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

该项目的目标是系统地开发一套数据探索和可视化工具，用于一种新型的回归分析数据形式，这种数据形式在最近的应用中变得越来越普遍。这些被称为随机对象的数据不具备传统数据的一些基本属性：例如，它们没有传统分析中认为理所当然的方向或角度。例子包括死亡率分布、大协方差矩阵和对球体的观察。许多现有的统计工具，如最小二乘、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fracimchet回归，用于处理这种数据类型。目前的项目旨在填补新数据类型与传统方法之间的空白，将随机对象转换为可被传统方法高效访问的形式。该项目将重点关注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，以协助初步的数据探索、数据可视化、模型诊断和提高估计精度。该项目还将包括对现代统计科学的研究生进行培训和指导。该项目旨在开发灵活和计算可扩展的方法，以便为一种新的回归类型提供足够的降维，其中预测器和响应都可以是度量空间值随机对象。所得结果适用于线性和非线性两种情况。其基本思想可用于将现有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是观测值之间没有内积或外积，而大多数传统统计工具（如协方差矩阵、相关、投影、回归和ANOVA分解）都需要这些内积或外积。为了克服这个困难，该项目采用了一个通用核，它弥合了度量空间和希尔伯特空间之间的差距，它允许通过正交化过程在希尔伯特空间的框架内构建独立结构。一般来说，通用核提供的桥梁在度量空间值数据分析中是至关重要的，它远远超出了当前充分降维的设置，因为当前大量的多变量和泛函数据分析方法只能在希尔伯特空间设置中使用。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。

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.