Multivariate Distribution-Free Nonparametric Testing Using Optimal Transportation

近年来，非参数方法在统计理论和实践中变得越来越流行，主要是因为它们比参数方法具有更大的灵活性和更多“数据驱动”的特征。在这个项目中，首席研究员（PI）分析了非参数统计的两个重要领域的估计、计算和不确定性量化。特别强调适用于多变量数据的方法，这一领域受到的关注相对较少，但在进行实际数据分析时往往是必要的。所开发的方法将是无调优的，计算上可行的，并且在对底层模型的最小假设下定义良好。在合作方面，PI将继续在天文学领域进行跨学科研究，特别是，该项目中讨论的一些方法将解决天文学中出现的重要科学问题，并将形成哥伦比亚大学两名博士生的学位论文。PI还将继续指导暑期实习生的传统。研究生支持将用于跨学科研究和编写代码。PI研究了非参数统计研究的两个核心方向，特别强调多变量数据。本课题主要研究基于多元无分布的非参数排序方法，将经典的单变量排序方法推广到多元数据。这个新的通用框架关键地使用了最优运输理论的思想，这是应用数学/概率/机器学习中一个重要且非常活跃的研究领域。该项目的第二部分是多元非参数（异方差）混合模型，直接受到PI天文学合作的推动。重要的观测限制，特别是异方差测量误差，使对感兴趣的恒星群的统计推断变得复杂。事实上，天文学中几乎所有的数据集都包含每次观测的已知（异方差）误差测量。这自然导致可以将数据建模为非参数（异方差）混合模型。PI将与他的合作者一起研究这个问题的几个方面：(a)数据分布的估计，(b)对观测值进行去噪，(c)研究相关的反卷积和流形学习问题。针对上述两个问题，本文将对这些方法进行系统的理论研究。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。

Nonparametric methods have become increasingly popular in the theory and practice of statistics in recent times primarily because of their fundamental advantages over parametric methods: greater flexibility and more "data-driven'' features. In this project the Principal Investigator (PI) analyzes the estimation, computation and uncertainty quantification in two important areas of nonparametric statistics. Special emphasis is given to methods applicable to multivariate data, an area that has received relatively less attention, though often necessary in performing real data analyses. The methods developed will be tuning-free, computationally feasible, and well-defined under minimal assumptions on the underlying models. On the collaborative front, the PI will continue interdisciplinary research in astronomy and in particular, some of the methodology discussed in this project will address important scientific questions arising from astronomy and will form the dissertation theses of two PhD students at Columbia. The PI will also continue the tradition of mentoring summer interns. The graduate student support will be used on interdisciplinary research and writing codes.The PI investigates two core directions of research in nonparametric statistics with special emphasis to multivariate data. The main trust of this project is to study multivariate distribution-free nonparametric rank-methods based that generalize the classical univariate rank-based procedures to multivariate data. This new general framework crucially uses ideas from the theory of optimal transport – an important and very active research area in applied mathematics/probability/machine learning. The second part of the project is on multivariate nonparametric (heteroscedastic) mixture models and is directly motivated by astronomy collaborations involving the PI. Statistical inference of stellar populations of interest is complicated by significant observational limitations – in particular, by heteroscedastic measurement errors. Indeed, almost all data sets in astronomy contain known (heteroscedastic) error measurements on every observation. This naturally leads to data that can be modeled as nonparametric (heteroscedastic) mixture models. The PI, along with his collaborators, will investigate several aspects of this problem: (a) estimation of the data distributions, (b) denoising the observations, and (c) studying the associated deconvolution and manifold learning problems. For both the above problems, a systematic theoretical study of the methods will be undertaken.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.