Algebraic geometry of Poisson regression

TL;DR

This paper explores the algebraic and geometric structure of optimal experimental designs for Poisson regression models, focusing on the Rasch Poisson counts model and its regions of optimality in parameter space.

Contribution

It introduces a novel algebraic and geometric framework to characterize optimal designs in Poisson regression, extending previous results to models with interactions.

Findings

Regions of optimality are often semi-algebraic with symmetries.

Characterization of optimality regions for designs with interaction effects.

Application of polyhedral and spectrahedral geometry to design optimality analysis.

Abstract

Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate local optimality. We propose to study for a given design its region of optimality in parameter space. Often these regions are semi-algebraic and feature interesting symmetries. We demonstrate this with the Rasch Poisson counts model. For any given interaction order between the explanatory variables we give a characterization of the regions of optimality of a special saturated design. This extends known results from the case of no interaction. We also give an algebraic and geometric perspective on optimality of experimental designs for the Rasch Poisson counts model using polyhedral and spectrahedral geometry.