Estimation, Computation, and Uncertainty Quantification in Structured Regression Models

在统计建模中，回归是研究响应变量和一组预测因子之间关系的主要工具。在本研究计划中，将研究一些结构化回归模型中有关估计、计算和不确定性量化的问题。强加的“结构”指的是系统的已知特征（领域知识），并有助于降低拟合统计模型/过程的复杂性。此外，强加这样的结构会产生可解释（但灵活）的模型。特别强调适用于多变量数据的方法，这是一个受到相对较少关注的领域，尽管在进行有效的数据分析时往往是必要的。在这个项目中进行的一些方法发展将解决由天文数据引起的重要科学问题。研究者还计划继续指导本科暑期实习生的传统，并参加纽约大学GSTEM外展项目，这是一个为期六周的高中女生暑期项目。在这个项目中追求的三个主要主题是：(i)在多个假设检验问题中纳入协变量信息；（ii）回归模型中的凸性约束估计和推理；（iii）用于检测（多元）回归函数中分段常数/仿射结构的统计方法。随着高效计算算法的发展，将开发新的方法来解决这些问题。此外，将对这些程序进行系统的理论研究，重点关注它们的适应性（风险）属性，并解决推理和不确定性量化的重要问题。该研究的预期应用是多种多样的，从估计遥远星系中恒星的径向速度分布（天文学），到开发检测神经元对之间相互作用的方法（神经科学），到估计生产和效用函数（经济学），以及为连续多元分段仿射回归函数中的参数构建置信区间（工程学）。

In statistical modeling, regression is the primary tool to study the relationship between a response variable and a collection of predictors. In this research project, questions related to estimation, computation, and uncertainty quantification in some structured regression models will be investigated. The imposed "structure" refers to known features (domain knowledge) of a system and helps to reduce the complexity of the fitted statistical model/procedure. Further, imposing such structures yields interpretable (yet flexible) models. Special emphasis is given to methods applicable to multivariate data, an area that has received relatively less attention, though often necessary in performing effective data analysis. Some of the methodological development undertaken in this project will address important scientific questions arising from astronomical data. The investigator also plans to continue the tradition of mentoring undergraduate summer interns and to participate in the NYU GSTEM outreach program, a six-week summer program for high school girls.The three main topics pursued in this project are: (i) incorporating covariate information in multiple hypothesis testing problems; (ii) convexity constrained estimation and inference in regression models; and (iii) statistical methods that are geared towards detecting piecewise constant/affine structure in a (multivariate) regression function. New methodology will be developed to address these topics along with the development of efficient algorithms for computation. Further, a systematic theoretical study of these procedures, focusing on their adaptive (risk) properties, will be undertaken, and the important issues of inference and uncertainty quantification will be addressed. The intended applications of the research are diverse, ranging from estimation of radial velocity distribution of stars in a distant galaxy (astronomy), to developing methodology for detecting interactions between pairs of neurons (neuroscience), to estimating production and utility functions (economics), and to constructing confidence intervals for parameters in a continuous multivariate piecewise affine regression function (engineering).