  This paper builds Wasserstein ambiguity sets for the unknown probability
distribution of dynamic random variables leveraging noisy partial-state
observations. The constructed ambiguity sets contain the true distribution of
the data with quantifiable probability and can be exploited to formulate robust
stochastic optimization problems with out-of-sample guarantees. We assume the
random variable evolves in discrete time under uncertain initial conditions and
dynamics, and that noisy partial measurements are available. All random
elements have unknown probability distributions and we make inferences about
the distribution of the state vector using several output samples from multiple
realizations of the process. To this end, we leverage an observer to estimate
the state of each independent realization and exploit the outcome to construct
the ambiguity sets. We illustrate our results in an economic dispatch problem
involving distributed energy resources over which the scheduler has no direct
control.
