Homogeneous Poisson Process Research Articles

On a New Shot Noise Process and the Induced Survival Model

Traditionally, in applications, the shot noise processes have been studied under the assumption that the underlying arrival point process (shock process) is the homogeneous (or nonhomogeneous) Poisson process. However, most of the real life shock processes do not possess the independent increments property and the Poisson assumption is made just for simplicity. Recently, in the literature, a new point process, the generalized Polya process (GPP), has been proposed and characterized. The GPP is defined via the stochastic intensity that depends on the number of events in the previous interval and, therefore, does not possess the independent increments property. In this paper, we consider the GPP as an underlying shock process for the shot noise process. The corresponding survival model is considered and the survival probability and its failure rate are derived and thoroughly analyzed. Furthermore, a new concept, the history-dependent residual life time, is defined and discussed.

On a cumulative performance function with Poisson-driven changes in the mean and variance of a performance metric

A proper evaluation of the performance of a system over a specified time horizon must take into consideration the magnitudes of fluctuations in performance level over the time horizon. This paper presents a model to study the cumulative performance of a system over the time period [0,T] when performance fluctuations, whether caused by changes in the internal state of the system or by external factors such as traffic overloads, are reflected by changes in both the mean and variance of the metric being used to assess the system performance. It is assumed that the shifts in the mean and variance of the performance metric occur at epochs which follow a homogeneous Poisson process. The mean and variance of a cumulative performance function are computed in closed form. A performance index, essentially a signal-to-noise ratio, is introduced as a composite measure to account for both the mean and standard deviation of the cumulative performance function. A generalization of this index is given to deal with the case when several performance metrics must be considered simultaneously. The application of the model is illustrated through actual data on packet loss in the Access Grid (AG). The model and ideas of this paper may be useful in performance evaluation of a variety of systems such as computer networks and manufacturing systems.

Geometrical Characterization of Offloading through Wireless LANs

The offloading of cellular traffic through WLAN APs (wireless local area network access points) distributed in a homogeneous Poisson process (HPP) is theoretically evaluated. The probability $P_w$ that a user can use WLAN and the expected number of vertical handovers $N_h$ are evaluated as the basic performance metrics of WLAN AP coverage. Explicit formulas are derived for the metrics, and the fundamental relationships between the metrics and many parameters such as the shape of each WLAN coverage region $D_1,D_2,\ldots$ are described. These metrics depend on the size and perimeter length of $D_i$ but do not depend on their other shape parameters for a convex $D_i$ . In addition, it is proven that a disk-shaped $D_i$ minimizes $N_h$ for a fixed WLAN coverage size and that $P_w$ is often insensitive to the perimeter length of $D_i$ . It is also proven that the $P_w$ of a user at a random location is equal to that moving along a random straight line or a random bounded curve. One hundred empirical location data sets of WLAN APs in Japan, Korea, and the US were used to confirm the theoretical results. Although these locations do not follow an HPP, many theoretical results are shown to be valid. For example, $N_h$ is minimized by a disk-shaped $D_i$ . Simultaneously, we find that $P_w$ slightly increases when a slender $D_i$ is used for highly clustered AP locations.

Serial Spike Time Correlations Affect Probability Distribution of Joint Spike Events.

Detecting the existence of temporally coordinated spiking activity, and its role in information processing in the cortex, has remained a major challenge for neuroscience research. Different methods and approaches have been suggested to test whether the observed synchronized events are significantly different from those expected by chance. To analyze the simultaneous spike trains for precise spike correlation, these methods typically model the spike trains as a Poisson process implying that the generation of each spike is independent of all the other spikes. However, studies have shown that neural spike trains exhibit dependence among spike sequences, such as the absolute and relative refractory periods which govern the spike probability of the oncoming action potential based on the time of the last spike, or the bursting behavior, which is characterized by short epochs of rapid action potentials, followed by longer episodes of silence. Here we investigate non-renewal processes with the inter-spike interval distribution model that incorporates spike-history dependence of individual neurons. For that, we use the Monte Carlo method to estimate the full shape of the coincidence count distribution and to generate false positives for coincidence detection. The results show that compared to the distributions based on homogeneous Poisson processes, and also non-Poisson processes, the width of the distribution of joint spike events changes. Non-renewal processes can lead to both heavy tailed or narrow coincidence distribution. We conclude that small differences in the exact autostructure of the point process can cause large differences in the width of a coincidence distribution. Therefore, manipulations of the autostructure for the estimation of significance of joint spike events seem to be inadequate.

Pulse Jitter, Delay Spread, and Doppler Shift in Mode-Stirred Reverberation

A time-domain characterization and stochastic model are developed for analyzing the evolution of the early randomization of fields inside mode-stirred reverberation chambers. The chamber is excited by a pulse amplitude modulated RF signal and stirred mechanically by a rotational stirrer. Phase noise (jitter) in the arrival times of a received pulse train is found to contain a systematic contribution that depends on stir speed and stir sense, as well as a random contribution governed by the rate of field depolarization. For the systematic component, a model of the phase delay for a uniformly orbiting scatterer inside a cavity is derived. For the random component, a nonhomogeneous Poisson model for arrivals of wave fronts in a nonstationary stir process exhibits good agreement with the empirical distribution of the measured jitter. At increasing stir velocities, the rate of field mixing increases less rapidly. The excess delay approaches asymptotically an Erlang distribution, in the limit of a homogeneous Poisson process for uniformly distributed late arrival times.

On the statistical properties of viral misinformation in online social media

The massive diffusion of online social media allows for the rapid and uncontrolled spreading of conspiracy theories, hoaxes, unsubstantiated claims, and false news. Such an impressive amount of misinformation can influence policy preferences and encourage behaviors strongly divergent from recommended practices. In this paper, we study the statistical properties of viral misinformation in online social media. By means of methods belonging to Extreme Value Theory, we show that the number of extremely viral posts over time follows a homogeneous Poisson process, and that the interarrival times between such posts are independent and identically distributed, following an exponential distribution. Moreover, we characterize the uncertainty around the rate parameter of the Poisson process through Bayesian methods. Finally, we are able to derive the predictive posterior probability distribution of the number of posts exceeding a certain threshold of shares over a finite interval of time.

On the vanishing of homology in random Čech complexes

We compute the homology of random Čech complexes over a homogeneous Poisson process on the d-dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erdős -Rényi phase transition, where the Čech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14–51, 2017

An imperfect maintenance policy for mission-oriented systems subject to degradation and external shocks

This paper develops a maintenance model for mission-oriented systems subject to natural degradation and external shocks. For mission-oriented systems which are used to perform safety-critical tasks, maintenance actions need to satisfy a range of constraints such as availability/reliability, maintenance duration and the opportunity of maintenance. Additionally, in developing maintenance policy, one needs to consider the natural degradation due to aging and wearing along with the external shocks due to variations of the operating environment. In this paper, the natural degradation is modeled as a Wiener process and the arrival of random shock as a homogeneous Poisson process. The damage caused by shocks is integrated into the degradation process, according to the cumulative shock model. Improvement factor model is used to characterize the impact of maintenance actions on system restoration. Optimal maintenance policy is obtained by minimizing the long-run cost rate. Finally, an example of subsea blowout preventer system is presented to illustrate the effectiveness of the proposed model.

Sojourning With the Homogeneous Poisson Process

ABSTRACTIn this pedagogical article, distributional properties, some surprising, pertaining to the homogeneous Poisson process (HPP), when observed over a possibly random window, are presented. Properties of the gap-time that covered the termination time and the correlations among gap-times of the observed events are obtained. Inference procedures, such as estimation and model validation, based on event occurrence data over the observation window, are also presented. We envision that through the results in this article, a better appreciation of the subtleties involved in the modeling and analysis of recurrent events data will ensue, since the HPP is arguably one of the simplest among recurrent event models. In addition, the use of the theorem of total probability, Bayes’ theorem, the iterated rules of expectation, variance and covariance, and the renewal equation could be illustrative when teaching distribution theory, mathematical statistics, and stochastic processes at both the undergraduate and graduate levels. This article is targeted toward both instructors and students.

Stochastic Pruning and Its Application for Fast Estimation of the Expected Total Output of Complex Systems

A powerful method referred to as stochastic pruning is introduced for analysing the performance of common complex systems whose component failures follow a homogeneous Poisson process. The method has been applied to create a very fast solver for estimating the production availability of large repairable flow networks with complex topology. It is shown that the key performance measures production availability and system reliability are all properties of a stochastically pruned network with corresponding pruning probabilities. The high-speed solver is based on an important result regarding the average total output of a repairable system including components characterised by constant failure/hazard rates. The average output over a specified operation time interval is given by the ratio of the expected momentary output of the stochastically pruned system, where the separate components are pruned with probabilities equal to their unavailabilities, and the maximum momentary output in the absence of component failures. The running time of the algorithm for determining the expected total output of the system over a specified time interval is independent of the length of the operational interval and the failure frequencies of the edges. The high-speed solver has been embedded in a software tool, with graphics user interface by which a flow network topology is drawn on screen and the parameters characterising the edges and the nodes are easily specified. The software tool has been used to analyse a gas production network and to study the impact of the network topology on the network performance. It is shown that two networks built with identical type and number of components may have very different performance levels, because of slight differences in their topology.

Inhomogeneity of the density of Parascaris spp. eggs in faeces of individual foals and the use of hypothesis testing for treatment decision making

Faecal egg counts (FEC) are used widely for monitoring of parasite infection in animals, treatment decision-making and estimation of anthelmintic efficacy. When a single count or sample mean is used as a point estimate of the expectation of the egg distribution over some time interval, the variability in the egg density is not accounted for. Although variability, including quantifying sources, of egg count data has been described, the spatiotemporal distribution of nematode eggs in faeces is not well understood. We believe that statistical inference about the mean egg count for treatment decision-making has not been used previously. The aim of this study was to examine the density of Parascaris eggs in solution and faeces and to describe the use of hypothesis testing for decision-making. Faeces from two foals with Parascaris burdens were mixed with magnesium sulphate solution and 30 McMaster chambers were examined to determine the egg distribution in a well-mixed solution. To examine the distribution of eggs in faeces from an individual animal, three faecal piles from a foal with a known Parascaris burden were obtained, from which 81 counts were performed. A single faecal sample was also collected daily from 20 foals on three consecutive days and a FEC was performed on three separate portions of each sample. As appropriate, Poisson or negative binomial confidence intervals for the distribution mean were calculated. Parascaris eggs in a well-mixed solution conformed to a homogeneous Poisson process, while the egg density in faeces was not homogeneous, but aggregated. This study provides an extension from homogeneous to inhomogeneous Poisson processes, leading to an understanding of why Poisson and negative binomial distributions correspondingly provide a good fit for egg count data. The application of one-sided hypothesis tests for decision-making is presented.

The study of boiling-up onset of highly superheated n-pentane in a glass capillary at different pressures. II. Data processing

The paper presents the methods and results of statistical processing of measurements of the lifetimes of superheated n-pentane with right censoring. The data were obtained in a temperature range of 84.7–162.8°C and at a pressure on fluid of 0.10–1.28MPa. Simple statistical models of the superheated state lifetime distribution, such as exponential, Weibull, gamma and mixture thereof are discussed. In the experiment, both homogeneous and heterogeneous Poisson processes are observed.

Role of blockages in particle transport through homogeneous granular assemblies

This letter deals with the transport of particles through granular assemblies and, specifically, with the intermittent formation of blockages originated from collective and purely mechanical clogging of constrictions. We perform numerical experiments with a micro-hydromechanical model that is able to reproduce the complex interplay between the carrier fluid, the transported particles and the granular assembly. The probability distribution functions (PDFs) of the duration of blockages and displacements give the time scale on which the effect of blockages is erased and the advection-dispersion paradigm is valid. Our experiments show that these PDFs fit exponential laws, reinforcing the idea that the formation and destruction of blockages are homogeneous Poisson processes.

Diminishing blocking interval in particle-conveying channel bundles

We consider a channel bundle consisting of N c parallel channels conveying a particulate flux. Particles enter these channels according to a homogeneous Poisson process and exit after a fixed transit time, τ. An individual channel blocks if N particles are simultaneously present. When a channel is blocked the flux previously entering it is redistributed evenly over the remaining open channels. We perform event driven simulations to examine the behaviour of an initially empty channel bundle with a total entering flux of intensity Λ. The mean blockage time of the kth channel is denoted by ⟨ t k ⟩ ,k = 1, ... , N c . For N = 1, as shown previously, the interval between successive blockages is constant, while for N > 1 an accelerating cascade, i.e. one in which the interval between successive blockages decreases, is observed. After an initial transient regime we observe a well-defined universal regime that is characterized by \hbox{$\Delta_k^{(N)} = (-1)^{N-1}\frac{[(N-1)!]^2}{(\Lambda\tau)^N}$} Δ k ( N ) = ( − 1 ) N − 1 [ ( N − 1 ) ! ] 2 ( Λ τ ) N where \hbox{$\Delta_k^{(1)}=\langle t_k \rangle-\langle t_{k-1}\rangle$}Δk(1)=⟨tk⟩−⟨tk−1⟩ and \hbox{$\Delta_k^{(j)}=\Delta_k^{(j-1)}-\Delta_{k-1}^{(j-1)}$}Δk(j)=Δk(j−1)−Δk−1(j−1) denotes the jth order difference.

Characterizations of discrete compound Poisson distributions

ABSTRACTThe aim of this paper is to give some new characterizations of discrete compound Poisson distributions. Firstly, we give a characterization by the Lévy–Khintchine formula of infinitely divisible distributions under some conditions. The second characterization need to present by row sum of random triangular arrays converges in distribution. And we give an application in probabilistic number theory, the strongly additive function converging to a discrete compound Poisson in distribution. The next characterization, is an extension of Watanabe’s theorem of characterization of homogeneous Poisson process. The last characterization will be illustrated by waiting time distributions, especially the matrix-exponential representation.

Identifiability issues in dynamic stress–strength modeling

In many real-life scenarios, system reliability depends on dynamic stress–strength interference, where strength degrades and stress accumulates concurrently over time. In some other cases, shocks appear at random time points, causing damage which is only effective at the instant of shock arrival. In this paper, we consider the identifiability problem of a system under deterministic strength degradation and stochastic damage due to shocks arriving according to a homogeneous Poisson process. We provide conditions under which the models are identifiable with respect to lifetime data only. We also consider current status data and suggest to collect additional information and discuss the issues of model identifiability under different data configurations.

Temporal clustering of floods in Germany: Do flood-rich and flood-poor periods exist?

The repeated occurrence of exceptional floods within a few years, such as the Rhine floods in 1993 and 1995 and the Elbe and Danube floods in 2002 and 2013, suggests that floods in Central Europe may be organized in flood-rich and flood-poor periods. This hypothesis is studied by testing the significance of temporal clustering in flood occurrence (peak-over-threshold) time series for 68 catchments across Germany for the period 1932–2005. To assess the robustness of the results, different methods are used: Firstly, the index of dispersion, which quantifies the departure from a homogeneous Poisson process, is investigated. Further, the time-variation of the flood occurrence rate is derived by non-parametric kernel implementation and the significance of clustering is evaluated via parametric and non-parametric tests. Although the methods give consistent overall results, the specific results differ considerably. Hence, we recommend applying different methods when investigating flood clustering. For flood estimation and risk management, it is of relevance to understand whether clustering changes with flood severity and time scale. To this end, clustering is assessed for different thresholds and time scales. It is found that the majority of catchments show temporal clustering at the 5% significance level for low thresholds and time scales of one to a few years. However, clustering decreases substantially with increasing threshold and time scale. We hypothesize that flood clustering in Germany is mainly caused by catchment memory effects along with intra- to inter-annual climate variability, and that decadal climate variability plays a minor role.

Block modelling in dynamic networks with non-homogeneous Poisson processes and exact ICL

We develop a model in which interactions between nodes of a dynamic network are counted by non homogeneous Poisson processes. In a block modelling perspective, nodes belong to hidden clusters (whose number is unknown) and the intensity functions of the counting processes only depend on the clusters of nodes. In order to make inference tractable we move to discrete time by partitioning the entire time horizon in which interactions are observed in fixed-length time sub-intervals. First, we derive an exact integrated classification likelihood criterion and maximize it relying on a greedy search approach. This allows to estimate the memberships to clusters and the number of clusters simultaneously. Then a maximum-likelihood estimator is developed to estimate non parametrically the integrated intensities. We discuss the over-fitting problems of the model and propose a regularized version solving these issues. Experiments on real and simulated data are carried out in order to assess the proposed methodology.

The Poisson Maximum Entropy Model for Homogeneous Poisson Processes

Our main interest is parameter estimation using maximum entropy methods in the prediction of future events for Homogeneous Poisson Processes when the distribution governing the distribution of the parameters is unknown. We intend to use empirical Bayes techniques and the maximum entropy principle to model the prior information. This approach has also been motivated by the success of the gamma prior for this problem, since it is well known that the gamma maximizes Shannon entropy under appropriately chosen constraints. However, as an alternative, we propose here to apply one of the often used methods to estimate the parameters of the maximum entropy prior. It consists of moment matching, that is, maximizing the entropy subject to the constraint that the first two moments equal the empirical ones and we obtain the truncated normal distribution (truncated below at the origin) as a solution. We also use maximum likelihood estimation (MLE) methods to estimate the parameters of the truncated normal distribution for this case. These two solutions, the gamma and the truncated normal, which maximize the entropy under different constraints are tested as to their effectiveness for prediction of future events for homogeneous Poisson processes by measuring their coverage probabilities, the suitably normalized lengths of their prediction intervals and their goodness-of-fit measured by the Kullback–Leibler criterion and a discrepancy measure. The estimators obtained by these methods are compared in an extensive simulation study to each other as well as to the estimators obtained using the completely noninformative Jeffreys’ prior and the usual frequency methods. We also consider the problem of choosing between the two maximum entropy methods proposed here, that is, the gamma prior and the truncated normal prior, estimated both by matching of the first two moments and, by maximum likelihood, when faced with data and we advocate the use of the sample skewness and kurtosis. The methods are also illustrated on two examples: one concerning the occurrence of mammary tumors in laboratory animals taking part in a carcinogenicity experiment and the other, a warranty dataset from the automobile industry.

Soil Elements Influencing Community Structure in an Old-Growth Forest in Northeastern China

This study uses detailed soil and vegetation data collected in a 30-ha old-growth broad-leaved Korean pine forest to study the effect of soil properties on tree community structures. Spatial distribution patterns are simulated using a homogeneous Poisson process (HomP) and a homogeneous Thomas process (HomT). The simulated distributions are compared with the observed ones to explore correlations between certain tree species and several soil elements. The HomP model shows that all tested tree species are significantly correlated with at least one principal component in the upper-layer soil elements. The HomT model shows that only 36.4% of tree species are significantly correlated with the principal component of at least one upper-layer soil element. This result shows that the impact of dispersal limitation is greater than impact of environmental heterogeneity on species spatial distributions. The spatial autocorrelation of species induced by the dispersal limitation will largely conceal the plant-soil relationships caused by the heterogeneity of soil elements. An additional analysis shows that the elements in the upper soil layer which have the greatest impact on community niche structure are Pb, total phosphorus (TP), total nitrogen (TN), Cu, Cr, Zn and available nitrogen (AN). The corresponding elements in the lower soil layers are Pb, TP, Cu, organic carbon (OC), Mn, total potassium (TK) and AN. Different species seem to be complementary regarding the demands on the available soil resources. The results of this study show that the tree species in the different growth groups have different habitat preferences. Compared with subcanopy and shrub species, the canopy species have more significant correlations with the soil elements.