Fuzzy measure theory
Encyclopedia
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 the fuzzy measure (also capacity, see ) which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals.
A fuzzy measure g is called normalized
if
.
, a fuzzy measure is:
Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral
, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging
(OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
,
The equivalent axioms in Möbius representation are:
A fuzzy measure in Möbius representation M is called normalized
if
Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure g in standard representation can be recovered from the Möbius form using the Zeta transform:
), even in discrete cases the number of variables can be quite high (
). For this reason, in the context of multi-criteria decision analysis
and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that
and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or 
-fuzzy measure and k-additive measures, introduced by Sugeno and Grabisch respectively.
Sugeno
The Sugeno 
-measure is a special case of fuzzy measures defined iteratively. It has the following definition:
be a finite set and let 
. A Sugeno 
-measure is a function g from 
to [0, 1] with properties:
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 as well as Wang and Klir have showed that once the densities are known, it is possible to use the previous polynomial
to obtain the values of
uniquely.
to size 
. This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to 
, it allows for a compromise between modelling ability and simplicity.
) if its Möbius representation verifies 
, whenever 
for any 
, and there exists a subset F with k elements such that 
.
theory, the Shapley value
or Shapley index is used to indicate the weight of a game. Shapley values can calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton.
For a given fuzzy measure g, and
, the Shapley index for every 
is:
The Shapley value is the vector
Fuzzy set
Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to...
membership function
Membership function (mathematics)
The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy...
and the classical probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
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 the fuzzy measure (also capacity, see ) which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals.
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 E is the empty setEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
then
. - When E is a subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of F, then
.
A fuzzy measure g is called normalized
if
.Properties of fuzzy measures
For any, a fuzzy measure is:- additive if
for all 
; - supermodular if
; - submodular if
; - superadditive if
for all 
; - subadditive if
for all 
; - symmetric if
implies 
; - Boolean if
or 
.
Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral
Choquet integral
In decision theory, a Choquet integral is a way of measuring the expected utility of an uncertain event. It is applied specifically to membership functions and capacities. In imprecise probability theory, the Choquet integral is also used to calculate the lower expectation induced by a 2-monotone...
, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging
Ordered Weighted Averaging (OWA) Aggregation Operators
Introduced by Ronald R. Yager, the Ordered Weighted Averaging operators, commonly called OWA operators, provide a parameterized class of mean type aggregation operators. Many notable mean operators such as the max, arithmetic average, median and min, are members of this class...
(OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
Möbius representation
Let g be a fuzzy measure, the Möbius representation of g is given by the set function M, where for every,The equivalent axioms in Möbius representation are:
-
. -
, for all 
and all 
A fuzzy measure in Möbius representation M is called normalized
if
Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure g in standard representation can be recovered from the Möbius form using the Zeta transform:
Simplification assumptions for fuzzy measures
Since fuzzy measures are defined on the power set (or, more formally, on the sigma algebra associated with), even in discrete cases the number of variables can be quite high (). For this reason, in the context of multi-criteria decision analysisMulti-Criteria Decision Analysis
Multiple-criteria decision-making or multiple-criteria decision analysis is a sub-discipline of operations research that explicitly considers multiple criteria in decision-making environments. Whether in our daily lives or in professional settings, there are typically multiple conflicting criteria...
and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that
and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or -fuzzy measure and k-additive measures, introduced by Sugeno and Grabisch respectively.Sugeno 
-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
(alternatively 
) 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 as well as Wang and Klir have showed that once the densities are known, it is possible to use the previous polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
to obtain the values of
uniquely.k-additive fuzzy measure
The k-additive fuzzy measure limits the interaction between the subsets to size . This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to , it allows for a compromise between modelling ability and simplicity.Definition
A discrete fuzzy measure g on a set X is called k-additive () if its Möbius representation verifies , whenever for any , and there exists a subset F with k elements such that .Shapley and interaction indices
In gameGame
A game is structured playing, usually undertaken for enjoyment and sometimes used as an educational tool. Games are distinct from work, which is usually carried out for remuneration, and from art, which is more often an expression of aesthetic or ideological elements...
theory, the Shapley value
Shapley value
In game theory, the Shapley value, named in honour of Lloyd Shapley, who introduced it in 1953, is a solution concept in cooperative game theory. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players...
or Shapley index is used to indicate the weight of a game. Shapley values can calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton.
For a given fuzzy measure g, and
, the Shapley index for every is:The Shapley value is the vector