Poisson Process Research Articles

Generalized Iterated Poisson Process and Applications

Generalized Iterated Poisson Process and Applications

Catalyzing teacher moves in small-group problem solving: a quantitative discourse analysis

ABSTRACT This report validates a novel quantitative methodology for analyzing moment-to-moment interactions in classrooms called the Poisson Process Methodology (PPM). PPM differs from moment-to-moment qualitative and quantitative analyzes typically used in education by using time-series data to quantify the degree to which the presence of facilitator moves affect the future rate of student actions. Using PPM in the context of small-group problem solving further validates the use of the method in several settings in education. We applied the method to data from a larger study about students’ engagement in reasoning abstractly and quantitatively as they worked on mathematics problems with a facilitator. We present empirical results of PPM performed on two semi-structured interviews involving a facilitator and two students, which led to some interpretable results that extend previous understandings of forms of talk that promote quantitative reasoning.

Poisson Process with Linear Drift and Related Function Series

Poisson Process with Linear Drift and Related Function Series

Impact of dust and temperature on photovoltaic panel performance: A model-based approach to determine optimal cleaning frequency

Enhancing the reliability of photovoltaic (PV) systems is of paramount importance, given their expanding role in sustainable energy production, carbon emissions reduction, and supporting industrial growth. However, PV panels commonly encounter issues that significantly impact their performance. Specifically, the accumulation of dust and the rise in internal temperature lead to a drop in energy production efficiency. The primary issue addressed in this paper is using mathematical modeling to determine the optimal cleaning frequency. This paper first focuses on stochastic modeling for dust accumulation and temperature changes in PV panels, considering varying environmental conditions and proposing a model-based approach to determine the optimal cleaning frequency. Dust accumulation is described using a Non-homogeneous compound Poisson process (NHCPP), while temperature evolution is modeled using Markov chains. Within this framework, we consider the impact of wind speed and rainfall on dust accumulation and temperature. These factors, treated as covariates, are modeled using a two-dimensional time-continuous Markov chain with a finite state space. A Condition-based cleaning policy is proposed and assessed based on the degradation model. Optimal preventive cleaning thresholds and cleaning frequency (periodic and non-periodic) are determined to minimize the long-term average maintenance cost. The gain achieved by non-periodic inspections compared to periodic inspections ranges from 3.83% to 9.37%. Numerical experiments demonstrate the performance of the proposed cleaning policy, highlighting its potential to improve PV system efficiency and reliability.

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 patterns 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 maximises the discounted total expected reward 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.

Pricing of insurance-linked securities: a multi-peril approach

In this paper we build a methodology for pricing of insurance-linked securities which are tied to multiple natural catastrophe perils. As a representative example, we construct a multi-peril catastrophe (CAT) bond which can be linked to the industry loss indices or actual losses incurred by an insurer. We provide pricing formulas for such CAT bonds. We illustrate the introduced methodology on the US natural catastrophe data obtained from Property Claim Services (PCS). Within this dataset, we specifically examine two types of risks: losses associated with wind and thunderstorm events, and those linked to winter storm events. Then, we fit and validate the underlying compound non-homogeneous Poisson processes taking into account the fact that the data are left-truncated. The best fitted loss distributions appear to be Burr and Generalised Extreme Value and for the first peril and log-normal for the second. Finally, we visualise the zero-coupon CAT bond prices for the selected best-fitted models.

Pricing a product with network effects for sale to a fixed population of customers

AbstractWe consider a deterministic dynamic pricing problem for a product that exhibits network effects and that is sold to a fixed heterogeneous population of customers. We begin by introducing a demand model wherein those customers are arrayed over two‐dimensional space according to a bivariate probability distribution. Each customer's location in space provides a description of that customer's intrinsic value for the product as well as the extent to which the customer is influenced by the network effect. In the pricing problem, as sales accumulate over time, the set of customers who have already purchased the product grows, while the set of customers who have not yet purchased the product shrinks. The total customer population remains fixed. Those who have not yet purchased constitute the remaining population of potential buyers of the product. As time moves forward, the mix of customers that remain as potential buyers evolves endogenously. The demand model yields a geometric interpretation of the remaining population of potential buyers, and gives rise to a dynamic program with states that are sets in two‐dimensional space. It is not practical to solve the dynamic pricing problem to optimality, so we present bounds and comparative statics results that help us identify tractable heuristics and obtain rigorous performance guarantees. In numerical experiments, we find that fixed‐price policies may perform poorly, especially when the network effect is strong or the time horizon is long. We also introduce a stochastic version of the problem that uses a spatial Poisson process to describe the customers, and we develop and analyze a heuristic approach for that formulation.

Using the Non-Homogeneous Poisson Process (Duane's Model) to Analyze the Number of Failures in Industrial Equipment

Objective: The aim of this paper is to show the application of Duane's Chart to analyze the time between failures in a company in the South of Rio de Janeiro. Theoretical framework: A Duane chart is a scatter plot of the cumulative failure rate versus time. Use a Duane chart to do the following: •Assess whether the data follows a power-law process or a homogeneous Poisson process. •Determine whether a repairable system is improving, getting worse or remaining stable. The fitted line on the Duane chart plots the best fitted line when the power-law process assumption is valid and the shape and scale are estimated using the least squares method. The Duane plot should be approximately linear if the power-law process or homogeneous Poisson process is appropriate. A negative slope shows an improvement in reliability, a positive slope shows a deterioration in reliability and no slope (a horizontal line) shows a stable system (Zanin et al., 2010). Method: Data was taken from a company in the South of Rio de Janeiro and a case study was carried out using Duane's Chart. Final Considerations: This result indicates that Duane's Model is a very viable option when the aim is to monitor the failure rate as a function of time to check whether a system is improving or not. In this specific Case Study, it is clear that there is a significant improvement in the system over time and that the work carried out in this company is being very effective in reducing and even eliminating failures in the long term. Implications of the research: The use cases of Duane's Chart are multiplying in scientific literature and are proving to be highly effective for interpreting data in the area of reliability. Originality/value: Despite being a well-known Statistical Tool, Duane's Chart has a specific application in monitoring the number of failures over time in a system.

Processes with catastrophes: Large deviation point of view

In this paper, we propose a new definition of catastrophes and present our results on large deviations for Poisson processes with catastrophes that satisfy this definition. Our earlier work focused on (almost) uniformly distributed catastrophes, but the current paper extends the results to a larger class of catastrophes. We show that the rate function remains the same regardless of the distribution of catastrophic events. Additionally, we extend and generalize our previous results on the limiting behavior of the supremum of the considered processes.

A general inspection and replacement policy for protection systems subject to shocks with state dependent effect

This paper proposes a new maintenance policy for protection systems which fail due to competing causes: internal deterioration and fatal external shocks. The study is motivated by a real problem in a power distribution company, in which a circuit breaker is designed to protect an electrical network from critical events such as overloads and short circuits. The failure of the system is hidden and can lead to a disaster if it is not replaced before a demand is received. So, to improve its readiness, we have developed a new two-phase (inspection & replacement) policy for a protection system with a heterogeneous population. Heterogeneity reflects factors such as variability in the quality of spare parts or in the installation of the components, which can be poorly or properly executed, resulting in some having a short life, while others may achieve a long life. Inspections are imperfect, which induces misclassification. The deterioration process is modeled via the delay-time concept. Furthermore, the shock rate depends upon the state of the system and follows a non-homogeneous Poisson process. Thus, using a numerical approach, the objective of this study is to determine the number of inspections during the first phase, the interval between consecutive inspections, and the age for scheduled preventive replacement. We compared this policy to two other classical policies via a case study and found that it has significant cost-saving potential. Generally, our results indicate when inspections should be prioritized over age-preventive replacement, and vice versa. Multi-phase policies are particularly effective for components with different characteristic lives, with inspections being crucial for maintaining readiness, often requiring investment in quality assurance to prevent misclassification. Conversely, age-based policies are more advantageous for strong (long-lived) components or when the probability of misclassification is high.

Evaluating energy harvesting UAV-NOMA network with random user pairing in the finite blocklength regime

This study proposes an integrated system that combines energy harvesting (EH) enabled unmanned aerial vehicles (UAVs) with non-orthogonal multiple access (NOMA) to enhance communication system performance within a cellular network. Addressing the limitations of existing analyses that often assume an infinite blocklength scenario, we explore EH-enabled UAV-NOMA systems within a cellular framework under a finite blocklength (FBL) scenario. The study investigates the complex interactions and advantages resulting from the integration of EH, NOMA, and UAV technologies, aiming to assess whether EH can sustain communication within this framework. The network model considers base stations (BSs), UAVs, and terrestrial devices distributed with independent Poisson point processes (PPPs) over a large area. In this network, BSs employ NOMA to serve cell center devices directly, while cell edge devices, which are nor in direct contact with BS, are served via simultaneous wireless information and power transfer (SWIPT) enabled UAVs. The study derives metrics including joint harvesting and decoding probability for a randomly selected UAV, coverage probability (CP) for cell devices, and end-to-end block error rate (BLER) probabilities for typical device pairs. The findings demonstrate that the proposed scheme effectively supplies all the necessary transmit power for communication purposes through EH, achieving reasonable reliability. Additionally, the study highlights the importance of considering a combination of blocklengths from different phases to achieve optimal performance, rather than solely relying on an increment in blocklength. Finally, the effects of parameter variations on network performance are examined.

Modelling Salmonella Typhi in high-density urban Blantyre neighbourhood, Malawi, using point pattern methods

Salmonella Typhi is a human-restricted pathogen that is transmitted by the faecal–oral route and causative organism of typhoid fever. Using health facility data from 2016 to 2020, this study focuses on modelling the spatial variation in typhoid risk in Ndirande township in Blantyre. To pursue this objective, we developed a marked inhomogeneous Poisson process model that allows us to incorporate both individual-level and environmental risk factors. The results from our analysis indicate that typhoid cases are spatially clustered, with the incidence decreasing by 54% for a unit increase in the water, sanitation, and hygiene (WASH) score. Typhoid intensity was also higher in children aged below 18 years than in adults. However, our results did not show evidence of a strong temporal variation in typhoid incidence. We also discuss the inferential benefits of using point pattern models to characterise the spatial variation in typhoid risk and outline possible extensions of the proposed modelling framework.

The Wasserstein Metric between a Discrete Probability Measure and a Continuous One

This paper examines the Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform measure in the d-dimensional ball, providing an asymptotic estimation of their expectations as n approaches infinity. Furthermore, we investigate this problem within a mixed process framework, where n discrete random variables are generated by the Poisson process.

Vapor bubble nucleation in flowing liquids

In this work, a stochastic diffuse interface model coupled with the Navier–Stokes equations has been exploited to numerically investigate vapor nucleation in a non-equilibrium system (a flowing liquid). Both homogeneous and heterogeneous nucleation is addressed and the influence of macroscopic flows on nucleation observables is discussed. The extended mesoscale simulations allow us to infer the spatial distributions of the nucleated bubbles via Voronoi tessellation analysis and to represent the nucleation phenomenon as a stochastic Random Poisson Point process. The findings open the way to design multiscale fluid simulations experiencing phase change.

A novel prognostic method for degrading devices with nonlinear degradation processes indexed by both continuous and discrete time scales

Remaining useful life (RUL) prediction techniques based on degradation modeling to ensure the safety and reliability of stochastic degraded devices are of critical value. However, although the performance degradation of various devices in engineering practice is affected by multi-timescales, there are few RUL prediction methods for stochastic degraded devices with dual time scales in the existing literature. The research of prediction methods considering continuous and discrete time scales is blank. Toward this end, this paper proposes a novel prognostic method for nonlinear degradation devices based on the continuous-discrete dual time scales. First, the performance degradation with a discrete timescale is determined with a compound Poisson process and integrated into the diffusion process model to establish a nonlinear dual-timescale degradation model. Subsequently, the joint expression of the dual-timescale RUL distribution of the device in the first hitting time sense solves with threshold transformation. Finally, the effectiveness and superiority of the proposed method are verified through a numerical simulation and a gyroscope example. The experimental results show that the proposed method can provide more comprehensive information about the life distribution and effectively improve the accuracy of the RUL prediction.

Copula-based Bayesian estimation of probabilities for two dependent non homogenous compound Poisson processes

Bivariate copulas for two dependent non homogeneous compound Poisson processes (NCPP) are considered. In this scenario, the arrival times of events are assumed to follow two independent non homogeneous Poisson processes (NHPP), but the corresponding renewal variables are dependent through a bivariate copula. The article aims to provide a computational tool for the estimation of the probability that the random sum associated with the first NCPP is less than the other random sum by a truncated time. The proposed ML and Bayesian estimation methods in a general setting can be applied using bivariate copulas such as Gumbel (for upper tail dependency), Clayton (for lower tail dependency), and Frank (for symmetric dependency). Power Law (PL) intensity function for the NHPPs and Inverse-Gaussian as renewal distribution are considered as examples. Bayes estimates of NHPPs parameters use gamma priors, and for Bayes estimates of Inverse-Gaussian distribution, an MCMC algorithm is employed. Our method has a broad range of practical applications, especially in the insurance sector, where the renewal variable may represent the dollar amount associated with a claim. To illustrate our approach, we analyze injury claims in North Carolina and South Carolina over a specific period, using a dataset sourced from docs.databricks.com.

Finding the area and perimeter distributions for flat Poisson processes of a straight line and Voronoi mosaics

Finding the area and perimeter distributions for flat Poisson processes of a straight line and Voronoi mosaics

Fractional Duals of the Poisson Process on Time Scales with Applications in Cryptography

Fractional Duals of the Poisson Process on Time Scales with Applications in Cryptography

Convolutional Non-Homogeneous Poisson Process and its Application to Wildfire Ignition Risk Quantification for Power Delivery Networks

To quantify wildfire ignition risks on power delivery networks, the current practice predominantly relies on the empirically calculated fire danger indices, which may not well capture the effects of dynamically changing environmental factors. This article proposes a spatio-temporal point process model, known as the Convolutional Non-homogeneous Poisson Process (cNHPP), and applies the model to quantify wildfire ignition risks for power delivery networks. The proposed model captures both the current (i.e., instantaneous) and cumulative (i.e., historical) effects of key environmental processes (i.e., covariates) on wildfire risks, as well as the spatio-temporal dependency among different segments of the power delivery network. The computation and interpretation of the intensity function are thoroughly investigated. We apply the proposed approach to estimate wildfire ignition risks on major transmission lines in California, using historical fire data, meteorological and vegetation data obtained from the National Oceanic and Atmospheric Administration and National Aeronautics and Space Administration. A comprehensive comparison study is performed to show the applicability and predictive capability of the proposed approach.

Stochastic filtering of reaction networks partially observed in time snapshots

Stochastic reaction network models arise in intracellular chemical reactions, epidemiological models and other population process models, and are a class of continuous time Markov chains which have the nonnegative integer lattice as state space. We consider the problem of estimating the conditional probability distribution of a stochastic reaction network given exact partial state observations in time snapshots. We propose a particle filtering method called the targeting method. Our approach takes into account that the reaction counts in between two observation snapshots satisfy linear constraints and also uses inhomogeneous Poisson processes as proposals for the reaction counts to facilitate exact interpolation. We provide rigorous analysis as well as numerical examples to illustrate our method and compare it with other alternatives.