Simultaneous State and Parameter Estimation Using Maximum Relative Entropy with Nonhomogenous Differential Equation Constraints

In this paper, we continue our efforts to show how maximum relative entropy (MrE) can be used as a universal updating algorithm. Here, our purpose is to tackle a joint state and parameter estimation problem where our system is nonlinear and in a non-equilibrium state, i.e., perturbed by varying external forces. Traditional parameter estimation can be performed by using filters, such as the extended Kalman filter (EKF). However, as shown with a toy example of a system with first order non-homogeneous ordinary differential equations, assumptions made by the EKF algorithm (such as the Markov assumption) may not be valid. The problem can be solved with exponential smoothing, e.g., exponentially weighted moving average (EWMA). Although this has been shown to produce acceptable filtering results in real exponential systems, it still cannot simultaneously estimate both the state and its parameters and has its own assumptions that are not always valid, for example when jump discontinuities exist. We show that by applying MrE as a filter, we can not only develop the closed form solutions, but we can also infer the parameters of the differential equation simultaneously with the means. This is useful in real, physical systems, where we want to not only filter the noise from our measurements, but we also want to simultaneously infer the parameters of the dynamics of a nonlinear and non-equilibrium system. Although there were many assumptions made throughout the paper to illustrate that EKF and exponential smoothing are special cases ofMrE, we are not “constrained”, by these assumptions. In other words, MrE is completely general and can be used in broader ways