Hawkes Process with Stochastic Triggering Kernel

The impact from past to future is a vital feature in modelling time series data, which has been described by many point processes, e.g. the Hawkes process. In classical Hawkes process, the triggering kernel is assumed to be a deterministic function. However, the triggering kernel can vary with time due to the system uncertainty in real applications. To model this kind of variance, we propose a Hawkes process variant with stochastic triggering kernel, which incorporates the variation of triggering kernel over time. In this model, the triggering kernel is considered to be an independent multivariate Gaussian distribution. We derive and implement a tractable inference algorithm based on variational auto-encoder. Results from synthetic and real data experiments show that the underlying mean triggering kernel and variance band can be recovered, and using the stochastic triggering kernel is more accurate than the vanilla Hawkes process in capacity planning.