Advanced Bayesian Modeling for Spatial Functional Data: Non-Stationary Dynamics, Irregular Domains, and Scalable Inference

• Grant number: 26/09173-5
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: October 01, 2026
• End date: September 30, 2027
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Statistics
• Principal Investigator: Luiz Koodi Hotta
• Grantee: Alvaro Alexander Burbano Moreno
• Supervisor: Jorge Mateu
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Universitat Jaume I, Spain
• Associated to the scholarship: 23/18036-3 - Hierarchical Bayesian model for predicting spatially correlated curves and with non-equidistant spacing, BP.PD

Abstract

The integration of Functional Data Analysis and Spatial Statistics has significantly advanced the modeling of continuous curves in fields such as environmental science, meteorology, and epidemiology. However, Spatial Functional Data Analysis (SFDA) currently faces critical theoretical and computational barriers: the inability to handle irregular domains in full function-on-function regressions efficiently, the restrictive assumption of strictly stationary and isotropic spatial covariance structures, and the severe dimensionality bottlenecks inherent to complex hierarchical models estimated via traditional Markov Chain Monte Carlo (MCMC) algorithms.This research proposes an advanced Bayesian hierarchical framework to overcome these challenges. First, we develop a spatial function-on-function regression model designed for non-equidistant observations. By leveraging B-splines and Bernstein polynomials, this approach evaluates functional integrals algebraically via inner products, effectively bypassing the numerical instability of standard quadrature methods on irregular domains. Second, the statistical formulation is extended to capture non-stationary spatio-temporal behavior and geometric anisotropy by implementing a dynamic Matérn covariance structure. Finally, to address the high computational burden of MCMC sampling, we transition to Variational Inference (VI) within the Stan ecosystem. This scalable analytical optimization is theoretically bridged with Stochastic Partial Differential Equations (SPDEs), laying a rigorous mathematical foundation for future integration with Gaussian Markov Random Field (GMRF) solvers.Validated through high-dimensional environmental datasets (e.g., $PM_{10}$ and meteorological fluctuations), this project aims to consolidate robust predictive methodologies and deliver highly scalable, open-source computational tools to the scientific community. (AU)