Problems in Commutative Algebra and Functional Data Analysis

Abstract

This dissertation consists of three papers spanning two areas of mathematics and statistics: combinatorial commutative algebra and functional data analysis. The first paper examines the weak Lefschetz property (WLP) for Artinian algebras arising from hyperplane arrangements associated to graphs. In particular, we study the Artinian Orlik-Terao algebra of graphic arrangements and investigate when it satisfies WLP. We provide classifications for several families of graphs, including forests, cycle graphs, and all graphs on five vertices. We also identify structural features that force Lefschetz behavior.

The second and third papers lie in functional data analysis (FDA) and focus on regression problems with functional inputs. The second paper develops a unified, measure-theoretic framework for regression, showing that several classical functional and multivariate regression methods can be interpreted as estimating a common underlying object with respect to different measures. Within this framework, we conduct a comprehensive simulation study comparing classical methods and two types of modern functional neural networks, identifying settings in which different approaches perform well.

The final paper studies the Tecator spectrometric data set and extends existing functional neural network methods, previously defined only for single scalar responses, to the multiple scalar response setting. We review existing approaches in the literature, most of which treat responses separately or only consider one response, and we develop models that jointly predict multiple responses. Our results demonstrate the advantages of joint modeling and highlight the effectiveness of functional neural network methods in this context.

Together, this work contributes new theoretical results to commutative algebra and provides a unified perspective, empirical analysis, and modern methodological extensions in functional data analysis.