Fuzzy measure theory
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, fuzzy measure theory considers generalized
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
s in which the additive property is replaced by the weaker property of monotonicity. 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. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
measures, which are a subset of classical measures.


Definitions

Let \mathbf be a
universe of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The domain ...
, \mathcal be a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...
of
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of \mathbf, and E,F\in\mathcal. A
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
g:\mathcal\to\mathbb where # \emptyset \in \mathcal \Rightarrow g(\emptyset)=0 # E \subseteq F \Rightarrow g(E)\leq g(F) is called a ''fuzzy measure''. A fuzzy measure is called ''normalized'' or ''regular'' if g(\mathbf)=1.


Properties of fuzzy measures

A fuzzy measure is: * additive if for any E,F \in \mathcal such that E \cap F = \emptyset , we have g(E \cup F) = g(E) + g(F). ; * supermodular if for any E,F \in \mathcal , we have g(E \cup F) + g(E \cap F) \geq g(E) + g(F); * submodular if for any E,F \in \mathcal , we have g(E \cup F) + g(E \cap F) \leq g(E) + g(F); * superadditive if for any E,F \in \mathcal such that E \cap F = \emptyset , we have g(E \cup F) \geq g(E) + g(F); * subadditive if for any E,F \in \mathcal such that E \cap F = \emptyset , we have g(E \cup F) \leq g(E) + g(F); * symmetric if for any E,F \in \mathcal , we have , E, = , F, implies g(E) = g(F); * Boolean if for any E \in \mathcal , we have g(E) = 0 or g(E) = 1 . 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 A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical mechanics and potential theory, but found its way into decision theory in the 1980s, wher ...
, 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 mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
. 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.


Möbius representation

Let ''g'' be a fuzzy measure, the Möbius representation of ''g'' is given by the set function ''M'', where for every E,F \subseteq X , :M(E) = \sum_ (-1)^ g(F). The equivalent axioms in Möbius representation are: # M(\emptyset)=0. # \sum_ M(F) \geq 0, for all E \subseteq \mathbf and all i \in E A fuzzy measure in Möbius representation ''M'' is called ''normalized'' if \sum_M(E)=1. 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: : g(E) = \sum_ M(F), \forall E \subseteq \mathbf .


Simplification assumptions for fuzzy measures

Fuzzy measures are defined on a
semiring of sets In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
or monotone class, which may be as granular as the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of X, and even in discrete cases the number of variables can be as large as 2, X, . For this reason, in the context of
multi-criteria decision analysis Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making (both in daily life and in settings ...
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 g(E) = \sum_ g(\) 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 \lambda-fuzzy measure and ''k''-additive measures, introduced by Sugeno and Grabisch respectively.


Sugeno ''λ''-measure

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


Definition

Let \mathbf = \left\lbrace x_1,\dots,x_n \right\rbrace be a finite set and let \lambda \in (-1,+\infty). A Sugeno \lambda-measure is a function g:2^X\to ,1/math> such that # g(X) = 1. # if A, B\subseteq \mathbf (alternatively A, B\in 2^) with A \cap B = \emptyset then g(A \cup B) =g(A)+g(B)+\lambda g(A)g(B). As a convention, the value of g at a singleton set \left\lbrace x_i \right\rbrace is called a density and is denoted by g_i = g(\left\lbrace x_i \right\rbrace). In addition, we have that \lambda satisfies the property : \lambda +1 = \prod_^n (1+\lambda g_i) . 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 In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
to obtain the values of \lambda uniquely.


''k''-additive fuzzy measure

The ''k''-additive fuzzy measure limits the interaction between the subsets E \subseteq X to size , E, =k. 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 X, it allows for a compromise between modelling ability and simplicity.


Definition

A discrete fuzzy measure ''g'' on a set X is called ''k-additive'' ( 1 \leq k \leq , \mathbf, ) if its Möbius representation verifies M(E) = 0 , whenever , E, > k for any E \subseteq \mathbf , and there exists a subset ''F'' with ''k'' elements such that M(F) \neq 0 .


Shapley and interaction indices

In
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, the
Shapley value The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uniq ...
or Shapley index is used to indicate the weight of a game. Shapley values can be 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 , \mathbf, =n, the Shapley index for every i,\dots,n \in X is: : \phi (i) = \sum_ \frac (E \cup \) - g(E) The Shapley value is the vector \mathbf(g) = (\psi(1),\dots,\psi(n)).


See also

*
Probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
*
Possibility theory Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessa ...


References


Further reading

* Beliakov, Pradera and Calvo, ''Aggregation Functions: A Guide for Practitioners'', Springer, New York 2007. * Wang, Zhenyuan, and,
George J. Klir George Jiří Klir (April 22, 1932 – May 27, 2016) was a Czech-American computer scientist and professor of systems sciences at Binghamton University in Binghamton, New York. Biography George Klir was born in 1932 in Prague, Czechoslovakia. ...
, ''Fuzzy Measure Theory'', Plenum Press, New York, 1991.


External links


Fuzzy Measure Theory at Fuzzy Image Processing
{{Webarchive, url=https://web.archive.org/web/20190630034036/http://pami.uwaterloo.ca/tizhoosh/measure.htm , date=2019-06-30 Exotic probabilities Measure theory Fuzzy logic