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Many Markov chain Monte Carlo techniques currently available rely on discrete-time reversible Markov processes whose transition kernels are variations of the Metropolis-Hastings algorithm. We explore and generalize an alternative scheme recently introduced in the physics literature (Peters and de With 2012) where the target distribution is explored using a continuous-time nonreversible piecewise-deterministic Markov process. In the Metropolis-Hastings algorithm, a trial move to a region of lower target density, equivalently of higher \"energy,\" than the current state can be rejected with positive probability. In this alternative approach, a particle moves along straight lines around the space and, when facing a high energy barrier, it is not rejected but its path is modified by bouncing against this barrier. By reformulating this algorithm using inhomogeneous Poisson processes, we exploit standard sampling techniques to simulate exactly this Markov process in a wide range of scenarios of interest. Additionally, when the target distribution is given by a product of factors dependent only on subsets of the state variables, such as the posterior distribution associated with a probabilistic graphical model, this method can be modified to take advantage of this structure by allowing computationally cheaper \"local\" bounces, which only involve the state variables associated with a factor, while the other state variables keep on evolving. In this context, by leveraging techniques from chemical kinetics, we propose several computationally efficient implementations. Experimentally, this new class of Markov chain Monte Carlo schemes compares favorably to state-of-the-art methods on various Bayesian inference tasks, including for high-dimensional models and large datasets. Supplementary materials for this article are available online.

A call center is a service network in which agents provide telephone-based services. Customers who seek these services are delayed in tele-queues. This article summarizes an analysis of a unique record of call center operations. The data comprise a complete operational history of a small banking call center, call by call, over a full year. Taking the perspective of queueing theory, we decompose the service process into three fundamental components: arrivals, customer patience, and service durations. Each component involves different basic mathematical structures and requires a different style of statistical analysis. Some of the key empirical results are sketched, along with descriptions of the varied techniques required. Several statistical techniques are developed for analysis of the basic components. One of these techniques is a test that a point process is a Poisson process. Another involves estimation of the mean function in a nonparametric regression with lognormal errors. A new graphical technique is introduced for nonparametric hazard rate estimation with censored data. Models are developed and implemented for forecasting of Poisson arrival rates. Finally, the article surveys how the characteristics deduced from the statistical analyses form the building blocks for theoretically interesting and practically useful mathematical models for call center operations.

We study a class of Markov processes that combine local dynamics, arising from a fixed Markov process, with regenerations arising at a state-dependent rate. We give conditions under which such processes possess a given target distribution as their invariant measures, thus making them amenable for use within Monte Carlo methodologies. Since the regeneration mechanism can compensate the choice of local dynamics, while retaining the same invariant distribution, great flexibility can be achieved in selecting local dynamics, and the mathematical analysis is simplified. We give straightforward conditions for the process to possess a central limit theorem, and additional conditions for uniform ergodicity and for a coupling from the past construction to hold, enabling exact sampling from the invariant distribution. We further consider and analyse a natural approximation of the process which may arise in the practical simulation of some classes of continuous-time dynamics.

We consider Gaussian approximation in a variant of the classical Johnson–Mehl birth–growth model with random growth speed. Seeds appear randomly in $\\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The locations, birth times, and growth speeds of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed, the time distribution, and a weight function $h\\;:\\;\\mathbb{R}^d \\times [0,\\infty) \\to [0,\\infty)$ , we prove a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with fixed growth speed. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.

Conservation management becomes complicated when globally threatened species reach high densities locally, exceeding the carrying capacity of the ecosystem and causing damage. Managing high‐profile native species is particularly challenging, because ethical debates and public opposition to traditional control methods often prompt shifts toward strategies that prevent environmental harm rather than reducing populations. The koala (Phascolarctos cinereus) in South Australia exemplifies these challenges because, although it can damage the vegetation from high browsing pressure, culling is avoided due to public resistance. Therefore, managers have to consider costly and logistically constrained alternatives such as fertility control and translocation. Demographic models are valuable tools for predicting population dynamics, but their effectiveness depends on reliable population density estimates, often biased by expert‐elicited and citizen‐science data. We combined a point‐process model, an ensemble species distribution model, and a demographic model to project koala populations in the Mount Lofty Ranges over the next 25 years to assess the efficiency and cost‐effectiveness of fertility‐control interventions while accounting for sampling biases, habitat suitability, and local densities. We tested two hypotheses: (1) koala distribution is driven by rainfall, temperature, and soil acidity, with summer rainfall boosting habitat suitability, and (2) spatially targeted fertility interventions in high‐suitability areas are more cost‐effective than generalised strategies due to subpopulation connectivity. Our models confirmed that these three environmental factors shape koala distribution and that, in the absence of intervention, the koala population could increase by ~17‐25% in 25 years. Fertility control focusing on adult females emerged as the most cost‐effective (~AU $34 million) strategy, although it was slower at reducing population size compared to an intervention also sterilising female back young. While the choice of sterilisation scenario has minimal impact on overall costs, ethical considerations and long‐term conservation goals such as population density thresholds will have more influence on managing expenses effectively. The koala population in South Australia's Mount Lofty Ranges is increasing, raising concerns about overbrowsing and the need for sustainable management. Using combined demographic, point‐process, and species distribution models, we projected koala populations over 25 years to evaluate fertility‐control strategies. Our findings highlight rainfall, temperature, and vegetation as key drivers of habitat suitability, with targeted fertility control for adult females emerging as the most cost‐effective intervention (~AU$ 34 million).

In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.

Motivated by the development of optimal dispatching strategies for prehospital resources, we model the spatial distribution of ambulance call events in the Swedish municipality Skellefteå during 2014–2018 in order to identify important spatial covariates and discern hotspot regions. Our large-scale multivariate data point pattern of call events consists of spatial locations and marks containing the associated priority levels and sex labels. The covariates used are related to road network coverage, population density, and socio-economic status. For each marginal point pattern, we model the associated intensity function by means of a log-linear function of the covariates and their interaction terms, in combination with lasso-like elastic-net regularized composite/Poisson process likelihood estimation. This enables variable selection and collinearity adjustment as well as reduction of variance inflation from overfitting and bias from underfitting. To incorporate mobility adjustment, reflecting people’s movement patterns, we also include a nonparametric (kernel) intensity estimate as an additional covariate. The kernel intensity estimation performed here exploits a new heuristic bandwidth selection algorithm. We discover that hotspot regions occur along dense parts of the road network. A mean absolute error evaluation of the fitted model indicates that it is suitable for designing prehospital resource dispatching strategies. Supplementary materials accompanying this paper appear online.

This work is devoted to the problem of estimation of the localization of Poisson source. The observations are inhomogeneous Poisson processes registered by more than three detectors on the plane. We study the behavior of the Bayes estimators in the asymptotic of large intensities. It is supposed that the intensity functions of the signals arriving in the detectors have cusp-type singularity. We show the consistency, limit distributions, the convergence of moments and asymptotic efficiency of these estimators.

Preferential sampling models have garnered significant attention in recent years. Although the original model was developed for geostatistics, it founds applications in other types of data, such as point processes in the form of presence-only data. While this has been recognized in the Statistics literature, there is value in incorporating ideas from both presence-only and preferential sampling literature. In this paper, we propose a novel model that extends existing ideas to handle a continuous variable collected through opportunistic sampling. To demonstrate the potential of our approach, we apply it to sardine biomass data collected during commercial fishing trips. While the data is intuitively understood, it poses challenges due to two types of preferential sampling: fishing events (presence data) are non-random samples of the region, and fishermen tend to set their nets in areas with a high quality and value of catch (i.e., bigger schools of the target species). We discuss theoretical and practical aspects of the problem, and propose a well-defined probabilistic approach. Our approach employs a data augmentation scheme that predicts the number of unobserved fishing locations and corresponding biomass (in kg). This allows for evaluation of the Poisson Process likelihood without the need for numerical approximations. The results of our case study may serve as an incentive to use data collected during commercial fishing trips for decision-making aimed at benefiting both ecological and economic aspects. The proposed methodology has potential applications in a variety of fields, including ecology and epidemiology, where marked point process model are commonly used.

Traditional insurance’s earthquake contingency costs are insufficient for earthquake funding due to extreme differences from actual losses. The earthquake bond (EB) links insurance to capital market bonds, enabling higher and more sustainable earthquake funding, but challenges persist in pricing EBs. This paper presents zero-coupon and coupon-paying EB pricing models involving the inconstant event intensity and maximum strength of extreme earthquakes under the risk-neutral pricing measure. Focusing on extreme earthquakes simplifies the modeling and data processing time compared to considering infinite earthquake frequency occurring over a continuous time interval. The intensity is accommodated using the inhomogeneous Poisson process, while the maximum strength is modeled using extreme value theory (EVT). Furthermore, we conducted model experiments and variable sensitivity analyses on EB prices using earthquake data from Indonesia’s National Disaster Management Authority from 2008 to 2021. The sensitivity analysis results show that choosing inconstant intensity rather than a constant one implies significant EB price differences, and the maximum strength distribution based on EVT matches the data distribution. The presented model and its experiments can guide EB issuers in setting EB prices. Then, the variable sensitivities to EB prices can be used by investors to choose EB according to their risk tolerance.