Estimating Differential Equations from Temporal Point Processes

Abstract

Ordinary differential equations (ODEs) allow interpretation of phenomena in various scientific fields. They have mostly been applied to numerical data observed at regular intervals, but not to irregularly observed discrete events, also known as point processes. In this study, we introduce an ODE modeling of such events by combining ODEs with log-Gaussian Cox processes (Møller et al., 1998). In the experiments with different types of ODEs regarding infectious disease, predator-prey interaction, and competition among participants, our method outperformed existing baseline methods assuming regularly observed continuous data with respect to the accuracy of recovering the latent parameters of ODEs. Through both synthetic and actual examples, we also showed the ability of our method to extrapolate, model latent events that cannot be observed, and offer interpretability of phenomena from the viewpoint of the estimated parameters of ODE.