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预测变量集合之间关系的主要工具。在本研究计画中，将探讨结构化回归模型中的估计、计算与不确定性量化等相关问题。 强加的“结构”是指系统的已知特征（领域知识），有助于降低拟合统计模型/程序的复杂性。 此外，强加这样的结构会产生可解释的（但灵活的）模型。 特别强调适用于多变量数据的方法，这是一个相对较少关注的领域，尽管在进行有效的数据分析时经常是必要的。 在这个项目中进行的一些方法发展将涉及天文数据所引起的重要科学问题。研究者还计划继续指导本科生暑期实习生的传统，并参加纽约大学GSTEM外展计划，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课题是：（i）将协变量信息纳入多个假设检验问题;（ii）凸性约束估计和回归模型中的推理;（iii）将协变量信息纳入多个假设检验问题。以及（iii）适合于检测（多变量）回归函数中的分段常数/仿射结构的统计方法。新的方法将开发，以解决这些问题沿着的发展，有效的算法计算。此外，这些程序的系统的理论研究，侧重于他们的适应性（风险）属性，将进行，推理和不确定性量化的重要问题将得到解决。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），到开发用于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），以及构建连续多变量分段仿射回归函数（工程学）中参数的置信区间。

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).