Non-homogeneous Poisson Process Models Research Articles

Modeling Recurrent Failures on Large Directed Networks

Many lifeline infrastructure systems consist of thousands of components configured in a complex directed network. Disruption of the infrastructure constitutes a recurrent failure process over a directed network. Statistical inference for such network recurrence data is challenging because of the large number of nodes with irregular connections among them. Motivated by 16 years of Scottish Water operation records, we propose a network Gamma-Poisson Autoregressive NHPP (GPAN) model for recurrent failure data from large-scale directed physical networks. The model consists of two layers: the temporal layer applies a Non-Homogeneous Poisson Process (NHPP) with node-specific frailties, and the spatial layer uses a well-orchestrated gamma-Poisson autoregressive scheme to establish correlations among the frailties. Under the network-GPAN model, we develop a sum-product algorithm to compute the marginal distribution for each frailty conditional on the recurrence data. The marginal conditional frailty distributions are useful for predicting future failures based on historical data. In addition, the ability to rapidly compute these marginal distributions allows adoption of an EM type algorithm for estimation. Through a Bethe approximation, the output from the sum-product algorithm is used to compute maximum log-likelihood estimates. Applying the methods to the Scottish Water network, we demonstrate utility in aiding operation management and risk assessment of the water utility. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.

A finite failure software reliability model using extended log-logistic distribution

Software reliability engineering has been developed to examine software systems over the years. Non-homogeneous Poisson process (NHPP) models are widely employed for analyzing software reliability data, as they have the ability to describe the expected number of failures over time using the mean value function. In addition, these models have an intensity function representing the failure rate over time. This article proposes a new non-homogeneous Poisson process model based on extended log-logistic (NHPP ELL) distribution with three parameters. The model properties were derived and represented graphically. Furthermore, the maximum likelihood estimation method was utilized to estimate the model parameters, which were computed numerically using the Newton–Raphson method. Three real software reliability datasets were utilized to evaluate the performance of the new NHPP ELL model based on three distinct criteria: the mean square error, the coefficient of determination (R2), the Theil statistic, the predictive-ratio risk, and the predictive power. Moreover, a comparative study was conducted between the proposed model and several well-known NHPP models regarding goodness-of-fit and predictive performance. All the evaluation measures indicate that the proposed model outperforms the other models across all the considered datasets. In addition, the graphical plots depicting the actual data and the prediction results for all the fitted models show that the proposed model performs well in all three considered software reliability datasets compared to other models.

An updated software reliability model using the shanker model and failure data

AbstractSoftware developers' goal is to develop reliable and superior software. Due to the fact that software errors frequently generate large societal or financial losses, software reliability is essential. Software reliability growth models are a widely used technique for software reliability assessment. This study examines various nonhomogeneous Poisson process models with the newly developed software reliability distribution and evaluates the unknown model parameters based on frequentist and Bayesian methods of estimation. Finally, we conduct evaluations on real datasets using a variety of evaluation criteria to compare the results of previous software reliability growth models and show how the proposed model may be applied under both approaches in a practical setting. According to this study, the innovative model's mean square error, R2, , bias, predicted relative variation, Theil statistic, and mean error of prediction values show the lowest values under the Bayesian approach for data sets II to IV, and both approaches perform well for data set I. These implementation findings demonstrate the effectiveness of our specific approach based on our examination of failure data.

Spatio-Temporal Analysis and Prediction of Mass Telecommunication Base Station Failure Events

Large-scale telecommunication systems are lifeline infrastructure in modern society. A telecommunication system typically consists of a huge number of base stations with diverse geographical locations across a country, which highly complicates maintenance operations. To allocate maintenance resource properly, it is important to have a good understanding on the failure pattern of these base stations. Statistical inference of recurrent failures of these base stations is challenging because of the large number of base stations and the spatial correlation of their failure processes. Based on eight-month failure data of telecommunication base stations in Harbin, China, we propose a customized nonhomogeneous Poisson process (NHPP) model for recurrent failure data from telecommunication systems. The model consists of two layers, where the temporal layer applies an NHPP with station-specific frailty for failures of each base station, and the spatial layer uses a multivariate lognormal distribution to characterize the correlation among the frailties. The Monte Carlo EM (MCEM) algorithm is applied to estimate parameters included in the proposed model. We demonstrate the proposed model using the Harbin telecommunication system example with 7725 base stations and 4615 failure records.

Can Customer Arrival Rates Be Modelled by Sine Waves?

Customer arrival patterns observed in the real world typically exhibit strong seasonal effects. It is therefore natural to ask, can a nonhomogeneous Poisson process (NHPP) with a rate function that is the simple sum of sinusoids provide an adequate description of reality? If so, how can the sinusoidal NHPP be used to improve the performance of service systems? We empirically validate that the sinusoidal NHPP is consistent with arrival data from two settings of great interest in service operations: patient arrivals to an emergency department and customer calls to a bank call centre. This finding provides rigorous justification for the use of the sinusoidal NHPP assumption in many existing queuing models. We also clarify why a sinusoidal NHPP model is more suitable than the standard NHPP when the underlying arrival pattern is aperiodic (e.g., does not follow a weekly cycle). This is illustrated using data from a car dealership and also via a naturalistic staffing simulation based on the call centre. On the other hand, if the arrival pattern is periodic, we explain why both models should perform comparably. Even then, the sinusoidal NHPP is still necessary for managers to use to verify that the arrival pattern is indeed periodic, a step that is seldom performed in applications. Code for fitting the sinusoidal NHPP to data is provided on GitHub. History: This paper has been accepted for the Service Science/Stochastic Systems Joint Special Issue. Funding: N. Chen was supported by Natural Sciences and Engineering Research Council of Canada Discovery [Grant RGPIN-2020-04038]. H. Shen was supported by China Ministry and Science and Technology [Grant 2022SQGH10861], Hong Kong Special Administrative Region University Grants Council Collaborative Research Fund [Grant C7162-20G], University of Hong Kong Budget and Resources Committee Grant, University of Hong Kong Business School, Shenzhen Research Institutes [Grant SZRI2023-TBRF-03].

Prediction and Comparative Analysis of Software Reliability Model Based on NHPP and Deep Learning

Over time, software has become increasingly important in various fields. If the current software is more dependent than in the past and is broken owing to large and small issues, such as coding and system errors, it is expected to cause significant damage to the entire industry. To address this problem, the field of software reliability is crucial. In the past, efforts in software reliability were made to develop models by assuming a nonhomogeneous Poisson-process model (NHPP); however, as models became more complex, there were many special cases in which models fit well. Hence, this study proposes a software reliability model using deep learning that relies on data rather than mathematical and statistical assumptions. A software reliability model based on recurrent neural networks (RNN), long short-term memory (LSTM), and gated recurrent units (GRU), which are the most basic deep and recurrent neural networks, was constructed. The dataset was divided into two, Datasets 1 and 2, which both used 80% and 90% of the entire data, respectively. Using 11 criteria, the estimated and learned results based on these datasets proved that the software reliability model using deep learning has excellent capabilities. The software reliability model using GRU showed the most satisfactory results.

Prediction of Tool Remaining Useful Life Based on NHPP-WPHM

A tool remaining useful life prediction method based on a non-homogeneous Poisson process and Weibull proportional hazard model (WPHM) is proposed, taking into account the grinding repair of machine tools during operation. The intrinsic failure rate model is built according to the tool failure data. The WPHM is established by collecting vibration information during operation and introducing covariates to describe the failure rate of the tool operation. In combination with the tool grinding repair, the NHPP-WPHM under different repair times is established to describe the tool comprehensive failure rate. The failure threshold of the tool life is determined by the maximum availability, and the remaining tool life is predicted. Take the cylindrical turning tool of the CNC lathe as an example, the root mean square error, mean absolute error, mean absolute percentage error, and determination coefficient (R2) are used as indicators. The proposed method is compared with the actual remaining useful life and the remaining useful life prediction model based on the WPHM to verify the effectiveness of the model.

Software reliability model for open-source software that considers the number of finite faults and dependent faults.

Software has become a vital factor in the fourth industrial revolution. Owing to the increase in demand for software products in various fields (big data, artificial intelligence, the Internet of Things, etc.), the software industry has expanded more than ever before. Therefore, software reliability has become very important, and efforts are being made to increase it. One of these efforts is the development of software reliability models (SRMs). SRMs have been studied for a long time as a model that predicts software reliability by using the number of software faults. Software failures can occur for several reasons, including independent software faults such as code errors and software hangs, as well as dependent cases where code errors lead to other software faults. Recently, due to the diversity of software operating environments, software faults are more likely to occur in a dependent manner, and, for this reason, they are likely to increase rapidly from the beginning and progress slowly to the maximum number thereafter. In addition, many large companies have focused on open-source software (OSS) development, and OSS is being developed by many users. In this study, we propose a new SRM that considers the number of finite faults and dependent faults, and examine the goodness-of-fit of a new SRM and other existing non-homogeneous Poisson process models based on the OSS datasets. Through numerical examples, the proposed model demonstrated a significantly better goodness-of-fit when compared to other existing models, and it also exhibited better results on the newly proposed integrated criteria.

Detecting clusters of transcription factors based on a nonhomogeneous poisson process model

Rapidly growing genome-wide ChIP-seq data have provided unprecedented opportunities to explore transcription factor (TF) binding under various cellular conditions. Despite the rich resources, development of analytical methods for studying the interaction among TFs in gene regulation still lags behind. In order to address cooperative TF binding and detect TF clusters with coordinative functions, we have developed novel computational methods based on clustering the sample paths of nonhomogeneous Poisson processes. Simulation studies demonstrated the capability of these methods to accurately detect TF clusters and uncover the hierarchy of TF interactions. A further application to the multiple-TF ChIP-seq data in mouse embryonic stem cells(ESCs) showed that our methods identified the cluster of core ESC regulators reported in the literature and provided new insights on functional implications of transcrisptional regulatory modules. Effective analytical tools are essential for studying protein-DNA relations. Information derived from this research will help us better understand the orchestration of transcription factors in gene regulation processes.

An analysis modality for vascular structures combining tissue-clearing technology and topological data analysis

The blood and lymphatic vasculature networks are not yet fully understood even in mouse because of the inherent limitations of imaging systems and quantification methods. This study aims to evaluate the usefulness of the tissue-clearing technology for visualizing blood and lymphatic vessels in adult mouse. Clear, unobstructed brain/body imaging cocktails and computational analysis (CUBIC) enables us to capture the high-resolution 3D images of organ- or area-specific vascular structures. To evaluate these 3D structural images, signals are first classified from the original captured images by machine learning at pixel base. Then, these classified target signals are subjected to topological data analysis and non-homogeneous Poisson process model to extract geometric features. Consequently, the structural difference of vasculatures is successfully evaluated in mouse disease models. In conclusion, this study demonstrates the utility of CUBIC for analysis of vascular structures and presents its feasibility as an analysis modality in combination with 3D images and mathematical frameworks.

Bayesian criterion-based assessments of recurrent event models with applications to commercial truck driver behavior studies.

Multitype recurrent events are commonly observed in transportation studies, since commercial truck drivers may encounter different types of safety critical events (SCEs) and take different lengths of on-duty breaks in a driving shift. Bayesian nonhomogeneous Poisson process models are a flexible approach to jointly model the intensity functions of the multitype recurrent events. For evaluating and comparing these models, the deviance information criterion (DIC) and the logarithm of the pseudo-marginal likelihood (LPML) are studied and Monte Carlo methods are developed for computing these model assessment measures. We also propose a set of new concordance indices (C-indices) to evaluate various discrimination abilities of a Bayesian multitype recurrent event model. Specifically, the within-event C-index quantifies adequacy of a given model in fitting the recurrent event data for each type, the between-event C-index provides an assessment of the model fit between two types of recurrent events, and the overall C-index measures the model's discrimination ability among multiple types of recurrent events simultaneously. Moreover, we jointly model the incidence of SCEs and on-duty breaks with driving behaviors using a Bayesian Poisson process model with time-varying coefficients and time-dependent covariates. An in-depth analysis of a real dataset from the commercial truck driver naturalistic driving study is carried out to demonstrate the usefulness and applicability of the proposed methodology.

Forecasting COVID19 Reliability of the Countries by Using Non-Homogeneous Poisson Process Models.

Reliability is the probability that a system or a product fulfills its intended function without failure over a period of time and it is generally used to determine the reliability, release and testing stop time of the system. The primary objective of this study is to predict and forecast COVID19 reliabilities of the countries by utilizing this definition of the reliability. To our knowledge, this study is the first carried out in the direction of this objective. The major contribution of this study is to model the COVID19 data by considering the intensity functions with different types of functional shapes, including geometric, exponential, Weibull, gamma and identifying best fit (BF) model for each country, separately. To achieve the objective determined, cumulative number of confirmed cases are modelled by eight Non-Homogenous Poisson Process (NHPP) models. BF models are selected based on three comparison criteria, including Root Mean Square Error (RMSE), Normalized Root Mean Square Error (NRMSE), and Theil Statistics (TS). The results can be summarized as follows: S-shaped models provide better fit for 56 of 70 countries. Current outbreak may continue in 43 countries and a new outbreak may occur in 27 countries. 50 countries have the reliability smaller than 75%, 9 countries between 75% and 90%, and 11 countries a 90% or higher on 11 August 2021.Supplementary InformationThe online version contains supplementary material available at 10.1007/s00354-022-00183-1.

A Comprehensive Fault Prediction Model for Improving Software Reliability

We present a Comprehensive Fault Prediction Model (FPM) for Software Reliability in this article, which can estimate the greatest quantity of faults in a software. The proposed model implemented on a “Non-Homogeneous Poisson Process Model (NHPPM)” and includes fault reliant identification, rate of failure, and the software defect present after release. We looked at programmers' abilities, software performance, and flawless debugging as deciding factors for FPM. The Software Engineering Team Assessment and Prediction (SETAP) dataset is used for analysis of proposed FPM. The selected dataset is composed of sequential value which are linearly arranged over a given time duration. The attributes are analyzed to establish software reliability prediction model and comparison of proposed model is carried out with similar algorithms. The proposed FPM is executed in “Jira” and is compared with the present FPMs proposed in the literature.Results demonstrate comparatively less cumulative faults, and reduced residual errors which depicts high prediction accuracy and improved software reliability.

Non-homogeneous Poisson and linear regression models as approaches to study time series with change-points

In this study we consider some stochastic models and a Bayesian approach to analyze count time series data in the presence of one or more change-points. When the observed count data are all different of zero, it is possible to perform an analysis using linear regression models with normal distribution errors for the log-transformed data. When we have the presence of zero counts at different times, the statistical model based on the log-transformation is not suitable. Hence, in this case, it is possible to consider non-homogeneous Poisson processes (NHPP) models with a suitable rate function. In the present work we consider both models (linear regression and Poisson) to analyze three data sets. These sets are data recording the monthly visitors to New Zealand with the purpose of education, yearly tuberculosis incidence data from New York City, and monthly measles incidence data from Brazil. When the NHPP model is used a PLP (power law process) model is assumed for the intensity function. Additionally, in both models, the presence of change-points will be allowed.

Nonhomogeneous poisson process model of summer high temperature extremes over China

In this study, nonhomogeneous Poisson process (NHPP) models arising from the extreme value theory have been fitted to summer high temperature extremes (HTEs) at 321 meteorological stations over China. The seasonality and six prominent atmospheric teleconnection patterns in Northern Hemisphere are incorporated in the NHPP models reflecting the non-stationarity of occurrence rate in Poisson process of HTEs. In addition, Poisson regression model has also been applied to link HTEs and these teleconnection patterns. The linkages of HTEs and teleconnection patterns have been identified in both NHPP modeling and Poisson regression. Composite maps of differences of 500-hPa geopotential height and wind fields in the positive and negative phases of teleconnection patterns are constructed to show the impacts of atmospheric circulation patterns on extreme heat events. The spatial pattern of the associated anticyclonic or cyclonic circulations with teleconnection patterns partly explains the spatial variability of the occurrences of summer HTEs over China.

Using the Methods of Statistical Data Analysis to Improve the Trustworthiness of Software Reliability Modeling

In the past, several <i>software reliability growth models</i> (SRGMs) have been proposed to predict various aspects of software reliability, and non-homogeneous Poisson process (NHPP)-based SRGMs have drawn much attention. However, it should be noted that, in practice, blindly applying SRGMs might not lead to meaningful results when the characteristics of the data (i.e., data type) do not fit the characteristics of selected SRGMs. Moreover, there currently exist no guidelines for project managers or developers to follow in selecting the most appropriate SRGMs for a particular failure dataset. On the other hand, waiting to collect a substantial amount of failure data before fitting SRGMs might be unfeasible in many cases. In this paper, we propose useing statistical data analysis methods to investigate the characteristics of software failure data and to improve the accuracy of software reliability growth modeling. Here, SRGMs will be broadly partitioned into two categories: &#x201C;NHPP models&#x201D; and &#x201C;non-NHPP models.&#x201D; Similarly, software failure data can be classified into two categories: &#x201C;NHPP-type data&#x201D; and &#x201C;non-NHPP-type data.&#x201D; We propose a two-phase method to verify the hypothesis that failure data follow an NHPP. Through the proposed two-phase and Laplace trend tests, we will be able to infer whether a dataset is the NHPP type and whether it is properly used by NHPP-based SRGMs. In this case, the final choice for the most suitable NHPP model for the software failure process would be made, and the software reliability prediction result can be deemed trustworthy and helpful after correctly judging whether the dataset can be classified or not as being NHPP based. Experiments based on 25 real software failure data were presented and analyzed. Using our proposed method, project managers and/or developers will be able to analyze and use collected failure data to help improve both planning accuracy and software quality.

Assessment of the effectiveness of corrective maintenance of an oil pump using the proportional intensity model (PIM)

PurposeComplex repairable systems (CRSs) are generally modeled by stochastic processes called “point processes.” These are generally summed up in the nonhomogeneous Poisson process (NHPP) and the renewal process (RP), which represent the minimum and maximum repair, respectively. However, the industrial environment affects systems in some way. This is why the main objective of this work is to model the CRS with a concept reflecting the real state of the system by incorporating an indicator in the form of covariate. This type of model, known as the proportional intensity model (PIM), will be analyzed with simulated failure data to understand the behavior of the failure process, and then it will be tested for real data from a petroleum company to evaluate the effectiveness of corrective actions carried out.Design/methodology/approachTo solve the partial repair modeling problem, the PIM was used by introducing, on the basis of the NHPP model, a multiplicative scaling factor, which reflects the degree of efficiency after each maintenance action. Several values of this multiplicative factor will be considered to generate data. Then, based on the reliability and maintenance history of 12-year pump's operation obtained from the SONATRACH Company (south industrial center (CIS), Hassi Messaoud, Algeria), the performance of the PIM will be judged and compared with the model of NHPP and RP in order to demonstrate its flexibility in modeling CRS. Using the maximum likelihood approach and relying on the Matlab software, the best fitting model should have the largest likelihood value.FindingsThe use of the PIM allows a better understanding of the physical situation of the system by allowing easy modeling to apply in practice. This is expressed by the value which, in this case, represents an improvement in the behavior of the system provided by a good quality of the corrective maintenance performed. This result is based on the hypothesis that modeling with the PIM can provide more clarification on the behavior of the system. It can indicate the effectiveness of the maintenance crew and guide managers to confirm or revise their maintenance policy.Originality/valueThe work intends to reflect the real situation in which the system operates. The originality of the work is to allow the consideration of covariates influencing the behavior of the system during its lifetime. The authors focused on modeling the degree of repair after each corrective maintenance performed on an oil pump. Since PIM does not require a specific reliability distribution to apply it, it allows a wide range of applications in the various industrial environments. Given the importance of this study, the PIM can be generalized for more covariates and working conditions.

Prediction of Membrane Failure in a Water Purification Plant Using Nonhomogeneous Poisson Process Models.

The prediction of membrane failure in full-scale water purification plants is an important but difficult task. Although previous studies employed accelerated laboratory-scale tests of membrane failure, it is not possible to reproduce the complex operational conditions of full-scale plants. Therefore, we aimed to develop prediction models of membrane failure using actual membrane failure data. Because membrane filtration systems are repairable systems, nonhomogeneous Poisson process (NHPP) models, i.e., power law and log-linear models, were employed; the model parameters were estimated using the membrane failure data from a full-scale plant operated for 13 years. Both models were able to predict cumulative failures for forthcoming years; nonetheless, the power law model showed higher stability and narrower confidence intervals than the log-linear model. By integrating two membrane replacement criteria, namely deterioration of filtrate water quality and reduction of membrane permeability, it was possible to predict the time to replace all the membranes on a water purification plant. Finally, the NHPP models coupled with a nonparametric bootstrap method provided a method to select membrane modules for earlier replacement than others. Although the criteria for membrane replacement may vary among membrane filtration plants, the NHPP models presented in this study could be applied to any other plant with membrane failure data.

Shrinkage Priors for Nonparametric Bayesian Prediction of Nonhomogeneous Poisson Processes

We consider nonparametric Bayesian estimation and prediction for nonhomogeneous Poisson process models with unknown intensity functions. We propose a class of improper priors for intensity functions. Nonparametric Bayesian inference with kernel mixture based on the class improper priors is shown to be useful, although improper priors have not been widely used for nonparametric Bayes problems. Several theorems corresponding to those for finite-dimensional independent Poisson models hold for nonhomogeneous Poisson process models with infinite-dimensional parameter spaces. Bayesian estimation and prediction based on the improper priors are shown to be admissible under the Kullback-Leibler loss. Numerical methods for Bayesian inference based on the priors are investigated.

A Reliability Growth Process Model with Time-Varying Covariates and Its Application

The nonhomogeneous Poisson process model with power law intensity, also known as the Army Materiel Systems Analysis Activity (AMSAA) model, is commonly used to model the reliability growth process of many repairable systems. In practice, it is necessary to test the reliability of the product under different operational environments. In this paper we introduce an AMSAA-based model considering the covariate effects to measure the influence of the time-varying environmental condition. The parameter estimation of the model is typically performed using maximum likelihood on the failure data. The statistical properties of the estimation in the model are comprehensively derived by the martingale theory. Further inferences including confidence interval estimation and hypothesis tests are designed for the model. The performance and properties of the method are verified in a simulation study, compared with the classical AMSAA model. A case study is used to illustrate the practical use of the model. The proposed approach can be adapted for a wide class of nonhomogeneous Poisson process based models.