A Markov switching discrete-time Hawkes process: application to the monitoring of bats behavior

TL;DR

This paper introduces a novel Markov switching discrete-time Hawkes process model to analyze bat echolocation call sequences, capturing behavioral changes with an EM algorithm for inference, demonstrated through simulations and real data.

Contribution

It combines a Hawkes process with a latent Markov chain to model behavioral state changes, reformulating as a Hidden Markov Model for efficient inference.

Findings

Model is identifiable and reformulated as HMM with Poisson emissions.

EM algorithm effectively estimates parameters and hidden states.

Application successfully distinguishes different bat behaviors.

Abstract

Over the past few decades, the Hawkes process has become a popular framework for modeling temporal events thanks to its flexibility to capture different dependency structures. The objective of this work is to model call sequences emitted by bats for echolocation, whose patterns are known to change depending on the animal's activity. The novelty of the model lies in the combination of a Hawkes-type dependency from past events, as well as a latent variable that encodes changes in bat behavior. More precisely, we consider a discrete-time version of the Hawkes process, with an exponential kernel, where the immigration term varies according to a latent Markov chain. We prove that this model is identifiable and can be reformulated in terms of a Hidden Markov Model, with Poisson emissions. Based on these properties, we show that maximum likelihood inference of the model parameters can be performed using an EM algorithm, which involves a recursive M-step. A simulation study demonstrates the performance of our approach method for estimating the parameters, recovering the number of hidden states and classifying each bin of the trajectory. Finally, we illustrate the use of the proposed modeling to distinguish different behaviors of bats, based on the recording of their cries.