Distance Measure Concepts for Bayesian Inference of Chaotic Dynamical System Parameters

The problem of determining the parameters of dynamical systems often reduces to developing some notion of a "distance" between observed and simulated system trajectory data. The best parameter fit can then be found by adjusting the parameters that generate the simulated trajectory until the distance is minimized. However, in the case of chaotic dynamical systems, traditional distance measures such as the Mean Square Error (MSE) often fail to produce good results, a consequence of these systems' inherent sensitivity to changes in parameters and initial conditions. In this paper, we adopt the perspective that more robust distance measures can be formulated when the trajectories of these chaotic systems are treated as samples from probability distributions, rather than as time series data. Within this perspective, we evaluate the efficacy of three candidate distance measure concepts: the correlation integral likelihood (proposed by Haario et al.), the Wasserstein metric, and a family of information-theoretic distances based on the Kullback-Leibler divergence. We give particular emphasis to the performance of these methods on the Lorenz63 system, a canonical chaotic system with applications in modeling atmospheric convection.