  Event occurrence is not only subject to the environmental changes, but is
also facilitated by the events that have occurred in a system. Here, we develop
a method for estimating such extrinsic and intrinsic factors from a single
series of event-occurrence times. The analysis is performed using a model that
combines the inhomogeneous Poisson process and the Hawkes process, which
represent exogenous fluctuations and endogenous chain-reaction mechanisms,
respectively. The model is fit to a given dataset by minimizing the free
energy, for which statistical physics and a path-integral method are utilized.
Because the process of event occurrence is stochastic, parameter estimation is
inevitably accompanied by errors, and it can ultimately occur that exogenous
and endogenous factors cannot be captured even with the best estimator. We
obtained four regimes categorized according to whether respective factors are
detected. By applying the analytical method to real time series of debate in a
social-networking service, we have observed that the estimated exogenous and
endogenous factors are close to the first comments and the follow-up comments,
respectively. This method is general and applicable to a variety of data, and
we have provided an application program, by which anyone can analyze any series
of event times.
