基于试验损失的Bayes装备可靠性统计验证试验设计
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摘要
根据统计学原理,装备可靠性统计验证试验设计是按照可靠性统计验证试验的目的和战术技术指标评价中的风险、精度或置信水平等要求,研究如何确定试验样本量或试验时间,从而制定出一个可行的试验方案的过程。一个好的试验方案可以使得装备试验机构投入最少的资源,得出能够切实反映装备可靠性特征的试验数据,准确的判断装备的可靠性,从而最大限度的避免风险,更好的保护装备的生产方和使用方的利益。本文以试验方案的损失为目标函数,运用Bayes方法,基于两类风险和试验环境条件的约束,对装备的可靠性统计验证试验设计技术进行了研究。
论文的主要研究工作和创新点如下:
在Bayes理论的基础上,引用损失函数,按照验后期望损失最小的原则,推导出了抽样检验的决策不等式。综合考虑装备在可靠性统计验证试验过程中的成本费用(主要包括进行试验所需的初始费用、试验装备的成本和试验所需的人力费用等)和由于假设检验两类风险的存在而造成的期望损失(主要包括由于弃真所造成的生产方损失和由于采伪所造成的使用方损失),设计了进行可靠性统计验证试验的损失评估函数,以可靠性假设检验中的两类风险和现场的试验环境条件为约束条件,构建了基于试验损失的Bayes装备可靠性统计验证试验设计模型。以Bayes方法为基础,分别基于平均风险准则和验后风险准则,对两类风险的计算方法进行了论述和比较分析。结合装备可靠性参数的验前分布和现场试验数据的似然函数,对弃真风险和采伪风险的计算公式进行了推导。对于装备的可靠性统计验证问题,基于Bayes装备可靠性统计验证试验设计模型,给出了基于Matlab软件的通用求解算法。
对于寿命服从正态分布的装备可靠性统计验证试验设计进行了建模与分析,将对平均寿命的检验转化为对正态分布期望值的检验,构建了针对正态分布的Bayes可靠性统计验证试验设计的N模型。对于给定的抽样样本量,根据Bayes决策不等式,推导出了装备平均寿命决策阀值的计算公式,给出了正态型指标可靠性统计验证试验方案中接受域和拒绝域的确定方法。基于两种风险计算准则,分别推导了Bayes可靠性统计验证试验设计的N模型中两类风险的计算公式,通过仿真数据,对不同风险准则下两类风险和样本量的函数关系进行了讨论,在一定程度上为装备的试验机构提供决策依据。针对Bayes可靠性统计验证试验设计的N模型,给出了最优试验样本量的求解步骤,并通过示例对其建模与求解过程进行了分析。
针对成败型产品可靠性统计验证试验设计,以成功率作为可靠性的检验参数,构造了针对二项分布的Bayes可靠性统计验证试验设计的B模型。对于给定的试验样本量,按照验后期望损失最小的原则,推导出了验后概率比与接受数之间的函数关系,并通过具体的算例展示了接受数的临界值的计算方法。基于平均风险准则和验后风险准则,推导出了Bayes可靠性统计验证试验设计的B模型中弃真风险和采伪风险的计算公式,并且结合算例分别讨论了不同风险准则下两类风险在给定试验样本量(接受数)的情况下随着接受数(样本量)的变化规律。运用Matlab软件,借助仿真算例展示了Bayes可靠性统计验证试验设计的B模型的搜索迭代算法。
对于寿命服从指数分布的装备可靠性统计验证试验设计,将对装备可靠性的验证转化为对其失效率的检验,在充分考虑两类风险约束,产品失效个数和试验持续时间的约束基础上,构造了针对指数分布的Bayes可靠性统计验证试验设计的E模型。根据Bayes决策法则,对于给定的失效数,推导出了验后概率比和试验持续时间之间的约束关系,给出了试验持续时间的临界值的计算方法。基于两种风险准则,分别推导了Bayes可靠性统计验证试验设计的E模型中弃真风险和采伪风险的计算公式,给出了在给定失效数(试验持续时间)的基础上,不同试验持续时间(失效数)所对应的两类风险的变化规律。针对Bayes可靠性统计验证试验设计的E模型,结合指数型装备的可靠性特征,给出了最优试验方案的通用求解步骤,并通过示例验证了该方法的有效性。
According to statistical theory, the equipment reliability statistical demonstration test design is a process that studies on how to find the test sample size or test time, and then establishes a feasible test plan based on the requirement of reliability hypothesis, risks and precision of reliability evaluation. A good test plan could help equipment test bureau spend the least resource for testing data describing the character of the equip-ment reliability, and estimate equipment reliability exactly with the least test risk for the benefits of equipment’s producer and user. In this paper, based on the reliability test loss, using Bayesian method, the reliability statistical demonstration test design method was studied with the constraction on two types of risk and testing condition.
The main contributions and innovation of the thesis are as follows:
On the basis of Bayesian theory, decision inequation for sampling was deduced us-ing the loss function based on the principle of minimize the posterior expect loss. Con-sidering the cost which will be spent during reliability demonstration test (consists of initial test fee, cost of product, salary, and so on), and the expected loss caused by the risks in hypothesis test (consists of the producer’s loss caused by the first risk and the user’s loss caused by the second risk), the test loss function in reliability statistical demonstration test was established, and the Bayesian reliability statistical demonstration test design model based on test loss (BTDM) was carried out. The calculating method for the two types of risks based on average risk criteria and posterior risk criteria was discussed and comparatively analyzed based on the Bayesian method. The calculating formulas for the first risk and the second risk were given using the prior distribution of equipment reliability parameter and the likelihood function of the test data. To the problem of equipment reliability statistical demonstration, the common algorithm based on the Matlab software for BTDM was present.
Reliability statistical demonstration test design for the equipment whose lifetime was normal distributed was modeled and analyzed. The hypothesis for mean lifetime was converted to the one for the expectation of normal distribution, and the Bayesian reliability statistical demonstration test design model for normal distribution (BTDM-N) was constructed. According to the Bayesian decision inequation, to the certain sample number, the calculating formula of equipment mean lifetime critical value was deduced, acceptance region and rejection region in the Bayesian reliability statistical demonstra-tion test plan for normal distribution were given. The calculating formulas for two types of risks in the BTDM-N were present based on two kinds of risk criteria, the relation-ship of two types of risks and sample number based on the different risk criteria was discussed through the simulation data, which were used for equipment test bureau’s de-cision. The solving steps of the best sample number were present based on BTDM-N, the constructing and the calculating process for BTDM-N were illustrated using an ex-ample.
The successful ratio was treated as the reliability test parameter for binomial prod-uct reliability statistical demonstration test design, and the Bayesian reliability statistical demonstration test design model for binomial distribution (BTDM-B) was constructed. To certain sample number, the functional relationship of posterior probability ratio and acceptance number was deduced based on the principle of minimize expect posterior loss, and the calculating method of the critical value of acceptance number was present through a example. The calculating formulas of two types of risks in the BTDM-B was given based on average risk criteria and posterior risk criteria, and two types of risks based on the different sample number(acceptance number) for the certain acceptance number(sample number) under the different risk criteria was laid out through a example. The searching iteration of BTDM-B was present with a simulation example using Mat-lab software.
The failure ratio was treated as the parameter of the reliability test for exponential product reliability statistical demonstration test design, and the Bayesian reliability sta-tistical demonstration test design model for exponential distribution (BTDM-E) was constructed considering the constraint of the two types of risks, failure number and testing time. To certain failure number, the functional relationship of posterior probabil-ity ratio and testing time was deduced based on the Bayesian decision principle, and the calculating method of the critical value of testing time was given, The calculating for-mulas of two types of risks in BTDM-E based on the average risk criteria and the poste-rior risk criteria were deduced, and two types of risks based on the different failure number(testing time) for the certain testing time(failure number) under the different risk criteria was laid out through a example. The solution method of BTDM-E was illus-trated and validated follow a example based on the character of the exponential distrib-uted product.
论文的主要研究工作和创新点如下:
在Bayes理论的基础上,引用损失函数,按照验后期望损失最小的原则,推导出了抽样检验的决策不等式。综合考虑装备在可靠性统计验证试验过程中的成本费用(主要包括进行试验所需的初始费用、试验装备的成本和试验所需的人力费用等)和由于假设检验两类风险的存在而造成的期望损失(主要包括由于弃真所造成的生产方损失和由于采伪所造成的使用方损失),设计了进行可靠性统计验证试验的损失评估函数,以可靠性假设检验中的两类风险和现场的试验环境条件为约束条件,构建了基于试验损失的Bayes装备可靠性统计验证试验设计模型。以Bayes方法为基础,分别基于平均风险准则和验后风险准则,对两类风险的计算方法进行了论述和比较分析。结合装备可靠性参数的验前分布和现场试验数据的似然函数,对弃真风险和采伪风险的计算公式进行了推导。对于装备的可靠性统计验证问题,基于Bayes装备可靠性统计验证试验设计模型,给出了基于Matlab软件的通用求解算法。
对于寿命服从正态分布的装备可靠性统计验证试验设计进行了建模与分析,将对平均寿命的检验转化为对正态分布期望值的检验,构建了针对正态分布的Bayes可靠性统计验证试验设计的N模型。对于给定的抽样样本量,根据Bayes决策不等式,推导出了装备平均寿命决策阀值的计算公式,给出了正态型指标可靠性统计验证试验方案中接受域和拒绝域的确定方法。基于两种风险计算准则,分别推导了Bayes可靠性统计验证试验设计的N模型中两类风险的计算公式,通过仿真数据,对不同风险准则下两类风险和样本量的函数关系进行了讨论,在一定程度上为装备的试验机构提供决策依据。针对Bayes可靠性统计验证试验设计的N模型,给出了最优试验样本量的求解步骤,并通过示例对其建模与求解过程进行了分析。
针对成败型产品可靠性统计验证试验设计,以成功率作为可靠性的检验参数,构造了针对二项分布的Bayes可靠性统计验证试验设计的B模型。对于给定的试验样本量,按照验后期望损失最小的原则,推导出了验后概率比与接受数之间的函数关系,并通过具体的算例展示了接受数的临界值的计算方法。基于平均风险准则和验后风险准则,推导出了Bayes可靠性统计验证试验设计的B模型中弃真风险和采伪风险的计算公式,并且结合算例分别讨论了不同风险准则下两类风险在给定试验样本量(接受数)的情况下随着接受数(样本量)的变化规律。运用Matlab软件,借助仿真算例展示了Bayes可靠性统计验证试验设计的B模型的搜索迭代算法。
对于寿命服从指数分布的装备可靠性统计验证试验设计,将对装备可靠性的验证转化为对其失效率的检验,在充分考虑两类风险约束,产品失效个数和试验持续时间的约束基础上,构造了针对指数分布的Bayes可靠性统计验证试验设计的E模型。根据Bayes决策法则,对于给定的失效数,推导出了验后概率比和试验持续时间之间的约束关系,给出了试验持续时间的临界值的计算方法。基于两种风险准则,分别推导了Bayes可靠性统计验证试验设计的E模型中弃真风险和采伪风险的计算公式,给出了在给定失效数(试验持续时间)的基础上,不同试验持续时间(失效数)所对应的两类风险的变化规律。针对Bayes可靠性统计验证试验设计的E模型,结合指数型装备的可靠性特征,给出了最优试验方案的通用求解步骤,并通过示例验证了该方法的有效性。
According to statistical theory, the equipment reliability statistical demonstration test design is a process that studies on how to find the test sample size or test time, and then establishes a feasible test plan based on the requirement of reliability hypothesis, risks and precision of reliability evaluation. A good test plan could help equipment test bureau spend the least resource for testing data describing the character of the equip-ment reliability, and estimate equipment reliability exactly with the least test risk for the benefits of equipment’s producer and user. In this paper, based on the reliability test loss, using Bayesian method, the reliability statistical demonstration test design method was studied with the constraction on two types of risk and testing condition.
The main contributions and innovation of the thesis are as follows:
On the basis of Bayesian theory, decision inequation for sampling was deduced us-ing the loss function based on the principle of minimize the posterior expect loss. Con-sidering the cost which will be spent during reliability demonstration test (consists of initial test fee, cost of product, salary, and so on), and the expected loss caused by the risks in hypothesis test (consists of the producer’s loss caused by the first risk and the user’s loss caused by the second risk), the test loss function in reliability statistical demonstration test was established, and the Bayesian reliability statistical demonstration test design model based on test loss (BTDM) was carried out. The calculating method for the two types of risks based on average risk criteria and posterior risk criteria was discussed and comparatively analyzed based on the Bayesian method. The calculating formulas for the first risk and the second risk were given using the prior distribution of equipment reliability parameter and the likelihood function of the test data. To the problem of equipment reliability statistical demonstration, the common algorithm based on the Matlab software for BTDM was present.
Reliability statistical demonstration test design for the equipment whose lifetime was normal distributed was modeled and analyzed. The hypothesis for mean lifetime was converted to the one for the expectation of normal distribution, and the Bayesian reliability statistical demonstration test design model for normal distribution (BTDM-N) was constructed. According to the Bayesian decision inequation, to the certain sample number, the calculating formula of equipment mean lifetime critical value was deduced, acceptance region and rejection region in the Bayesian reliability statistical demonstra-tion test plan for normal distribution were given. The calculating formulas for two types of risks in the BTDM-N were present based on two kinds of risk criteria, the relation-ship of two types of risks and sample number based on the different risk criteria was discussed through the simulation data, which were used for equipment test bureau’s de-cision. The solving steps of the best sample number were present based on BTDM-N, the constructing and the calculating process for BTDM-N were illustrated using an ex-ample.
The successful ratio was treated as the reliability test parameter for binomial prod-uct reliability statistical demonstration test design, and the Bayesian reliability statistical demonstration test design model for binomial distribution (BTDM-B) was constructed. To certain sample number, the functional relationship of posterior probability ratio and acceptance number was deduced based on the principle of minimize expect posterior loss, and the calculating method of the critical value of acceptance number was present through a example. The calculating formulas of two types of risks in the BTDM-B was given based on average risk criteria and posterior risk criteria, and two types of risks based on the different sample number(acceptance number) for the certain acceptance number(sample number) under the different risk criteria was laid out through a example. The searching iteration of BTDM-B was present with a simulation example using Mat-lab software.
The failure ratio was treated as the parameter of the reliability test for exponential product reliability statistical demonstration test design, and the Bayesian reliability sta-tistical demonstration test design model for exponential distribution (BTDM-E) was constructed considering the constraint of the two types of risks, failure number and testing time. To certain failure number, the functional relationship of posterior probabil-ity ratio and testing time was deduced based on the Bayesian decision principle, and the calculating method of the critical value of testing time was given, The calculating for-mulas of two types of risks in BTDM-E based on the average risk criteria and the poste-rior risk criteria were deduced, and two types of risks based on the different failure number(testing time) for the certain testing time(failure number) under the different risk criteria was laid out through a example. The solution method of BTDM-E was illus-trated and validated follow a example based on the character of the exponential distrib-uted product.
引文
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