Statistics in Imaging: Dr. Carlos Llosa

Abstract:  We introduce Poisson Tensor-on-Tensor Regression (PToTR), a novel regression framework designed to handle tensor responses composed element-wise of random Poisson-distributed counts. Tensors 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 predictors, providing a robust and versatile tool for multi-dimensional data analysis. We propose algorithms for maximum likelihood estimation (MLE) under a canonical-polyadic (CP) structure on the regression coefficient tensor that satisfy the positivity of Poisson parameters. We demonstrate the utility of PToTR through three concrete applications. First, we implement an autoregressive PToTR on the Integrated Crisis Early Warning System database, showing improved performance when compared to Gaussian ToTR. Second, we utilize PToTR for positron emission tomography image reconstruction, leveraging CP regularization to achieve more robust and stable image reconstruction when compared to the traditional MLE algorithm. Third, we introduce the Poisson Tensor Analysis of Variance (PTANOVA) model as a special case of PToTR. A one-way PTANOVA is used for change-point detection of communication patterns in longitudinal dyadic data. These applications highlight the versatility and robustness of PToTR in addressing complex, structured count data across various domains.