  Cascades of Poisson processes are probabilistic models for spatio-temporal
phenomena in which (i) previous events may trigger subsequent events, and (ii)
both the background and triggering processes are conditionally Poisson. Such
phenomena are typically "data rich but knowledge poor", in the sense that large
datasets are available yet a mechanistic understanding of the background and
triggering processes which generate the data are unavailable. In these settings
nonparametric estimation plays a central role. However existing nonparametric
estimators have computational and storage complexity $\mathcal{O}(N^2)$,
precluding their application on large datasets. Here, by assuming the
triggering process acts only locally, we derive nonparametric estimators with
computational complexity $\mathcal{O}(N\log N)$ and storage complexity
$\mathcal{O}(N)$. Our approach automatically learns the domain of the
triggering process from data and is essentially free from hyperparameters. The
methodology is applied to a large seismic dataset where estimation under
existing algorithms would be infeasible.
