Homogeneous Poisson Process Research Articles

Poisson process modelling of spike occurrence in single particle inductively coupled plasma mass spectrometry time scans for very dilute nanoparticle dispersions

The time scan measured when a dispersion of nanoparticles is analyzed by single particle inductively coupled plasma mass spectrometry (sp-ICP-MS) is typically made of many intense short-lived spikes. On the basis of theoretical arguments backed by experimental data and statistical tests, I show that, in the very dilute limit, there is a one-to-one correspondence between the spikes present in a time scan and the points of discontinuity of the outcome of a homogeneous Poisson process. This relationship underlies the random nature of the sp-ICP-MS time scan. Furthermore, I establish that the stochastic processes describing the nanoparticle arrivals in the plasma and the time sequence of spikes in the time scan are fully identical, thus helping to place on firmer grounds a frequently made hypothesis regarding the Poissonian character of the sp-ICP-MS time scan.

Reliability evaluation of non-repairable systems with failure mechanism trigger effect

Trigger effect indicates that some events occur because of the trigger of other events. This kind of effect often appears in the operation of products and leads to failure. Due to the randomness of trigger events, it is difficult to predict the reliability of a product that embraces triggering processes. In this paper, we develop a new reliability model for non-repairable systems with the trigger effect. The effect is divided into two types: degradation-degradation trigger effect and degradation-overstress trigger effect. To model this effect, the degradation process is described by the continuous or discrete stochastic process, while the overstress process is described by the homogeneous Poisson process. Furthermore, a method integrating Petri Net and Monte Carlo is proposed to evaluate the complex system reliability. To demonstrate the proposed model, a conductive slip ring in an electromechanical orientation device is introduced as a case, where the trigger effect of one trigger mechanism and three dependent basic mechanisms are considered. It is shown that the trigger time model and system modeling method can describe the trigger failure behaviors and evaluate the reliability of complex systems with the trigger effect.

Performance analysis of an improved forked communication network model

The utilisation of communication network models is growing day by day in a drastic way around the world. In order to communicate the data round the devices, the utilisation or the establishment of communication networks is mandatory. In this paper, a forked communication network model with the base of a queuing model and the equations developed for a better understanding of the model. The results calculated with the help of MATLAB and MathCAD Software. The results observed in detail in the results section. To better understand the performance of the current forked model of communication network, the performance metrics of the current network model had compared with the performance metrics of the previous two three-node communication networks. The comparison done with the arrival process of packets to the network in one model follows the Duane process. In other model follows the Poisson process, whereas in the current model follows the Homogeneous Poisson process of arrivals. The comparison had given in detail in the comparison section.

An optimal preventive maintenance policy for a two-stage competing-risk system with hidden failures

In many complex industrial systems, failures are not always evident to the operating crew, thus inspections are used to identify such hidden failures and then make necessary maintenance actions. In this paper, we study a two-stage system that is subject to two competing risks of degradation and random shocks. Failures of this two-stage competing-risk system are hidden and could only be detected by periodic inspections. For such a system, a delay-time model is introduced to describe the evolution of the system behavior. Random shocks are assumed to follow a homogeneous Poisson process. A preventive maintenance policy is presented in response to the observed system state, based on which the expected cost rate of the system is derived. This paper aims to find the minimal expected cost rate by determining the optimal inspection interval. A numerical example is presented to demonstrate the applicability of the proposed method and results.

On Some Statistical Properties of the Spatio-Temporal Product Density

We present an extension of the non-parametric edge-corrected Ohser-type kernel estimator for the spatio-temporal product density function. We derive the mean and variance of the estimator and give a closed-form approximation for a spatio-temporal Poisson point process. Asymptotic properties of this second-order characteristic are derived, using an approach based on martingale theory. Taking advantage of the convergence to normality, confidence surfaces under the homogeneous Poisson process are built. A simulation study is presented to compare our approximation for the variance with Monte Carlo estimated values. Finally, we apply the resulting estimator and its properties to analyse the spatio-temporal distribution of the invasive meningococcal disease in the Rhineland Regional Council in Germany.

System Analysis and Reliability Analysis for Environmental Test Chambers

The purpose of this paper is to present the results of the application of various models to estimate the reliability of environmental test chambers, especially, the methodology proposed by the International Electrotechnical Commission (IEC), using the Homogeneous Poisson process (HPP) and Non-Homogeneous Poisson process (NHPP) models, is adopted first, and then, a non-monotonic trend test and bathtub curve intensity function not covered by IEC are used, and supplementary analysis is used to characterize the resulting failure intensity. For the first time, the stochastic process model was applied to evaluate the reliability of 20 environmental test chambers. The results show that the IEC standards process is suitable for the reliability evaluation of a single chamber, and 16 chambers conform to the HPP model and 4 chambers conform to the NHPP model. However, there is the power-law model (PLP) rejection cases among the overall description of multiple chambers by the IEC model. The rejected cases were analyzed again by using a non-monotonic trend test and constructing a double Weibull process tub curve strength function, and the 3-stage time interval of the bathtub curve failure is obtained, which is in line with the actual operation data. The Ward clustering method is used for the mean time between failures of 20 chambers, resulting in 4 types of chamber groups with different reliability values (71,52,100,130 days), which is of great significance for studying the reliability of the environmental test chamber and carrying out customized maintenance.

Modified one-sided EWMA charts for monitoring time between events

Control charts based on Time Between Events (TBE) have been shown to be effective in high quality processes, where the defects or failures (events) occur at a very low probability. The occurrences of events are usually modeled as a homogeneous Poisson process. The TBE data is likely to follow a skewed distribution, such as the gamma distribution. In order to monitor high quality processes, this study focuses on the development of modified one-sided Exponentially Weighted Moving Average (EWMA) TBE charts. The Monte-Carlo simulation method is used to evaluate the Run Length (RL) properties of the proposed schemes. The results show that the proposed schemes are more sensitive to process shifts than some competitive procedures. Finally, a real data example is provided to illustrate the implementation of the proposed charts.

Homogeneous Poisson process in daily case of covid-19

Since December 2019, an increasing number of new pneumonia cases have emerged in Wuhan, China. The rise of the spread of diseases caused by the Corona Virus Disease (covid-19) which has been established as a pandemic by WHO on March 12, 2020, gave rise to so much anxiety and speculation from various parties. The case of covid-19 positive patients Daily can be calculated by the homogeneous Poisson process. A Poisson process with a constant rate (λ) is called a homogeneous Poisson process. The average number of positive patients of Covid-19 from January 24, 2020, to April 16, 2020, is still very large. The chances of not having cases of covid-19 positive patients from January 24, 2020 to April 16, 2020 are very small so there will always be covid-19 cases every. Therefore, elements of society and government must consider handling and preventing the Covid-19 case.

Nonhomogeneous bivariate compound Poisson process with short-term periodicity

This paper presents new results on the nonhomogeneous bivariate compound Poisson process with a short-term periodic intensity function. The dependence between margins is modeled using the Lévy copula. The model’s parameters are estimated by the maximum likelihood method. Finally, a set of real data on automobile insurance is analyzed using the methodology of this study. The empirical results show that the nonhomogeneous bivariate compound Poisson process with the Clayton Lévy copula is a good model for describing real data in comparison with the homogeneous bivariate compound Poisson process.

Comparison of Poisson process and machine learning algorithms approach for credit card fraud detection

This article describes the financial fraud detection in imbalanced data. We compare various approaches for credit card fraud detection problem. On the one hand, we use homogeneous and heterogeneous Poisson process to determine the probability of predicting fraud with the various intensity parametric functions. On the other hand, we solve classification problem using machine learning algorithms and different family of ensemble methods like boostings. The results of both methods are compared. The “false positive” problem is also discussed in the article.

Solving a Joint Pricing and Inventory Control Problem for Perishables via Deep Reinforcement Learning

We study a joint pricing and inventory control problem for perishables with positive lead time in a finite horizon periodic‐review system. Unlike most studies considering a continuous density function of demand, in our paper the customer demand depends on the price of current period and arrives according to a homogeneous Poisson process. We consider both backlogging and lost‐sales cases, and our goal is to find a simultaneously ordering and pricing policy to maximize the expected discounted profit over the planning horizon. When there is no fixed ordering cost involved, we design a deep reinforcement learning algorithm to obtain a near‐optimal ordering policy and show that there are some monotonicity properties in the learned policy. We also show that our deep reinforcement learning algorithm achieves a better performance than tabular‐based Q‐learning algorithms. When a fixed ordering cost is involved, we show that our deep reinforcement learning algorithm is effective and efficient, under which the problem of “curse of dimension” is circumvented.

Bayesian hierarchical models for linear networks

The purpose of this study is to highlight dangerous motorways via estimating the intensity of accidents and study its pattern across the UK motorway network. Two methods have been developed to achieve this aim. First, the motorway-specific intensity is estimated by using a homogeneous Poisson process. The heterogeneity across motorways is incorporated using two-level hierarchical models. The data structure is multilevel since each motorway consists of junctions that are joined by grouped segments. In the second method, the segment-specific intensity is estimated. The homogeneous Poisson process is used to model accident data within grouped segments but heterogeneity across grouped segments is incorporated using three-level hierarchical models. A Bayesian method via Markov Chain Monte Carlo is used to estimate the unknown parameters in the models and the sensitivity to the choice of priors is assessed. The performance of the proposed models is evaluated by a simulation study and an application to traffic accidents in 2016 on the UK motorway network. The deviance information criterion (DIC) and the widely applicable information criterion (WAIC) are employed to choose between models.

Inferring extinction date of a species using non‐homogeneous Poisson processes with a change‐point

Abstract Bayesian methods have been developed for inferring the true year of extinction of a species from sighting records that have both certain and uncertain sightings. These methods typically make the restrictive assumption that all sighting types (i.e. certain, valid uncertain, invalid uncertain) derive from constant rate Poisson processes. In this study, the constant rate assumption in the homogeneous Poisson process is relaxed by allowing certain and uncertain sightings to follow independent non‐homogeneous Poisson processes. The model can thus identify whether any of the sighting rates were increasing, decreasing or constant. In addition, a change‐point is introduced to model the uncertain sightings, where the sighting rates before and after the change‐point vary. We have used Markov Chain Monte Carlo (MCMC) sampling to generate the posterior distributions for model parameters including species extinction time. The proposed method was applied to the sighting records of the black‐footed ferret Mustela nigripes and the ivory‐billed woodpecker Campephilus principalis (IBW) species. Based on a hypothesis test, the results of the model indicate that the species both went extinct in the years 1988 and 1956 respectively. Moreover, a decline in the certain sighting rate was also inferred for both these species, possibly indicating the decrease in the species abundance as it converges to extinction. Thus, earlier models that assume a constant sighting rate may well be biased. Uncertain sighting rates for the IBW were found to increase before extinction (indicating possibly some additional ecological attention received near extinction) and remained constant after extinction.

The Post-Fire Assembly Processes of Tree Communities Based on Spatial Analysis of a Sierra Nevada Mixed-Conifer Forest

Understanding the mechanisms underlying tree spatial arrangements may provide significant insights into the processes in the maintenance of species coexistence. We examined the potential role of habitat heterogeneity, dispersal limitation, negative density dependence, fire history, and unilateral intraspecific and interspecific interactions of adults on juveniles in shaping the spatial patterns of four dominant tree species (Abies concolor, Pinus lambertiana, Calocedrus decurrens, and Quercus kelloggii) after fire in the Yosemite Forest Dynamic Plot, California, USA. We used the univariate pair correlation function and implemented three point pattern processes (homogeneous Poisson process, inhomogeneous Poisson process, and homogeneous Thomas process) to evaluate the potential contributions of habitat filtering and dispersal limitation. We used a bivariate null model to evaluate unilateral intraspecific and interspecific interactions of adults on juveniles. We also used the pairwise correlation function to investigate the spatial patterns of density dependence. To understand the effect of fire, we used the univariate pair correlation function to investigate pattern changes during the six years following fire. We compared spatial pattern changes in both sprouting species (Quercus kelloggii) and seeding species (Abies concolor), and also examined the changes in patterns of large-diameter individuals of Abies concolor, Pinus lambertiana, and Calocedrus decurrens in 2013 (pre-fire), 2016 (two years post-fire), and 2019. Comparing the contributions of the homogeneous Thomas process and the inhomogeneous Poisson process at different spatial scales showed the importance of dispersal limitation and habitat heterogeneity at finer scales (0 m to 5 m) and coarser scales (5 m to 60 m), respectively, which suggests that the joint effects of dispersal limitation and habitat heterogeneity contribute to the spatial patterns of these three dominant tree species. Furthermore, the results showed that the young individuals of Abies concolor and Pinus lambertiana were more commonly found around the conspecific adults. Juvenile regeneration to the 1 cm diameter threshold was highly aggregated following the fire. Large-diameter trees of Abies concolor, Pinus lambertiana, and Calocedrus decurrens generally did not exhibit patterns different from complete spatial randomness (Calocedrus decurrens), or displayed only slight aggregation (Abies concolor and Pinus lambertiana). In addition, Abies concolor and Pinus lambertiana showed positive and negative conspecific density dependence in the immediate post-fire period, respectively.

Statistical properties of Poisson-Voronoi tessellation cells in bounded regions

Many spatial statistics methods require neighbourhood structures such as the one determined by a Voronoi tessellation, so understanding statistical properties of Voronoi cells is crucial. While distributions of cell properties when data locations follow an unbounded homogeneous Poisson process have been studied, little attention has been given to how these properties change when a boundary is imposed. This is important when geographical data are gathered within a restricted study area, such as a national boundary or a coastline. We study the effects of imposing a boundary on the cell properties of a Poisson Voronoi tessellation. The area, perimeter and number of edges of individual cells with and without boundary conditions are investigated by simulation. Distributions of these properties differ substantially when boundaries are imposed, and these differences are affected by proximity to the boundary. We also investigate how changes in such properties when boundaries are imposed vary over two-dimensional space.

On assurance testing for repairable systems

We consider assurance testing for repairable systems when supplementary information is available in addition to the data collected in the assurance test. The supplementary information is incorporated using a Bayesian inferential framework. Assurance testing is considered for both a homogeneous Poisson process and a nonhomogenous Poisson process and is illustrated with examples.

Software reliability modeling based on NHPP for error occurrence in each fault with periodic debugging schedule

In this article, we discuss a continuous time software reliability model under the non homogeneous Poisson process (NHPP) assumption for error occurrence in each fault to suit periodic debugging, in which errors are not corrected at the instants of their detection but at some pre-specified debugging times. We describe maximum likelihood estimation for the model parameters and provide a computational method to estimate those parameters. This in turn helps to estimate the reliability of the software. We also discuss some asymptotic properties of the estimated model parameters, specially the number of errors initially present in the software. Finally, we investigate the finite sample properties of the estimates under a specific family of NHPP models, specially that of the initial number of errors, through simulation.

Exploring probabilistic seismic risk assessment accounting for seismicity clustering and damage accumulation: Part I. Hazard analysis

Probabilistic seismic hazard analysis (PSHA), as a tool to assess the probability that ground motion of a given intensity or larger is experienced at a given site and time span, has historically comprised the basis of both building design codes in earthquake-prone regions and seismic risk models. The PSHA traditionally refers solely to mainshock events and typically employs a homogeneous Poisson process to model their occurrence. Nevertheless, recent disasters, such as the 2010–2011 Christchurch sequence or the 2016 Central Italy earthquakes, to name a few, have highlighted the potential pitfalls of neglecting the occurrence of foreshocks, aftershocks, and other triggered events, and pinpointed the need to revisit the current practice. Herein, we employ the epidemic-type aftershock sequence (ETAS) model to describe seismicity in Central Italy, investigate the model’s capability to reproduce salient features of observed seismicity, and compare ETAS-derived one-year hazard estimates with ones obtained with a standard mainshock-only Poisson-based hazard model. A companion paper uses the hazard models derived herein to compare and contrast loss estimates for the residential exposure of Umbria in Central Italy.

Specialist care in rural hospitals: From Emergency Department consultation to hospital discharge

In urban and rural hospitals, congested Emergency Departments (EDs) are filled with patients boarding in the ED awaiting admission to inpatient wards. We study this problem beyond the walls of the ED, examining the multi-departmental process managed by specialists. In rural hospitals, an Internal Medicine Specialist (Internist) commonly serves simultaneously as both the Intensive Care Unit (ICU) physician and Internist on call. We develop a stochastic dynamic programming framework for specialists’ workflow decisions and apply it to data sets developed from two rural hospitals. One uses the dual role approach and the other, similar to urban hospitals, staffs the ICU with another physician, each with a single role. Our empirical results show that, excluding an overnight batch, arrivals of ED consultation requests for rural specialists follow a homogeneous Poisson process. Our models help identify better policies and determine how much better off a hospital is with two rather than one Internist on call. Although current guidelines suggest an early inpatient discharge strategy, we find that specialists should give higher priority to ED consultations unless a threshold number of patients are boarding in the ED or until a threshold time of day when specialists should give higher priority to inpatient discharges.

Estimation of the expected value of the arrival epochs given a bounded period of observation for the homogeneous and non-homogeneous Poisson processes

In this paper, we derive analytical expressions of the expected value of the arrival epochs given a bounded period of observation for the homogeneous Poisson process (HPP) and for the general case of a non-homogeneous Poisson process (NHPP). These expressions are exact, but their applicability require some computational adaptations. We illustrate the results obtained through the proposed expression by comparing them with Monte Carlo sampling. The results show that these expressions outperform Monte Carlo in precision and computational effort.