基于证据理论的不确定性处理研究及其在测试中的应用
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摘要
基于证据理论的不确定性处理研究是一个前沿性研究课题，在国内外受到广泛关注。进行该方面的研究，不仅有助于完善证据理论的理论体系，有助于解决在测量、建模与仿真和可靠性评估等方面出现的信息不完整，实验数据缺乏、小子样数据和认识性不确定性等技术问题，还有助于提高和丰富测量技术的理论基础和实际应用性。
本文在装备预研重点基金《复杂装备有限样本可靠性评估研究》(9140A19030908ZW0401)的支持下开展研究，主要是研究解决在数据不充分和信息不完整条件下，产品性能评估和实验数据处理的许多不确定性问题。本文主要进行了两方面的工作。首先进行了证据理论的数学理论研究，研究了证据理论对不确定性的表达方法，研究了证据合成方法及其应用，研究了证据体的精确化方法。第二是进行了证据理论在不确定性数据处理中的应用研究，主要包括研究了证据理论处理不确定性数据的基本方法，研究了概率包络计算方法及其应用，研究了不等精度测试数据的处理方法，研究了不精确性测试数据的不确定度评估方法，研究了测量数据的可能性表达和建模。在完成理论研究的同时，本文提供了靶膜厚度的多传感器数据融合，中子测试数据处理，安保系统失效概率的计算，概率包络在复杂系统性能评估中的应用等若干试验和仿真结果，两者相符性很好。
本文的主要创新之处：
1．研究了证据理论在进行不确定性表达和量化的基本方法，提出了一种基于证据理论的不等精度测试数据处理方法，并应用到中子测试数据处理中。
2．在分析各种证据合成方法特性和实验对比的基础上，提出了一种新的证据合成方法。该方法兼容Dempster证据合成方法，继承了与运算证据聚焦性的优点，反映了证据间的交叉融合程度，解决了冲突性证据处理问题，放宽了证据合成条件。同时研究了多传感器数据融合的测量信息特征，并用本文提出的证据合成方法进行了靶膜厚度多传感器测量数据融合实验，融合效果很好。
3．研究了证据体的不确定性计算方法，在此基础上，提出了不精确性测试数据的不确定度评估方法，并应用到专家数据、中子测试数据和局部放电数据等处理中，取得了很好的效果。
4．研究了概率包络的平均离散方法和外离散方法，研究了不确定性变量的概率包络的计算方法，利用概率包络与证据体相互转换和扩展原理，实现了不确定性函数概率包络的算法，结合QMU方法，研究了概率包络分析在复杂系统性能评估中的应用，并进行了仿真实验。
5．提出了基于最大不确定性和不确定性不变的两种证据体精确化方法，并应用到安保系统的失效概率的仿真计算中。研究了测量数据的可能性分布表达及其模型建立方法，研究了概率分布到可能性分布的最优转换以及截性三角形近似算法，并应用到测量不确定度的计算、传递和评估中。
The study on uncertainties processing based on evidence theory is focused by many scientists and engineers all over the world. The work is very useful to help to enrich the theoretical mechanism of evidence theory, to solve the problems in which exists imprecise information, simple sample data and epistemic uncertainty, but also to enhance the foundation and application of measurement technology.
In this paper, the study is presented based on the evidence theory and the information process of the uncertainties. And the study is concentrated in 2 aspects. Firstly, the mathematic theory related to the evidence theory is discussed, including uncertainty quantification, relationship among the evidence theory, probability theory and possibility theory, and finally a modified combination rule for evidence. As for the second aspects, the study is focused on the application of the evidence theory into the uncertain data processing. The basic method for the uncertain data processing is presented. Based on the study of the probability boundary analysis, the computational equations for several uncertain input probability boxes are given. In addition, the computational method for probability boxes of the uncertain function is presented based on the conversion of probability boxes and evidence body. As for the third aspects, the study is made for the measured uncertainty data and information processing technology. The main task for this is to carryout the data statistics of the dynamic measured data based on the evidence theory and the expression of the measured data of the possibility distribution, based on which, several evaluation methods for the uncertainty measurement are put forward. Moreover, the fusion of the measurement data of multiple sensors are made with the modified evidence composition rule presented in this paper. In addition to the theoretical study, experimental results and simulation results are presented in this paper with fairly good accordance with each other.
The major innovative of this paper is given as the follows.
The unique performance of the evidence theory in dealing with the imprecise probability is described. And the related deductions are made to identify the similarity and difference of the evidence theory with other theories in dealing with uncertainty. Such concepts as basic mass assign, belief function, plausible function, the upper and lower probability are used for the process and statistics of the measurement data, so is applied to process neutrons data.
Based on the discussion of the advantages and disadvantages of the various combination rules for evidence, a modified combination rule is presented to deal with consistence or inconsistence evidences obtained from multiple sources. The modified rule adapts AND-operation to combine consistent evidences and reflects the intersection of focal elements, and allocates the conflict probability to very inconsistence focus element according to its average supported degree. Experiments show that the new combination rule is very reliable and rational for all kinds of evidences including highly conflicting evidences.
The measurement uncertainty is often evaluated by a probabilistic approach, but such approach is not always adapted to imprecise measurement data. After discussing the relation between Shannon entropy and measurement uncertainty, a general formula for evaluation of measurement uncertainty is proposed, which can be applied in both precise data and imprecise data.
The conversion between the evidence body and probability boxes is discussed provide two methods for the probability boxes namely the average discretization and the external discretization. In accordance with the three principles in dealing with the probability boxes as rigor-preserving, best possible and sample uncertainty, the computational methods for the probability boxes with various uncertainty variables are given with known type of distribution or partially limited information. And the computational methods of the probability boxes of the uncertainty function are presented with conversion method of the probability boxes and evidence body.
The study covers the possibility expression of the measurement error, the evaluation of the measurement uncertainty, the transformation from probability distribution into possibility distribution, and the modeling of the possibility distribution of the measurement data.
本文在装备预研重点基金《复杂装备有限样本可靠性评估研究》(9140A19030908ZW0401)的支持下开展研究，主要是研究解决在数据不充分和信息不完整条件下，产品性能评估和实验数据处理的许多不确定性问题。本文主要进行了两方面的工作。首先进行了证据理论的数学理论研究，研究了证据理论对不确定性的表达方法，研究了证据合成方法及其应用，研究了证据体的精确化方法。第二是进行了证据理论在不确定性数据处理中的应用研究，主要包括研究了证据理论处理不确定性数据的基本方法，研究了概率包络计算方法及其应用，研究了不等精度测试数据的处理方法，研究了不精确性测试数据的不确定度评估方法，研究了测量数据的可能性表达和建模。在完成理论研究的同时，本文提供了靶膜厚度的多传感器数据融合，中子测试数据处理，安保系统失效概率的计算，概率包络在复杂系统性能评估中的应用等若干试验和仿真结果，两者相符性很好。
本文的主要创新之处：
1．研究了证据理论在进行不确定性表达和量化的基本方法，提出了一种基于证据理论的不等精度测试数据处理方法，并应用到中子测试数据处理中。
2．在分析各种证据合成方法特性和实验对比的基础上，提出了一种新的证据合成方法。该方法兼容Dempster证据合成方法，继承了与运算证据聚焦性的优点，反映了证据间的交叉融合程度，解决了冲突性证据处理问题，放宽了证据合成条件。同时研究了多传感器数据融合的测量信息特征，并用本文提出的证据合成方法进行了靶膜厚度多传感器测量数据融合实验，融合效果很好。
3．研究了证据体的不确定性计算方法，在此基础上，提出了不精确性测试数据的不确定度评估方法，并应用到专家数据、中子测试数据和局部放电数据等处理中，取得了很好的效果。
4．研究了概率包络的平均离散方法和外离散方法，研究了不确定性变量的概率包络的计算方法，利用概率包络与证据体相互转换和扩展原理，实现了不确定性函数概率包络的算法，结合QMU方法，研究了概率包络分析在复杂系统性能评估中的应用，并进行了仿真实验。
5．提出了基于最大不确定性和不确定性不变的两种证据体精确化方法，并应用到安保系统的失效概率的仿真计算中。研究了测量数据的可能性分布表达及其模型建立方法，研究了概率分布到可能性分布的最优转换以及截性三角形近似算法，并应用到测量不确定度的计算、传递和评估中。
The study on uncertainties processing based on evidence theory is focused by many scientists and engineers all over the world. The work is very useful to help to enrich the theoretical mechanism of evidence theory, to solve the problems in which exists imprecise information, simple sample data and epistemic uncertainty, but also to enhance the foundation and application of measurement technology.
In this paper, the study is presented based on the evidence theory and the information process of the uncertainties. And the study is concentrated in 2 aspects. Firstly, the mathematic theory related to the evidence theory is discussed, including uncertainty quantification, relationship among the evidence theory, probability theory and possibility theory, and finally a modified combination rule for evidence. As for the second aspects, the study is focused on the application of the evidence theory into the uncertain data processing. The basic method for the uncertain data processing is presented. Based on the study of the probability boundary analysis, the computational equations for several uncertain input probability boxes are given. In addition, the computational method for probability boxes of the uncertain function is presented based on the conversion of probability boxes and evidence body. As for the third aspects, the study is made for the measured uncertainty data and information processing technology. The main task for this is to carryout the data statistics of the dynamic measured data based on the evidence theory and the expression of the measured data of the possibility distribution, based on which, several evaluation methods for the uncertainty measurement are put forward. Moreover, the fusion of the measurement data of multiple sensors are made with the modified evidence composition rule presented in this paper. In addition to the theoretical study, experimental results and simulation results are presented in this paper with fairly good accordance with each other.
The major innovative of this paper is given as the follows.
The unique performance of the evidence theory in dealing with the imprecise probability is described. And the related deductions are made to identify the similarity and difference of the evidence theory with other theories in dealing with uncertainty. Such concepts as basic mass assign, belief function, plausible function, the upper and lower probability are used for the process and statistics of the measurement data, so is applied to process neutrons data.
Based on the discussion of the advantages and disadvantages of the various combination rules for evidence, a modified combination rule is presented to deal with consistence or inconsistence evidences obtained from multiple sources. The modified rule adapts AND-operation to combine consistent evidences and reflects the intersection of focal elements, and allocates the conflict probability to very inconsistence focus element according to its average supported degree. Experiments show that the new combination rule is very reliable and rational for all kinds of evidences including highly conflicting evidences.
The measurement uncertainty is often evaluated by a probabilistic approach, but such approach is not always adapted to imprecise measurement data. After discussing the relation between Shannon entropy and measurement uncertainty, a general formula for evaluation of measurement uncertainty is proposed, which can be applied in both precise data and imprecise data.
The conversion between the evidence body and probability boxes is discussed provide two methods for the probability boxes namely the average discretization and the external discretization. In accordance with the three principles in dealing with the probability boxes as rigor-preserving, best possible and sample uncertainty, the computational methods for the probability boxes with various uncertainty variables are given with known type of distribution or partially limited information. And the computational methods of the probability boxes of the uncertainty function are presented with conversion method of the probability boxes and evidence body.
The study covers the possibility expression of the measurement error, the evaluation of the measurement uncertainty, the transformation from probability distribution into possibility distribution, and the modeling of the possibility distribution of the measurement data.
引文
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