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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 classical Probability 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 when ''X'' is finite:

1. When ''E'' is the Empty Set then
g(E)=0.

2. When ''E'' is a Subset of ''F'',
then g(E)\leq g(F).

A fuzzy measure ''g'' is called ''normalized''
if g(X)=1.




EXAMPLES OF FUZZY MEASURES


Sugeno \lambda-measure

The Sugeno \lambda-measure is a special case of fuzzy measures defined iteratively. It has the following definition

Definition

Let X = \left\lbrace x_1,...,x_n ight brace be a finite set and let \lambda \in (-1,+\infty). A Sugeno \lambda-measure is a function ''g'' from 2^X to 1 with properties:

# g(X) = 1.
# if ''A'', ''B'' \subseteq 2^X with A \cap B = \emptyset then g(A \cup B) =g(A)+g(B)+\lambda g(A)g(B).

As a convention, the measure of a singleton set \left\lbrace x_i ight brace
is called a density and is denoted by g_i = g(\left\lbrace x_i ight brace). In addition, we have that \lambda satisfies the property

\lambda +1 = \prod_{i=1}^n (1+\lambda g_i) .


SEE ALSO



EXTERNAL LINKS

  • http://pami.uwaterloo.ca/tizhoosh/measure.htm



REFERENCES


  • Wang, Zhenyuan, and , George J. Klir , ''Fuzzy Measure Theory'', Plenum Press, New York, 1991.