  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 paper
explores the construction of Wasserstein ambiguity sets in dynamic scenarios
where data is 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. Further, 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 a UAV detection application show the superior
performance resulting from the proposed dynamic ambiguity sets.
