Homogeneity Pursuit in Regression Analysis: Statistical Theory, Integer Optimization, and Algorithms

该研究项目是一个跨学科领域，涉及统计学、运筹学和机器学习，其动机是研究未知量和未知的相关结构，称为同质性追求。同质性追求统计范式的新颖性在于它能够在关联分析中同时进行参数聚类和参数估计。该项目希望为从业者提供新的统计工具，从数据中产生新的知识。首席研究员将应用开发的方法来推导有毒物质的环境暴露混合物，表观遗传学中的DNA甲基化整合，以及社会科学中的调查问卷总结。该项目还包括涉及研究生的大量教育倡议，并使受训者接触与研究活动有关的主题的最新研究。该项目为关联分析开发了一个新的统计框架，其中相似的模型参数在估计时被融合到子组中。所开发的方法利用混合整数规划（MIO）扩展最佳子集正则化，在回归分析中同时执行参数聚类和估计操作。它还提供了分析和算法工具来改进现有的统计解决方案。首先，该项目建立了一个新的MIO公式，用于高维模型参数的同时聚类和估计。该框架灵活有效地拟合了广泛的重要统计模型，包括用于横截面数据的广义线性模型（GLMs）、用于非线性相互作用的变系数指数模型和用于纵向数据的混合效应模型。其次，为了解决MIO问题，本课题开发并实现了一种新的快速可靠的算法，称为l -零损耗优化交替惩罚算子（APOLLO）。第三，该项目计划建立glm和半参数模型中群体成员和群体参数的MIO估计器的有限和大样本性质，并研究整数优化理论来证明MIO求解器APOLLO的合理性。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 research project lies in a cross-disciplinary field that intersects statistics, operations research, and machine learning with a motivation to study unknown quantities and associated structures of the unknowns, referred to as homogeneity pursuit. The novelty of the statistical paradigm of homogeneity pursuit pertains to its capacity for simultaneous operation of parameter clustering and estimation in association analyses. The project expects to deliver new statistical tools to the hands of practitioners to generate new knowledge from data. The principal investigator will apply the developed methodology for the derivation of environmental exposure mixtures of toxic agents, DNA methylation integration in epigenetics, and survey questionnaire summarization in social sciences. The project also includes substantial educational initiatives involving graduate students and exposing trainees to state-of-the-art research in the topics related to the research activities. The project develops a new statistical framework for association analyses in which similar model parameters are fused into subgroups while being estimated. The developed methodology harnesses mixed integer programming (MIO) to extend the best-subset regularization to perform a simultaneous operation of parameter clustering and estimation in regression analysis. It also provides both analytic and algorithmic tools to improve the existing statistical solutions. First, the project builds a new MIO formulation of simultaneous clustering and estimation for high-dimensional model parameters. The framework is flexible and efficient to fit a wide range of important statistical models, including generalized linear models (GLMs) for cross-sectional data, varying coefficient index models for nonlinear interactions, and mixed-effects models for longitudinal data. Second, to solve MIO problems, the project develops and implements a new fast and reliable algorithm, termed as Alternating Penalized Operator for L-zero Loss Optimization (APOLLO). Third, the project plans to establish both finite and large sample properties of the MIO estimator for group memberships and group parameters in GLMs and semiparametric models and investigate the theory of integer optimization to justify the MIO solver, APOLLO.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.