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Membership function
About: Membership function is a research topic. Over the lifetime, 15795 publications have been published within this topic receiving 418366 citations.
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14 Feb 2011
TL;DR: The purpose of this monograph is to explore the relationship between Fuzzy Data, Bayes' Theorem, and Problems, and the role that these models play in the development of knowledge and understanding of fuzzy data.
Abstract: Preface. Part I FUZZY INFORMATION. 1. Fuzzy Data. 1.1 One-dimensional Fuzzy Data. 1.2 Vector-valued Fuzzy Data. 1.3 Fuzziness and Variability. 1.4 Fuzziness and Errors. 1.5 Problems. 2. Fuzzy Numbers and Fuzzy Vectors. 2.1 Fuzzy Numbers and Characterizing Functions. 2.2 Vectors of Fuzzy Numbers and Fuzzy Vectors. 2.3 Triangular Norms. 2.4 Problems. 3. Mathematical Operations for Fuzzy Quantities. 3.1 Functions of Fuzzy Variables. 3.2 Addition of Fuzzy Numbers. 3.3 Multiplication of Fuzzy Numbers. 3.4 Mean Value of Fuzzy Numbers. 3.5 Differences and Quotients. 3.6 Fuzzy Valued Functions. 3.7 Problems. Part II DESCRIPTIVE STATISTICS FOR FUZZY DATA. 4. Fuzzy Samples. 4.1 Minimum of Fuzzy Data. 4.2 Maximum of Fuzzy Data. 4.3 Cumulative Sum for Fuzzy Data. 4.4 Problems. 5. Histograms for Fuzzy Data. 5.1 Fuzzy Frequency of a Fixed Class. 5.2 Fuzzy Frequency Distributions. 5.3 Axonometric Diagram of the Fuzzy Histogram. 5.4 Problems. 6. Empirical Distribution Functions. 6.1 Fuzzy Valued Empirical Distribution Function. 6.2 Fuzzy Empirical Fractiles. 6.3 Smoothed Empirical Distribution Function. 6.4 Problems. 7. Empirical Correlation for Fuzzy Data. 7.1 Fuzzy Empirical Correlation Coefficient. 7.2 Problems. Part III FOUNDATIONS OF STATISTICAL INFERENCE WITH FUZZY DATA. 8. Fuzzy Probability Distributions. 8.1 Fuzzy Probability Densities. 8.2 Probabilities based on Fuzzy Probability Densities. 8.3 General Fuzzy Probability Distributions. 8.4 Problems. 9. A Law of Large Numbers. 9.1 Fuzzy Random Variables. 9.2 Fuzzy Probability Distributions induced by Fuzzy Random Variables. 9.3 Sequences of Fuzzy Random Variables. 9.4 Law of Large Numbers for Fuzzy Random Variables. 9.5 Problems. 10. Combined Fuzzy Samples. 10.1 Observation Space and Sample Space. 10.2 Combination of Fuzzy Samples. 10.3 Statistics of Fuzzy Data. 10.4 Problems. Part IV CLASSICAL STATISTICAL INFERENCE FOR FUZZY DATA. 11. Generalized Point Estimations. 11.1 Estimations based on Fuzzy Samples. 11.2 Sample Moments. 11.3 Problems. 12. Generalized Confidence Regions. 12.1 Confidence Functions. 12.2 Fuzzy Confidence Regions. 12.3 Problems. 13. Statistical Tests for Fuzzy Data. 13.1 Test Statistics and Fuzzy Data. 13.2 Fuzzy p-Values. 13.3 Problems. Part V BAYESIAN INFERENCE AND FUZZY INFORMATION. 14. Bayes' Theorem and Fuzzy Information. 14.1 Fuzzy a-priori Distributions. 14.2 Updating Fuzzy a-priori Distributions. 14.3 Problems. 15. Generalized Bayes' Theorem. 15.1 Likelihood Function for Fuzzy Data. 15.2 Bayes' Theorem for Fuzzy a-priori Distribution and Fuzzy Data. 15.3 Problems. 16. Bayesian Confidence Regions. 16.1 Confidence Regions based on Fuzzy Data. 16.2 Fuzzy HPD-Regions. 16.3 Problems. 17. Fuzzy Predictive Distributions. 17.1 Discrete Case. 17.2 Discrete Models with Continuous Parameter Space. 17.3 Continuous Case. 17.4 Problems. 18. Bayesian Decisions and Fuzzy Information. 18.1 Bayesian Decisions. 18.2 Fuzzy Utility. 18.3 Discrete State Space. 18.4 Continuous State Space. 18.5 Problems. References. Part VI REGRESSION ANALYSIS AND FUZZYINFORMATION. 19 Classical regression analysis. 19.1 Regression models. 19.2 Linear regression models with Gaussian dependent variables. 19.3 General linear models. 19.4 Nonidentical variances. 19.5 Problems. 20 Regression models and fuzzy data. 20.1 Generalized estimators for linear regression models based on the extension principle. 20.2 Generalized confidence regions for parameters. 20.3 Prediction in fuzzy regression models. 20.4 Problems. 21 Bayesian regression analysis. 21.1 Calculation of a posteriori distributions. 21.2 Bayesian confidence regions. 21.3 Probabilities of hypotheses. 21.4 Predictive distributions. 21.5 A posteriori Bayes estimators for regression parameters. 21.6 Bayesian regression with Gaussian distributions. 21.7 Problems. 22 Bayesian regression analysis and fuzzy information. 22.1 Fuzzy estimators of regression parameters. 22.2 Generalized Bayesian confidence regions. 22.3 Fuzzy predictive distributions. 22.4 Problems. Part VII FUZZY TIME SERIES. 23 Mathematical concepts. 23.1 Support functions of fuzzy quantities. 23.2 Distances of fuzzy quantities. 23.3 Generalized Hukuhara difference. 24 Descriptive methods for fuzzy time series. 24.1 Moving averages. 24.2 Filtering. 24.2.1 Linear filtering. 24.2.2 Nonlinear filters. 24.3 Exponential smoothing. 24.4 Components model. 24.4.1 Model without seasonal component. 24.4.2 Model with seasonal component. 24.5 Difference filters. 24.6 Generalized Holt-Winter method. 24.7 Presentation in the frequency domain. 25 More on fuzzy random variables and fuzzy random vectors. 25.1 Basics. 25.2 Expectation and variance of fuzzy random variables. 25.3 Covariance and correlation. 25.4 Further results. 26 Stochastic methods in fuzzy time series analysis. 26.1 Linear approximation and prediction. 26.2 Remarks concerning Kalman filtering. Part VIII APPENDICES. A1 List of symbols and abbreviations. A2 Solutions to the problems. A3 Glossary. A4 Related literature. References. Index.
177 citations
TL;DR: This paper presents a method for ranking fuzzy numbers based on the comparison of expected intervals of these numbers, and it verifies the distinguishability, rationality and robustness qualities.
Abstract: This paper presents a method for ranking fuzzy numbers based on the comparison of expected intervals of these numbers. This relation is fuzzy and it verifies the distinguishability, rationality and robustness qualities. The term expected interval is extended to no normal fuzzy numbers, and this method then it allows to compare these type sets.
176 citations
TL;DR: The proposed method is simpler than the methods presented in Chen and Lee (2010a, 2010b) and provides a useful way for dealing with fuzzy multiple attributes group decision-making problems based on interval type-2 fuzzy sets.
Abstract: In this paper, we present a new method to deal with fuzzy multiple attributes group decision-making problems based on ranking interval type-2 fuzzy sets. First, we propose a new method for ranking interval type-2 fuzzy sets. Then, we propose a new method for fuzzy multiple attributes group decision-making based on the proposed ranking method of interval type-2 fuzzy sets. We also use some examples to illustrate the fuzzy multiple attributes group decision-making process of the proposed method. The proposed method is simpler than the methods presented in Chen and Lee (2010a, 2010b) for fuzzy multiple attributes group decision-making based on interval type-2 fuzzy sets. It provides us with a useful way for dealing with fuzzy multiple attributes group decision-making problems based on interval type-2 fuzzy sets.
176 citations
TL;DR: This paper proposes a new approach of weights of evidence method based on fuzzy sets and fuzzy probabilities for mineral potential mapping, which allows objective or subjective definition of a fuzzy membership function of evidence augmented by objective definition of fuzzy or conditional probabilities.
Abstract: This paper proposes a new approach of weights of evidence method based on fuzzy sets and fuzzy probabilities for mineral potential mapping. It can be considered as a generalization of the ordinary weights of evidence method, which is based on binary or ternary patterns of evidence and has been used in conjunction with geographic information systems for mineral potential mapping during the past few years. In the newly proposed method, instead of separating evidence into binary or ternary form, fuzzy sets containing more subjective genetic elements are created; fuzzy probabilities are defined to construct a model for calculating the posterior probability of a unit area containing mineral deposits on the basis of the fuzzy evidence for the unit area. The method can be treated as a hybrid method, which allows objective or subjective definition of a fuzzy membership function of evidence augmented by objective definition of fuzzy or conditional probabilities. Posterior probabilities calculated by this method would depend on existing data in a totally data-driven approach method, but depend partly on expert's knowledge when the hybrid method is used. A case study for demonstration purposes consists of application of the method to gold deposits in Meguma Terrane, Nova Scotia, Canada.
176 citations
TL;DR: A new way of looking at fuzzy intervals is introduced, which enables interval analysis to be directly extended to fuzzy intervals, without resorting to alpha-cuts, in agreement with Zadeh's extension principle.
Abstract: In this paper, we introduce a new way of looking at fuzzy intervals. Instead of considering them as fuzzy sets, we see them as crisp sets of entities we call gradual (real) numbers. They are a gradual extension of real numbers, not of intervals. Such a concept is apparently missing in fuzzy set theory. Gradual numbers basically have the same algebraic properties as real numbers, but they are functions. A fuzzy interval is then viewed as a pair of fuzzy thresholds, which are monotonic gradual real numbers. This view enables interval analysis to be directly extended to fuzzy intervals, without resorting to alpha-cuts, in agreement with Zadeh's extension principle. Several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end- points of intervals are changed into left and right fuzzy bounds. Our approach is illustrated on two known problems: computing fuzzy weighted averages and determining fuzzy floats and latest starting times in activity network scheduling.
176 citations