The study of real-life dynamical problems is associated with complex mathematical models, which are described by means of stochastic dynamical systems. In these models it is essential to estimate states and parameters by means of observations. However, these observations are usually imperfect, so the parameters and states obtained are not optimal. Another important problem within mathematical modeling by means of stochastic differential equations is the study of methods to find solutions to said equations.
The project “Bayesian Estimation of a Mixed Effects Model Defined by a Stochastic Differential Equation” led by Professor Saba Infante, PhD., four researchers, and one undergraduate student, studied these problems for the past two years. He was in charge of creating the structure of a mixed effects model, carrying out theoretical calculations of the likelihood of the model, and the study of methods to find approximate solutions of the equations associated with the model. To do this, algorithms such as Markov Monte Carlo (MCMC) were implemented in the R language, which allowed the estimation quality of the model to be calibrated.
As a result of this project, both a doctoral thesis and an undergraduate thesis were written. In addition, their results were presented at national and international conferences, and resulted in the publication of three articles: Ensemble Kalman filter and extended Kalman filter for state-parameter dual estimation in Mixed effects Models defined; Unscented Kalman Filter and Gauss-Hermite Kalman Filter for Range-Bearing Target Tracking; and Stimulation of a mixed effects model using a partially observed diffusion process.
