Bayesian filters for parameter estimations in istochastic differential equations mixed-effects models - Quito Mendoza, Brandon Estéfano (2021)
Estimation of parameters in Stochastic Differential Equations (SDE) models is not straightforward. Mathematical models that describes real life dynamic systems usually are nonlinear type and involve several parameters. A natural approach would be the maximum likelihood methods, however, the transition densities are rarely known, and therefore is not possible to explicitly obtain the likelihood. The inference and analysis of dynamic systems that includes stochastic process allows to estimate unknown states and parameters, including a observation process (state-space models) that usually contains errors. The practical use of this estimation methods is common, specially due to its application on different research topics such as: finance, telecommunications, audio signal processing, optimum process control, machine learning, global position systems, physical phenomena systems, infectious disease outbreak modeling. The last one have important interest in this project. This thesis first show a review of the background of Bayesian filtering, a review of the literature and basic concepts around the same topic and numerical approximation methods e.g., Markov chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC). This recursive filters allow to perform inference on unknown parameters and states from imperfect observations and with a finite number of iterations. We propose a methodology based on the Bayesian paradigm that implements the mentioned recursive filters in SDE models, in specific the stochastic SEIR epidemic model and a Mixed-Effects model version of this epidemic model is proposed. Results of the inference of the particle filter in the SEIR model with real data from COVID-19 epidemic in Ecuador is presented. Computational cost and other features are discussed, with a general performance conclusion.