1  Introduction

Warning

This book is a work in progress, please bear with me as I update and fix the issues.

If you have any particular examples you’d like to be included please let me know @ c.jonestodd@auckland.ac.nz

The stelfi package fits Hawkes and log-Gaussian Cox Point Process models, with extensions, using Template Model Builder (Kristensen et al. (2016)).

Overview

A Hawkes process is a self-exciting temporal point process where the occurrence of an event immediately increases the chance of another (see Hawkes (1971)). stelfi also offers functionality to fit self-inhibiting process and a non-homogeneous background rate.

A log-Gaussian Cox process is a Poisson point process where the log-intensity is given by a Gaussian random field. stelfi also offers functionality to extend this to a joint likelihood formulation fitting a marked log-Gaussian Cox model.

In addition, the stelfi offers functionality to fit self-exciting spatiotemporal point processes. Models are fitted via maximum likelihood using ‘TMB’ (Template Model Builder) (Kristensen et al. (2016)). Where included 1) random fields are assumed to be Gaussian and are integrated over using the Laplace approximation and 2) a stochastic partial differential equation model, introduced by Lindgren, Rue, and Lindström (2011), is defined for the field(s).

Hawkes, AG. 1971. “Spectra of Some Self-Exciting and Mutually Exciting Point Processes.” Biometrika.

Kristensen, Kasper, Anders Nielsen, Casper W. Berg, Hans Skaug, and Bradley M. Bell. 2016. “TMB: Automatic Differentiation and Laplace Approximation.” Journal of Statistical Software. https://doi.org/10.18637/jss.v070.i05.

Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach (with Discussion).” Journal of the Royal Statistical Society B 73 (4): 423–98.