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.     

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.     

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.     

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。     

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     

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.     

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.     

Soft rough fuzzy sets and soft fuzzy rough sets

Fuzzy set theory, soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. Feng et al. introduced the notions of rough soft set, soft rough set and soft rough fuzzy set by combining fuzzy set, rough set and soft set all together. This paper is devoted to the further discussion of the combinations of fuzzy set, rough set and soft set. A new soft rough set model is proposed and its properties are derived. Furthermore, fuzzy soft set is employed to granulate the universe of discourse and a more general model called soft fuzzy rough set is established. The lower and upper approximation operators are presented and their related properties are surveyed.     

The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their application in multi-criteria decision-making

In this paper, the cross-entropy of intuitionistic hesitant fuzzy sets (IHFSs) is developed by integrating the cross-entropy of intuitionistic fuzzy sets (IFSs) and hesitant fuzzy sets (HFSs). First, several measurement formulae are discussed and their properties are studied. Then, two approaches, which are based on the developed intuitionistic hesitant fuzzy cross-entropy, are proposed for solving multi-criteria decision-making (MCDM) problems within an intuitionistic hesitant fuzzy environment. For both methods, an optimisation model is established in order to determine the weight vector for MCDM problems with incomplete information on criteria weights. Finally, an example is provided in order to illustrate the practicality and effectiveness of the proposed approaches.     

A novel Likert scale based on fuzzy sets theory

The Likert method is commonly used as a standard psychometric scale to measure responses. This measurement scale has a procedure that facilitates survey construction and administration, and data coding and analysis. However, there are some drawbacks in the Likert scaling. This paper addresses the information distortion and information lost arising from the closed-form scaling and the ordinal nature of this measurement method. To overcome these problems, a novel fuzzy Likert scale developed based on the fuzzy sets theory has been proposed. The major contribution of the fuzzy Likert approach is that it permits partial agreement of a scale point. By incorporating this capability into the measurement process, the new scale can capture the lost information and regulate the distorted information. A quantitative analysis based on the concept Consensus has proven that the new scale can provide a more accurate measurement. The implementation feasibility and the improved measurement performance of the fuzzy Likert scale have been demonstrated via a simulation study on a low birth weight analysis.     

Robustness of model-based fault diagnosis: Decisions with information-gap models of uncertainty

Fault diagnosis is analysed here as a decision between alternative hypotheses, based on uncertain evidence. W e consider severe lack of information, and perceive the uncertainty as an information gap between what is known, and what needs to be known for a perfect decision. This uncertainty is quantified with info-gap models of uncertainty, which require less information than probabilistic models. Previous work with convex set-models is extended to linear info-gap models which are not necessarily convex, as well as to more general info-gap models with arbitrary expansion properties. We define a decision algorithm based on info-gap models and prove three theorems, one establishing the connection with the earlier work on convex models, the other two showing that the algorithm is maximally robust for linear info-gap models as well as for general infogap models of uncertainty. An illustrative example is presented which shows how these results can be used for optimizing the design of a model-based fault diagnosis algorithm.     

A new correlation coefficient of mass function in evidence theory and its application in fault diagnosis

Applied Intelligence - Evidence theory as a generalization of probability theory has an advantage of handling uncertainty. The mass function, also named as basic probability assignment (BPA), plays...     

A novel fuzzy linear regression model based on a non-equality possibility index and optimum uncertainty

Various kinds of fuzzy regression models are introduced in the literature and many different methods are proposed to estimate fuzzy parameters of the models. In this study, a new approach is introduced to find the parameters of a linear fuzzy regression, with fuzzy outputs, the input data of which is measured by crisp numbers. Based on a non-equality possibility index, a new objective function is designed and solved, by which a minimum degree of acceptable uncertainty (the h-level or h-cut) is found. Four numerical examples are presented to compare the proposed approach with some other methods. Results show superiority of the new approach based on the criterion used by Kim and Bishu in the cases studied here. A realistic application of the proposed method is also presented, by which the total energy consumption of the Residential-Commercial sector in Iran is modeled using three variables of the GDP, number of the Households and an Energy Price index as inputs (exogenous variables) to the model.