Bayesian Reliability——Growth Analysis for Statistical of Diverse Population Based on Non-homogeneous Poisson Process

New armament systems are subjected to the method for dealing with multi-stage system reliability-growth statistical problems of diverse population in order to improve reliability before starting mass production. Aiming at the test process which is high expense and small sample-size in the development of complex system, the specific methods are studied on how to process the statistical information of Bayesian reliability growth regarding diverse populations. Firstly, according to the characteristics of reliability growth during product development, the Bayesian method is used to integrate the testing information of multi-stage and the order relations of distribution parameters. And then a Gamma-Beta prior distribution is proposed based on non-homogeneous Poisson process（NHPP） corresponding to the reliability growth process. The posterior distribution of reliability parameters is obtained regarding different stages of product, and the reliability parameters are evaluated based on the posterior distribution. Finally, Bayesian approach proposed in this paper for multi-stage reliability growth test is applied to the test process which is small sample-size in the astronautics filed. The results of a numerical example show that the presented model can make use of the diverse information synthetically, and pave the way for the application of the Bayesian model for multi-stage reliability growth test evaluation with small sample-size. The method is useful for evaluating multi-stage system reliability and making reliability growth plan rationally.

Immune Clone Maximum Likelihood Estimation of Improved Non-homogeneous Poisson Process Model Parameters

Aiming at the solving problem of improved nonhomogeneous Poisson process( NHPP) model in engineering application,the immune clone maximum likelihood estimation( MLE)method for solving model parameters was proposed. The minimum negative log-likelihood function was used as the objective function to optimize instead of using iterative method to solve complex system of equations,and the problem of parameter estimation of improved NHPP model was solved by immune clone algorithm. And the interval estimation of reliability indices was given by using fisher information matrix method and delta method. An example of failure truncated data from multiple numerical control( NC) machine tools was taken to prove the method. and the results show that the algorithm has a higher convergence rate and computational accuracy, which demonstrates the feasibility of the method.

Analysis of Ozone Behaviour in the City of Puebla-Mexico Using Non-Homogeneous Poisson Models with Multiple Change-Points

In this work, some non-homogeneous Poisson models are considered to study the behaviour of ozone in the city of Puebla, Mexico. Several functions are used as the rate function for the non-homogeneous Poisson process. In addition to their dependence on time, these rate functions also depend on some parameters that need to be estimated. In order to estimate them, a Bayesian approach will be taken. The expressions for the distributions of the parameters involved in the models are very complex. Therefore, Markov chain Monte Carlo algorithms are used to estimate them. The methodology is applied to the ozone data from the city of Puebla, Mexico.

Comparing the Adequacy of Some Non-Homogeneous Poisson Models to Estimate Ozone Exceedances in Mexico City

We consider some non-homogeneous Poisson models to estimate the mean number of times that a given environmental threshold of interest is surpassed by a given pollutant. Seven different rate functions for the Poisson processes describing the models are taken into account. The rate functions considered are the Weibull, exponentiated-Weibull, and their generalisation the Beta-Weibull rate function. We also use the Musa-Okumoto, the Goel-Okumoto, a generalised Goel- Okumoto and the Weibull-geometric rate functions. Whenever thought justifiable, the model allowing the presence of change-points is also going to be considered. The different models are applied to the daily maximum ozone measurements data provided by the monitoring network of the Metropolitan Area of Mexico City. The aim is to compare the adjustment of different rate functions to the data. Even though, some of the rate functions have been considered before, now we are applying them to the same data set. In previous works they were used in different data sets and therefore a comparison of the adequacy of those models were not possible. The measurements considered here were obtained after a series of environmental measures were implemented in Mexico City. Hence, the data present a different behaviour from that of earlier studies.

Non-Homogeneous Poisson Processes Applied to Count Data:A Bayesian Approach Considering Different Prior Distributions

This article discusses the Bayesian approach for count data using non-homogeneous Poisson processes, considering different prior distributions for the model parameters. A Bayesian approach using Markov Chain Monte Carlo (MCMC) simulation methods for this model was first introduced by [1], taking into account software reliability data and considering non-informative prior distributions for the parameters of the model. With the non-informative prior distributions presented by these authors, computational difficulties may occur when using MCMC methods. This article considers different prior distributions for the parameters of the proposed model, and studies the effect of such prior distributions on the convergence and accuracy of the results. In order to illustrate the proposed methodology, two examples are considered: the first one has simulated data, and the second has a set of data for pollution issues at a region in Mexico City.

One-Sample Bayesian Predictive Analyses for an Exponential Non-Homogeneous Poisson Process in Software Reliability

The Goel-Okumoto software reliability model, also known as the Exponential Nonhomogeneous Poisson Process,is one of the earliest software reliability models to be proposed. From literature, it is evident that most of the study that has been done on the Goel-Okumoto software reliability model is parameter estimation using the MLE method and model fit. It is widely known that predictive analysis is very useful for modifying, debugging and determining when to terminate software development testing process. However, there is a conspicuous absence of literature on both the classical and Bayesian predictive analyses on the model. This paper presents some results about predictive analyses for the Goel-Okumoto software reliability model. Driven by the requirement of highly reliable software used in computers embedded in automotive, mechanical and safety control systems, industrial and quality process control, real-time sensor networks, aircrafts, nuclear reactors among others, we address four issues in single-sample prediction associated closely with software development process. We have adopted Bayesian methods based on non-informative priors to develop explicit solutions to these problems. An example with real data in the form of time between software failures will be used to illustrate the developed methodologies.

Two-Sample Bayesian Predictive Analyses for an Exponential Non-Homogeneous Poisson Process in Software Reliability

The Goel-Okumoto software reliability model is one of the earliest attempts to use a non-homogeneous Poisson process to model failure times observed during software test interval. The model is known as exponential NHPP model as it describes exponential software failure curve. Parameter estimation, model fit and predictive analyses based on one sample have been conducted on the Goel-Okumoto software reliability model. However, predictive analyses based on two samples have not been conducted on the model. In two-sample prediction, the parameters and characteristics of the first sample are used to analyze and to make predictions for the second sample. This helps in saving time and resources during the software development process. This paper presents some results about predictive analyses for the Goel-Okumoto software reliability model based on two samples. We have addressed three issues in two-sample prediction associated closely with software development testing process. Bayesian methods based on non-informative priors have been adopted to develop solutions to these issues. The developed methodologies have been illustrated by two sets of software failure data simulated from the Goel-Okumoto software reliability model.

A transient process observation method based on the non-homogeneous Poisson process model

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.

二维线性化Navier-Stokes-Poisson方程解的逐点估计

本文研究了二维空间线性化的等熵可压缩Navier-Stokes-Poisson方程柯西问题.通过把方程组转变成关于单个函数的方程,求解出各个函数,得到方程组的格林函数.利用对格林函数的详细分析,获得了方程组解的逐点估计.结果显示方程组中电流密度以热核的速度衰减,动量密度衰减慢得多,且其L2范数不衰减.

通过嵌合管来增强负Poisson比结构的耐撞性能

为了提高结构耐撞性,从而实现车辆轻量化,提出了一种嵌合管(由薄壁方形管嵌入双箭头格子而组成的)具有负Poisson比的增强型结构。通过负Poisson比的收缩效应,来增强负Poisson比结构和嵌合管的耦合作用。建立了增强型结构的有限元模型,并通过压缩试验,来验证其准确性;通过数值方法,在轴向载荷下,比较双箭头点阵结构、结构填充管和嵌合管负Poisson比增强型结构的耐撞性能,并进行了参数分析。结果表明:与双箭头点阵结构和填充管结构相比,增强型结构的总吸收能量为1.12 MJ,提升了约40%;比吸能13.2 kJ/kg,提升了约30%;短梁的夹角70°、长梁的夹角40°、长梁厚度1.4~1.6 mm、短梁厚度1.2 mm的结构填充管结构,具有较大的比吸能和较小的碰撞峰值力;增加嵌管厚度显著提升耐撞性能,但结构质量急剧增加,嵌管厚度达到1.2 mm左右时,兼顾耐撞性和轻量化的需求。

基于DP算法的Poisson回归模型的变量选择

利用已有的DP算法,对Poisson回归模型的似然函数进行优化估计出模型的参数,分别写出Lasso、Adaptive Lasso、SCAD、MCP 4种惩罚似然函数,用DP算法进行求解,实现变量选择.为了实现算法的自动变量选择,在对惩罚参数的选择上,使用了AIC、BIC信息准则.为了验证DP算法的可行性,通过随机模拟的方式生成数据进行求解.在各项指标下,将DP算法与已有的相关算法进行对比.

广义结构Poisson括号与Casimir函数

证明了广义结构Poisson括号恒等于零的相关结果并得到验证,改进了广义Poisson括号的Casimir函数的结果,并得到了一个推论。

任意特征域上sl(2)的Poisson结构

利用李代数上Poisson代数的定义与双线性决定了任意特征域上李代数sl(2)的所有Poisson结构,得到结论:特征不为2的域上,sl(2)的Poisson结构由一个参数完全决定,都是非交换且非结合的;特征为2的域上,sl(2)的Poisson结构由8个参数完全决定,而且既有结合的Poisson结构、交换的Poisson结构,也有非交换且非结合的Poisson结构.

一种三维Poisson-Nernst-Planck方程的虚单元计算

利用虚单元方法在多面体网格上求解一种三维稳态Poisson-Nernst-Planck(PNP)方程,并给出PNP方程的虚单元离散形式,推导电势方程及离子浓度方程的刚度矩阵与荷载向量的矩阵表达式.数值实验结果表明,在3种不同的多面体网格下实现了PNP方程的虚单元计算,数值解在L^(2)和H^(1)范数下均达到最优阶.

基于重心插值配点法求解变系数广义Poisson方程

提出了一种求解二维变系数广义Poisson方程的重心插值配点法,重心插值配点法是一种真正的无网格配点方法,其在计算域内不用划分网格,得出的数值解在一定误差范围内可以无限接近精确解,同时具有很好的数值稳定性.本文中的数值算例均采用重心有理插值配点法和重心Lagrange插值配点法来计算,然后通过与解析解的比较,可以得出该方法具有快速、准确和易于数值计算的优点,是一种较好的数值计算方法.

线性Poisson-Boltzmann方程的虚单元L^(2)误差估计

针对一类线性Poisson-Boltzmann方程的虚单元L^(2)误差估计进行分析.首先引入正则化的线性Poisson-Boltzmann方程,将原问题转化为非奇性Poisson-Boltzmann方程.然后给出L^(2)范数的误差估计.最后在四边形和五边形混合多边形网格上进行数值实验,数值结果验证了理论分析的正确性.

Topology Optimization of Metamaterial Microstructures for Negative Poisson’s Ratio under Large Deformation Using a Gradient-Free Method

Negative Poisson’s ratio(NPR)metamaterials are attractive for their unique mechanical behaviors and potential applications in deformation control and energy absorption.However,when subjected to significant stretching,NPR metamaterials designed under small strain assumption may experience a rapid degradation in NPR performance.To address this issue,this study aims to design metamaterials maintaining a targeted NPR under large deformation by taking advantage of the geometry nonlinearity mechanism.A representative periodic unit cell is modeled considering geometry nonlinearity,and its topology is designed using a gradient-free method.The unit cell microstructural topologies are described with the material-field series-expansion(MFSE)method.The MFSE method assumes spatial correlation of the material distribution,which greatly reduces the number of required design variables.To conveniently design metamaterials with desired NPR under large deformation,we propose a two-stage gradient-free metamaterial topology optimization method,which fully takes advantage of the dimension reduction benefits of the MFSE method and the Kriging surrogate model technique.Initially,we use homogenization to find a preliminary NPR design under a small deformation assumption.In the second stage,we begin with this preliminary design and minimize deviations in NPR from a targeted value under large deformation.Using this strategy and solution technique,we successfully obtain a group of NPR metamaterials that can sustain different desired NPRs in the range of[−0.8,−0.1]under uniaxial stretching up to 20% strain.Furthermore,typical microstructure designs are fabricated and tested through experiments.The experimental results show good consistency with our numerical results,demonstrating the effectiveness of the present gradientfree NPR metamaterial design strategy.

稀疏过程在常利率环境下的广义Poisson风险模型中的应用

风险理论是连接数学与金融问题的一座桥梁,它是保险精算学(即实际寿险精算)的重要组成部分,是保险精算人员对保险业务进行风险管理的重要理论工具,同时也是保险公司发展业务进行有效营运的理论保证。破产概率是风险理论研究的核心问题,而利息是影响破产概率的重要因素。本文讨论了常利率环境下离散时间风险模型的破产概率,给出了模型的破产概率的上界。首先构造了风险模型,通过鞅方法得到模型的破产概率满足的Lundberg不等式和相关公式;然后给出了离散时间情形下的双险种风险模型,得到了与破产相关的一些变量的表达式和性质。

A gated recurrent unit model to predict Poisson’s ratio using deep learning

Static Poisson’s ratio(vs)is crucial for determining geomechanical properties in petroleum applications,namely sand production.Some models have been used to predict vs;however,the published models were limited to specific data ranges with an average absolute percentage relative error(AAPRE)of more than 10%.The published gated recurrent unit(GRU)models do not consider trend analysis to show physical behaviors.In this study,we aim to develop a GRU model using trend analysis and three inputs for predicting n s based on a broad range of data,n s(value of 0.1627-0.4492),bulk formation density(RHOB)(0.315-2.994 g/mL),compressional time(DTc)(44.43-186.9 μs/ft),and shear time(DTs)(72.9-341.2μ s/ft).The GRU model was evaluated using different approaches,including statistical error an-alyses.The GRU model showed the proper trends,and the model data ranges were wider than previous ones.The GRU model has the largest correlation coefficient(R)of 0.967 and the lowest AAPRE,average percent relative error(APRE),root mean square error(RMSE),and standard deviation(SD)of 3.228%,1.054%,4.389,and 0.013,respectively,compared to other models.The GRU model has a high accuracy for the different datasets:training,validation,testing,and the whole datasets with R and AAPRE values were 0.981 and 2.601%,0.966 and 3.274%,0.967 and 3.228%,and 0.977 and 2.861%,respectively.The group error analyses of all inputs show that the GRU model has less than 5% AAPRE for all input ranges,which is superior to other models that have different AAPRE values of more than 10% at various ranges of inputs.