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

TL;DR

This paper investigates parameter estimation in multi-dimensional stochastic switching systems inspired by the Buridan's ass paradox, comparing methods and proposing a geometric approach for higher dimensions.

Contribution

It introduces a geometric estimation method for multi-dimensional Markov processes and analyzes its robustness and limitations compared to traditional techniques.

Findings

Likelihood and moment methods become intractable in higher dimensions

The geometric approach effectively estimates parameters in multi-dimensional systems

The method shows robustness to noise presence

Abstract

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.