Abstract
Point process is a common statistical model used to describe the pattern of event occurrence for many real-world applications, such as earthquake prediction and financial modelling. The prompting characteristics of past events on future ones are a vital factor in the clustering effects in point processes. Hawkes process is the most extensively used point process model for modelling self-exciting phenomena.
One of the key challenges for the Hawkes process is the selection of the function for modelling baseline intensity and the triggering kernel. The vanilla Hawkes process assumes a constant valued function for the baseline intensity and a parametric stationary function for the triggering kernel. The parametric and stationary assumption makes inference convenient but limits the model expression.
To generalize the classical Hawkes process, various nonparametric and nonstationary approaches for the Hawkes process are proposed in this thesis. Specifically, three different nonparametric and nonstationary approaches for Hawkes processes are proposed.
The model independent stochastic declustering (MISD) algorithm is a classical frequentist nonparametric inference algorithm for Hawkes process with triggering kernel and it uses a bin-based histogram function. However, the number of bins, which is fixed manually, usually leads to underfitting or overfitting when improperly chosen. In this thesis, a refined MISD algorithm is proposed to ease the choice of bin number.
Next, a Bayesian nonparametric Hawkes process model is proposed, with Gaussian process as prior for baseline intensity and triggering kernel. Correspondingly, a variational Gaussian approximation, Polya-Gamma based Gibbs sampling, expectation-maximization (EM) and mean-field variational inference algorithms are proposed.
As the next step, nonstationarity is introduced into the classical Hawkes process. A fast cumulant-based multi-resolution segmentation algorithm is proposed that partitions the process into segments to capture the time-varying characteristics.
Finally, future research directions for the nonparametric and nonstationary Hawkes process are discussed.