Dimension Reduction and Data Visualization for Regression Analysis of Metric-Space-Valued Data

用于度量空间值数据回归分析的降维和数据可视化

• 批准号: 2210775
• 负责人: Bing Li
• 金额: $ 29万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2210775&HistoricalAwards=false
• 关键词: Dimension; Reduction; Data; Visualization; Regression

The goal of this project is to systematically develop a set of data exploration and visualization tools for a new type of regression analysis for a form of data that has become increasingly common in recent applications. Such data, known as random objects, do not possess some basic properties of conventional data: for example, they do not have directions or angles that are taken for granted in conventional analysis. Examples include mortality distributions, large-covariance matrices, and observations on spheres. Many existing statistical tools, such as least squares, regression, R-squares, and dimension reduction, cannot be directly applied. A new type of regression, called Fréchet regression, has recently been developed to handle this data type. The current project aims to fill the gap between the new data type and conventional methods by transforming random objects into forms that are accessible by conventional methods with high efficiency. The project will focus on sufficient dimension reduction for the new data type. The results are expected to provide data analysis tools and related computer packages for the new type of regression, to assist preliminary data exploration, data visualization, model diagnostics, and improved estimation accuracy. The project will also involve training and mentoring for graduate students in modern statistical sciences.The project aims to develop flexible and computationally scalable methods for sufficient dimension reduction for a new type of regression where both the predictor and the response can be metric-space-valued random objects. The results are intended to apply in both linear and nonlinear cases. The underlying idea can be used convert existing methods from the multivariate setting to metric-space-valued random elements. The main difficulty in dealing with metric-space-valued random objects is that there are no inner or outer products between observations, which are required by most of the traditional statistical tools, such as covariance matrices, correlation, projection, regression, and ANOVA decomposition. To circumvent this difficulty, the project employs a universal kernel that bridges the gap between metric spaces and Hilbert spaces, which allows construction of an independence structure within the framework of Hilbert spaces though the process of orthogonalization. The bridge provided by the universal kernel is of fundamental importance in metric-space-valued data analysis in general, going far beyond the current setting of sufficient dimension reduction, because a great number of current methods for multivariate and functional data analysis can only be used in the Hilbert space setting.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.

该项目的目标是系统地开发一套数据探索和可视化工具，用于对最近应用中越来越常见的数据形式进行新型回归分析。这种数据被称为随机对象，不具备常规数据的一些基本属性：例如，它们不具有常规分析中认为理所当然的方向或角度。示例包括死亡率分布、大协方差矩阵和球体上的观测。许多现有的统计工具，如最小二乘法，回归，R平方和降维，不能直接应用。最近开发了一种新的回归类型，称为Fréchet回归，用于处理这种数据类型。目前的项目旨在填补新的数据类型和传统的方法之间的差距，通过转换随机对象的形式，可以通过传统的方法，以高效率访问。该项目将侧重于新数据类型的充分降维。研究结果有望为新型回归提供数据分析工具和相关的计算机软件包，以帮助初步的数据探索、数据可视化、模型诊断和提高估计精度。该项目还将涉及对现代统计科学研究生的培训和指导，旨在为一种新型回归开发灵活的、可在计算上扩展的方法，以便充分降低维数，其中预测器和响应都可以是度量空间值随机对象。结果适用于线性和非线性的情况。其基本思想可以用于将现有的方法从多变量设置转换为度量空间值随机元素。处理度量空间值随机对象的主要困难在于观测值之间没有内积或外积，这是大多数传统统计工具所需要的，例如协方差矩阵，相关性，投影，回归和ANOVA分解。为了克服这一困难，该项目采用了一个通用的核，桥梁之间的差距差距度量空间和希尔伯特空间，这使得建设一个独立的结构内的希尔伯特空间的框架内，通过正交化的过程。由通用核提供的桥梁在一般的度量空间值数据分析中具有根本的重要性，远远超出了充分降维的当前设置，因为目前大量的多元和函数数据分析方法只能在希尔伯特空间环境中使用。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准。