Poisson Process Research Articles

Furutsu-Novikov–like Cross-Correlation–Response Relations for Systems Driven by Shot Noise

We consider a dynamic system that is driven by an intensity-modulated Poisson process with intensity Λ(t)=λ(t)+ϵν(t). We derive an exact relation between the input-output cross-correlation in the spontaneous state (ϵ=0) and the linear response to the modulation (ϵ>0). If ϵ is sufficiently small, linear-response theory captures the full response. The relation can be regarded as a variant of the Furutsu-Novikov theorem for the case of shot noise. As we show, the relation is still valid in the presence of additional independent noise. Furthermore, we derive an extension to Cox-process input, which provides an instance of colored shot noise. We discuss applications to particle detection and to neuroscience. Using the new relation, we obtain a fluctuation-response relation for a leaky integrate-and-fire neuron. We also show how the new relation can be used in a remote control problem in a recurrent neural network. The relations are numerically tested for both stationary and nonstationary dynamics. Lastly, extensions to marked Poisson processes and to higher-order statistics are presented. Published by the American Physical Society 2024

Stochastic model of seed dispersal with homogeneous and non-homogeneous Poisson processes under habitat reduction conditions.

This study presents a stochastic model of seed dispersal based on a branching random walk (BRW) framework, incorporating both homogeneous and non-homogeneous Poisson point processes (PPP). Building on the model introduced by Coletti et al. (2023), we examine the effects of habitat reduction on seed dispersal dynamics. We analyze the phase transition behavior of the BRW model under varying conditions of habitat fragmentation, focusing on how these conditions influence the critical dispersal rate. Specifically, we study a BRW on the real line with a non-homogeneous PPP driven by a log-normal density, constrained between spatial barriers. Our simulations localize the critical dispersal rate with respect to barrier positions and compare this dependence between homogeneous and non-homogeneous models.

Clustering in point processes on linear networks using nearest neighbour volumes

This study introduces a novel method specifically designed to detect clusters of points within linear networks. This method extends a classification approach used for point processes in spatial contexts. Unlike traditional methods that operate on planar spaces, our approach adapts to the unique geometric challenges of linear networks, where classical properties of point processes are altered, and intuitive data visualisation becomes more complex. Our method utilises the distribution of the Kth nearest neighbour volumes, extending planar-based clustering techniques to identify regions of increased point density within a network. This approach is particularly effective for distinguishing overlapping Poisson processes within the same linear network. We demonstrate the practical utility of our method through applications to road traffic accident data from two Colombian cities, Bogota and Medellin. Our results reveal distinct clusters of high-density points in road segments where severe traffic accidents (resulting in injuries or fatalities) are most likely to occur, highlighting areas of increased risk. These clusters were primarily located on major arterial roads with high traffic volumes. In contrast, low-density points corresponded to areas with fewer accidents, likely due to lower traffic flow or other mitigating factors. Our findings provide valuable insights for urban planning and road safety management.

Approaches for Behavior Intensity Estimation in Groups of Heterogeneous Individuals: Precision and Applicability for Data with Uncertainty

In socially oriented areas, there arises the problem of assessing the cumulative characteristics of behavior, such as intensity, that are realized in groups of individuals. All individuals vary in their behavior and the available data is limited and may be associated with significant uncertainty: only a few episodes may be known and only a few individualsi the group may be observed. Mathematical models of behavior are used for estimation of key characteristics of the behavior. One of them is based on the gamma–Poisson point process, that reflects the heterogeneity of individuals in a form of a mixing distribution. This general model allows to formulate several methods of frequency estimation: the Cox regression, estimation of the copula parameter, and a posteriori inference in Bayesian belief networks. The aim of the paper is to assess their determine the precision of these methods based on the Kantorovich–Rubinstein distance between estimates and the true distribution of the desired parameter. The analysis of assumptions of those methods allows to formulate rules, that allow to chose the appropriate method in various sutuations of data availability. It has been shown that the copula-based approach provides the most accurate estimates and has the mild assumptions for the number of observed objects, but it cannot take into account external factors that may influence the behavior. Among methods that can take into account process covariants, estimates based on a posteriori inference in hybrid Bayesian belief networks have the highest precision. The paper considers a method for quantification of a hybrid BBNs with the approximation of mixtures of truncated exponents, that is data-demanding at the stage of calculating a priori estimates. However, it is noted that there are other approaches to setting hybrid BSDs in which a priori estimates can be set completely expertly.

Multiple Speciation and Extinction Rate Shifts Shaped the Macro-Evolutionary History of the Genus Lycium Towards a Rather Gradual Accumulation of Species Within the Genus

The Neotropics are the most species-rich region on Earth, and spectacular diversification rates in plants are reported in plants, mostly occurring in oceanic archipelagos, making Neotropical and island plant lineages a model for macro-evolutionary studies. The genus Lycium in the Solanaceae family, originating from the Neotropics and exhibiting a unique disjunct geography across several islands, is therefore expected to experience exceptional diversification events. In this study, we aimed to quantify the diversification trajectories of the genus Lycium to elucidate the diversification events within the genus. We compiled a DNA matrix of six markers on 75% of all the species in the genus to reconstruct a dated phylogeny. Based on this phylogeny, we first revisited the historical biogeography of the genus. Then, we fitted a Compound Poisson Process on Mass Extinction Time model to investigate the following key evolutionary events: speciation rate, extinction rate, as well as mass extinction events. Our analysis confirmed that South America is the origin of the genus, which may have undergone a suite of successive long-distance dispersals. Also, we found that most species arose as recently as 5 million years ago, and that the diversification rate found is among the slowest rates in the plant kingdom. This is likely shaped by the multiple speciation and extinction rate shifts that we also detected throughout the evolutionary history of the genus, including one mass extinction at the early stage of its evolutionary history. However, both speciation and extinction rates remain roughly constant over time, leading to a gradual species accumulation over time.

A novel life-cycle analysis framework to assess the performances of tall buildings considering the climate change

Structural performance against wind hazards is necessary in modern tall building design, especially for structures in the region prone to tropical cyclones. In fact, wind-related failures and building life-cycle analysis (LCA) may be impacted by climate change on the increase of frequency and intensity of tropical cyclones. In this study, a novel LCA framework considering the climate change is proposed, which includes taking into account the climate change via the serviceability failure cost for the cases where the building vibrates exceeding the corresponding thresholds. In detail, the non-stationary Poisson process is used to model the occurrence and intensity of tropical cyclones, which in turn leads to the financial loss of the coastal building. Specifically, the vulnerability of the building is derived considering both the uncertainty of the extreme wind speeds in the tropical cyclone boundary layer and the fragility curve which includes the uncertainty from the structural damping ratio. To estimate the maintenance cost, the financial values associated with maintenance-related operations and the emission of greenhouse gas are combined. In order to illustrate the proposed LCA framework, a case study targeting a 240 m high benchmark building located close to the coast of Hong Kong is conducted. In addition, the sensitivity of the damping ratio, limit threshold, and climate change are assessed via the LCA results of the benchmark building.

NHPP Software Reliability Model with Rayleigh Fault Detection Rate and Optimal Release Time for Operating Environment Uncertainty

Software is used in diverse settings and depends on development and testing environments. Software development should improve the reliability, quality, cost, and stability of software, making the software testing period crucial. We proposed a software reliability model (SRM) that considers the uncertainty of software environments and the fault detection rate function as a Rayleigh distribution, with an explicit mean value function solution in the model. The goodness-of-fit of the proposed model relative to several existing nonhomogeneous Poisson process (NHPP) SRMs is presented based on three software application failure datasets. Further, a cost model is also presented that addresses the error-removal risk level and required time. The optimal testing release policy for minimizing the expected total cost (ETC) is also determined for NHPP SRMs. The impact of the software environment is studied by varying it, and the optimal release times and minimum ETCs are compared. The goodness-of-fit comparison confirmed that the proposed model has more accurate prediction values than other models. Further, whereas the existing models applied to the cost model do not change after a certain operation period, the proposed model yields changes in release time even for long operating periods.

A novel method for rapidly simulating X-ray pulsar signals at a spacecraft

This paper addresses the challenges posed by the low photon flux and limited atmospheric penetration capabilities of X-ray pulsar signals, which are further exacerbated by technical, economic, and temporal constraints during in-orbit observations. These challenges result in a lack of continuous and high-fidelity data support for ground-based X-ray pulsar navigation (XPNAV) research and validation. To overcome these obstacles, we propose an innovative method for the efficient and rapid simulation of X-ray pulsar signals as received by spacecraft. Building on the physical process that, although the arrival times of the same pulsar signal differ between the spacecraft and the Solar System Barycenter (SSB), these times can be calibrated to the same phase at the SSB using a timing model, we leverage the relationship between the pulsar signal’s arrival rate at the SSB and its arrival phase. Using the Order Statistical Method (OSM), we directly and effectively generate the arrival phase series of pulsar signals received by the spacecraft, which follow a non-homogeneous Poisson process (NHPP). Subsequently, from these phase series, we derive the arrival time series of spacecraft-received pulsar signals, incorporating multiple physical effects based on a relationship model between arrival time series and phases. This method significantly reduces the required simulation time while maintaining high accuracy, thereby greatly enhancing simulation efficiency. Validation against data from the Rossi X-ray Timing Explorer (RXTE) and the Neutron Star Interior Composition Explorer (NICER) shows that the correlation coefficients between the accumulated profiles of simulated and measured data exceed 0.99. Furthermore, our method reduces the simulation time of high-flux pulsar signals by 25.7%, high spin frequency signals by 40.8%, and long observation duration signals by 25.38% compared to the latest fast simulation techniques, highlighting its advantages in accuracy, speed, and versatility for simulating pulsar signals across varying flux rates, spin frequencies, and observation durations. This approach provides continuous and effective data support for XPNAV research, significantly advancing theoretical and practical developments in the field.

Joint Distribution of the Supremum and the Moment of Its Attainment by a Poisson Process with Linear Drift

Joint Distribution of the Supremum and the Moment of Its Attainment by a Poisson Process with Linear Drift

Growing integer partitions with uniform marginals and the equivalence of partition ensembles

We present an explicit construction of a Markovian random growth process on integer partitions such that given it visits some level n, it passes through any partition λ of n with equal probabilities. The construction has continuous time, but we also investigate its discrete time jump chain. The jump probabilities are given by explicit but complicated expressions, so we find their asymptotic behavior as the partition becomes large. This allows us to explain how the limit shape is formed.Using the known connection of the considered probabilistic objects to Poisson point processes, we give an alternative description of the partition growth process in these terms. Then we apply the constructed growth process to find sufficient conditions for a phenomenon known as equivalence of two ensembles of random partitions for a finite number of partition characteristics. This result allows to show that counts of odd and even parts in a random partition of n are asymptotically independent as n→∞ and to find their limiting distributions, which are, somewhat surprisingly, different.

Longitudinal Fluvial Dispersion of Coarse Particles: Insights From Field Observations and Model Simulations

AbstractIn this study, we use field observations augmented with model simulations to examine gravel dispersion over nine years (2007–2015) in Halfmoon Creek. The observations of flow, entrainment, and dispersion were used to develop a forward model utilizing the Einstein‐Hubbell‐Sayre (EHS) compound Poisson process. The observed mean virtual velocity of the tracer population slows down with cumulative excess energy after the 2010 large event. The forward model deviates from the observations in representation of tails, overpredicts mean displacements, and shows a narrower spatial distribution. The heavy‐tailed resting times indicate prolonged immobilization of some grains, suggesting the preferential movement of other most mobile grains. As such, 34% of most mobile grains constitute 50% of the total entrainments. The consideration of preferential movement explains the longitudinal spread but still overpredicts the displacement after the 2010 event. The model was then explored to consider additional transport‐related mechanisms causing deviations, such as reduction in virtual velocity, entrainment probability, and morphological trapping of meander bends, which helps to adequately recreate the observed dispersive behavior. The available historical flow records used for simulating dispersive behavior over multiple decades reveal an abrupt increase in displacements for exceptionally large events, suggesting the exhumation of deeply buried grains back in transport. The simulation results highlight the need for tracer studies with large sample sizes and improved recovery rates for longer time frames experiencing floods of widely varying magnitudes. Such models, inspired by Einstein's stochastic theory can be valuable for various river research applications.

Constraining the Duration and Ages of Stratigraphic Unconformities on Mars Using Exhumed Craters

AbstractCrater counting is a widely applied methodology for dating large areas of planetary surfaces, but is difficult to apply the method to constrain the durations of stratigraphic unconformities. Unconformities with exhumed craters are thought to indicate long hiatuses that can only be indirectly dated through stratigraphic relationships with other surfaces with uniform exposure ages. On Mars, sedimentary deposits with prominent unconformities with exhumed craters are found in layered deposits in the Arabia Terra region as well as Gale crater within Mount Sharp. In this work, we present a Linear Crater Counting methodology and apply it to constrain these unconformities observed in Arabia Terra and in Mount Sharp. The method applies a linear sampling domain correction to conventional two‐dimensional crater size frequency distributions and Bayesian Poisson process statistics in order to constrain the likely durations of these unconformities. We found that unconformities in Arabia Terra were on the order of 0.1–1 Gyr in length and that the unconformity preserved at Mount Sharp is at least 0.2 Gyr in length given estimates of the ages of the host craters. Hiatuses of these lengths constrain the age of the overlying deposits to be Late Hesperian or Amazonian in age. Two utility plots are also provided, along with the derivation, for researchers to apply this method to dating arbitrary geologic contacts on Mars and to adapt it to other bodies.

A simplified approach to counting statistics with an imperfect pileup rejector

A simplified theoretical model is developed to predict counting statistics for a stationary Poisson process passing through a spectrometer with pulse-pileup rejection. The model is applicable to digital counters used for spectrometry as well as set-ups utilising analogue electronics for pulse shaping and pileup rejection. In comparison with an existing model for a perfect pileup rejector, the new model addresses the common imperfection of having a finite time resolution of the fast channel, allowing quasi-coincident signals to pass through the pile-up rejector. From the Laplace transform of a simplified interval-density distribution, approximate expressions are derived for the throughput factor and the variance of the number of counted events. The results are compared with computer simulations of a cascade of extending dead time and subsequent pileup rejection. In addition, a rigorous throughput factor is derived from probabilistic reasoning, as well as an effective throughput factor for singular and coincident events.

Statistical Mapping of PFOA and PFOS in Groundwater throughout the Contiguous United States.

Per-and polyfluoroalkyl substances (PFAS) are synthetic chemicals that are increasingly being detected in groundwater. The negative health consequences associated with human exposure to PFAS make it essential to quantify the distribution of PFAS in groundwater systems. Mapping PFAS distributions is particularly challenging because a national patchwork of testing and reporting requirements has resulted in sparse and spatially biased data. In this analysis, an inhomogeneous Poisson process (IPP) modeling approach is adopted from ecological statistics to continuously map PFAS distributions in groundwater across the contiguous United States. The model is trained on a unique data set of 8910 PFAS groundwater measurements, using combined concentrations of two PFAS analytes. The IPP model predictions are compared with results from random forest models to highlight the robustness of this statistical modeling approach on sparse data sets. This analysis provides a new approach to not only map PFAS contamination in groundwater but also prioritize future sampling efforts.

The last-success stopping problem with random observation times

AbstractSuppose N independent Bernoulli trials with success probabilities $$p_1, p_2,\ldots $$ p 1 , p 2 , … are observed sequentially at times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. The case $$p_k=1/k$$ p k = 1 / k has been studied by many authors as a version of the familiar best choice problem, where both N and the observation times are random. We consider a more general profile $$p_k=\theta /(\theta +k-1)$$ p k = θ / ( θ + k - 1 ) and assume that the prior distribution of N is negative binomial with shape parameter $$\nu $$ ν , so the arrivals occur at times of a mixed Poisson process. The setting with two parameters offers a high flexibility in understanding the nature of the optimal strategy, which we show is intrinsically related to monotonicity properties of the Gaussian hypergeometric function. Using this connection, we find that the myopic stopping strategy is optimal if and only if $$\nu \ge \theta $$ ν ≥ θ . Furthermore, we derive formulas to assess the winning probability and discuss limit forms of the problem for large N.

Generalized Itô’s lemma and the stochastic thermodynamics of diffusion with resetting

Abstract Methods from the theory of stochastic processes are increasingly being used to extend classical thermodynamics to mesoscopic non-equilibrium systems. One characteristic feature of these systems is that averaging the stochastic entropy with respect to an ensemble of stochastic trajectories leads to a second law of thermodynamics that quantifies the degree of departure from thermodynamic equilibrium. A well known mechanism for maintaining a diffusing particle out of thermodynamic equilibrium is stochastic resetting. In its simplest form, the position of the particle instantaneously resets to a fixed position x 0 at a sequence of times generated from a Poisson process of constant rate r. Within the context of stochastic thermodynamics, instantaneous resetting to a single point is a unidirectional process that has no time-reversed equivalent. Hence, the average rate of entropy production calculated using the Gibbs–Shannon entropy cannot be related to the degree of time-reversal symmetry breaking. The problem of unidirectionality can be avoided by considering resetting to a random position or diffusion in an intermittent confining potential. In this paper we show how stochastic entropy production along sample paths of diffusion processes with resetting can be analyzed in terms of extensions of Itô’s formula for stochastic differential equations (SDEs) that include both continuous and discrete processes. First, we use the stochastic calculus of jump-diffusion processes to calculate the rate of stochastic entropy production for instantaneous resetting, and show how previous results are recovered upon averaging over sample trajectories. Second, we formulate single-particle diffusion in a switching potential as a hybrid SDE and develop a hybrid extension of Itô’s stochastic calculus to derive a general expression for the rate of stochastic entropy production. We illustrate the theory by considering overdamped Brownian motion in an intermittent harmonic potential. Finally, we calculate the average rate of entropy production for a population of non-interacting Brownian particles moving in a common switching potential. In particular, we show that the latter induces statistical correlations between the particles, which means that the total entropy is not given by the sum of the 1-particle entropies.

Transient and Steady-State Analysis of an M/PH2/1 Queue with Catastrophes

In the paper, we consider the PH2-distribution, which is a particular case of the PH-distribution. In other words, The first service phase is exponentially distributed, and the service rate is μ. After the first service phase, the customer can to go away with probability p or continue the service with probability (1−p) and service rate μ′. We study an analysis of an M/PH2/1 queue model with catastrophes, which is regarded as a generalization of an M/M/1 queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system’s behavior, and conclusions and future directions of the model are given.

Choosing between additive and conventional manufacturing of spare parts: On the impact of failure rate uncertainties and the tools to reduce them

Recently, there has been great interest in using additive manufacturing (AM) to produce spare parts: allowing to produce spare parts with short lead times and close to the point of use, AM reduces the need for large inventories otherwise required by conventional manufacturing (CM) techniques to deal with intermittent spare parts demand. However, using AM to produce spare parts is limited by two main drawbacks: high production costs and uncertain failure rates arising from AM still being a relatively new production technique. While the former can be counterbalanced by inventory cost reductions, it is unclear how the latter impacts the sourcing option decision (i.e. AM or CM). We aim to fill this gap by studying a periodic review model in which spare parts demands follow a Poisson process. To make our analysis accurate and reliable, we leverage a material science approach to obtain realistic values for the failure rate uncertainties. From the results, it emerges that AM is heavily penalized by failure rate uncertainties much higher than those of CM. Consequently, we then focus on some recent tools developed to reduce AM failure rate uncertainties: porosity assessment and in-situ monitoring. We find that the failure rate uncertainty should be reduced by five and six percentage points to make it worthwhile to invest in porosity assessment and in-situ monitoring, respectively. Finally, we determine under which circumstances each tool is preferred over the other and found that porosity assessment is typically the most competitive uncertainty reduction tool.

The poisson property of extreme events in optics

Statistics of extreme events in optics, defined as above-threshold counts of an optical signal, is shown to converge, in the large-sample-size limit, to a generalized Poisson distribution whose mean is found via the exponent of the respective extreme-value distribution. Specifically, extreme-event readouts from polynomial and exponential optical nonlinearities are shown to converge in their statistics to Poisson distributions whose means are, respectively, exponential and slower-than-exponential functions of the extreme-event-counter threshold. Extreme-event counts of a phase readout, on the other hand, converge to a Poisson process whose mean is a light-tailed function of the threshold. The Poisson-limit property of extreme events in optics suggests a powerful resource for a unified treatment of a vast variety of extreme-event phenomena, ranging from optical rogue waves to laser-induced damage.

Mixed Poisson Processes with Dropout for Consumer Studies

We adapt the classical mixed Poisson process models for investigation of consumer behaviour in a situation where after a random time we can no longer identify a customer despite the customer remaining in the panel and continuing to perform buying actions. We derive explicit expressions for the distribution of the number of purchases by a random customer observed at a random subinterval for a given interval. For the estimation of parameters in the gamma–Poisson scheme, we use the estimator minimizing the Hellinger distance between the sampling and model distributions, and demonstrate that this method is almost as efficient as the maximum likelihood being much simpler. The results can be used for modelling internet user behaviour where cookies and other user identifiers naturally expire after a random time.