The influence of some fuzzy implication operators on the accuracy of a fuzzy model-part II

The influence of fuzzy implication operators and the connective Also on the accuracy of a fuzzy model of a d.c. series motor is considered. Some typical fuzzy implication operators are applied to the construction of a fuzzy model of a d.c. series motor. A root-mean-square error is used as the criterion of the fuzzy model's adequacy to the real system. A number of mathematical operations necessary for the implementation of the fuzzy model are used as the criterion by which the fuzzy model's applicability if estimated from the point of view of computing techniques. The best types of fuzzy relations, representing fuzzy models of a real system, are chosen in order to secure the least root-mean-square error with minimal number of mathematical operations necessary for computer implementation.     

Generalizing the modeling of fuzzy logic controllers by parameterized aggregation operators

We describe the type of reasoning used in the typical fuzzy logic controller, the Mamdani reasoning method. We point out the basic assumptions in this model. We discuss the S-OWA operators which provide families of parameterized “andlike” and “orlike” operators. We generalize the Mamdani model by introducing these operators. We introduce a method, which we call Direct Fuzzy Reasoning (DFR), which results from one choice of the parameters. We develop some learning algorithms for the new method. We show how the Takagi-Sugeno-Kang (TSK) method of reasoning is an example of this DFR method.     

Modeling holistic fuzzy implication using co-copulas

Two related aggregation operators called copulas and co-copulas are introduced and various properties are described. The relationship, of these operators to t-norms and t-conorms is noted. Generalizations of these, respectively, called conjunctors and disjunctors, are introduced. We suggest the use of disjunctor operators for modeling the multi-valued implication operator in fuzzy logic. We point out that the selection of operators used in fuzzy logic, in addition to having appropriate pointwise properties, should be holistic, this requires consideration of the nature of the resulting fuzzy set as a whole. Focusing on the protoform of fuzzy modus ponens and looking at the information contained in the inferred fuzzy set we show that the use of co-copulas has some desirable properties. Taking advantage of the fact that the weighted sum of co-copulas is a co-copula we consider the problem of constructing customized implication operators.     

Two fuzzier implication operators in the theory of fuzzy power sets

The theory of fuzzy power sets requires the use of an implication operator acting within the set of values taken by the membership functions of the fuzzy sets. Two such operators and resulting relationships between fuzzy sets are studied here, and the results compared with previous ones obtained with other implication operators.     

A new class of fuzzy implications. Axioms of fuzzy implication revisited

Many different fuzzy implication operators have been proposed; most of them fit into one of the two classes: implication operations that are based on an explicit representation of implication A → B in terms of &, , and ¬ (e.g., S-implications that are based on the formula B ¬ A), and R-implications that are based on an implicit representation of implication A → B as the weakest C for which C&B implies A. However, some fuzzy implication operations (such as b^a) cannot be naturally represented in this form. To describe such operations, we propose a new (third) class of implication operations called A-implications whose relation to &, , and ¬ is described by (implicit) axioms.     

Fuzzy power sets and fuzzy implication operators

The theory of fuzzy power sets, which has hitherto been insufficiently developed, is shown very naturally to require the use of a fuzzy implication operator (Section 1). Six such operators are gathered from the literature on multiple-valued logic (Section 2), and their effects on fuzzy power-set theory are compared throughout the rest of the paper. After certain fundamental definitions of set characteristics (Section 3), the six operators are carried in parallel while working out basic aspects of power-set theory. Among these are the properties of the set-inclusion relation and the set-equivalence relation (Section 4), two distinct concepts of disjointness (Section 5), questions of consistency in the relations between a set and its complement (Section 6), and a very concrete theorem on a difference among the operators with regard to the derivation of crisp conclusions from fuzzy premises (Section 7). Finally (Section 8), emphasis is placed on the dependence of the choice of operators upon the purposes the user has in hand.     

Implication in fuzzy logic

Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.     

Symmetric implicational method of fuzzy reasoning

Fuzzy reasoning should take into account the factors of both the logic system and the reasoning model, thus a new fuzzy reasoning method called the symmetric implicational method is proposed, which contains the full implication inference method as its particular case. The previous full implication inference principles are improved, and unified forms of the new method are respectively established for FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) to let different fuzzy implications be used under the same way. Furthermore, reversibility properties of the new method are analyzed from some conditions that many fuzzy implications satisfy, and it is found that its reversibility properties seem fine. Lastly, the more general α-symmetric implicational method is put forward, and its unified forms are achieved.     

Forecasting portfolio returns using weighted fuzzy time series methods

We propose using weighted fuzzy time series (FTS) methods to forecast the future performance of returns on portfolios. We model the uncertain parameters of the fuzzy portfolio selection models using a possibilistic interval-valued mean approach, and approximate the uncertain future return on a given portfolio by means of a trapezoidal fuzzy number. Introducing some modifications into the classical models of fuzzy time series, based on weighted operators, enables us to generate trapezoidal numbers as forecasts of the future performance of the portfolio returns. This fuzzy forecast makes it possible to approximate both the expected return and the risk of the investment through the value and ambiguity of a fuzzy number.We incorporate our proposals into classical fuzzy time series methods and analyze their effectiveness compared with classical weighted fuzzy time series models, using historical returns on assets from the Spanish stock market. When our weighted FTS proposals are used to point-wise forecast portfolio returns the one-step ahead accuracy is improved, also with respect to non-fuzzy forecasting methods.     

Hesitant fuzzy information aggregation in decision making

As a generalization of fuzzy set, hesitant fuzzy set is a very useful tool in situations where there are some difficulties in determining the membership of an element to a set caused by a doubt between a few different values. The aim of this paper is to develop a series of aggregation operators for hesitant fuzzy information. We first discuss the relationship between intutionistic fuzzy set and hesitant fuzzy set, based on which we develop some operations and aggregation operators for hesitant fuzzy elements. The correlations among the aggregation operators are further discussed. Finally, we give their application in solving decision making problems.     

模糊推理的变权综合算法及其构造的模糊系统响应能力分析

将变权综合原理应用于模糊推理,提出一种新的模糊推理算法-变权综合推理算法,并给出利用模糊蕴涵算子确定规则变权的方法。证明了由一些正常蕴涵算子如Lukasiewicz蕴涵、Goguen蕴涵、Godel蕴涵、Dubois-Prade蕴涵等确定的变权综合算法是相容的。进一步分析了该变权综合算法构造的模糊系统的响应能力,结果表明:基于这些正常蕴涵算子的变权综合算法构造的模糊系统具有泛逼近性。     

Multi-criteria decision-making methods based on intuitionistic fuzzy sets

In this paper we present new methods for solving multi-criteria decision-making problem in an intuitionistic fuzzy environment. First, we define an evaluation function for the decision-making problem to measure the degrees to which alternatives satisfy and do not satisfy the decision-maker’s requirement. Then, we introduce and discuss the concept of intuitionistic fuzzy point operators. By using the intuitionistic fuzzy point operators, we can reduce the degree of uncertainty of the elements in a universe corresponding to an intuitionistic fuzzy set. Furthermore, a series of new score functions are defined for multi-criteria decision-making problem based on the intuitionistic fuzzy point operators and the evaluation function and their effectiveness and advantage are illustrated by examples.     

Application of fuzzy implication to probe nonsymmetric relations: part I

Nine fuzzy implication operators were studied to determine their ability to identify nonsymmetric relations. The fuzzy degrees of the antecedent and consequent propositions of each implication were evaluated using the cumulative distribution function, which has the property that fuzzy degrees are uniformly distributed between 0 and 1 and hence are as fuzzy as possible.Under these conditions, average implications from the nine operators are confined to small subintervals of [0, 1]. For eight out of nine implication operators, the average implication largely duplicates information from the correlation coefficient. These points are explained theoretically.Confidence limits for fuzzy implication are also discussed.     

Theoretical methods of constructing fuzzy inference relations

In this paper, a theoretical method is presented to select fuzzy implication operators for the fuzzy inference sentence “if x is A, then y is B”. By applying representation theorems, thirty-two fuzzy implication operators are obtained. It is shown that the obtained operators are generalizations of classical inference rule A → B, A ^c → B, A → B ^c and A ^c → B ^c respectively and can be divided into four classes. By discussion, it is found that thirty of them among 420 fuzzy implication operators presented by Li can be derived by applying representation theorems and another two new ones are obtained by the use of our methods.     

A neural fuzzy control system with structure and parameter learning

A general connectionist model, called neural fuzzy control network (NFCN), is proposed for the realization of a fuzzy logic control system. The proposed NFCN is a feedforward multilayered network which integrates the basic elements and functions of a traditional fuzzy logic controller into a connectionist structure which has distributed learning abilities. The NFCN can be constructed from supervised training examples by machine learning techniques, and the connectionist structure can be trained to develop fuzzy logic rules and find membership functions. Associated with the NFCN is a two-phase hybrid learning algorithm which utilizes unsupervised learning schemes for structure learning and the backpropagation learning scheme for parameter learning. By combining both unsupervised and supervised learning schemes, the learning speed converges much faster than the original backpropagation algorithm. The two-phase hybrid learning algorithm requires exact supervised training data for learning. In some real-time applications, exact training data may be expensive or even impossible to obtain. To solve this problem, a reinforcement neural fuzzy control network (RNFCN) is further proposed. The RNFCN is constructed by integrating two NFCNs, one functioning as a fuzzy predictor and the other as a fuzzy controller. By combining a proposed on-line supervised structure-parameter learning technique, the temporal difference prediction method, and the stochastic exploratory algorithm, a reinforcement learning algorithm is proposed, which can construct a RNFCN automatically and dynamically through a reward-penalty signal (i.e., “good” or “bad” signal). Two examples are presented to illustrate the performance and applicability of the proposed models and learning algorithms.     

Fuzzy implication operators for difference operations for fuzzy sets and cardinality-based measures of comparison

In this paper, we determine by means of fuzzy implication operators, two classes of difference operations for fuzzy sets and two classes of symmetric difference operations for fuzzy sets which preserve properties of the classical difference operation for crisp sets and the classical symmetric difference operation for crisp sets respectively. The obtained operations allow us to construct as in [B. De Baets, H. De Meyer, Transitivity-preserving fuzzification schemes for cardinality-based similarity measures, European Journal of Operational Research 160 (2005) 726–740], cardinality-based similarity measures which are reflexive, symmetric and transitive fuzzy relations and, to propose two classes of distances (metrics) which are fuzzy versions of the well-known distance of cardinality of the symmetric difference of crisp sets.     

Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval

The problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well-known metric, which preserves the expected interval of the fuzzy number, is completely solved. The previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number. Properties of this new trapezoidal approximation operator are studied.     

模糊推理三Ⅰ算法的还原性

模糊推理算法的还原性是判断蕴涵算子与推理方法配合效果的一个重要标准,只有蕴涵算子与推理方法搭配适当,才能使模糊推理有一个好的效果。本文对模糊推理三I算法具备还原性的条件进行了研究。首先,当与蕴涵算子相伴随的三角模为连续三角模时,给出了FM P问题三I算法具有还原性的充要条件;其次,当蕴涵算子为连续的正则蕴涵算子时,给出了FM T问题的三I算法具有还原性的充要条件;最后,当正则蕴涵算子关于补运算满足对合律时,给出了FM T问题三I算法满足还原性的一个充分条件。     

模糊信息度量及推理中蕴涵运算的信息度约束

根据胡固定教授经典集信息量定义给出一种新的模糊糊信息量度量方法,研究了其性质,并与Yager的特征测度进行了比较,表明该潮度是合理的.提出了模糊推理中蕴涵运算的信息度约束,给出了基于合成算法的模糊推理中,模糊蕴涵运算满足该信息度约束的充分条件.     

Intuitionistic fuzzy implication operators: Expressions and properties

The expressions of 32 fuzzy coimplication operators(FCO) and 32 intuitionistic fuzzy implication operators(IFIO) are given in this paper. The Co-D-P properties which the FCOs should satisfy are presented. The FCOs and IFIOs’situation of satisfying the properties which they should satisfy, respectively, are discussed in details.