arXiv subjects

Tom Stindl

Publications and source records attributed to Tom Stindl.

5 recordsLinked to original sources

arXiv2025

Neural Networks for Parameter Estimation of the Discretely Observed Hawkes Process

When the sample path of a Hawkes process is observed discretely, such that only the total event counts in disjoint time intervals are known, the likelihood function becomes intractable. To overcome the challenge of likelihood-based inference in this setting, we propose to use a likelihood-free approach that uses simulated data to train a fully connected neural network (NN) to estimate the parameters of the Hawkes process from a summary statistic of the count data. A naive imputation estimate of the parameters forms the basis for our summary statistic, which is fast to generate and requires minimal expert knowledge to design. The resulting NN estimator is comparable to the best extant approximate likelihood estimators in terms of mean-squared error but requires significantly less computational time. We implement NN quantile estimation for fast uncertainty quantification. The proposed estimation procedure is applied to weekly count data for two infectious diseases, with a time-varying background rate used to capture seasonal fluctuations in infection risk.

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arXiv2025

Parametric inference for the discretely observed multivariate Hawkes process using particle Markov Chain Monte Carlo

The multivariate Hawkes process (MHP) is a useful statistical model for analysing multidimensional event time sequences that exhibit self-excitation and cross-excitation. When the MHP is monitored discretely, only the total number of events for each dimension in disjoint time intervals is observed. The likelihood function relative to this data is intractable, so traditional inference techniques are not available. To address this, we design an unbiased estimate of the intractable likelihood function using sequential Monte Carlo (SMC) based on a representation of the unobserved event times as latent variables in a state-space model. The unbiasedness of the SMC estimate allows for its use in place of the true likelihood in a Metropolis-Hastings algorithm, enabling the construction of a Markov Chain Monte Carlo sample from the posterior distribution over the parameters of the MHP. Using simulated data, we assess the performance of our method and demonstrate that it outperforms existing approaches in terms of mean squared error and computational efficiency. Terrorist activity in Afghanistan and Pakistan from 2018 to 2021 is analysed based on daily count data to examine the dynamics of terrorism in the region.

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arXiv2024

Estimating the Hawkes process from a discretely observed sample path

The Hawkes process is a widely used model in many areas, such as finance, seismology, neuroscience, epidemiology, and social sciences. Estimation of the Hawkes process from continuous observations of a sample path is relatively straightforward using either the maximum likelihood or other methods. However, estimating the parameters of a Hawkes process from observations of a sample path at discrete time points only is challenging due to the intractability of the likelihood with such data. In this work, we introduce a method to estimate the Hawkes process from a discretely observed sample path. The method takes advantage of a state-space representation of the incomplete data problem and use the sequential Monte Carlo (aka particle filtering) to approximate the likelihood function. As an estimator of the likelihood function the SMC approximation is unbiased, and therefore it can be used together with the Metropolis-Hastings algorithm to construct Markov Chains to approximate the likelihood distribution, or more generally, the posterior distribution of model parameters. The performance of the methodology is assessed using simulation experiments and compared with other recently published methods. The proposed estimator is found to have a smaller mean square error than the two benchmark estimators. The proposed method has the additional advantage that confidence intervals for the parameters are easily available. We apply the proposed estimator to the analysis of weekly count data on measles cases in Tokyo Japan and compare the results to those by one of the benchmark methods.

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arXiv2022

Stochastic declustering of earthquakes with the spatiotemporal RETAS model

Epidemic-Type Aftershock Sequence (ETAS) models are point processes that have found prominence in seismological modeling. Its success has led to the development of a number of different versions of the ETAS model. Among these extensions is the RETAS model which has shown potential to improve the modeling capabilities of the ETAS class of models. The RETAS model endows the main-shock arrival process with a renewal process which serves as an alternative to the homogeneous Poisson process. Model fitting is performed using likelihood-based estimation by directly optimizing the exact likelihood. However, inferring the branching structure from the fitted RETAS model remains a challenging task since the declustering algorithm that is currently available for the ETAS model is not directly applicable. This article solves this problem by developing an iterative algorithm to calculate the smoothed main and aftershock probabilities conditional on all available information contained in the catalog. Consequently, an objective estimate of the spatial intensity function can be obtained and an iterative semi-parametric approach is implemented to estimate model parameters with information criteria used for tuning the smoothing parameters. The methods proposed herein are illustrated on simulated data and a New Zealand earthquake catalog.

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arXiv2021

Spatiotemporal ETAS model with a renewal main-shock arrival process

This article proposes a spatiotemporal point process model that enhances the classical Epidemic-Type Aftershock Sequence (ETAS) model by incorporating a renewal main-shock arrival process, which we term the renewal ETAS (RETAS) model. This modification is similar in spirit to the renewal Hawkes (RHawkes) process but the conditional intensity process supports a spatial domain. It empowers the main-shock intensity with the capability to reset upon the arrival of main-shocks and therefore allows for heavier clustering of earthquakes than the spatiotemporal ETAS model introduced by Ogata (1998). We introduce a likelihood evaluation algorithm for parameter estimation and provide a novel procedure to evaluated the fitted model's goodness-of-fit based on a sequential application of the Rosenblatt transformation. A simulation algorithm for the RETAS model is developed and applied to validate the numerical performance of the likelihood evaluation algorithm and goodness of fit test procedure. We illustrate the proposed model and procedures on various earthquake catalogs around the world each with distinctly different seismic activity. These catalogs will demonstrate that the RETAS model affords additional flexibility in comparison to the classical spatiotemporal ETAS model and has the potential for superior modeling and forecasting of seismicity.

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