  Interval-censored data solely records the aggregated counts of events during
specific time intervals - such as the number of patients admitted to the
hospital or the volume of vehicles passing traffic loop detectors - and not the
exact occurrence time of the events. It is currently not understood how to fit
the Hawkes point processes to this kind of data. Its typical loss function (the
point process log-likelihood) cannot be computed without exact event times.
Furthermore, it does not have the independent increments property to use the
Poisson likelihood. This work builds a novel point process, a set of tools, and
approximations for fitting Hawkes processes within interval-censored data
scenarios. First, we define the Mean Behavior Poisson process (MBPP), a novel
Poisson process with a direct parameter correspondence to the popular
self-exciting Hawkes process. We fit MBPP in the interval-censored setting
using an interval-censored Poisson log-likelihood (IC-LL). We use the parameter
equivalence to uncover the parameters of the associated Hawkes process. Second,
we introduce two novel exogenous functions to distinguish the exogenous from
the endogenous events. We propose the multi-impulse exogenous function - for
when the exogenous events are observed as event time - and the latent
homogeneous Poisson process exogenous function - for when the exogenous events
are presented as interval-censored volumes. Third, we provide several
approximation methods to estimate the intensity and compensator function of
MBPP when no analytical solution exists. Fourth and finally, we connect the
interval-censored loss of MBPP to a broader class of Bregman divergence-based
functions. Using the connection, we show that the popularity estimation
algorithm Hawkes Intensity Process (HIP) is a particular case of the MBPP. We
verify our models through empirical testing on synthetic data and real-world
data.
