Showing papers on "Membership function published in 2002"
TL;DR: Establishing a small set of terms that let us easily communicate about type-2 fuzzy sets and also let us define such sets very precisely, and presenting a new representation for type- 2 fuzzy sets, and using this new representation to derive formulas for union, intersection and complement of type-1 fuzzy sets without having to use the Extension Principle.
Abstract: Type-2 fuzzy sets let us model and minimize the effects of uncertainties in rule-base fuzzy logic systems. However, they are difficult to understand for a variety of reasons which we enunciate. In this paper, we strive to overcome the difficulties by: (1) establishing a small set of terms that let us easily communicate about type-2 fuzzy sets and also let us define such sets very precisely, (2) presenting a new representation for type-2 fuzzy sets, and (3) using this new representation to derive formulas for union, intersection and complement of type-2 fuzzy sets without having to use the Extension Principle.
2,382 citations
TL;DR: This paper applies a fuzzy membership to each input point and reformulate the SVMs such that different input points can make different contributions to the learning of decision surface.
Abstract: A support vector machine (SVM) learns the decision surface from two distinct classes of the input points. In many applications, each input point may not be fully assigned to one of these two classes. In this paper, we apply a fuzzy membership to each input point and reformulate the SVMs such that different input points can make different contributions to the learning of decision surface. We call the proposed method fuzzy SVMs (FSVMs).
1,374 citations
TL;DR: This paper defines a broad family of fuzzy rough sets, each one of which, called an (I, J)-fuzzy rough set, is determined by an implicator I and a triangular norm J.
911 citations
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TL;DR: The complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number, and a major part of this work is dedicated to a discussion of the intuitive interpretation of complex-valued grades of membership.
Abstract: The objective of this paper is to investigate the innovative concept of complex fuzzy sets. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0, 1], but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. The inherent difficulty in acquiring intuition for the concept of complex-valued membership presents a significant obstacle to the realization of its full potential. Consequently, a major part of this work is dedicated to a discussion of the intuitive interpretation of complex-valued grades of membership. Examples of possible applications, which demonstrate the new concept, include a complex fuzzy representation of solar activity (via measurements of the sunspot number), and a signal processing application. A comprehensive study of the mathematical properties of the complex fuzzy set is presented. Basic set theoretic operations on complex fuzzy sets, such as complex fuzzy complement, union, and intersection, are discussed at length. Two novel operations, namely set rotation and set reflection, are introduced. Complex fuzzy relations are also considered. Index Terms-Complex fuzzy intersection, complex fuzzy relations, complex fuzzy sets, complex fuzzy union, complex-valued grades of membership, fuzzy complex numbers.
579 citations
04 Nov 2002
TL;DR: It is shown that the k-harmonic means method is superior on simple low-dimensional synthetic datasets and image segmentation tasks, and that having a soft membership function is essential for finding high-quality clusterings, but having a non-constant data weight function is useful also.
Abstract: We investigate here the behavior of the standard k-means clustering algorithm and several alternatives to it: the k-harmonic means algorithm due to Zhang and colleagues, fuzzy k-means, Gaussian expectation-maximization, and two new variants of k-harmonic means. Our aim is to find which aspects of these algorithms contribute to finding good clusterings, as opposed to converging to a low-quality local optimum. We describe each algorithm in a unified framework that introduces separate cluster membership and data weight functions. We then show that the algorithms do behave very differently from each other on simple low-dimensional synthetic datasets and image segmentation tasks, and that the k-harmonic means method is superior. Having a soft membership function is essential for finding high-quality clusterings, but having a non-constant data weight function is useful also.
371 citations
TL;DR: This work presents a fuzzy TOPSIS model under group decisions for solving the facility location selection problem, where the ratings of various alternative locations under different subjective attributes and the importance weights of all attributes are assessed in linguistic values represented by fuzzy numbers.
Abstract: This work presents a fuzzy TOPSIS model under group decisions for solving the facility location selection problem, where the ratings of various alternative locations under different subjective attributes and the importance weights of all attributes are assessed in linguistic values represented by fuzzy numbers. The objective attributes are transformed into dimensionless indices to ensure compatibility with the linguistic ratings of the subjective attributes. Furthermore, the membership function of the aggregation of the ratings and weights for each alternative location versus each attribute can be developed by interval arithmetic and α -cuts of fuzzy numbers. The ranking method of the mean of the integral values is applied to help derive the ideal and negative-ideal fuzzy solutions to complete the proposed fuzzy TOPSIS model. Finally, a numerical example demonstrates the computational process of the proposed model.
290 citations
TL;DR: A fuzzy TOPSIS approach for selecting plant location is proposed, where the ratings of various alternative locations under various criteria and the weights of various criteria are assessed in linguistic terms represented by fuzzy numbers.
Abstract: A fuzzy TOPSIS approach for selecting plant location is proposed, where the ratings of various alternative locations under various criteria and the weights of various criteria are assessed in linguistic terms represented by fuzzy numbers. In the proposed method, the ratings and weights assigned by decision makers are averaged and normalised into a comparable scale. The membership function of each normalised weighted rating can be developed by interval arithmetic of fuzzy numbers. To avoid complicated aggregation of fuzzy numbers, these normalised weighted ratings are defuzzified into crisp values. A closeness coefficient is defined to determine the ranking order of alternative locations by calculating the distances to both the ideal and negative-ideal solutions. Using the suggested method, the decision makers' fuzzy assessments with different rating viewpoints and the trade-off among different criteria are considered in the aggregation procedure to assure more convincing decision making. A numerical example demonstrates the feasibility of the proposed method.
275 citations
05 Nov 2002
TL;DR: Different types of membership functions are evaluated in the fuzzy control of an induction motor drive using triangular, trapezoidal, Gaussian, bell, sigmoidal and polynomial types.
Abstract: In this paper, different types of membership functions are evaluated in the fuzzy control of an induction motor drive The general membership functions under consideration are triangular, trapezoidal, Gaussian, bell, sigmoidal and polynomial types In the beginning, fuzzy controller sensitivity has been analyzed and compared for different membership functions with the triangular function as the base The performance of fuzzy control is then evaluated with each type of membership function for a speed-controlled induction motor drive with indirect vector control in the inner loop
268 citations
TL;DR: In the new millennium more and more researchers will attempt to capture Type 2 representation and develop reasoning with Type 2 formulas that reveal the rich information content available in information granules, as well as expose the risk associated with the graded representation of words and computing with words.
219 citations
TL;DR: An improved fuzzy TOPSIS model is suggested, where membership functions for the weighted normalized fuzzy ratings are presented and a simple method is also proposed for ranking fuzzy numbers with mean of relative areas.
Abstract: Chen [2] extended the TOPSIS to a fuzzy environment. In his work, a vertex method was proposed to measure the distance between two given triangular fuzzy numbers. He further applied the vertex method to measure the distance between the weighted normalized fuzzy ratings and the fuzzy positive (negative}-ideal solutions to complete the fuzzy TOPSIS model. Despite the merits of his work, this application is not reasonable. Because the weighted normalized fuzzy ratings are truly not triangular fuzzy numbers. To overcome the above shortcomings, we suggest an improved fuzzy TOPSIS model, where membership functions for the weighted normalized fuzzy ratings are presented. A simple method is also proposed for ranking fuzzy numbers with mean of relative areas. This ranking method is further applied to establish the proposed model. Illustrative examples demonstrate the merits of the proposed ranking method and the feasibility of the improved fuzzy TOPSIS model, respectively.
171 citations
TL;DR: The theoretical framework of FDT in continuous space is extended to digital cubic spaces and it is shown that for any fuzzy digital object, fuzzy distance is a metric for the support of the object.
TL;DR: An equivalent multi objective linear programming form of the problem has been formulated in the proposed methodology using fuzzy set theory approach and the proposed solution procedure has been used to solve numerical examples.
Proceedings Article•
01 Jan 2002
TL;DR: The generalization ability of the fuzzy support vector machine is the same with or better than that of the support vectors machine for pair- wise classification.
Abstract: Since support vector machines for pattern classification are based on two-class classification problems, unclassifiable regions ex- ist when extended to n(> 2)-class problems. In our previous work, to solve this problem, we developed fuzzy support vector machines for one- to-(n −1) classification. In this paper, we extend our method to pairwise classification. Namely, using the decision functions obtained by training the support vector machines for classes i and j (j �= i,j =1 ,...,n), for class i we define a truncated polyhedral pyramidal membership function. The membership functions are defined so that, for the data in the classi- fiable regions, the classification results are the same for the two methods. Thus, the generalization ability of the fuzzy support vector machine is the same with or better than that of the support vector machine for pair- wise classification. We evaluate our method for four benchmark data sets and demonstrate the superiority of our method.
TL;DR: It is demonstrated that the Kalman filter can be an effective tool for improving the performance of a fuzzy system and is compared with gradient descent and adaptive neuro-fuzzy inference system (ANFIS) based optimization of fuzzy membership functions.
TL;DR: In this paper, an equivalence class of probability distributions compatible with available data and corresponding upper and lower cumulative density functions are used to define the membership functions of fuzzy numbers starting from available information is also described.
TL;DR: A taxonomy of fuzzy graphs is presented that treats fuzziness in vertex existence, edgeexistence, edge connectivity, and edge weight, and provides algorithmic solutions to some standard graph-theoretic problems for fuzzy graphs.
TL;DR: A decision-based fusion system based on the uncertainty approach utilizing an extension of the Choquet fuzzy integral, GCFI, which shows a huge improvement in the probability of detection and a reduction in the false alarm rate over the best algorithm and two numeric fusion schemes.
TL;DR: This paper provides a spectrum of random fuzzy dependent-chance programming in which the underlying philosophy is based on selecting the decision with maximal chance to meet the event and trains a neural network to approximate chance functions based on the training data generated by the random fuzzy simulation.
TL;DR: The correlation coefficient calculated in this paper is a fuzzy number, rather than a crisp value, based on Zadeh's extension principle, and a pair of nonlinear programs are formulated to find the α-cut of the fuzzy correlation coefficient.
TL;DR: The proposed weighted fuzzy reasoning algorithm can allow the rule-based systems to perform fuzzy reasoning in a more flexible and more intelligent manner.
Abstract: This paper presents a Weighted Fuzzy Petri Net model (WFPN) and proposes a weighted fuzzy reasoning algorithm for rule-based systems based on Weighted Fuzzy Petri Nets. The fuzzy production rules in the knowledge base of a rule-based system are modeled by Weighted Fuzzy Petri Nets, where the truth values of the propositions appearing in the fuzzy production rules and the certainty factors of the rules are represented by fuzzy numbers. Furthermore, the weights of the propositions appearing in the rules are also represented by fuzzy numbers. The proposed weighted fuzzy reasoning algorithm can allow the rule-based systems to perform fuzzy reasoning in a more flexible and more intelligent manner.
01 Aug 2002
TL;DR: This paper introduces a new learning methodology to quickly generate accurate and simple linguistic fuzzy models: the cooperative rules (COR) methodology, which acts on the consequents of the fuzzy rules to find those that are best cooperating.
Abstract: This paper introduces a new learning methodology to quickly generate accurate and simple linguistic fuzzy models: the cooperative rules (COR) methodology. It acts on the consequents of the fuzzy rules to find those that are best cooperating. Instead of selecting the consequent with the highest performance in each fuzzy input subspace, as ad-hoc data-driven methods usually do, the COR methodology considers the possibility of using another consequent, different from the best one, when it allows the fuzzy model to be more accurate thanks to having a rule set with the best cooperation. Our proposal has shown good results in solving three different applications when compared to other methods.
TL;DR: Two estimation methods along with a fuzzy least-squares approach are proposed that can be effectively used for the parameter estimation of fuzzy regression models.
TL;DR: New forms of fuzzy membership curves are introduced, designed to consider the uncertainty range that represents the degree of uncertainties involved in both probabilistic parameter estimates and subjective judgments.
TL;DR: It is shown that the first model reduces to the classical newsboy problem, and therefore an optimal solution is easily available, and the second model shows that the objective function is concave and hence present a characterization of the optimal solution.
01 Feb 2002
TL;DR: It is shown that a large class of well-established types of uncertainty representations can be modeled within this framework and its use as a unifying structure for modeling knowledge about an uncertain variable is discussed.
Abstract: We introduce the fuzzy measure and discuss its use as a unifying structure for modeling knowledge about an uncertain variable. We show that a large class of well-established types of uncertainty representations can be modeled within this framework. A view of the Dempster-Shafer (D-S) belief structure as an uncertainty representation corresponding to a set of possible fuzzy measures is discussed. A methodology for generating this set of fuzzy measures from a belief structure is described. A measure of entropy associated with a fuzzy measure is introduced and its manifestation for different fuzzy measures is described. The problem of uncertain decision making for the case in which the uncertainty represented by a fuzzy measure is considered. The Choquet integral is introduced as providing a generalization of the expected value to this environment.
TL;DR: A fuzzy multipurpose decision making model integrating different preference representations: preference orderings, utility functions and fuzzy preference relations is presented and its internal consistency is studied.
TL;DR: Three sets of membership functions, with different degrees of fuzziness, are proposed for fuzzy approaches, based on fuzzy and probability theory, for constructing control charts for linguistic data.
Abstract: In this article, different procedures of constructing control charts for linguistic data, based on fuzzy and probability theory, are discussed. Three sets of membership functions, with different degrees of fuzziness, are proposed for fuzzy approaches. A comparison between fuzzy and probability approaches, based on the Average Run Length and samples under control, is conducted for real data. Contrary to the conclusions of Raz and Wang (1990b) the choice of degree of fuzziness affected the sensitivity of control charts.
TL;DR: A generalization of the concept of symmetric fuzzy measure based in a decomposition of the universal set in what is called subsets of indifference, which is based on the Choquet integral.
Abstract: In this paper we propose a generalization of the concept of symmetric fuzzy measure based in a decomposition of the universal set in what we have called subsets of indifference. Some properties of these measures are studied, as well as their Choquet integral. Finally, a degree of interaction between the subsets of indifference is defined.
TL;DR: This paper presents sufficient conditions on the parameters of the Takagi-Sugeno-Kang (TSK) fuzzy system under which the output of the TSK fuzzy system becomes monotonic with respect to its input.
TL;DR: The sum normal constraint is applied in this paper to both gradient descent optimization and Kalman filter optimization of fuzzy membership functions.
Abstract: Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a certain shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a small number of variables and the membership optimization problem can be reduced to a parameter optimization problem. This is the approach that is typically taken, but it results in membership functions that are not (in general) sum normal. That is, the resulting membership function values do not add up to one at each point in the domain. This optimization approach is modified in this paper so that the resulting membership functions are sum normal. Sum normality is desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The sum normal constraint is applied in this paper to both gradient descent optimization and Kalman filter optimization of fuzzy membership functions. The methods are illustrated on a fuzzy automotive cruise controller.