arXiv subjects

M. N. M. van Lieshout

Publications and source records attributed to M. N. M. van Lieshout.

At least 19 recordsLinked to original sources

arXiv2026

Optimal local interventions in the two-dimensional Abelian sandpile model

The Abelian sandpile model serves as a canonical example of self-organized criticality. This critical behavior manifests itself through large cascading events triggered by small perturbations. Such large-scale events, known as avalanches, are often regarded as stylized representations of catastrophic phenomena, such as earthquakes or forest fires. Motivated by this perspective, we study strategies to reduce avalanche sizes. We provide a first rigorous analysis of the impact of interventions in the Abelian sandpile model, considering a setting in which an external controller can perturb a configuration by removing sand grains at selected locations. We first develop and formalize an extended method to compute the expected size of an avalanche originating from a connected component of critical vertices, i.e., vertices at maximum height. Using this method, we characterize the structure of avalanches starting from square components and explicitly analyze the effect of interventions in such components. Our results show that the optimal intervention locations strike an interesting balance between reduction of largest avalanche sizes and increasing the number of mitigated avalanches.

math.OC↗

arXiv2026

Resource Allocation Based on Past Incident Patterns

We formulate and solve two resource allocation problems motivated by a preparedness question of emergency response services. First, we consider the assignment of vehicles to stations, and, in a second step, assign crews to vehicles. In both cases, we work in a minimax framework and define the objective function for a spatial catchment area as the total risk in this area per resource unit allocated to it. The solutions are explicit and can be calculated in practice by a greedy algorithm that successively allocates a resource unit to an area having maximal relative risk, with suitable tie breaker rules. The approach is illustrated on a data set of incidents reported to the Twente Fire Brigade.

math.OC↗

arXiv2025

Controlling the low-temperature Ising model using spatiotemporal Markov decision theory

We introduce the spatiotemporal Markov decision process (STMDP), a special type of Markov decision process that models sequential decision-making problems which are not only characterized by temporal, but also by spatial interaction structures. To illustrate the framework, we construct an STMDP inspired by the low-temperature two-dimensional Ising model on a finite, square lattice, evolving according to the Metropolis dynamics. We consider the situation in which an external decision maker aims to drive the system towards the all-plus configuration by flipping spins at specified moments in time. In order to analyze this problem, we construct an auxiliary MDP by means of a reduction of the configuration space to the local minima of the Hamiltonian. Leveraging the convenient form of this auxiliary MDP, we uncover the structure of the optimal policy by solving the Bellman equations in a recursive manner. Finally, we conduct a numerical study on the performance of the optimal policy obtained from the auxiliary MDP in the original Ising STMDP.

math.OC↗

arXiv2024

A Cox rate-and-state model for monitoring seismic hazard in the Groningen gas field

To monitor the seismic hazard in the Groningen gas field, we modify the rate-and-state model that relates changes in pore pressure to induced seismic hazard by allowing for noise in pore pressure measurements and by explicitly taking into account gas production volumes. We analyse the first and second-moment structure of the resulting Cox process, propose an unbiased estimating equation approach for the unknown model parameters and derive the posterior distribution of the driving random measure. We use a parallel Metropolis adjusted Langevin algorithm for sampling from the posterior and to monitor the hazard.

cs.CE↗

arXiv2024

A non-homogeneous Semi-Markov model for Interval Censoring

Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on semi-Markov processes in which interval generation is assumed to be time-dependent and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to homogeneous approaches by way of a parametric example.

math.PR↗

arXiv2024

XGBoostPP: Tree-based Estimation of Point Process Intensity Functions

We propose a novel tree-based ensemble method, named XGBoostPP, to nonparametrically estimate the intensity of a point process as a function of covariates. It extends the use of gradient-boosted regression trees (Chen & Guestrin, 2016) to the point process literature via two carefully designed loss functions. The first loss is based on the Poisson likelihood, working for general point processes. The second loss is based on the weighted Poisson likelihood, where spatially dependent weights are introduced to further improve the estimation efficiency for clustered processes. An efficient greedy search algorithm is developed for model estimation, and the effectiveness of the proposed method is demonstrated through extensive simulation studies and two real data analyses. In particular, we report that XGBoostPP achieves superior performance to existing approaches when the dimension of the covariate space is high, revealing the advantages of tree-based ensemble methods in estimating complex intensity functions.

stat.ME↗

arXiv2023

Optimal decision rules for marked point process models

We study a Markov decision problem in which the state space is the set of finite marked point configurations in the plane, the actions represent thinnings, the reward is proportional to the mark sum which is discounted over time, and the transitions are governed by a birth-death-growth process. We show that thinning points with large marks is optimal when births follow a Poisson process and marks grow logistically. Explicit values for the thinning threshold and the discounted total expected reward over finite and infinite horizons are also provided. When the points are required to respect a hard core distance, upper and lower bounds on the discounted total expected reward are derived.

math.PR↗

arXiv2022

Non-parametric adaptive bandwidth selection for kernel estimators of spatial intensity functions

We propose a new fully non-parametric two-step adaptive bandwidth selection method for kernel estimators of spatial point process intensity functions based on the Campbell-Mecke formula and Abramson's square root law. We present a simulation study to assess its performance relative to the Cronie-Van Lieshout global bandwidth selector and apply the technique to data on induced earthquakes in theGroningen gas field.

stat.ME↗

arXiv2022

Infill asymptotics for logistic regression estimators for spatio-temporal point processes

This paper discusses infill asymptotics for logistic regression estimators for spatio-temporal point processes whose intensity functions are of log-linear form. We establish strong consistency and asymptotic normality for the parameters of a Poisson point process model and demonstrate how these results can be extended to general point process models. Additionally, under proper conditions, we also extend our central limit theorem to other unbiased estimating equations that are based on the Campbell--Mecke theorem.

math.ST↗

arXiv2022

Exploring seismic hazard in the Groningen gas field using adaptive kernel smoothing and inhomogeneous summary statistics

The discovery of gas in Groningen in 1959 has been a massive boon to the Dutch economy. From the 1990s onwards though, gas production has led to induced seismicity. In this paper, we carry out a comprehensive exploratory analysis of the spatio-temporal earthquake catalogue. We develop a non-parametric adaptive kernel smoothing technique to estimate the spatio-temporal hazard map and to interpolate monthly well-based gas production statistics. Second- and higher-order inhomogeneous summary statistics are used to show that the state of the art rate-and-state models for the prediction of seismic hazard fail to capture inter-event interaction in the earthquake catalogue. Based on these findings, we suggest a modified rate-and-state model that also takes into account changes in gas production volumes and uncertainty in the pore pressure field.

physics.geo-ph↗

arXiv2021

Data-driven chimney fire risk prediction using machine learning and point process tools

Chimney fires constitute one of the most commonly occurring fire types. Precise prediction and prompt prevention are crucial in reducing the harm they cause. In this paper, we develop a combined machine learning and statistical modeling process to predict chimney fires. Firstly, we use random forests and permutation importance techniques to identify the most informative explanatory variables. Secondly, we design a Poisson point process model and apply associated logistic regression estimation to estimate the parameters. Moreover, we validate the Poisson model assumption using second-order summary statistics and residuals. We implement the modeling process on data collected by the Twente Fire Brigade and obtain plausible predictions. Compared to similar studies, our approach has two advantages: i) with random forests, we can select explanatory variables non-parametrically considering variable dependence; ii) using logistic regression estimation, we can fit the statistical model efficiently by tuning it to focus on important regions and times of the fire data.

stat.ME↗

arXiv2021

State estimation for aoristic models

Aoristic data can be described by a marked point process in time in which the points cannot be observed directly but are known to lie in observable intervals, the marks. We consider Bayesian state estimation for the latent points when the marks are modelled in terms of an alternating renewal process in equilibrium and the prior is a Markov point point process. We derive the posterior distribution, estimate its parameters and present some examples that illustrate the influence of the prior distribution.

math.ST↗

arXiv2020

Testing biodiversity using inhomogeneous summary statistics

McGill's theory of biodiversity is based upon three axioms: individuals of the same species cluster together, many rare species co-exist with a few common ones and individuals of different species grow independently of each other. Over the past decade, classical point pattern analyses have been employed to verify these axioms based on the false assumption of stationarity. In this paper, we use inhomogeneous versions of the classical summary statistics for spatial point patterns to assess the validity of McGill's first and third axioms for data obtained from a 50 hectare plot on Barro Colorado Island.

stat.ME↗

arXiv2019

Infill asymptotics and bandwidth selection for kernel estimators of spatial intensity functions

We investigate the asymptotic mean squared error of kernel estimators of the intensity function of a spatial point process. We show that when $n$ independent copies of a point process in $\mathbb R^d$ are superposed, the optimal bandwidth $h_n$ is of the order $n^{-1/(d+4)}$ under appropriate smoothness conditions on the kernel and true intensity function. We apply the Abramson principle to define adaptive kernel estimators and show that asymptotically the optimal adaptive bandwidth is of the order $n^{-1/(d+8)}$ under appropriate smoothness conditions.

math.ST↗

arXiv2017

Nearest-neighbour Markov point processes on graphs with Euclidean edges

We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as the analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley--Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges.

math.ST↗

arXiv2017

A multi-scale area-interaction model for spatio-temporal point patterns

Models for fitting spatio-temporal point processes should incorporate spatio-temporal inhomogeneity and allow for different types of interaction between points (clustering or regularity). This paper proposes an extension of the spatial multi-scale area-interaction model to a spatio-temporal framework. This model allows for interaction between points at different spatio-temporal scales and the inclusion of covariates. We fit the proposed model to varicella cases registered during 2013 in Valencia, Spain. The fitted model indicates small scale clustering and regularity for higher spatio-temporal scales.

stat.ME↗

arXiv2016

Bandwidth selection for kernel estimators of the spatial intensity function

We discuss and compare various approaches to the problem of bandwidth selection for kernel estimators of intensity functions of spatial point processes. We also propose a new method based on the Campbell formula applied to the reciprocal intensity function. The new method is fully non-parametric, does not require knowledge of the product densities, and is not restricted to a specific class of point process models.

stat.ME↗

arXiv2016

Non-parametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets

We propose new summary statistics to quantify the association between the components in coverage-reweighted moment stationary multivariate random sets and measures. They are defined in terms of the coverage-reweighted cumulant densities and extend classic functional statistics for stationary random closed sets. We study the relations between these statistics and evaluate them explicitly for a range of models. Unbiased estimators are given for all statistics and applied to simulated examples.

math.ST↗