  We consider problems in which a system receives external \emph{perturbations}
from time to time. For instance, the system can be a train network in which
particular lines are repeatedly disrupted without warning, having an effect on
passenger behavior. The goal is to predict changes in the behavior of the
system at particular points of interest, such as passenger traffic around
stations at the affected rails. We assume that the data available provides
records of the system functioning at its "natural regime" (e.g., the train
network without disruptions) and data on cases where perturbations took place.
The inference problem is how information concerning perturbations, with
particular covariates such as location and time, can be generalized to predict
the effect of novel perturbations. We approach this problem from the point of
view of a mapping from the counterfactual distribution of the system behavior
without disruptions to the distribution of the disrupted system. A variant on
\emph{distribution regression} is developed for this setup.
