- Fuzzy measure theory
Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures,
fuzzy set membership function and the classicalprobability measures. In the fuzzy measure theory, the conditions are precise, but the information about an element alone is insufficient to determine which special classes of measure should be used. The central concept of fuzzy measure theory is fuzzy measure, which was introduced by Sugeno in 1974.Axioms
Fuzzy measure can be considered as generalization of the classical probability measure. A fuzzy measure "g" over a set "X" (the
universe of discourse with the subsets "E", "F", ...) satisfies the following conditions:1. When "E" is the
empty set then .2. When "E" is a
subset of "F", then .A fuzzy measure "g" is called "normalized"if .
Examples of Fuzzy Measures
ugeno -measure
The Sugeno -measure is a special case of fuzzy measures defined iteratively. It has the following definition
Definition
Let be a finite set and let . A Sugeno -measure is a function "g" from to [0, 1] with properties:
# .
# if "A", "B" with then .As a convention, the value of g at a singleton set is called a density and is denoted by . In addition, we have that satisfies the property
.
Tahani and Keller [cite journal
author = H. Tahani and J. Keller
title = Information Fusion in Computer Vision Using the Fuzzy Integral
journal = IEEE Transactions on Systems, Man and Cybernetic
volume = 20
number = 3
pages = 733–741
year = 1990
doi = 10.1109/21.57289] as well as Wang and Klir have showed that that once the densities are known, it is possible to use the previouspolynomial to obtain the values of uniquely.ee also
*
Probability theory
*Possibility theory External links
*http://pami.uwaterloo.ca/tizhoosh/measure.htm
References
* Wang, Zhenyuan, and ,
George J. Klir , "Fuzzy Measure Theory", Plenum Press, New York, 1991.
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