The treatment of uncertainty in knowledge-based systems using fuzzy sets and possibility theory

Knowledge representation issues related to the modelling of imprecision and uncertainty are discussed in the framework of possibility theory. Differences and relations between possibility theory, fuzzy sets, probability theory and Shafer evidence theory are presented. Then, patterns of reasoning and inference and control procedures are studied in presence of uncertainty and imprecision using a possibilistic approach. the basic ideas and the main trends are emphasized rather than the mathematical and logical foundations or the technical details of implementation which can be found in the references.     

Switching between type-2 fuzzy sets and intuitionistic fuzzy sets: an application in medical diagnosis

When dealing with vagueness, there are situations when there is insufficient information available, making it impossible to satisfactorily evaluate membership. The intuitionistic fuzzy set theory is more suitable than fuzzy sets to deal with such problem. In 1996, Atanassov proposed the mapping from intuitionistic fuzzy sets to fuzzy sets. Furthermore, intuitionistic fuzzy sets are isomorphic to interval valued fuzzy sets, and interval valued fuzzy sets are regarded as the special cases of type-2 fuzzy sets in recently studies. However, their discussions are not only hardly comprehending but also lacking the reliable applications. In this study, the advantage of type-2 fuzzy sets is employed, and the switching relation between type-2 fuzzy sets and intuitionistic fuzzy sets is defined axiomatically. The switching results are applied to show the usefulness of the proposed method in pattern recognition and medical diagnosis reasoning.     

基于分形和FNN的水轮机组振动故障在线诊断*

水轮发电机组的故障诊断具有模糊性和耦合性,提出一种基于模糊神经网络FNN的水轮发电机组振动故障在线诊断方法。首先,对反映转子振动状态的轴心轨迹用分形维数提取其结构特征,实现图形量化,以便FNN在线识别;接着,以6种典型振动故障为研究对象,在总结了包括轴心轨迹在内4类共14种故障征兆的基础上,分析各故障征兆的模糊属性,给出它们的模糊处理;然后,建立一种六层的前向FNN映射征兆到故障间的模糊推理,并给出学习算法修正网络参数;FNN通过自学习可保证良好的在线诊断精度。实例分析结果验证了其可行性。     

Level sets and the extension principle for interval valued fuzzy sets and its application to uncertainty measures

We describe the representation of a fuzzy subset in terms of its crisp level sets. We then generalize these level sets to the case of interval valued fuzzy sets and provide for a representation of an interval valued fuzzy set in terms of crisp level sets. We note that in this representation while the level sets are crisp the memberships are still intervals. Once having this representation we turn to its role in the extension principle and particularly to the extension of measures of uncertainty of interval valued fuzzy sets. Two types of extension of uncertainty measures are investigated. The first, based on the level set representation, leads to extensions whose values for the measure of uncertainty are themselves fuzzy sets. The second, based on the use of integrals, results in extensions whose value for the uncertainty of an interval valued fuzzy sets is an interval.     

Reversed fuzzy Petri nets and their application for fault diagnosis

An alterative approach to the backward reasoning is presented. In classical reasoning, both users and developers of many expert systems are dedicated to the forward reasoning. However, in many newly arising expert systems such as various diagnosis systems, the backward reasoning is of special interest and often preferable. In this paper, the fuzzy Petri nets are used to analytically represent the knowledge of fault diagnosis in manufacturing systems and an iterative algorithm based on max-algebra is used to deduce the consequence–antecedent relationship between their manifestation and antecedent. Finally, the legitimacy and efficiency of the proposed approach are proved and validated by an illustrative example.     

改进的故障树模糊诊断方法及其应用

针对故障诊断中存在的重复诊断问题和不确定性因素,提出了一种基于故障树的模糊故障诊断改进方法。该方法将模糊理论融入到故障诊断中,综合考虑了系统故障树的最小割集和基本组成单元两方面的因素,重点设计了故障树的诊断流程,改进了以往故障诊断中出现的重复诊断的缺陷,优化了诊断过程。以飞机液压子系统为平台,与其他方法进行比较,得出该方法可以保证结果的正确性,并能节省诊断时间。     

Modelling ellipsoidal uncertainty by multidimensional fuzzy sets

The paper addresses the problem of an aggregation of knowledge obtained from many sets of experiments each with different confidence level during identification of a linear dynamical system with uncertain parameters. The proposed approach describes the parameter space by a multidimensional fuzzy set specified by non-symmetric ellipsoidal α-cuts. An algorithm for determination of minimum confidence level guaranteeing stability of such a system is presented.     

Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis

The Pythagorean fuzzy set (PFS) which is an extension of intuitionistic fuzzy set, is more capable of expressing and handling the uncertainty under uncertain environments, so that it was broadly applied in a variety of fields. Whereas, how to measure PFSs’ distance appropriately is still an open issue. It is well known that the square root of Jensen–Shannon divergence is a true metric in the probability distribution space which is a useful measure of distance. On account of this point, a novel divergence measure between PFSs is proposed by taking advantage of the Jensen–Shannon divergence in this paper, called as PFSJS distance. This is the first work to consider the divergence of PFSs for measuring the discrepancy of data from the perspective of the relative entropy. The new PFSJS distance measure has some desirable merits, in which it meets the distance measurement axiom and can better indicate the discrimination degree of PFSs. Then, numerical examples demonstrate that the PFSJS distance can avoid generating counter-intuitive results which is more feasible, reasonable and superior than existing distance measures. Additionally, a new algorithm based on the PFSJS distance measure is designed to solve the problems of medical diagnosis. By comparing the different methods in the medical diagnosis application, it is found that the new algorithm is as efficient as the other methods. These results prove that the proposed method is practical in dealing with the medical diagnosis problems.     

Ranking fuzzy numbers in the setting of possibility theory

The arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood. Equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals, which is addressed in this paper. A complete set of comparison indices is proposed in the framework of Zadeh's possibility theory. It is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers. Moreover, this approach is related to previous ones, and its possible extension to the ranking of n fuzzy numbers is discussed at length. Finally, it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities.     

模糊规则挖掘及在裂解炉过程诊断中的应用

乙烯是重要的化工基础原料,乙烯裂解炉的安全平稳操作是提高乙烯产量的重要手段。首先利用小波分析对过程变量去噪,提出基于过程参数正态分布的模糊离散化方法,利用基于粒度熵的粗糙决策表最优约简方法、提取乙烯裂解过程中的模糊诊断规则。通过实际应用表明,提高了乙烯裂解炉的操作稳定性,为操作员及时发现过程的异常现象提供警示。     

A fuzzy temporal notation and its application to specify fault patterns for diagnosis

Many systems collect vast amounts of data over time, which is used to perform critical tasks like diagnosis, surveillance, resource management, planning and forecasting. To effectively use the historical data for these purposes, it is important to analyze the data and to gain insight into its significant aspects, by identifying the presence and characteristics of specific patterns. We describe a fuzzy logical notation, enhanced with facilities for expressing approximate temporal patterns, to build compositional and abstract models of syntactic structure of patterns. We present an algorithm, which detects where and how strongly the given pattern (i.e., a formula) is present. The approach is illustrated by specifying and detecting fault patterns for trace-based diagnosis of dynamic systems.     

动态模糊聚类及其在变压器故障诊断中的应用

理论分析和实践表明,电力变压器绝缘故障与油中特征气体组分含量及特征气体组分比值密切相关。提出一种基于遗传算法的动态模糊聚类算法,有效融合特征气体组分含量及组分比值两类故障信息,完成对变压器故障的动态聚类分析。该算法采用实数编码方案,染色体长度可变,不同的长度对应于不同的故障类别数;并采用一种新的适合于变长染色体的交叉和变异算子。与特征气体法、三比值法进行对比实验,表明该算法具有较高的判定正确率。     

Model-based fault diagnosis using fuzzy matching

A new fuzzy-model-based approach to fault detection and diagnosis is proposed. The scheme uses a set of fuzzy reference models which describe faulty and fault-free operation, and a classifier based on fuzzy matching for fault diagnosis. The reference models are obtained off-line from simulation data. A fuzzy model which describes the actual behavior of the plant is identified online from normal operating data and compared to each of the reference models. A degree of similarity is evaluated every time the online fuzzy model is identified. Dempster's rule of combination is used to combine new evidence with that already collected. The method of diagnosis accounts for any ambiguity (which may result from fault-free and faulty operation, or different faults, having similar symptoms at a given operating point) by comparing the fuzzy reference models with each other. Results are presented which demonstrate the effectiveness of the scheme when it is used to detect and identify faults in the cooling coil subsystem of the air-handling unit of both simulated and experimental air-conditioning plant     

Modelling uncertainty with generalized credal sets: application to conjunction and decision

To model conflict, non-specificity and contradiction in information, upper and lower generalized credal sets are introduced. Any upper generalized credal set is a convex subset of plausibility measures interpreted as lower probabilities whose bodies of evidence consist of singletons and a certain event. Analogously, contradiction is modelled in the theory of evidence by a belief function that is greater than zero at empty set. Based on generalized credal sets, we extend the conjunctive rule for contradictory sources of information, introduce constructions like natural extension in the theory of imprecise probabilities and show that the model of generalized credal sets coincides with the model of imprecise probabilities if the profile of a generalized credal set consists of probability measures. We give ways how the introduced model can be applied to decision problems.     

Modeling uncertainty in clinical diagnosis using fuzzy logic.

This paper describes a fuzzy approach to computer-aided medical diagnosis in a clinical context. It introduces a formal view of diagnosis in clinical settings and shows the relevance and possible uses of fuzzy cognitive maps. A constraint satisfaction method is introduced that uses the temporal uncertainty in symptom durations that may occur with particular diseases. The method results in an estimate of the stage of the disease if the temporal constraints of the disease in relation to the occurrence of the symptoms are satisfied. A lightweight fuzzy process is described and evaluated in the context of diagnosis of two confusable diseases. The process is based on the idea of an incremental simple additive model for fuzzy sets supporting and negating particular diseases. These are combined to produce an index of support for a particular disease. The process is developed to allow fuzzy symptom information on the intensity and duration of symptoms. Results are presented showing the effectiveness of the method for supporting differential diagnosis.     

减法聚类-ANFIS在网络故障诊断的应用研究

提出了一种基于减法聚类-自适应模糊神经网络（ANFIS）的网络故障诊断建模方法。减法聚类算法生成初始模糊推理系统,ANFIS建立网络故障诊断原始模型,应用混合算法对模糊规则的参数进行训练并建立最终的模型。仿真实验表明基于减法聚类-ANFIS的建模方法是有效的;通过仿真结果比较,减法聚类-ANFIS的网络故障诊断能力及收敛速度均优于BP神经网络,更适合作为网络故障诊断模型。     

Soft fuzzy rough sets and its application in decision making

Recently, the theory and applications of soft set has brought the attention by many scholars in various areas. Especially, the researches of the theory for combining the soft set with the other mathematical theory have been developed by many authors. In this paper, we propose a new concept of soft fuzzy rough set by combining the fuzzy soft set with the traditional fuzzy rough set. The soft fuzzy rough lower and upper approximation operators of any fuzzy subset in the parameter set were defined by the concept of the pseudo fuzzy binary relation (or pseudo fuzzy soft set) established in this paper. Meanwhile, several deformations of the soft fuzzy rough lower and upper approximations are also presented. Furthermore, we also discuss some basic properties of the approximation operators in detail. Subsequently, we give an approach to decision making problem based on soft fuzzy rough set model by analyzing the limitations and advantages in the existing literatures. The decision steps and the algorithm of the decision method were also given. The proposed approach can obtain a object decision result with the data information owned by the decision problem only. Finally, the validity of the decision methods is tested by an applied example.     

Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty

In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preference among feasible solutions. Moreover some constraints may have priority over others. Lastly, constraints may involve uncertain parameters. This paper advocates the use of fuzzy sets and possibility theory as a realistic approach for the representation of these three aspects. Fuzzy constraints encompass both preference relations among possible instantiations and priorities among constraints. In a Fuzzy Constraint Satisfaction Problem (FCSP), a constraint is satisfied to a degree (rather than satisfied or not satisfied) and the acceptability of a potential solution becomes a gradual notion. Even if the FCSP is partially inconsistent, best instantiations are provided owing to the relaxation of some constraints. Fuzzy constraints are thus flexible. CSP notions of consistency and k-consistency can be extended to this framework and the classical algorithms used in CSP resolution (e.g., tree search and filtering) can be adapted without losing much of their efficiency. Most classical theoretical results remain applicable to FCSPs. In the paper, various types of constraints are modelled in the same framework. The handling of uncertain parameters is carried out in the same setting because possibility theory can account for both preference and uncertainty. The presence of uncertain parameters leads to ill-defined CSPs, where the set of constraints which defines the problem is not precisely known.     

Some guidelines for fuzzy sets application in legal reasoning

As an introduction to our work, we emphasize the parallel interpretation of abstract tools and the concepts of undetermined and vague information. Imprecision, uncertainty and their relationships are inspected. Suitable interpretations of the fuzzy sets theory are applied to legal phenomena in an attempt to clearly circumscribe the possible applications of the theory. The fundamental notion of reference sets is examined in detail, hence highlighting their importance. A systematic and combinatorial classification of the relevant subsets of the legal field is supplied for practical application. Although the use of the fuzzy sets theory is sometimes suggested as a palliative measure (no competition exists), it can also be complementary (serve as a building block to improve modelisation). An Appendix gives a brief recall of the key-concepts of the axiomatic theory of fuzziness and its developments: fuzzy sets, fuzzy logic, fuzzy control and theory of possibility.