Inverse Modeling of Dynamical Systems: Multi-Dimensional Extensions of a Stochastic Switching Problem∗

The Buridan’s ass paradox is characterized by perpetual indecision between two states, which are never attained. When this problem is formulated as a dynamical system, indecision is modeled by a discrete-state Markov process determined by the system’s unknown parameters. Interest lies in estimating these parameters from a limited number of observations. We compare estimation methods and examine how well each can be generalized to multi-dimensional extensions of this system. By quantifying statistics such as mean, variance, frequency, and cumulative power, we construct both method of moments type estimators and likelihood-based estimators. We show, however, why these techniques become intractable in higher dimensions, and thus develop a geometric approach to reveal the parameters underlying the Markov process. We also examine the robustness of this method to the presence of noise. ∗This research is supported in part by NSA grant H98230-11-1-0222 and NSF grant DMS-1062817. †Michigan State University, [email protected] ‡Susquehanna University, [email protected] §Saint Vincent College, [email protected] 1 ar X iv :1 20 8. 35 38 v1 [ m at h. D S] 1 7 A ug 2 01 2