Measure of Heterogeneity for Complex Data Objects

技术的进步导致了大型和复杂数据对象的常规收集，如矩阵、张量、函数和流形。从庞大而复杂的数据来源中理解并得出具有合理科学原理的结论是具有挑战性的。难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设计来处理作为信息单位的数字，它们变得不充分，因为一个人的脸，例如，可以被数字化成许多数字，但必须作为一个整体来分析，以保留基本信息。该项目的目标是为这一需求开发适当的统计方法和软件。所开发的概念和方法将为复杂数据的分析提供重要的工具，在科学和工程领域都有广泛的应用。该项目不仅将推进统计方法和传播计算软件，而且还将为重要和具有挑战性的科学和公共卫生问题提供解决方案。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，通过吸引博士和博士后学生、初级教员和暑期实习生，PI将利用该项目培养新一代的统计学家和数据科学家。为了在统计分析中考虑复杂的数据结构并保留必要的信息，重要的是将某些结构作为观测点，例如在给定时间从人身上收集的功能磁共振成像（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得空间。该项目旨在开发统计方法来分析张量作为数据对象，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，该项目引入了球杂质的概念，作为复杂数据对象分布的异质性的度量，并将研究其在开发基于树的方法中对非欧几里得空间中的数据对象进行分类的用途。将努力深入了解球杂质的理论和经验性质，同时开发和分发软件。所开发的方法将用于分析来自UK 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效用，而且还可以导致深刻的和新的科学发现。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.