Classification and Regression using AddiVortes: (Bayesian) Additive Voronoi Tessellations

This thesis presents the development of AddiVortes, a novel algorithm that integrates Voronoi tessellations within a Bayesian additive ensemble framework. By leveraging the flexible spatial partitioning capabilities of Voronoi tessellations, the method is designed to capture the interactions between covariates more effectively, where traditional axis-parallel splits often prove suboptimal. Extensive empirical analysis on both real-world and simulated datasets demonstrates that AddiVortes performs competitively against leading ``black-box'' models, including random forests and Bayesian Additive Regression Trees (BART). To address the challenge of categorical data, the thesis introduces a new technique called permutation encoding for effectively incorporating nominal covariates. The framework is further extended to handle binary and nominal classification through probit link functions and data augmentation, as well as heteroscedastic regression by employing a secondary ensemble to model covariate-dependent noise. The work concludes with a discussion of the algorithm’s properties and outlines potential directions for future research.