Non-homogeneous Poisson Process Models Research Articles

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.

Multi-phase algorithm design for accurate and efficient model fitting

Recent research applies soft computing techniques to fit software reliability growth models. However, runtime performance and the distribution of the distance from an optimal solution over multiple runs must be explicitly considered to justify the practical utility of these approaches, promote comparison, and support reproducible research. This paper presents a meta-optimization framework to design stable and efficient multi-phase algorithms for fitting software reliability growth models. The approach combines initial parameter estimation techniques from statistical algorithms, the global search properties of soft computing, and the rapid convergence of numerical methods. Designs that exhibit the best balance between runtime performance and accuracy are identified. The approach is illustrated through nonhomogeneous Poisson process and covariate software reliability growth models, including a cross-validation step on data sets not used to identify designs. The results indicate the nonhomogeneous Poisson process model considered is too simple to benefit from soft computing because it incurs additional runtime with no increase in accuracy attained. However, a multi-phase design for the covariate software reliability growth model consisting of the bat algorithm followed by a numerical method achieves better performance and converges consistently, compared to a numerical method only. The proposed approach supports higher-dimensional covariate software reliability growth model fitting suitable for implementation in a tool.

A transient process observation method based on the non-homogeneous Poisson process model * *Support of the National Natural Science Foundation of China (1435010, 11575145, 11922507)

The current-mode-counting method is a new approach to observing transient processes, especially in transient nuclear fusion, based on the non-homogeneous Poisson process (NHPP) model. In this paper, a new measurement process model of the pulsed radiation field produced by transient nuclear fusion is built based on the NHPP. A simulated measurement is performed using the model, and the current signal from the detector is obtained by simulation based on Poisson process thinning. The neutron time spectrum is reconstructed and is in good agreement with the theoretical value, with its maximum error of a characteristic parameter less than 2.3%. Verification experiments were carried out on a CPNG-6 device at the China Institute of Atomic Energy, with a detection system with a nanosecond response time. The experimental charge amplitude spectra are in good agreement with those obtained by the traditional counting mode, and the characteristic parameters of the time spectrum are in good agreement with the theoretical values. This shows that the current-mode-counting method is effective for the observation of transient nuclear fusion processes.

System Analysis and Reliability Analysis for Environmental Test Chambers

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

Agricultural drought periods analysis by using nonhomogeneous poisson models and regionalization of appropriate model parameters

Precipitation has a dominant role in agriculture, and a regular rain pattern is usually vital to agriculture; excessive or inadequate rainfall can be harmful. In this paper, an agricultural drought index is utilized to study the agricultural drought periods and analyze them with their intensities at various locations. Some nonhomogeneous Poisson models are also used to calculate the probability of agricultural droughts (number of times occurred) in a time interval of interest. It is to be assumed that the number of agricultural drought event occurrences is a Nonhomogeneous Poisson Process (NHPP) has a rate function, which depends on some parameters that must be estimated. Two cases of these functions are considered: the Weibull and linear intensity function. The Bayesian approach with Gibbs sampling under the Markov Chain Monte Carlo (MCMC) algorithm is used to estimate the parameters of these functions. The most appropriate fitted model is selected by using Deviance Information Criteria (DIC) and use that appropriately fitted model to calculate the accumulated events of agricultural drought in a time interval of interest at each location. Ordinary Kriging (OK) is used to regionalize the parameters and present its spatial behavior. The results based on the DIC indicate that the Power Law Process (PLP) performs better than the linear intensity function, NHPP model. The interpolated parameter values for the appropriate model, their patterns, and fluctuations for the study area are efficiently presented using contour maps. It is a novel and straightforward approach to assess the selected model parameter values used to predict the accumulated drought events at un-sampled locations. The proposed framework might also help to analyze other spatial variables of interest and can be used for climate-change study, ecosystem modeling, etc. The findings can also help to make decisions for sustainable environmental management in Pakistan.

Mixed-Effects Nonhomogeneous Poisson Process Model for Multiple Repairable Systems

The nonhomogeneous Poisson process (NHPP) has become a useful approach for modeling failure patterns of recurrent failure data revealed by minimal repairs from an individual repairable system. Sometimes, multiple repairable systems may present system-to-system variability owing to operation environments or working intensities of individual systems. In this paper, we go over the application of generalized mixed-effects models to recurrent failure data from multiple repairable systems based on the NHPP. The generalized mixed-effects models explicitly involve between-system variation through randomeffects, along with a common baseline for all the systems through fixed-effects for non-normal data. Details on estimation of the parameters of the mixed-effects NHPP models and construction of their confidence intervals are examined. An applicative example shows prominent proof of the mixed-effects NHPP models for the purpose of reliability analysis.

A New Reliability Model Based on Lindley Distribution with Application to Failure Data

Software reliability is an important feature that influences systems’ reliability. Software reliability models are a common tool to evaluate software reliability quantitatively. Various reliability models have been suggested based on the NHPP (nonhomogeneous Poisson process). In this article, a new NHPP model based on the Lindley distribution is proposed. The mathematical formulas for its measures of reliability are obtained and graphically illustrated. The proposed model’s parameters are estimated using both the NLSE (nonlinear least squares estimation) and the WNLSE (weighted nonlinear least squares estimation) methods. The model is then validated based on several different reliability datasets. The methods of estimation are evaluated and compared using three different criteria. The performance of the new model is also evaluated and compared, both objectively and subjectively, with three previously suggested models. The application results show that our new model demonstrates good performance in our selected failure data.

Demand Forecast of Restoring Air Material of Helicopter Based on NHPP and Weibull Model

The demand forecast of spare parts for helicopter air material is one of the key links in the supply and guarantee of helicopter air material. In order to forecast the demand of helicopter restoring air material effectively, a method based on NHPP (non-homogeneous Poisson process) and Weibull model is proposed in this paper. In this method, the reparability of air material is fully considered, and the repaired air materials are divided into two categories: basically repairable and completely repairable. Then, reliability assessment is carried out based on NHPP and Weibull model, and the demand of spare parts for air material is predicted on this basis. The effectiveness is proved by an example.

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

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

Research on the software reliability quantitative evaluation of nuclear power plant digital control system based on non-homogeneous poisson process model

With the application of digital technology, the software used in various systems has become increasingly important, as well as complex. This increase in complexity and scale has led to an increase in the probability of software failure. Quantitative reliability evaluation of the digital instrumentation and control system software used in nuclear power plants represent a significant challenge. This study uses a non-homogeneous Poisson process model for quantitative evaluation of the software reliability of a nuclear power plant safety digital control system (DCS), which is the research object. To analyse the research object, a software test environment is established, and the research object is tested to obtain the software failure data. The software reliability growth model, which is widely used in other industries, is combined with data on nuclear power plant DCS software defects that obey a non-homogeneous Poisson process, to construct a non-homogeneous Poisson process model. The formulas for software reliability indicators such as total failures, remaining failures, failure rate (failures per week), reliability, and time to next failure(s) are derived. The software failure data obtained from the test are used to determine the variables of the model based on maximum likelihood estimation; the estimates are then used to quantitatively evaluate software reliability-related indicators. Finally, the minimum correlation error method is used to analyse the error by comparing the actual and model prediction results. The model can evaluate the current reliability level of software and predict the impact of subsequent reliability test results. The findings from this study can provide data support for evaluating the reliability of nuclear power plant DCS software. Therefore, this study can be significant for further research on the reliability of nuclear power plant DCS software.

An early fault elimination method of computerized numerical control machine tools

Early faults of product have plagued many machine tool companies, especially middle-small enterprises limited by capital. To eliminate the early faults and improve the reliability of computerized numerical control (CNC) machine tools, we propose a systematic early fault elimination method based on the after-sale data (customer field data) of CNC machine tools. The proposed method includes four steps: fault data collection; a four-parameter non-homogenous Poisson process (NHPP) model; a mixed fault analysis method that is the combination of fault tree analysis (FTA) method and fault mode, effects, and criticality analysis (FMECA) method; and early fault elimination measures. The first step is the basis of the analysis. The second step is used to determine the early faults from the fault data. The third step is to analyze the early faults determined. Risk priority number (RPN) is calculated to assess the criticality of the fault causes determined by the FTA. The elimination measures are taken in the end. We apply our method to a machine tool company in China, and the results prove the practicality and effectiveness of the proposed method.

Distribution of waiting time between two successive aftershocks from modified Omori's law

The primary aim of this letter is the derivation of the distribution of the waiting time between two successive aftershocks from the modified Omori's law. We also propose two models. One model is the elapsed time-accumulation (ETA) model, and the Non-homogeneous Poisson Process (NPP) model. Using these two models, we confirm the conditions under which the distributions of the waiting time obtained follow the Zipf-Mandelbrot law.

Two-Dimensional NHPP Models Based on Several Testing-Time Functions and Their Applications

Enormous testing-time and testing-effort (central processing unit (CPU) time, execution time, man-hour, test coverage, executed test case, and so forth) are invested in testing-phase for enhancing software reliability. Although they are software reliability growth factors, existing software reliability growth models (SRGMs) are not introduced simultaneously. Also, the software reliability growth factors enable to substitute each other partly. Therefore, we develop new bivariate nonhomogeneous Poisson process (NHPP) models based on two types of testing-time functions in this paper. Concretely, we assume that the testing-time as a software reliability growth factor is composed of the testing-time and testing-effort factors. Then, the testing-time as the software reliability growth factor is expressed by the Cobb–Douglas and constant elasticity of substitution (CES) type testing-time functions. We conduct goodness-of-fit comparisons of existing models with proposed bivariate models. Also, we discuss estimation method of the optimal software release time and optimal amount of testing-effort as an application. Finally, we show numerical examples by using actual datasets.

Assessment of e-Social Activity in Psychiatric Patients.

This paper introduces a novel method to assess the social activity maintained by psychiatric patients using information and communication technologies. In particular, we model the daily usage patterns of phone calls and social and communication apps using point processes. We propose a novel nonhomogeneous Poisson process model with periodic (circadian) intensity function using a truncated Fourier series expansion, which is inferred using a trust-region algorithm. We also extend the model using a mixture of periodic intensity functions to cope with the different daily patterns of a person. The analysis of the usage of phone calls and social and communication apps of a cohort of 259 patients reveals common patterns shared among patients with relatively high homogeneity and differences among patient pathologies.

Semiparametric frailty models for zero-inflated event count data in the presence of informative dropout.

Recurrent events data are commonly encountered in medical studies. In many applications, only the number of events during the follow-up period rather than the recurrent event times is available. Two important challenges arise in such studies: (a) a substantial portion of subjects may not experience the event, and (b) we may not observe the event count for the entire study period due to informative dropout. To address the first challenge, we assume that underlying population consists of two subpopulations: a subpopulation nonsusceptible to the event of interest and a subpopulation susceptible to the event of interest. In the susceptible subpopulation, the event count is assumed to follow a Poisson distribution given the follow-up time and the subject-specific characteristics. We then introduce a frailty to account for informative dropout. The proposed semiparametric frailty models consist of three submodels: (a) a logistic regression model for the probability such that a subject belongs to the nonsusceptible subpopulation; (b) a nonhomogeneous Poisson process model with an unspecified baseline rate function; and (c) a Cox model for the informative dropout time. We develop likelihood-based estimation and inference procedures. The maximum likelihood estimators are shown to be consistent. Additionally, the proposed estimators of the finite-dimensional parameters are asymptotically normal and the covariance matrix attains the semiparametric efficiency bound. Simulation studies demonstrate that the proposed methodologies perform well in practical situations. We apply the proposed methods to a clinical trial on patients with myelodysplastic syndromes.

Confidence Intervals for the Failure Intensity and Number of Remaining Errors Functions Based on Non-Homogeneous Poisson Process Model

Confidence Intervals for the Failure Intensity and Number of Remaining Errors Functions Based on Non-Homogeneous Poisson Process Model

Entropy Based Software Reliability Analysis of Multi-Version Open Source Software

The number of issues fixed in the current release of the software is one of the factors which decides the next release of the software. The source code files get changed during fixing of these issues. The uncertainty arises due to these changes is quantified using entropy based measures. We developed a Non-Homogeneous Poisson Process model for Open Source Software to understand the fixing of issues across releases. Based on this model, optimal release-updating using entropy and maximizing the active user's satisfaction level subject to fixing of issues up to a desired level, is investigated as well. The proposed models have been validated on five products of the Apache open source project. The optimal release time estimated from the proposed model is close to the observed release time at different active user's satisfaction levels. The proposed decision model can assist management to appropriately determine the optimal release-update time. The proposed entropy based model for issues estimation shows improvement in performance for 21 releases out of total 23 releases, when compared with well-known traditional software reliability growth models, namely GO model <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> and S-shaped model <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</xref> . The proposed model is also found statistically significant.

Seasonal warranty prediction based on recurrent event data

Warranty return data from repairable systems, such as home appliances, lawn mowers, computers and automobiles, result in recurrent event data. The nonhomogeneous Poisson process (NHPP) model is used widely to describe such data. Seasonality in the repair frequencies and other variabilities, however, complicate the modeling of recurrent event data. Not much work has been done to address the seasonality, and this paper provides a general approach for the application of NHPP models with dynamic covariates to predict seasonal warranty returns. The methods presented here, however, can be applied to other applications that result in seasonal recurrent event data. A hierarchical clustering method is used to stratify the population into groups that are more homogeneous than the overall population. The stratification facilitates modeling the recurrent event data with both time-varying and time-constant covariates. We demonstrate and validate the models using warranty claims data for two different types of products. The results show that our approach provides important improvements in the predictive power of monthly events compared with models that do not take the seasonality and covariates into account.

Comparative Study of the Non-Homogeneous Poisson Process Type-I Generalized Half-Logistic Distribution

In this paper, Non-Homogeneous Poisson Process (NHPP) model are created based on Type-I Generalized Half-Logistic Distribution (GHLD- I). Three methods for estimating the parameters of the NHPP GHLD- I model are considered in the case of failure-occurrence time data, for this purpose the necessary likelihood equations are obtained. Confidence intervals are studied, the upper and the lower bounds of the parameters are constructed. An application based on the NHPP GHLD-I and using four published data sets are conducted. The performance of NHPP GHLD-I model is checked based on three evaluation criteria and useful results are obtained.