Quantifying model uncertainty for the observed non-Gaussian data by the Hellinger distance

TL;DR

This paper introduces a method to quantify uncertainty in stochastic models with non-Gaussian data by minimizing the Hellinger distance, effectively estimating parameters and their optimal intervals.

Contribution

It proposes a novel approach to infer non-Gaussian parameters in stochastic differential equations using the Hellinger distance, applicable to complex systems with non-Gaussian fluctuations.

Findings

Method successfully estimates single and multiple parameters.

Identifies optimal parameter estimation intervals.

Applicable to modeling abrupt climate changes.

Abstract

Mathematical models for complex systems under random fluctuations often certain uncertain parameters. However, quantifying model uncertainty for a stochastic differential equation with an $\alpha$-stable L\'evy process is still lacking. Here, we propose an approach to infer all the uncertain non-Gaussian parameters and other system parameters by minimizing the Hellinger distance over the parameter space. The Hellinger distance measures the similarity between an empirical probability density of non-Gaussian observations and a solution (as a probability density) of the associated nonlocal Fokker-Planck equation. Numerical experiments verify that our method is feasible for estimating single and multiple parameters. Meanwhile, we find an optimal estimation interval of the estimated parameters. This method is beneficial for extracting governing dynamical system models under non-Gaussian fluctuations, as in the study of abrupt climate changes in the Dansgaard-Oeschger events.