Fuzzy sets are constructs that come with a well defined meaning [36–38]. They capture the semantics of the framework they intend to operate within. Fuzzy sets are the building conceptual blocks (generic constructs) that are used in problem description, modeling, control, pattern classification tasks and the like. Their estimation and interpretation are of paramount relevance. Before discussing specific techniques of membership function estimation, it is worth casting the overall presentation in a certain context by emphasizing the aspect of the use of a finite number of fuzzy sets leading to some essential vocabulary reflective of the underlying domain knowledge. In particular, we are concerned with the related semantics, calibration capabilities of membership functions, and the locality of fuzzy sets.
The key objectives of this study revolve around the fundamental concept of membership function estimation, calibration, perception, and reconciliation of views at information granules. Being fully cognizant of the semantics of fuzzy sets, it becomes essential to come up with a suite of effective mechanisms for the development of fuzzy sets and offer a comprehensive view of their interpretation.
Given the main objectives of the study, its organization is reflective of them. We start with a discussion on the use of domain knowledge and the problem-oriented formation of fuzzy sets (Sect. 2). In Sect. 3, we focus on usercentric estimation of membership functions (including horizontal, vertical and pairwise comparison). In the following Section, we focus on the construction of fuzzy sets regarded as granular representatives of numeric data. Fuzzy clustering is covered in Sect. 5. Several design guidelines are presented in Sect. 6. Fuzzy sets are typically cast in some context, and in this regard we discuss the issue of nonlinear mappings (transformations) that provide a constructive view of the calibration of fuzzy sets. The crux of the calibration deals with a series of contractions and expansions of selected regions of the universe of discourse. Further on we discuss several ways of reconciliation of fuzzy sets and fuzzy mappings being a direct result of various views (perspectives) of the same fuzzy set/fuzzy mapping.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bezdek JC(1981) Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York, NY.
Bortolan G, Pedrycz W (2002) An interactive framework for an analysis of ECG signals. Artificial Intelligence in Medicine, 24(2): 109-132.
Buckley JJ, Feuring T, Hayashi Y (2001) Fuzzy hierarchical analysis revisited. European J. Operational Research, 129(1): 48-64.
Ciaramella A, Tagliaferri R, Pedrycz W, Di Nola A (2006) A Fuzzy relational neural network. Intl. J. Approximate Reasoning, 41(2): 146-163.
Civanlar MR, Trussell HJ (1986) Constructing membership functions using statistical data. Fuzzy Sets and Systems, 18(1): 1-13.
Chen MS, Wang SW (1999) Fuzzy clustering analysis for optimizing fuzzy membership functions. Fuzzy Sets and Systems, 103(2): 239-254.
Dishkant CH (1981) About membership functions estimation. Fuzzy Sets and Systems, 5(2): 141-147.
Dombi J (1990) Membership function as an evaluation. Fuzzy Sets and Systems, 35(1): 1-21.
Hong TP, Lee CY (1996) Induction of fuzzy rules and membership functions from training examples. Fuzzy Sets and Systems, 84(1): 389-404.
Klement E, Mesiar R, Pap E (2000) Triangular Norms. Kluwer Academic Publishers Dordrecht, The Netherlands.
Kulak O, Kahraman C (2005) Fuzzy multi-attribute selection among trans-portation companies using axiomatic design and analytic hierarchy process. Information Sciences, 170(2-4): 191-210.
Masson MH, Denoeux T (2006) Inferring a possibility distribution from empirical data. Fuzzy Sets and Systems, 157(3): 319-340.
Medaglia AL, Fang SC, Nuttle HLW, Wilson JR (2002) An efficient and flexi-ble mechanism for constructing membership functions. European J. Operational Research, 139(1): 84-95.
Medasani S, Kim J, Krishnapuram R (1998) An overview of membership function generation techniques for pattern recognition. Intl. J. Approximate Reasoning, 19(3-4): 391-417.
Mikhailov L, Tsvetinov P (2004) Evaluation of services using a fuzzy analytic hierarchy process. Applied Soft Computing, 5(1): 23-33.
Miller GA (1956) The magical number seven plus or minus two: some limits of our capacity for processing information. Psychological Review, 63: 81-97.
Pedrycz W, Rocha A (1993) Hierarchical FCM in a stepwise discovery of structure in data. Soft Computing, 10: 244-256.
Pedrycz A, Reformat M (2006) Knowledge-based neural networks. IEEE Trans. Fuzzy Systems, 1: 254-266.
Pedrycz W (1993) Fuzzy neural networks and neurocomputations. Fuzzy Sets and Systems, 56: 1-28.
Pedrycz W (1994) Why triangular membership functions? Fuzzy Sets and Systems, 64: 21-30.
Pedrycz W (1995) Fuzzy Sets Engineering. CRC Press, Boca Raton, FL.
Pedrycz W, Valente de Oliveira J(1996) An algorithmic framework for development and optimization of fuzzy models. Fuzzy Sets and Systems, 80: 37-55.
Pedrycz W, Gudwin R, Gomide F (1997) Nonlinear context adaptation in the calibration of fuzzy sets. Fuzzy Sets and Systems, 88: 91-97.
Pedrycz W, Gomide F (1998) An Introduction to Fuzzy Sets: Analysis and Design. MIT Press, Cambridge, MA.
Pedrycz W (2001) Fuzzy equalization in the construction of fuzzy sets. Fuzzy Sets and Systems 119: 329-335 of fuzzy sets. Fuzzy Sets and Systems, 88: 91-97.
Pedrycz W (ed) (2001) Granular Computing: An Emerging Paradigm. Physica Verlag, Heidelberg, Germany.
Pedrycz W, Vukovich G (2002) On elicitation of membership functions. IEEE Trans. Systems, Man, and Cybernetics - Part A, 32(6): 761-767.
Pendharkar PC (2003) Characterization of aggregate fuzzy membership functions using Saaty’s eigenvalue approach. Computers and Operations Research, 30(2): 199-212.
Saaty TL (1980) The Analytic Hierarchy Process. McGraw Hill, New York, NY.
Saaty TL (1986) Scaling the membership functions. European J. Operational Research, 25(3): 320-329.
Schweizer B, Sklar A (1983) Probabilistic Metric Spaces North-Holland, New York, NY.
Simon D (2005) H∞ Estimation for fuzzy membership function optimization. Intl. J. Approximate Reasoning 40(3): 224-242.
Turksen IB (1991) Measurement of membership functions and their acquisition. Fuzzy Sets and Systems, 40(1): 5-138.
van Laarhoven PJM, Pedrycz W (1983) A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11(1-3): 199-227.
Yang CC, Bose NK (2006) Generating fuzzy membership function with self-organizing feature map. Pattern Recognition Letters, 27(5): 356-365.
Zadeh LA (1996) Fuzzy logic = computing with words. IEEE Trans. Fuzzy Systems, 4: 103-111.
Zadeh LA (1997) Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets and Systems, 90: 111-117.
Zadeh LA (2005) Toward a generalized theory of uncertainty (GTU) - an outline. Information Sciences, 172: 1-40.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pedrycz, W. (2008). Semantics and Perception of Fuzzy Sets and Fuzzy Mappings. In: Fulcher, J., Jain, L.C. (eds) Computational Intelligence: A Compendium. Studies in Computational Intelligence, vol 115. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78293-3_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-78293-3_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78292-6
Online ISBN: 978-3-540-78293-3
eBook Packages: EngineeringEngineering (R0)