Distributionally Robust Probabilistic Prediction for Stochastic Dynamical Systems

TL;DR

This paper introduces a distributionally robust probabilistic prediction framework for stochastic dynamical systems, providing worst-case performance guarantees and practical suboptimal solutions.

Contribution

It develops a novel functional-maximin approach that transforms intractable optimization over probability measures into Euclidean space, enabling practical suboptimal predictors.

Findings

Proposes Noise-DRPP and Eig-DRPP as suboptimal predictors.

Derives bounds on the optimality gaps of the proposed predictors.

Numerical simulations demonstrate the effectiveness of the methods.

Abstract

Probabilistic prediction of stochastic dynamical systems (SDSs) aims to accurately predict the conditional probability distributions of future states. However, accurate probabilistic predictions tightly hinge on accurate distributional information from a nominal model, which is hardly available in practice. To address this issue, we propose a novel functional-maximin-based distributionally robust probabilistic prediction (DRPP) framework. In this framework, one can design probabilistic predictors that have worst-case performance guarantees over a pre-defined ambiguity set of SDSs. Nevertheless, DRPP requires optimizing over the space of probability measures with density functions with respect to the Lebesgue measure, which is generally intractable. We develop a methodology that equivalently transforms the original maximin from function spaces to Euclidean spaces. Although it remains intractable to seek a global optimal solution, two suboptimal solutions are derived. By relaxing the constraints on the ambiguity set, we obtain a suboptimal predictor called Noise-DRPP. Relaxing the constraints on the predictor yields another suboptimal predictor, Eig-DRPP. Moreover, optimality gaps between the proposed predictors and the global optimal predictor are derived. Finally, we conduct elaborate numerical simulations to compare the performance of different predictors under different SDSs.