A parameter estimation method for continuous time dynamical systems based on the unscented Kalman filter and maximum likelihood

J. Carlsson, C. Nordheim, Master thesis, Chalmers University of Technology, supervisors J. Hagmar, M. Anguelova, examiner J. Sjöberg, May 2011.


Acquiring good models of biological and biochemical systems is important in e.g. drug development. These systems are commonly modeled as continuous dynamical systems via ordinary di erential equations (ODEs). Measurements on these systems are often taken at discrete time instants, which together with the ODEs yields a system of both continuous and discrete equations. It is usually not realistic to expect the ODEs to provide an exact description of the system. Modeling the system with stochastic di erential equations, together with adding noise terms to the measurement equations, is a formal way of including the uncertainties in both the system and the measurement model. This thesis addresses the problem of estimating parameters in this class of models. The parameter estimation framework that we develop consists of maximum likelihood estimation of the parameters, where the likelihood is approximated via predictions from the unscented Kalman lter. The optimization in the parameter space is performed using a local gradient based method. The gradient is computed analytically by di erentiating the lter equations of the unscented Kalman lter with respect to the parameters. This framework is implemented in Mathematica and validated using two benchmark problems. The performance is compared to that of a corresponding, previously used, framework using the extended Kalman lter. The framework is also compared with frameworks using rst and second order nite di erence approximations of the gradient. For the two benchmark problems, no improvement is observed by using the unscented Kalman lter instead of the extended Kalman lter. The framework using analytical gradient is computationally faster than those using nite di erences for both of the benchmark problems. For one of the benchmark problems, the framework using the analytical gradient gives parameter estimates comparable to both the nite di erence frameworks; for the other benchmark problem, the rst order di erence framework gives considerably worse parameter estimates than the second order di erence framework and the framework using the analytical gradient.

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