An Efficient Class of Bayesian Generalized Quadratic Nonlinear Dynamic Models with Application to Birth Rate Monitoring

Many real-world spatio-temporal processes exhibit nonlinear dynamics that can often be described through stochastic partial differential equations. These models are flexible and scientifically motivated, however, implementing them in a fully Bayesian framework can be computationally challenging. We are motivated by birth rate data, which has important implications for public health and are known to follow nonlinear dynamics. We propose a covariance calibration strategy that specifies the covariance matrix of a linear mixed effects model to be close in Frobenius norm to that of a Generalized Quadratic Nonlinearity (GQN) model. We refer to this as Frobenius norm matching. This allows us to model nonlinear dynamics using an easier to implement linear framework. The calibrated linear model is efficiently implemented using Exact Posterior Regression (EPR), a recently proposed Bayesian model that enables sampling of fixed and random effects directly from the posterior distribution. We provide simulation studies that compare to implementations using MCMC. Finally, we use this approach to analyze Florida county-level birth rate data from 1990-2023. Our results indicate that our non-linear spatio-temporal model outperforms linear dynamic spatio-temporal models for this data, and identifies covariate effects consistent with existing literature, all while avoiding the computational difficulties of MCMC.