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

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.