Jia Xu - MSc Defence

TITLE: Robust Estimation for Differential Equation Models

ABSTRACT: Ordinary Differential Equations (ODEs) are widely used in Biology, Economics, Finance and other fields. In most cases, unknown parameters are involved in the ODEs. Given the data available, we need to estimate the values of these parameters. Methods such as maximum likelihood and least squares have their shortages to address this problem. We combine generalized profiling and robust methods to propose a new approach. The solution to an ODE is represented by a linear combination of basis functions, such as B-spline basis. We also define a penalty term to control the roughness of the solution and maintain the fidelity of the solution to the ODE model. The basis coefficients and ODE parameters are estimated in two levels of optimization. In the inner-optimization, the coefficient estimates are treated as an implicit function of the ODE parameters. In the outer-optimization, we estimate the ODE parameters based on the outcome of the inner-optimization. Simulation studies show that the robust method gives satisfactory estimates for the ODE parameters from noisy data with outliers. We also apply the robust method to a predator-prey ODE model with a real ecological data set.