Multivariate Distribution-Free Nonparametric Testing Using Optimal Transportation

近年来，非参数方法在统计学的理论和实践中越来越受欢迎，主要是因为它们比参数方法具有更大的灵活性和更多的“数据驱动”特征。在这个项目中，主要研究者（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，这一领域得到的关注相对较少，但在执行真实的数据分析时经常需要。所开发的方法将是无调谐的，计算上可行的，并在底层模型的最小假设下定义良好。在合作方面，PI将继续进行天文学的跨学科研究，特别是本项目中讨论的一些方法将解决天文学产生的重要科学问题，并将形成哥伦比亚两名博士生的论文。PI还将继续指导暑期实习生的传统。研究生的支持将用于跨学科的研究和编写代码。PI调查非参数统计研究的两个核心方向，特别强调多变量数据。本计画的主要目的是研究多元分布自由的非参数秩方法，将经典的单变量秩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传输理论的思想-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变量非参数（异方差）混合模型，并直接由天文学合作涉及PI的动机。由于观测上的限制，特别是异方差的测量误差，对感兴趣的恒星星族的统计推断变得复杂。事实上，几乎所有天文学数据集都包含每个观测的已知（异方差）误差测量。这自然会导致可以建模为非参数（异方差）混合模型的数据。PI和他的合作者沿着将研究这个问题的几个方面：（a）数据分布的估计，（B）对观察结果进行降噪，以及（c）研究相关的反卷积和流形学习问题。对于上述两个问题，将进行系统的理论研究的方法。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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.