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# Why triangular membership functions?

Fuzzy Sets and Systems, no. 1 (1994): 21-30

EI

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摘要

In this note we look at a certain theoretically sound motivation behind the common use of triangular (and trapezoidal) membership functions. The studies are completed within a conceptual framework of fuzzy modelling whose structure comprises of input and output interfaces linked with a single transformation module aimed at processing ling...更多

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简介

- The proposition below describes some relationships between specific forms of triangular or trapezoidal membership functions and their entropies.
- Let them investigate how the changes in the shapes of the triangular membership functions that are restricted to the same support affect the resulting entropies.
- Let A and B be fuzzy sets with triangular membership functions with the same support £2 = supp(A) = supp(B).

重点内容

- The triangular membership functions A1, A e, . . . , An defined in X c R with a specific overlap of 1⁄2 (namely the height of intersection of each two successive fuzzy sets equals ~, hgt(A1 t-lAb±l)=1⁄2, i = 1, 2. . . . , n - 1) have been frequently used in many applications of fuzzy sets including fuzzy controllers, fuzzy models, and classification schemes
- Assuming that p(x) is a uniform p.d.f., one can verify that the codebook consisting of the labels with the triangular membership functions of the same support leads to entropy equalization
- It is worth noting that the interface with triangular membership functions with 1 overlap between any two successive fuzzy sets implies no reconstruction error
- We have shown that the commonly used triangular membership functions constitute an immediate solution to the optimization problems emerging in fuzzy modelling
- It should be pinpointed that the triangular membership functions constitute one among some other possibilities yielding the optimal values of the introduced criteria
- While the simplicity of this partition becomes evident, it does no preclude that one could construct some other partitions being optimal in the sense of the assumed criteria for the input and output interface and simultaneously performing better when it comes to the optimization of the processing block itself

结果

- Let them show that the modal value (m) of the fuzzy set does not occur in the final entropy formula, see Figure 3.
- (i) The input interface consisting of triangular fuzzy sets of the same support size and a uniform p.d.f, exhibit a balanced entropy, namely H(Ai) = const for all i's.
- The proposition shows how the entropy level can be conveniently reduced by replacing the triangular fuzzy set by another one with a trapezoidal membership function.
- Let p(x) take on a constant value over the support of the triangular fuzzy set A, p(x) = c, x ~ supp(A).
- The basic idea exploited here is to distribute the supports of the fuzzy sets according to the modal values of the probability density function p(x) of the considered variable and subsequently equalize the entropies through the reduction of the highest entropy values by replacing the original membership functions by their trapezoidal counterparts.
- Arrange the resulting triangular fuzzy sets according to their entropy values, say Ai,, Ai2,
- Assuming that p(x) is a uniform p.d.f., one can verify that the codebook consisting of the labels with the triangular membership functions of the same support leads to entropy equalization.
- It is worth noting that the interface with triangular membership functions with 1 overlap between any two successive fuzzy sets implies no reconstruction error.
- Let A i be triangular membership functions with the overlap level equal to 1⁄2.
- For any x E X it is sufficient to consider only the two fuzzy sets the supports of whose include this value; note that the remaining ones do not contribute to the reconstruction formula.
- The authors have shown that the commonly used triangular membership functions constitute an immediate solution to the optimization problems emerging in fuzzy modelling.

结论

- Under some assumptions about the underlying density probability function the fuzzy partition built out of the triangular membership functions leads to entropy equalization.
- It should be pinpointed that the triangular membership functions constitute one among some other possibilities yielding the optimal values of the introduced criteria.
- While the simplicity of this partition becomes evident, it does no preclude that one could construct some other partitions being optimal in the sense of the assumed criteria for the input and output interface and simultaneously performing better when it comes to the optimization of the processing block itself

总结

- The proposition below describes some relationships between specific forms of triangular or trapezoidal membership functions and their entropies.
- Let them investigate how the changes in the shapes of the triangular membership functions that are restricted to the same support affect the resulting entropies.
- Let A and B be fuzzy sets with triangular membership functions with the same support £2 = supp(A) = supp(B).
- Let them show that the modal value (m) of the fuzzy set does not occur in the final entropy formula, see Figure 3.
- (i) The input interface consisting of triangular fuzzy sets of the same support size and a uniform p.d.f, exhibit a balanced entropy, namely H(Ai) = const for all i's.
- The proposition shows how the entropy level can be conveniently reduced by replacing the triangular fuzzy set by another one with a trapezoidal membership function.
- Let p(x) take on a constant value over the support of the triangular fuzzy set A, p(x) = c, x ~ supp(A).
- The basic idea exploited here is to distribute the supports of the fuzzy sets according to the modal values of the probability density function p(x) of the considered variable and subsequently equalize the entropies through the reduction of the highest entropy values by replacing the original membership functions by their trapezoidal counterparts.
- Arrange the resulting triangular fuzzy sets according to their entropy values, say Ai,, Ai2,
- Assuming that p(x) is a uniform p.d.f., one can verify that the codebook consisting of the labels with the triangular membership functions of the same support leads to entropy equalization.
- It is worth noting that the interface with triangular membership functions with 1 overlap between any two successive fuzzy sets implies no reconstruction error.
- Let A i be triangular membership functions with the overlap level equal to 1⁄2.
- For any x E X it is sufficient to consider only the two fuzzy sets the supports of whose include this value; note that the remaining ones do not contribute to the reconstruction formula.
- The authors have shown that the commonly used triangular membership functions constitute an immediate solution to the optimization problems emerging in fuzzy modelling.
- Under some assumptions about the underlying density probability function the fuzzy partition built out of the triangular membership functions leads to entropy equalization.
- It should be pinpointed that the triangular membership functions constitute one among some other possibilities yielding the optimal values of the introduced criteria.
- While the simplicity of this partition becomes evident, it does no preclude that one could construct some other partitions being optimal in the sense of the assumed criteria for the input and output interface and simultaneously performing better when it comes to the optimization of the processing block itself

引用论文

- J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms (Plenum Press, New York, 1981).
- A. De Luca and S. Termini, A definition of non-probabilistic entropy in the setting of fuzzy sets theory, Information & Control 20 (1972) 301-312.
- D. Dubois and H. Prade, Possibility Theory - An Approach to Computerized Processing of Uncertainty (Plenum Press, New
- P. Eyckhoff, System Identification: Parameter and State Estimation (J. Wiley, London, 1974).
- A. Gersho and R.M. Gray, Vector quantization and Signal Compression (Kluwer Academic Publishers, Dordrecht, 1992).
- W. Pedrycz, Fuzzy Control and Fuzzy Systems, 2rid extended edition (Research Studies Press/J. Wiley, Taunton/New York, 1993).
- W. Pedrycz and J. Valente de Oliveria, Optimization of fuzzy relational models, Proc. 5th IFSA World Contress, vol. 2, Seoul, Korea, July 4-9 (1993) 1187-1190.
- J. Valente de Oliveria, On optimal fuzzy systems with I/O interfaces, Proc. 2nd Int. Conf. on Fuzzy Systems, San Francisco
- L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.

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