Poisson-response Tensor-on-Tensor Regression and Applications

TL;DR

The paper introduces PToTR, a new tensor regression framework for modeling count data with Poisson responses across multiple dimensions, with applications in diverse fields.

Contribution

It develops a novel Poisson-response tensor-on-tensor regression model with algorithms and initial theoretical analysis for structured count data.

Findings

Demonstrated PToTR's effectiveness in longitudinal crisis data analysis

Applied PToTR to PET image reconstruction successfully

Used PToTR for change-point detection in communication patterns

Abstract

We introduce Poisson-response tensor-on-tensor regression (PToTR), a novel regression framework designed to handle tensor responses composed element-wise of random Poisson-distributed counts. Tensors, or multi-dimensional arrays, composed of counts are common data in fields such as international relations, social networks, epidemiology, and medical imaging, where events occur across multiple dimensions like time, location, and dyads. PToTR accommodates such tensor responses alongside tensor covariates, providing a versatile tool for multi-dimensional data analysis. We propose algorithms for maximum likelihood estimation under a canonical polyadic (CP) structure on the regression coefficient tensor that satisfy the positivity of Poisson parameters and then provide an initial theoretical error analysis for PToTR estimators. We also demonstrate the utility of PToTR through three concrete applications: longitudinal data analysis of the Integrated Crisis Early Warning System database, positron emission tomography (PET) image reconstruction, and change-point detection of communication patterns in longitudinal dyadic data. These applications highlight the versatility of PToTR in addressing complex, structured count data across various domains.