Homogeneous Poisson Process Research Articles

Monitoring Software Failure Process using Half Logistic Distribution

In this paper, Software reliability is the anticipation of operations which are free of error in the software in a stated environment during the detailed time duration. Statistical Process Control can survey the gauging of software failure and thereby devote significantly to the enhancement of software reliability. Such an assessment assists the software development team to pinpoint and diagnose their actions during software failure process and hence, assure superior software reliability. A control mechanism planted on the cumulative observations of interval domain failure data using mean value function of the Half Logistic Distribution (HLD) based on Non Homogeneous Poisson Process (NHPP) is proposed. The maximum likelihood estimation approach is used to estimate the unknown parameters of the model. A new mechanism is coded to analyze the observations instead of using regular control charts. General Terms Decision Rule, Software testing, Software failure data, Quality Software.

A generalised δ-shock model with two types of shocks

ABSTRACTIn the δ-shock model, a system fails if the interval time between two consecutive shocks is less than a fixed threshold δ, which can be seen as the system recovery time from a shock. In this paper, we generalise the δ-shock model by assuming that the system is subject to two types of shocks, and the system recovery time for a type i shock is δi, i = 1, 2. If a shock occurs while the system still has not recovered from the consequence of the previous shock, the system fails. The shock to the system is governed by a homogeneous Poisson process with intensity λ, and each shock is of type 1 with probability p, type 2 with probability q = 1 − p. The explicit expression of the reliability function for the lifetime T of the generalised δ-shock model is derived. Both the new better than used property and the asymptotic exponentiality of the lifetime T of the generalised δ-shock model are proven. Moreover, by assuming that the system can be repaired at failure and work again, the limiting distribution of the system state is obtained. Finally, some numerical results are also provided to illustrate the proposed model.

Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations

We consider exit problems for general Lévy processes, where the first passage over a threshold is detected either immediately or at an epoch of an independent homogeneous Poisson process. It is shown that the two corresponding one-sided problems are related through a surprisingly simple identity. Moreover, we identify a simple link between two-sided exit problems with one continuous and one Poisson exit. Finally, identities for reflected processes and a link between some Parisian type exit problems are established. For spectrally one-sided Lévy processes this approach enables alternative proofs for a number of previously established identities, providing additional insight.

Underwater Minefield Detection in Clutter Data Using Spatial Point-Process Models

In this paper, we study the problem of detection of underwater minefields amidst dense clutter as that of statistical inference under a spatial point-process model. Specifically, we model the locations (mine and clutter) as samples of a Thomas point process with parent locations representing mines and children representing clutter. Accordingly, the parents are distributed according to a homogeneous Poisson process and, given the parent locations, the children are distributed as independent Poisson processes with intensity functions that are Gaussian densities centered at the parents. This provides a likelihood function for parent locations given the observed clutter (children). Under this model, we develop a framework for penalized maximum-likelihood (ML) estimation of model parameters and parent locations. The optimization is performed using a combination of analytical and Monte Carlo methods; the Monte Carlo part relies on a birth–death–move procedure for adding/removing points in the parent set. This framework is illustrated using both simulated and real data sets, the latter obtained courtesy of Naval Surface Warfare Center Panama City Division (NSWC-PCD), Panama City, FL, USA. The results, evaluated both qualitatively and quantitatively, underscore success in estimating parent locations and other parameters, at a reasonable computation cost.

Obligasi Bencana Alam Dengan Suku Bunga Stokastik Dan Pendekatan Campuran

This study contains the group claims model as discussed by (Lee, 2007) for the pricing of natural disaster bonds. This research was conducted with several stages. First make the formula of bond price with stochastic interest rate and disaster event following non homogeneous poisson process. It further estimates the parameters of disaster loss data from the Insurance Information Institute (III) from 1989 to 2012 and interest rates from the Federal Reserve Bank. Because the determination of aggregate distribution is difficult to be exact, numerical calculation is done by mixed approach method (Gamma and Inverse Gaussian) to determine the solution of natural disaster bond price. Finally, shows how the impact of financial risk and disaster risk on the price of natural disaster bonds.

Inferring the demographic history from DNA sequences: An importance sampling approach based on non-homogeneous processes

In Ait Kaci Azzou et al. (2015) we introduced an Importance Sampling (IS) approach for estimating the demographic history of a sample of DNA sequences, the skywis plot. More precisely, we proposed a new nonparametric estimate of a population size that changes over time. We showed on simulated data that the skywis plot can work well in typical situations where the effective population size does not undergo very steep changes. In this paper, we introduce an iterative procedure which extends the previous method and gives good estimates under such rapid variations. In the iterative calibrated skywis plot we approximate the effective population size by a piecewise constant function, whose values are re-estimated at each step. These piecewise constant functions are used to generate the waiting times of non homogeneous Poisson processes related to a coalescent process with mutation under a variable population size model. Moreover, the present IS procedure is based on a modified version of the Stephens and Donnelly (2000) proposal distribution. Finally, we apply the iterative calibrated skywis plot method to a simulated data set from a rapidly expanding exponential model, and we show that the method based on this new IS strategy correctly reconstructs the demographic history.

An optimal replacement policy for complex multi-component systems

Given a reward structure, this paper addresses an optimal replacement problem for complex multi-component systems. To maintain revenue stream resulting from system, the system is inspected according to a homogeneous Poisson process and certain actions are carried out in response to the system state. Decisions are based on a performance measure described by a Squared Bessel process. Given some assumption, we explore the inherent relation between the Squared Bessel process and an extended Gamma (EG) process. Since there are some flow of income and increasing costs due to inspections, the problem is to optimally stop processing the system and carrying out a renewal to maximize the reward functional. To this end, using the local characteristics of the EG process as a stopping criterion and the expected total discounted reward as a measure of policy, this paper aims at determining an optimal operating (stopping) time which truly balances both income and cost and so maximizes the expected discounted reward over a cycle. In support of the model a numerical example is provided to show feasibility of this programme in real application. Attention is restricted to perfect repair and inspection, but the paper provides the structure so that different scenarios can be explored.

Optimal Approximation of Stochastic integrals with Respect to a Homogeneous Poisson Process

We consider strong approximation in the asymptotic setting of stochastic integrals with respect to a homogeneous Poisson process. An integrand \({f:[0,T] \to \mathbb{R}}\) is assumed to be an r-smooth function from \({\mathcal{C}^r([0,T])}\). We show that the \({L^p}\)-error (\({p \in [1,+\infty)}\)) of any approximation method, which uses n values of f, cannot converge to zero faster than \({n^{-r}}\) as \({n\to+\infty}\). This holds except for a subset of \({\mathcal{C}^r([0,T])}\) with empty interior. The best speed of convergence is achieved by the Ito–Taylor algorithm.

Loss estimation of steel buildings to earthquake mainshock–aftershock sequences

Following a large earthquake numerous aftershocks can be triggered due to the complex stress interaction between and within tectonic plates. Although aftershocks are normally smaller in magnitude, their ground motion intensity can be large and have different energy content than the mainshock. Even seemingly undamaged buildings may be damaged as a result of aftershocks. The mainshock-damaged buildings with deteriorated structural properties are more susceptible to damage.This paper proposes a framework for loss estimation of steel structures subjected to mainshock–aftershock sequences. The analysis is based on a typical 4-story steel frame with a deterioration model. Mainshocks are modeled as a homogeneous Poisson process, while aftershocks are simulated from non-homogeneous Poisson process, magnitudes of which are characterized by the Gutenberg–Richter relationship. The proposed framework is applied to examine the effects of aftershocks on seismic loss. The expected seismic loss of the building subjected to two levels of earthquakes, the Design Earthquake (DE) and the Maximum Considered Earthquake (MCE), followed by aftershocks are examined considering both transition cost and downtime cost. Monte Carlo Simulation (MCS) with Latin Hypercube Sampling (LHS) is applied to examine the uncertainty in the loss estimation. Uncertainty in earthquake ground motions, structural model, damage and loss are all considered. It was found that even if aftershocks have little effect on structural response, they may still have a significant impact on seismic loss due to the uncertainty of the damage state and cost estimation. This methodology can be used to mitigate seismic loss and evaluate the current building design.

The Ostomachion Process

Stochastic partition processes for exchangeable graphs produce axis-aligned blocks on a product space. In relational modeling, the resulting blocks uncover the underlying interactions between two sets of entities of the relational data. Although some flexible axis-aligned partition processes, such as the Mondrian process, have been able to capture complex interacting patterns in a hierarchical fashion, they are still in short of capturing dependence between dimensions. To overcome this limitation, we propose the Ostomachion process (OP), which relaxes the cutting direction by allowing for oblique cuts. The partitions generated by an OP are convex polygons that can capture inter-dimensional dependence. The OP also exhibits interesting properties: 1) Along the time line the cutting times can be characterized by a homogeneous Poisson process, and 2) on the partition space the areas of the resulting components comply with a Dirichlet distribution. We can thus control the expected number of cuts and the expected areas of components through hyper-parameters. We adapt the reversible-jump MCMC algorithm for inferring OP partition structures. The experimental results on relational modeling and decision tree classification have validated the merit of the OP.

Optimal global approximation of stochastic differential equations with additive Poisson noise

We consider strong global approximation of SDEs driven by a homogeneous Poisson process with intensity ź > 0. We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson process. We consider two classes of methods using equidistant or nonequidistant sampling of the Poisson process, respectively. We provide a construction of optimal schemes, based on the classical Euler scheme, which asymptotically attain the established minimal errors. It turns out that methods based on nonequidistant mesh are more efficient than those based on the equidistant mesh.

Dynamic clustering of hazard functions: an application to disease progression in chronic heart failure.

We analyse data collected from the administrative datawarehouse of an Italian regional district (Lombardia) concerning patients affected by Chronic Heart Failure. The longitudinal data gathering for each patient hospital readmissions in time, as well as patient-specific covariates, is studied as a realization of non homogeneous Poisson process. Since the aim behind this study is to identify groups of patients behaving similarly in terms of disease progression and then healthcare consumption, we conjectured the time segments between two consecutive hospitalizations to be Weibull distributed in each hidden cluster. Adding a frailty term to take into account the within subjects unknown variability, the corresponding patient-specific hazard functions are reconstructed. Therefore, the comprehensive distribution for each time to event variable is modelled as a Weibull Mixture. We are then able to easily interpret the related hidden groups as healthy, sick, and terminally ill subjects.

Do Suicides From the Golden Gate Bridge Cluster?

Suicides from popular venues (known as "hotspots") are often publicized and may result in imitation by subsequent suicides that may lead to clustering of the suicides over time. In order to examine whether the suicides from the Golden Gate Bridge showed clustering, data from the 224 suicides during 1999-2009 were analyzed using the Anderson-Darling Test was run on the data against a null hypothesis of a negative exponential distribution (as would be generated by a homogenous Poisson process). It was found that the data were almost a perfect fit for the Poisson distribution and so showed no evidence of clustering beyond that expected to occur by chance alone. This indicates that there was no imitation or contagion in the suicides from the Golden Gate Bridge.

High quality process monitoring using a class of inter-arrival time distributions of the renewal process

For high-quality processes where the defect rate is very low, e.g., parts per million (ppm), time-between-events (TBE) control charts have several advantages over the ordinary control charts. Most existing TBE control charts are based on the homogeneous Poisson process assumption, so that the distribution of TBE can only be exponential. However, the exponential distribution is not suitable in many applications, especially when the failure rate is not constant. In this article, we introduce a new TBE control chart, based on the renewal process, where the distribution of the TBE belongs to a parametric class of absolutely continuous distributions, which includes some well-known and commonly used lifetime distributions, i.e., exponential, Rayleigh, Weibull, Burr type XII, Pareto and Gompertz. The control structure of the proposed chart is derived analytically in general and numerical examples are presented for the Weibull distribution, due to its relevance in reliability. The performance of the proposed control charts is evaluated in terms of some standard useful measures, including average run length (ARL), the standard deviation of run length, the coefficient of variation of run length, expected quality loss (EQL) and relative ARL (RARL). The effect of parameter estimation, using both maximum likelihood and Bayesian methods, is also discussed. This study also presents an illustrative example and four case studies to highlight the practical aspects of the new TBE chart.

Stochastic renewal process model for maintenance (case study: thermal electricity generation in Sudan)

<p>The renewal process defines as a counting process where the times between the count renewals is a random variables and their distribution is identical. In the electricity generation machines there are spare parts replaced due to damage or expired and replacement process occur repeatedly and the renewal process of here assume that times between replacements are independent random variables and it has identical probability distribution. In this paper, renewal process model has applied on the time of fault for machine in Bahri Thermal Station for electricity generation, which is belong to the National Electricity Authority in Sudan during the period (2011-2015). Through the renewal process model estimation is clear that, the failure time (renewal) for the machines follow Weibull distribution with 2-parameters and when the time trend has been tested it is clear that no trend exist which mean that the renewal process represent Homogeneous Poisson Process (HPP), and the repair rate (renewal) is occurred constantly. In addition, the findings approve that whenever the repair rate (renewal) increase the mean time between failures (MTBF) increases too and this clear in machine no (6).</p>

Warranty claim analysis according to good usage policy

Warranty is a term provided by the producer in order to overcome the loss of a company's goodwill if some products fail to perform as expected due to error while in production. In previous works on warranty, it is noted that frequently some products are claimed for warranty due to damage caused by bad usage or lack of knowledge by the user. The cost related to these claims is high and needs to be considered. In this work two models are developed for determining two types of warranty claims using homogeneous Poisson process (HPP), learning rate factor (Chase et al., 1999) and simulation. The two types are fatal failure and intermittent failure of the product. In the last section of the work there is a brief description of a new model which may bring the concept of a variable warranty period of product.

Poisson Process and Its Application to the Storm Water Overflows

The homogenous Poisson process is often used to describe the event arrivals. Such Poisson process has been applied in various areas. This study focuses on the arrival pattern of storm water overflows. A set of overflow data was obtained from the storm water pipeline of a municipality. The aim is to verify the overflow arrival pattern and check whether the Poisson process can be applied. The adopted method is the analysis over the inter-arrival times. The exponential distribution test is conducted on the annual data set as well as the entire data set. The results show that all data sets follow the exponential distribution. With the verification of Poisson process, specific examples are also given to show how the Poisson process properties can be used in the management of storm water pipeline management. For other data that are featured with various heterogeneities, the homogenous Poisson process might not be able to be verified and used. Under such circumstances, non-homogenous survival model can be used to simulate the arrival process.

A generalization of the space-fractional Poisson process and its connection to some Lévy processes

This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.

Detection and spatial characterization of minicolumnarity in the human cerebral cortex.

Spatial characterization of vertical organization of neurons in human cerebral cortex, cortical columnarity or minicolumns, and its possible association with various psychiatric and neurological diseases has been investigated for many years. In this study, we obtained 3D coordinates of disector sampled cells from layer III of Brodmann area 4 of the human cerebral cortex using light microscopy and 140-μm-thick glycolmethacrylate sections. A new analytical tool called cylindrical K-function was applied for spatial point pattern analysis of 3D datasets to see whether there is a spatially organized columnar structure. In order to demonstrate the behaviour of the cylindrical K-function, the result from brain tissues was compared with two models: A homogeneous Poisson process exhibiting complete spatial randomness, and a Poisson line cluster point process. The latter is a point process model in 3D space, which exhibits spatial structure of points similar to minicolumns. The data show in three out of four samples nonrandom patterns in the 3D neuronal positions with the direction of minicolumns perpendicular to the pial surface of the brain - without a priori assuming the existence of minicolumns. Studies on columnarity are difficult and have mainly been based on two-dimensional images analysis of thin sections of the cerebral cortex with the a priori assumption that minicolumns existed. A clear difference from complete spatial randomness in the data could be detected with the new tool, the cylindrical K-function, although classical functional summary statistics are less useful in this connection.

A statistical model for seismic hazard assessment of hydraulic‐fracturing‐induced seismicity

AbstractWe analyze the interevent time distribution of hydraulic‐fracturing‐induced seismicity collected during 18 stages at four different regions. We identify a universal statistical process describing the distribution of hydraulic‐fracturing‐induced events in time. The distribution of waiting times between subsequently occurring events is given by the exponential probability density function of the homogeneous Poisson process. Our findings suggest that hydraulic‐fracturing‐induced seismicity is directly triggered by the relaxation of stress and pore pressure perturbation initially created by the injection. Therefore, compared to this relaxation, the stress transfer caused by the occurrence of preceding seismic events is mainly insignificant for the seismogenesis of subsequently occurring events. We develop a statistical model to compute the occurrence probability of hydraulic‐fracturing‐induced seismicity. This model can be used to assess the seismic hazard associated with hydraulic fracturing operations. No aftershock triggering has to be included in the statistical model.