Estimation for Non-Linear Hawkes Processes with Erlang Memory Kernels

Specialeforsvar: Carl Gustav Rømer

Titel: Estimation for Non-Linear Hawkes Processes with Erlang Memory Kernels

Abstract:  Ditlevsen and Löcherbach 2017 introduces a non-linear Hawkes process for modelling the firing times of neurons interacting in a monocyclic network of groups. The interactions are expressed using an Erlang memory kernel, parameterized by the so-called memory parameters. We propose efficient methods for simulation and estimation in this model. For the purpose of simulation, we derive an exact method as well as a faster approximate method. We then turn to the problem of estimation. Assuming a fixed interaction network, we propose a maximum likelihood procedure for the memory parameters. Under reasonable regularity assumptions, we derive that the maximum likelihood estimator of the continuous memory parameters is asymptotically normally distributed in the large population limit. For a known number of interacting groups, we then present a method for clustering the neurons using a non-model-based distance measure. For the same purpose, we propose a faster expectation-maximization algorithm, based on a non-homogenous Poisson process latent variable model. Lastly, we introduce a likelihood-based method for estimating the interaction network. We show that this can be rephrased as a traveling salesman problem. All methods are shown to be reasonably effective in simulation studies.

Vejleder: Susanne Ditlevsen
Censor:   Mads Stehr, CBS