对这些数据对象进行分类。特别是，该项目引入了球状杂质的概念，作为复杂数据对象分布的异质性的衡量标准，并将研究其在开发基于树的方法以对非欧几里德空间中的数据对象进行分类方面的使用。将努力深入了解球状杂质的理论和经验性质，并将同时开发和分发软件。所开发的方法将用于分析来自英国Biobank和DBGaP的大型公共数据库，如Human Connectome Project。对这些重要数据集的分析不仅可以评估新方法的实用性，还可以带来有洞察力的新科学发现。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Technological advances have led to the routine collection of large and complex data objects such as matrices, tensors, functions, and manifolds. Understanding and drawing conclusions from large and complex data sources with a sound scientific rationale is challenging. The level of difficulty generally increases with the size and complexity of the data. Most existing statistical tools are designed to deal with a number as the unit of information, and they become inadequate, because a human face, for example, can be digitalized into many numbers but must be analyzed as a whole, to retain the essential information. The goal of this project is to develop proper statistical methods and software for this need. The concept and methods to be developed will provide vital tools for the analysis of complex data, which have broad applications in both science and engineering. This project will not only advance statistical methodology and disseminate computing software but also offer solutions to important and challenging scientific and public health problems. Applications in understanding and diagnosing Alzheimer's disease and breast cancer are two examples of major public health significance. Furthermore, by engaging doctoral and postdoctoral students, junior faculty members, and summer interns, the PI will take advantage of this project in training new generations of statisticians and data scientists. To consider the complex data structures and to retain the essential information during the statistical analysis, it is important to treat certain structures as the observational point such as the functional resonance imaging (MRI) collected from a person at a given time. Such high dimensional points are referred to as tensors which may or may not fit in a traditionally defined Euclidean space. This project aims to develop statistical methods to analyze tensors as data objects, and classify such data objects in a possibly non-Euclidean space. In particular, this project introduces the concept of ball impurity as a measure of heterogeneity in the distribution of complex data objects and will investigate its use in developing tree-based methods to classify data objects in non-Euclidean spaces. The efforts will be made for an in-depth understanding of the theoretical and empirical properties of the ball impurity, and software will be developed and distributed simultaneously. The developed methods will be used to analyze large-scale, public databases from UK Biobank and dbGaP such as Human Connectome Project. The analyses of these important datasets can not only assess the utility of the novel methods, but also lead to insightful and new scientific discoveries.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.