Showing papers on "Membership function published in 2001"
Book•
01 Jan 2001
TL;DR: This chapter discusses Type-2 Fuzzy Sets, a New Direction for FLSs, and Relations and Compositions on different Product Spaces on Different Product Spaces, as well as operations on and Properties of Type-1 Non-Singleton Type- 2 FuzzY Sets.
Abstract: (NOTE: Each chapter concludes with Exercises.) I: PRELIMINARIES. 1. Introduction. Rule-Based FLSs. A New Direction for FLSs. New Concepts and Their Historical Background. Fundamental Design Requirement. The Flow of Uncertainties. Existing Literature on Type-2 Fuzzy Sets. Coverage. Applicability Outside of Rule-Based FLSs. Computation. Supplementary Material: Short Primers on Fuzzy Sets and Fuzzy Logic. Primer on Fuzzy Sets. Primer on FL. Remarks. 2. Sources of Uncertainty. Uncertainties in a FLS. Words Mean Different Things to Different People. 3. Membership Functions and Uncertainty. Introduction. Type-1 Membership Functions. Type-2 Membership Functions. Returning to Linguistic Labels. Multivariable Membership Functions. Computation. 4. Case Studies. Introduction. Forecasting of Time-Series. Knowledge Mining Using Surveys. II: TYPE-1 FUZZY LOGIC SYSTEMS. 5. Singleton Type-1 Fuzzy Logic Systems: No Uncertainties. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Defuzzification. Possibilities. Fuzzy Basis Functions. FLSs Are Universal Approximators. Designing FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. A Final Remark. Computation. 6. Non-Singleton Type-1 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Possibilities. FBFs. Non-Singleton FLSs Are Universal Approximators. Designing Non-Singleton FLSs. Case Study: Forecasting of Time-Series. A Final Remark. Computation. III: TYPE-2 FUZZY SETS. 7. Operations on and Properties of Type-2 Fuzzy Sets. Introduction. Extension Principle. Operations on General Type-2 Fuzzy Sets. Operations on Interval Type-2 Fuzzy Sets. Summary of Operations. Properties of Type-2 Fuzzy Sets. Computation. 8. Type-2 Relations and Compositions. Introduction. Relations in General. Relations and Compositions on the Same Product Space. Relations and Compositions on Different Product Spaces. Composition of a Set with a Relation. Cartesian Product of Fuzzy Sets. Implications. 9. Centroid of a Type-2 Fuzzy Set: Type-Reduction. Introduction. General Results for the Centroid. Generalized Centroid for Interval Type-2 Fuzzy Sets. Centroid of an Interval Type-2 Fuzzy Set. Type-Reduction: General Results. Type-Reduction: Interval Sets. Concluding Remark. Computation. IV: TYPE-2 FUZZY LOGIC SYSTEMS. 10. Singleton Type-2 Fuzzy Logic Systems. Introduction. Rules. Fuzzy Inference Engine. Fuzzification and Its Effect on Inference. Type-Reduction. Defuzzification. Possibilities. FBFs: The Lack Thereof. Interval Type-2 FLSs. Designing Interval Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Case Study: Knowledge Mining Using Surveys. Computation. 11. Type-1 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-1 Non-Singleton Type-2 FLSs. Designing Interval Type-1 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Final Remark. Computation. 12. Type-2 Non-Singleton Type-2 Fuzzy Logic Systems. Introduction. Fuzzification and Its Effect on Inference. Interval Type-2 Non-Singleton Type-2 FLSs. Designing Interval Type-2 Non-Singleton Type-2 FLSs. Case Study: Forecasting of Time-Series. Computation. 13. TSK Fuzzy Logic Systems. Introduction. Type-1 TSK FLSs. Type-2 TSK FLSs. Example: Forecasting of Compressed Video Traffic. Final Remark. Computation. 14. Epilogue. Introduction. Type-2 Versus Type-1 FLSs. Appropriate Applications for a Type-2 FLS. Rule-Based Classification of Video Traffic. Equalization of Time-Varying Non-linear Digital Communication Channels. Overcoming CCI and ISI for Digital Communication Channels. Connection Admission Control for ATM Networks. Potential Application Areas for a Type-2 FLS. A. Join, Meet, and Negation Operations For Non-Interval Type-2 Fuzzy Sets. Introduction. Join Under Minimum or Product t-Norms. Meet Under Minimum t-Norm. Meet Under Product t-Norm. Negation. Computation. B. Properties of Type-1 and Type-2 Fuzzy Sets. Introduction. Type-1 Fuzzy Sets. Type-2 Fuzzy Sets. C. Computation. Type-1 FLSs. General Type-2 FLSs. Interval Type-2 FLSs. References. Index.
2,555 citations
TL;DR: The centroid and generalized centroid of a type-2 fuzzy set are introduced, and how to compute them is explained, and examples are given that compare the exact computational results with the approximate results.
1,141 citations
TL;DR: This paper studies the Sanchez's approach for medical diagnosis and extends this concept with the notion of intuitionistic fuzzy set theory (which is a generalization of fuzzySet theory).
848 citations
TL;DR: The paper analyzes the main methods for automatic rule generation and structure optimization and grouped them into several families and compared according to the rule interpretability criterion.
Abstract: Fuzzy inference systems (FIS) are widely used for process simulation or control. They can be designed either from expert knowledge or from data. For complex systems, FIS based on expert knowledge only may suffer from a loss of accuracy. This is the main incentive for using fuzzy rules inferred from data. Designing a FIS from data can be decomposed into two main phases: automatic rule generation and system optimization. Rule generation leads to a basic system with a given space partitioning and the corresponding set of rules. System optimization can be done at various levels. Variable selection can be an overall selection or it can be managed rule by rule. Rule base optimization aims to select the most useful rules and to optimize rule conclusions. Space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters. Structure optimization is of a major importance: selecting variables, reducing the rule base and optimizing the number of fuzzy sets. Over the years, many methods have become available for designing FIS from data. Their efficiency is usually characterized by a numerical performance index. However, for human-computer cooperation another criterion is needed: the rule interpretability. An implicit assumption states that fuzzy rules are by nature easy to be interpreted. This could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts. The paper analyzes the main methods for automatic rule generation and structure optimization. They are grouped into several families and compared according to the rule interpretability criterion. For this purpose, three conditions for a set of rules to be interpretable are defined.
709 citations
TL;DR: All this is needed to implement a type-2 fuzzy logic system (FLS) is discussed, including join and meet under minimum/product t-norm, algebraic operations, properties of membership grades oftype-2 sets, andType-2 relations and their compositions.
700 citations
TL;DR: In this article, a genetic algorithm was used to automatically learn the knowledge base by finding an appropiate data base by means of a GA while using a simple generation method to derive the rule base.
Abstract: A method is proposed to automatically learn the knowledge base by finding an appropiate data base by means of a genetic algorithm while using a simple generation method to derive the rule base. Our genetic process learns the number of linguistic terms per variable and the membership function parameters that define their semantics, while a rule base generation method learns the number of rules and their composition.
281 citations
TL;DR: The fuzzy models developed in this paper provide a novel approach to support decisions regarding sustainable development based on context-dependent economic, ecological, and societal sustainability indicators.
230 citations
15 Jul 2001
TL;DR: Using the decision functions obtained by training the SVM, for each class, a truncated polyhedral pyramidal membership function is defined and, for the data in the classifiable regions, the classification results are the same for the two methods.
Abstract: In conventional support vector machines (SVMs), an n-class problem is converted into n two-class problems. For the ith two-class problem we determine the optimal decision function which separates class i from the remaining classes. In classification, a datum is classified into class i only when the value of the ith decision function is positive. In this architecture, the datum is unclassifiable if the values of more than one decision function are positive or all the valves are negative. In the paper, to overcome this problem, we propose fuzzy support vector machines (FSVMs). Using the decision functions obtained by training the SVM, for each class, we define a truncated polyhedral pyramidal membership function. Since, for the data in the classifiable regions, the classification results are the same for the two methods, the generalization ability of the FSVM is the same as or better than that of the SVM. We evaluate our method for three benchmark data sets and demonstrate the superiority of the FSVM over the SVM.
229 citations
TL;DR: This paper proposes a new adaptive control method in an effort to tune all the RBF parameters thereby reducing the approximation error and improving control performance.
Abstract: Recently, through the use of parameterized fuzzy approximators, various adaptive fuzzy control schemes have been developed to deal with nonlinear systems whose dynamics are poorly understood. An important class of parameterized fuzzy approximators is constructed using radial basis function (RBF) as a membership function. However, some tuneable parameters in RBF appear nonlinearly and the determination of the adaptive law for such parameters is a nontrivial task. In this paper, we propose a new adaptive control method in an effort to tune all the RBF parameters thereby reducing the approximation error and improving control performance. Global boundedness of the overall adaptive system and tracking to within a desired precision are established with the new adaptive controller. Simulations performed on a simple nonlinear system illustrate the approach.
198 citations
TL;DR: In this paper, the authors proposed an approximate approach for ranking fuzzy numbers based on the left and right dominance of one fuzzy number over the other, which is useful in ranking the fuzzy numbers when membership functions cannot be acquired.
Abstract: This study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance. The proposed approach only requires a few left and right spreads at some α-levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other. The total dominance is then determined by combining the left and right dominance based on a decision maker's optimistic perspectives. Such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired. The approach proposed herein is relatively simple in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers. By using a few left and right spreads, two groups of examples demonstrate the accuracy and applicability of the proposed approach.
169 citations
TL;DR: A ranking method for fuzzy numbers in which fuzzy numbers are measured point by point and at each point the most preferred number is identified and these numbers are ranked on the basis of their preference ratio.
TL;DR: This paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average based on the α-cut representation of fuzzy sets and the extension principle, and a pair of fractional programs is formulated to find theα-cut of fuzzy weightedAverage.
TL;DR: A laser computer output microfilmer wherein vertical and horizontal misalignment errors between the superimposed data character images and a format slide image are electrically corrected by appropriately incrementing or decrementing the presetting inputs of respective vertical andizontal counters pulsed by line scan synchronizing and clock signals.
TL;DR: It is proved that some of these sets still suffer from spikiness, and a method is presented for constructing such unfactorable joint sets from scalar distance measures.
Abstract: The shape of if-part fuzzy sets affects how well feedforward fuzzy systems approximate continuous functions. We explore a wide range of candidate if-part sets and derive supervised learning laws that tune them. Then we test how well the resulting adaptive fuzzy systems approximate a battery of test functions. No one shape emerges as the best. The sine function often does well and has tractable learning, but its undulating side-lobes may have no linguistic meaning. This suggests that function-approximation accuracy may sometimes have to outweigh linguistic or philosophical interpretations. We divide the if-part sets into two large classes. The first consists of n-dimensional joint sets that factor into n scalar sets. These sets ignore the correlations among input vector components. Fuzzy systems suffer in general from exponential rule explosion in high dimensions when they blindly approximate functions. The factorable fuzzy sets themselves also suffer from a curse of dimensionality: they tend to become binary spikes in high dimension. The second class consists of the more general but less common n-dimensional joint sets that do not factor into n scalar fuzzy sets. We present a method for constructing such unfactorable joint sets from scalar distance measures. Fuzzy systems that use unfactorable sets need not suffer from exponential rule explosion but their increased complexity may lead to intractable learning and inscrutable if-then rules. We prove that some of these sets still suffer from spikiness.
01 Feb 2001
TL;DR: The proposed linguistic hedge fuzzy logic controller has the following advantages: it needs only the simple-shape membership functions rather than the carefully designed ones for characterizing the related variables, and it performs better than the conventional fuzzy logic controllers do.
Abstract: In this paper, we propose a novel fuzzy logic controller, called linguistic hedge fuzzy logic controller, to simplify the membership function constructions and the rule developments. The design methodology of linguistic hedge fuzzy logic controller is a hybrid model based on the concepts of the linguistic hedges and the genetic algorithms. The linguistic hedge operators are used to adjust the shape of the system membership functions dynamically, and ran speed up the control result to fit the system demand. The genetic algorithms are adopted to search the optimal linguistic hedge combination in the linguistic hedge module, According to the proposed methodology, the linguistic hedge fuzzy logic controller has the following advantages: 1) it needs only the simple-shape membership functions rather than the carefully designed ones for characterizing the related variables; 2) it is sufficient to adopt a fewer number of rules for inference; 3) the rules are developed intuitionally without heavily depending on the endeavor of experts; 4) the linguistic hedge module associated with the genetic algorithm enables it to be adaptive; 5) it performs better than the conventional fuzzy logic controllers do; and 6) it can be realized with low design complexity and small hardware overhead. Furthermore, the proposed approach has been applied to design three well-known nonlinear systems. The simulation and experimental results demonstrate the effectiveness of this design,.
TL;DR: An approach for selecting and blending bio-optical algorithms is demonstrated using an ocean color satellite image of the northwest Atlantic shelf based on a fuzzy logic classification scheme applied to the satellite-derived water-leaving radiance data and it is used to select and blend class-specific algorithms.
Abstract: An approach for selecting and blending bio-optical algorithms is demonstrated using an ocean color satellite image of the northwest Atlantic shelf. This approach is based on a fuzzy logic classification scheme applied to the satellite-derived water-leaving radiance data, and it is used to select and blend class-specific algorithms. Local in situ bio-optical data were used to characterize optically-distinct water classes a priori and to parameterize algorithms for each class. Although the algorithms can be of any type (empirical or analytical), this demonstration involves class-specific semi-analytic algorithms, which are the inverse of a radiance model. The semi-analytic algorithms retrieve three variables related to the concentrations of optically active constituents. When applied to a satellite image, the fuzzy logic approach involves three steps. First, a membership function is computed for each pixel and each class. This membership function expresses the likelihood that the measured radiance belongs to a class, with a known reflectance distribution. Thus, for each pixel, class memberships are assigned to the predetermined classes on the basis of the derived membership functions. Second, three variables are retrieved from each of the class-specific algorithms for which the pixel has membership. Third, the class memberships are used to weight the class specific retrievals to obtain a final blended retrieval for each pixel. This approach allows for graded transitions between water types, and blends separately tuned algorithms for different water masses without suffering from the "patchwork quilt" effect associated with hard-classification schemes.
01 Jul 2001
TL;DR: The design problem of output feedback controllers for Takagi–Sugeno fuzzy models is considered and sufficient conditions for the asymptotic convergence of the fuzzy observers are given.
Abstract: In this paper the design problem of output feedback controllers for Takagi–Sugeno fuzzy models is considered. As for the premise variables, we consider two cases: the outputs and the state variables. In each case, we first consider the design of observers. In the first case we give sufficient conditions for the asymptotic convergence of the fuzzy observers. In the second case we give observers for an approximation of the original system. We then propose the output feedback controllers based on state feedback controllers and observers. Two design examples are given to illustrate the theory.
28 May 2001
TL;DR: A framework for modeling driver behavior within driving simulators serves as a basis for building human- like driving behavior models for autonomous vehicles operating within the virtual environment of a driving simulator.
Abstract: A framework for modeling driver behavior within driving simulators is described in this paper. This framework serves as a basis for building human- like driving behavior models for autonomous vehicles operating within the virtual environment of a driving simulator. The framework consists of four units, the Perception Unit, the Emotions Unit, the Decision- making Unit (DMU), and the Decision- implementation Unit (DIU). The Perception Unit defines how the model perceives its environment in local and global terms. The Emotions Unit defines how the model responds emotionally to its environment. The DMU investigates the environment for possible actions that might potentially serve the model's emotional demands. And finally the DIU tries to implement these decisions when a traffic condition, perceived as safe enough for such an implementation, emerges. Each of these units has its own set of fuzzy variables and fuzzy ifthen rules. Any driving model, that is based on this framework, should provide membership function parameters for these fuzzy variables in accordance with the category of human driving behavior this model is targeting. Our framework addresses decision making and implementation at the maneuvering and operational levels of the driving task. Decisions at the planning level are addressed through a script- based traffic controller. The present model is limited to simulating human behaviors when driving in a two- lane rural environment.
TL;DR: A new method to automatically learn the Knowledge Base of a Fuzzy Rule-Based System is proposed by finding an appropriate Data Base using a Genetic Algorithm and considering a simple generation method to derive the Rule Base.
TL;DR: This proposed fuzzy approach allows QFD users to avoid subjective and arbitrary quantification of linguistic data and to develop corresponding procedures to deal with the fuzzy data.
Abstract: Quality function deployment (QFD) is a customer-driven quality management and product development system for achieving higher customer satisfaction. The QFD process involves various inputs in the form of linguistic data, e.g., human perception, judgment, and evaluation on importance or relationship strength. Such data are usually ambiguous and uncertain. An aim of this paper is to examine the implementation of QFD under a fuzzy environment and to develop corresponding procedures to deal with the fuzzy data. It presented a process model using linguistic variables, fuzzy arithmetic, and defuzzification techniques. Based on an example, this paper further examined the sensitivity of the ranking of technical characteristics to the defuzzification strategy and the degree of fuzziness of fuzzy numbers. Results indicated that selection of the defuzzification strategy and membership function are important. This proposed fuzzy approach allows QFD users to avoid subjective and arbitrary quantification of linguistic data. The paper also presents a scheme to represent and interprete the results.
TL;DR: The proposed approach demonstrates the potential for formalizing the inclusion of learning effects into the LOB scheduling of repetitive-unit construction by incorporating relevant factors such as, number of operations in one unit, activity complexity, and job and management conditions.
03 Sep 2001
TL;DR: The basic idea is that the individual encoding scheme incorporates several syntactical restrictions that facilitate the handling of rule sets in disjunctive normal form, which has some innovative ideas with respect to the encoding of GP individuals representing rule sets.
Abstract: In essence, data mining consists of extracting knowledge from data. This paper proposes a co-evolutionary system for discovering fuzzy classification rules. The system uses two evolutionary algorithms: a genetic programming (GP) algorithm evolving a population of fuzzy rule sets and a simple evolutionary algorithm evolving a population of membership function definitions. The two populations co-evolve, so that the final result of the co-evolutionary process is a fuzzy rule set and a set of membership function definitions which are well adapted to each other. In addition, our system also has some innovative ideas with respect to the encoding of GP individuals representing rule sets. The basic idea is that our individual encoding scheme incorporates several syntactical restrictions that facilitate the handling of rule sets in disjunctive normal form. We have also adapted GP operators to better work with the proposed individual encoding scheme.
TL;DR: It is shown that the model has a unique solution and the solution can be given in an analytic expression and an index is given to evaluate the goodness of fit between the observed value and the estimated value.
TL;DR: The value of the correlation coefficient between interval-valued intuitionistic fuzzy sets lies in the interval [-1, 1], as computed from the formula proposed.
Abstract: In this paper, we propose a method to calculate the correlation coefficient of intuitionistic fuzzy sets by means of mathematical statistics. This value obtained from our formula tell us not only the strength of relationship between the intuitionistic fuzzy sets, but also whether the intuitionistic fuzzy sets are positively or negatively related. This approach looks better than previous methods which only evaluate the strength of the relation. Furthermore, we extend the proposed method to interval-valued intuitionistic fuzzy sets. The value of the correlation coefficient between interval-valued intuitionistic fuzzy sets lies in the interval [-1, 1], as computed from our formula.
25 Jul 2001
TL;DR: In this paper, the membership values for each pattern are extended as type-2 fuzzy memberships by assigning membership grades to the type-1 memberships and cluster centers that are estimated bytype-2 memberships may converge to a more desirable location than cluster centers obtained by a type- 1 FCM method in the presence of noise.
Abstract: This paper presents a type-2 fuzzy C-means (FCM) algorithm that is an extension of the conventional fuzzy C-means algorithm. In our proposed method, the membership values for each pattern are extended as type-2 fuzzy memberships by assigning membership grades to the type-1 memberships. In doing so, cluster centers that are estimated by type-2 memberships may converge to a more desirable location than cluster centers obtained by a type-1 FCM method in the presence of noise. Experimental results are given to show the effectiveness of our method.
02 Dec 2001
TL;DR: The complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number, and is presented, including examples of possible applications, which demonstrate the new theory.
Abstract: The innovative concept of complex fuzzy sets is introduced. 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. A study of this original concept is presented, including examples of possible applications, which demonstrate the new theory.
TL;DR: The present investigation focuses on nonfuzzy input and fuzzy output data type and proposes approaches to handle the outlier problem and introduces a pre-assigned k -limiting value whose value must be determined based on the conditions of the current problem.
TL;DR: The study elaborates on the role of the fuzzy equalization in system design and establishes a detailed equalization algorithm developed for triangular fuzzy sets.
TL;DR: A generalized metric for fuzzy numbers is made use of and an exact study is developed for the case of normal fuzzy random variables and an asymptotic study for the cases of simple general fuzzyrandom variables.
TL;DR: In this article, the authors consider uncertain preferences for non-market goods, but they move away from a probabilistic representation of uncertainty and propose the use of fuzzy contingent valuation, where a decision maker never fully knows her own utility function and treats utility as a fuzzy number.
Abstract: In this article, we consider uncertain preferences for non-market goods, but we move away from a probabilistic representation of uncertainty and propose the use of fuzzy contingent valuation. We assume that a decision maker never fully knows her own utility function and we treat utility as a fuzzy number. The methodology is illustrated using data on forest valuation in Sweden. Fuzzy contingent valuation provides estimates of resource value in the form of a fuzzy number and includes estimates obtained using a standard probabilistic approach.