Data-Driven Ambiguity Sets With Probabilistic Guarantees for Dynamic Processes

Distributional ambiguity sets provide quantifiable ways to characterize the uncertainty about the true probability distribution of random variables of interest. This makes them a key element in data-driven robust optimization by exploiting high-confidence guarantees to hedge against uncertainty. This article explores the construction of Wasserstein ambiguity sets in dynamic scenarios, where data are collected progressively and may only reveal partial information about the unknown random variable. For random variables evolving according to known dynamics, we leverage assimilated samples to make inferences about their unknown distribution at the end of the sampling horizon. Under exact knowledge of the flow map, we provide sufficient conditions that relate the growth of the trajectories with the sampling rate to establish a reduction of the ambiguity set size as the horizon increases. Furthermore, we characterize the exploitable sample history that results in a guaranteed reduction of ambiguity sets under errors in the computation of the flow and when the dynamics is subject to bounded unknown disturbances. Our treatment deals with both full- and partial-state measurements and, in the latter case, exploits the sampled-data observability properties of linear time-varying systems under irregular sampling. Simulations on an unmanned aerial vehicle detection application show the superior performance resulting from the proposed dynamic ambiguity sets.