Parameter estimation using polynomial chaos and maximum likelihood

Abstract

When modelling biological processes, there are always errors, uncertainties and variations present. In this paper, we consider the coefficients in the mathematical model to be random variables, whose distribution and moments are unknown a priori, and need to be determined by comparison with experimental data. A stochastic spectral representation of the parameters and the solution stochastic process is used, based on polynomial chaoses. The polynomial chaos representation generates a system of equations of the same type as the original model. The inverse problem of finding the parameters is reduced to establishing the best-fit values of the random variables that represent them, and this is done using maximum likelihood estimation. In particular, in modelling biofilm growth, there are variations, measurement errors and uncertainties in the processes. The biofilm growth model is given by a parabolic differential equation, so the polynomial chaos formulation generates a system of partial differential equations. Examples are presented.