  An important part of system modeling is determining parameter values,
particularly for biomolecular systems, where direct measurements of individual
parameters are typically hard. While Extended Kalman Filters have been used for
this purpose, the choice of the process noise covariance is generally unclear.
In this chapter, we address this issue for biomolecular systems using a
combination of Monte Carlo simulations and experimental data, exploiting the
dependence of the process noise covariance on the states and parameters, as
given in the Langevin framework. We adapt a Hybrid Extended Kalman Filtering
technique by updating the process noise covariance at each time step based on
estimates. We compare the performance of this framework with different fixed
values of process noise covariance in biomolecular system models, including an
oscillator model, as well as in experimentally measured data for a negative
transcriptional feedback circuit. We find that the Extended Kalman Filter with
such process noise covariance update is closer to the optimality condition in
the sense that the innovation sequence becomes white and in achieving a balance
between the mean square estimation error and parameter convergence time. The
results of this chapter may help in the use of Extended Kalman Filters for
systems where process noise covariance depends on states and/or parameters.
