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Fuzzy set

From Wikipedia, the free encyclopedia
Sets whose elements have degrees of membership

Inmathematics,fuzzy sets (also known asuncertain sets) aresets whoseelements have degrees of membership. Fuzzy sets were introduced independently byLotfi A. Zadeh in 1965 as an extension of the classical notion of set.[1][2]At the same time,Salii (1965) defined a more general kind of structure called an "L-relation", which he studied in anabstract algebraic context; fuzzy relations are special cases ofL-relations whenL is theunit interval [0, 1].They are now used throughoutfuzzy mathematics, having applications in areas such aslinguistics (De Cock, Bodenhofer & Kerre 2000),decision-making (Kuzmin 1982), andclustering (Bezdek 1978).

In classicalset theory, the membership of elements in a set is assessed in binary terms according to abivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of amembership function valued in thereal unit interval [0, 1]. Fuzzy sets generalize classical sets, since theindicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1.[3] In fuzzy set theory, classical bivalent sets are usually calledcrisp sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such asbioinformatics.[4]

Definition

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A fuzzy set is a pair(U,m){\displaystyle (U,m)} whereU{\displaystyle U} is a set (often required to benon-empty) andm:U→[0,1]{\displaystyle m\colon U\rightarrow [0,1]} a membership function. The reference setU{\displaystyle U} (sometimes denoted byΩ{\displaystyle \Omega } orX{\displaystyle X}) is calleduniverse of discourse, and for eachx∈U,{\displaystyle x\in U,} the valuem(x){\displaystyle m(x)} is called thegrade of membership ofx{\displaystyle x} in(U,m){\displaystyle (U,m)}. The functionm=μA{\displaystyle m=\mu _{A}} is called themembership function of the fuzzy setA=(U,m){\displaystyle A=(U,m)}.

For a finite setU={x1,…,xn},{\displaystyle U=\{x_{1},\dots ,x_{n}\},} the fuzzy set(U,m){\displaystyle (U,m)} is often denoted by{m(x1)/x1,…,m(xn)/xn}.{\displaystyle \{m(x_{1})/x_{1},\dots ,m(x_{n})/x_{n}\}.}

Letx∈U{\displaystyle x\in U}. Thenx{\displaystyle x} is called

The (crisp) set of all fuzzy sets on a universeU{\displaystyle U} is denoted withSF(U){\displaystyle SF(U)} (or sometimes justF(U){\displaystyle F(U)}).[citation needed]

Crisp sets related to a fuzzy set

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For any fuzzy setA=(U,m){\displaystyle A=(U,m)} andα∈[0,1]{\displaystyle \alpha \in [0,1]} the following crisp sets are defined:

Note that some authors understand "kernel" in a different way; see below.

Other definitions

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∀{\displaystyle \forall }x∈U:μA(x)=m(x)=0{\displaystyle x\in U:\mu _{A}(x)=m(x)=0}
∀x∈U:μA(x)=μB(x){\displaystyle \forall x\in U:\mu _{A}(x)=\mu _{B}(x)}
∀x∈U:μA(x)≤μB(x){\displaystyle \forall x\in U:\mu _{A}(x)\leq \mu _{B}(x)}
μA(x)=0.5{\displaystyle \mu _{A}(x)=0.5}
is called acrossover point.
ΛA={α∈[0,1]:A=α≠∅}={α∈[0,1]:{\displaystyle \Lambda _{A}=\{\alpha \in [0,1]:A^{=\alpha }\neq \varnothing \}=\{\alpha \in [0,1]:{}}∃{\displaystyle \exists }x∈U(μA(x)=α)}=μA(U){\displaystyle x\in U(\mu _{A}(x)=\alpha )\}=\mu _{A}(U)}
Hgt⁡(A)=sup{μA(x)∣x∈U}=sup(μA(U)){\displaystyle \operatorname {Hgt} (A)=\sup\{\mu _{A}(x)\mid x\in U\}=\sup(\mu _{A}(U))}
wheresup{\displaystyle \sup } denotes thesupremum, which exists becauseμA(U){\displaystyle \mu _{A}(U)} is non-empty and bounded above by 1. IfU is finite, we can simply replace the supremum by the maximum.
Hgt⁡(A)=1{\displaystyle \operatorname {Hgt} (A)=1}
In the finite case, where the supremum is a maximum, this means that at least one element of the fuzzy set has full membership. A non-empty fuzzy setA{\displaystyle A} may be normalized with resultA~{\displaystyle {\tilde {A}}} by dividing the membership function of the fuzzy set by its height:
∀x∈U:μA~(x)=μA(x)/Hgt⁡(A){\displaystyle \forall x\in U:\mu _{\tilde {A}}(x)=\mu _{A}(x)/\operatorname {Hgt} (A)}
Besides similarities this differs from the usualnormalization in that the normalizing constant is not a sum.
Width⁡(A)=sup(Supp⁡(A))−inf(Supp⁡(A)){\displaystyle \operatorname {Width} (A)=\sup(\operatorname {Supp} (A))-\inf(\operatorname {Supp} (A))}
In the case whenSupp⁡(A){\displaystyle \operatorname {Supp} (A)} is a finite set, or more generally aclosed set, the width is just
Width⁡(A)=max(Supp⁡(A))−min(Supp⁡(A)){\displaystyle \operatorname {Width} (A)=\max(\operatorname {Supp} (A))-\min(\operatorname {Supp} (A))}
In then-dimensional case(U⊆Rn){\displaystyle (U\subseteq \mathbb {R} ^{n})} the above can be replaced by then-dimensional volume ofSupp⁡(A){\displaystyle \operatorname {Supp} (A)}.
In general, this can be defined given anymeasure onU, for instance by integration (e.g.Lebesgue integration) ofSupp⁡(A){\displaystyle \operatorname {Supp} (A)}.
∀x,y∈U,∀λ∈[0,1]:μA(λx+(1−λ)y)≥min(μA(x),μA(y)){\displaystyle \forall x,y\in U,\forall \lambda \in [0,1]:\mu _{A}(\lambda {x}+(1-\lambda )y)\geq \min(\mu _{A}(x),\mu _{A}(y))}.
Without loss of generality, we may takex ≤y, which gives the equivalent formulation
∀z∈[x,y]:μA(z)≥min(μA(x),μA(y)){\displaystyle \forall z\in [x,y]:\mu _{A}(z)\geq \min(\mu _{A}(x),\mu _{A}(y))}.
This definition can be extended to one for a generaltopological spaceU: we say the fuzzy setA{\displaystyle A} isconvex when, for any subsetZ ofU, the condition
∀z∈Z:μA(z)≥inf(μA(∂Z)){\displaystyle \forall z\in Z:\mu _{A}(z)\geq \inf(\mu _{A}(\partial Z))}
holds, where∂Z{\displaystyle \partial Z} denotes theboundary ofZ andf(X)={f(x)∣x∈X}{\displaystyle f(X)=\{f(x)\mid x\in X\}} denotes theimage of a setX (here∂Z{\displaystyle \partial Z}) under a functionf (hereμA{\displaystyle \mu _{A}}).

Fuzzy set operations

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Main article:Fuzzy set operations

Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity.

∀x∈U:μ¬A(x)=1−μA(x){\displaystyle \forall x\in U:\mu _{\neg {A}}(x)=1-\mu _{A}(x)}.
∀x∈U:μA∩B(x)=t(μA(x),μB(x)){\displaystyle \forall x\in U:\mu _{A\cap {B}}(x)=t(\mu _{A}(x),\mu _{B}(x))},
and theirunionA∪B{\displaystyle A\cup {B}} is defined by:
∀x∈U:μA∪B(x)=s(μA(x),μB(x)){\displaystyle \forall x\in U:\mu _{A\cup {B}}(x)=s(\mu _{A}(x),\mu _{B}(x))}.

By the definition of the t-norm, we see that the union and intersection arecommutative,monotonic,associative, and have both anull and anidentity element. For the intersection, these are ∅ andU, respectively, while for the union, these are reversed. However, the union of a fuzzy set and its complement may not result in the full universeU, and the intersection of them may not give the empty set ∅. Since the intersection and union are associative, it is natural to define the intersection and union of a finitefamily of fuzzy sets recursively. It is noteworthy that the generally accepted standard operators for the union and intersection of fuzzy sets are the max and min operators:

∀x∈U:μ¬A(x)=n(μA(x)).{\displaystyle \forall x\in U:\mu _{\neg {A}}(x)=n(\mu _{A}(x)).}
  • The triple of fuzzy intersection, union and complement form aDe Morgan Triplet. That is,De Morgan's laws extend to this triple.
Examples for fuzzy intersection/union pairs with standard negator can be derived from samples provided in the article aboutt-norms.
The fuzzy intersection is notidempotent in general, because the standard t-normmin is the only one which has this property. Indeed, if the arithmetic multiplication is used as the t-norm, the resulting fuzzy intersection operation is not idempotent. That is, iteratively taking the intersection of a fuzzy set with itself is not trivial. It instead defines them-th power of a fuzzy set, which can be canonically generalized for non-integer exponents in the following way:
∀x∈U:μAν(x)=μA(x)ν.{\displaystyle \forall x\in U:\mu _{A^{\nu }}(x)=\mu _{A}(x)^{\nu }.}

The case of exponent two is special enough to be given a name.

∀x∈U:μCON(A)(x)=μA2(x)=μA(x)2.{\displaystyle \forall x\in U:\mu _{CON(A)}(x)=\mu _{A^{2}}(x)=\mu _{A}(x)^{2}.}

Taking00=1{\displaystyle 0^{0}=1}, we haveA0=U{\displaystyle A^{0}=U} andA1=A.{\displaystyle A^{1}=A.}

∀x∈U:μA∖B(x)=t(μA(x),n(μB(x))),{\displaystyle \forall x\in U:\mu _{A\setminus {B}}(x)=t(\mu _{A}(x),n(\mu _{B}(x))),}
which meansA∖B=A∩¬B{\displaystyle A\setminus B=A\cap \neg {B}}, e. g.:
∀x∈U:μA∖B(x)=min(μA(x),1−μB(x)).{\displaystyle \forall x\in U:\mu _{A\setminus {B}}(x)=\min(\mu _{A}(x),1-\mu _{B}(x)).}[7]
Another proposal for a set difference could be:
∀x∈U:μA−B(x)=μA(x)−t(μA(x),μB(x)).{\displaystyle \forall x\in U:\mu _{A-{B}}(x)=\mu _{A}(x)-t(\mu _{A}(x),\mu _{B}(x)).}[7]
  • Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking theabsolute value, giving
∀x∈U:μA△B(x)=|μA(x)−μB(x)|,{\displaystyle \forall x\in U:\mu _{A\triangle B}(x)=|\mu _{A}(x)-\mu _{B}(x)|,}
or by using a combination of justmax,min, and standard negation, giving
∀x∈U:μA△B(x)=max(min(μA(x),1−μB(x)),min(μB(x),1−μA(x))).{\displaystyle \forall x\in U:\mu _{A\triangle B}(x)=\max(\min(\mu _{A}(x),1-\mu _{B}(x)),\min(\mu _{B}(x),1-\mu _{A}(x))).}[7]
Axioms for definition of generalized symmetric differences analogous to those for t-norms, t-conorms, and negators have been proposed by Vemur et al. (2014) with predecessors by Alsina et al. (2005) and Bedregal et al. (2009).[7]
  • In contrast to crisp sets, averaging operations can also be defined for fuzzy sets.

Disjoint fuzzy sets

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In contrast to the general ambiguity of intersection and union operations, there is clearness for disjoint fuzzy sets:Two fuzzy setsA,B{\displaystyle A,B} aredisjoint iff

∀x∈U:μA(x)=0∨μB(x)=0{\displaystyle \forall x\in U:\mu _{A}(x)=0\lor \mu _{B}(x)=0}

which is equivalent to

∄{\displaystyle \nexists }x∈U:μA(x)>0∧μB(x)>0{\displaystyle x\in U:\mu _{A}(x)>0\land \mu _{B}(x)>0}

and also equivalent to

∀x∈U:min(μA(x),μB(x))=0{\displaystyle \forall x\in U:\min(\mu _{A}(x),\mu _{B}(x))=0}

We keep in mind thatmin/max is a t/s-norm pair, and any other will work here as well.

Fuzzy sets are disjoint if and only if their supports aredisjoint according to the standard definition for crisp sets.

For disjoint fuzzy setsA,B{\displaystyle A,B} any intersection will give ∅, and any union will give the same result, which is denoted as

A∪˙B=A∪B{\displaystyle A\,{\dot {\cup }}\,B=A\cup B}

with its membership function given by

∀x∈U:μA∪˙B(x)=μA(x)+μB(x){\displaystyle \forall x\in U:\mu _{A{\dot {\cup }}B}(x)=\mu _{A}(x)+\mu _{B}(x)}

Note that only one of both summands is greater than zero.

For disjoint fuzzy setsA,B{\displaystyle A,B} the following holds true:

Supp⁡(A∪˙B)=Supp⁡(A)∪Supp⁡(B){\displaystyle \operatorname {Supp} (A\,{\dot {\cup }}\,B)=\operatorname {Supp} (A)\cup \operatorname {Supp} (B)}

This can be generalized to finite families of fuzzy sets as follows:Given a familyA=(Ai)i∈I{\displaystyle A=(A_{i})_{i\in I}} of fuzzy sets with index setI (e.g.I = {1,2,3,...,n}). This family is(pairwise) disjoint iff

for all x∈U there exists at most one i∈I such that μAi(x)>0.{\displaystyle {\text{for all }}x\in U{\text{ there exists at most one }}i\in I{\text{ such that }}\mu _{A_{i}}(x)>0.}

A family of fuzzy setsA=(Ai)i∈I{\displaystyle A=(A_{i})_{i\in I}} is disjoint, iff the family of underlying supportsSupp∘A=(Supp⁡(Ai))i∈I{\displaystyle \operatorname {Supp} \circ A=(\operatorname {Supp} (A_{i}))_{i\in I}} is disjoint in the standard sense for families of crisp sets.

Independent of the t/s-norm pair, intersection of a disjoint family of fuzzy sets will give ∅ again, while the union has no ambiguity:

⋃i∈I˙Ai=⋃i∈IAi{\displaystyle {\dot {\bigcup \limits _{i\in I}}}\,A_{i}=\bigcup _{i\in I}A_{i}}

with its membership function given by

∀x∈U:μ⋃i∈I˙Ai(x)=∑i∈IμAi(x){\displaystyle \forall x\in U:\mu _{{\dot {\bigcup \limits _{i\in I}}}A_{i}}(x)=\sum _{i\in I}\mu _{A_{i}}(x)}

Again only one of the summands is greater than zero.

For disjoint families of fuzzy setsA=(Ai)i∈I{\displaystyle A=(A_{i})_{i\in I}} the following holds true:

Supp⁡(⋃i∈I˙Ai)=⋃i∈ISupp⁡(Ai){\displaystyle \operatorname {Supp} \left({\dot {\bigcup \limits _{i\in I}}}\,A_{i}\right)=\bigcup \limits _{i\in I}\operatorname {Supp} (A_{i})}

Scalar cardinality

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For a fuzzy setA{\displaystyle A} with finite supportSupp⁡(A){\displaystyle \operatorname {Supp} (A)} (i.e. a "finite fuzzy set"), itscardinality (akascalar cardinality orsigma-count) is given by

Card⁡(A)=sc⁡(A)=|A|=∑x∈UμA(x){\displaystyle \operatorname {Card} (A)=\operatorname {sc} (A)=|A|=\sum _{x\in U}\mu _{A}(x)}.

In the case thatU itself is a finite set, therelative cardinality is given by

RelCard⁡(A)=‖A‖=sc⁡(A)/|U|=|A|/|U|{\displaystyle \operatorname {RelCard} (A)=\|A\|=\operatorname {sc} (A)/|U|=|A|/|U|}.

This can be generalized for the divisor to be a non-empty fuzzy set: For fuzzy setsA,G{\displaystyle A,G} withG ≠ ∅, we can define therelative cardinality by:

RelCard⁡(A,G)=sc⁡(A|G)=sc⁡(A∩G)/sc⁡(G){\displaystyle \operatorname {RelCard} (A,G)=\operatorname {sc} (A|G)=\operatorname {sc} (A\cap {G})/\operatorname {sc} (G)},

which looks very similar to the expression forconditional probability.Note:

Distance and similarity

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For any fuzzy setA{\displaystyle A} the membership functionμA:U→[0,1]{\displaystyle \mu _{A}:U\to [0,1]} can be regarded as a familyμA=(μA(x))x∈U∈[0,1]U{\displaystyle \mu _{A}=(\mu _{A}(x))_{x\in U}\in [0,1]^{U}}. The latter is ametric space with several metricsd{\displaystyle d} known. A metric can be derived from anorm (vector norm)‖‖{\displaystyle \|\,\|} via

d(α,β)=‖α−β‖{\displaystyle d(\alpha ,\beta )=\|\alpha -\beta \|}.

For instance, ifU{\displaystyle U} is finite, i.e.U={x1,x2,...xn}{\displaystyle U=\{x_{1},x_{2},...x_{n}\}}, such a metric may be defined by:

d(α,β):=max{|α(xi)−β(xi)|:i=1,...,n}{\displaystyle d(\alpha ,\beta ):=\max\{|\alpha (x_{i})-\beta (x_{i})|:i=1,...,n\}} whereα{\displaystyle \alpha } andβ{\displaystyle \beta } are sequences of real numbers between 0 and 1.

For infiniteU{\displaystyle U}, the maximum can be replaced by a supremum.Because fuzzy sets are unambiguously defined by their membership function, this metric can be used to measure distances between fuzzy sets on the same universe:

d(A,B):=d(μA,μB){\displaystyle d(A,B):=d(\mu _{A},\mu _{B})},

which becomes in the above sample:

d(A,B)=max{|μA(xi)−μB(xi)|:i=1,...,n}{\displaystyle d(A,B)=\max\{|\mu _{A}(x_{i})-\mu _{B}(x_{i})|:i=1,...,n\}}.

Again for infiniteU{\displaystyle U} the maximum must be replaced by a supremum. Other distances (like the canonical 2-norm) may diverge, if infinite fuzzy sets are too different, e.g.,∅{\displaystyle \varnothing } andU{\displaystyle U}.

Similarity measures (here denoted byS{\displaystyle S}) may then be derived from the distance, e.g. after a proposal by Koczy:

S=1/(1+d(A,B)){\displaystyle S=1/(1+d(A,B))} ifd(A,B){\displaystyle d(A,B)} is finite,0{\displaystyle 0} else,

or after Williams and Steele:

S=exp⁡(−αd(A,B)){\displaystyle S=\exp(-\alpha {d(A,B)})} ifd(A,B){\displaystyle d(A,B)} is finite,0{\displaystyle 0} else

whereα>0{\displaystyle \alpha >0} is a steepness parameter andexp⁡(x)=ex{\displaystyle \exp(x)=e^{x}}.[citation needed]

L-fuzzy sets

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Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable)algebra orstructureL{\displaystyle L} of a given kind; usually it is required thatL{\displaystyle L} be at least aposet orlattice. These are usually calledL-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizations were first considered in 1967 byJoseph Goguen, who was a student of Zadeh.[8] A classical corollary may be indicating truth and membership values by {f, t} instead of {0, 1}.

An extension of fuzzy sets has been provided byAtanassov. Anintuitionistic fuzzy set (IFS)A{\displaystyle A} is characterized by two functions:

1.μA(x){\displaystyle \mu _{A}(x)} – degree of membership ofx
2.νA(x){\displaystyle \nu _{A}(x)} – degree of non-membership ofx

with functionsμA,νA:U→[0,1]{\displaystyle \mu _{A},\nu _{A}:U\to [0,1]} with∀x∈U:μA(x)+νA(x)≤1{\displaystyle \forall x\in U:\mu _{A}(x)+\nu _{A}(x)\leq 1}.

This resembles a situation like some person denoted byx{\displaystyle x} voting

After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions.

For this situation, special "intuitive fuzzy" negators, t- and s-norms can be defined. WithD∗={(α,β)∈[0,1]2:α+β=1}{\displaystyle D^{*}=\{(\alpha ,\beta )\in [0,1]^{2}:\alpha +\beta =1\}} and by combining both functions to(μA,νA):U→D∗{\displaystyle (\mu _{A},\nu _{A}):U\to D^{*}} this situation resembles a special kind ofL-fuzzy sets.

Once more, this has been expanded by definingpicture fuzzy sets (PFS) as follows: A PFS A is characterized by three functions mappingU to [0, 1]:μA,ηA,νA{\displaystyle \mu _{A},\eta _{A},\nu _{A}}, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition∀x∈U:μA(x)+ηA(x)+νA(x)≤1{\displaystyle \forall x\in U:\mu _{A}(x)+\eta _{A}(x)+\nu _{A}(x)\leq 1}This expands the voting sample above by an additional possibility of "refusal of voting".

WithD∗={(α,β,γ)∈[0,1]3:α+β+γ=1}{\displaystyle D^{*}=\{(\alpha ,\beta ,\gamma )\in [0,1]^{3}:\alpha +\beta +\gamma =1\}} and special "picture fuzzy" negators, t- and s-norms this resembles just another type ofL-fuzzy sets.[9]

Pythagorean fuzzy sets

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One extension of IFS is what is known as Pythagorean fuzzy sets. Such sets satisfy the constraintμA(x)2+νA(x)2≤1{\displaystyle \mu _{A}(x)^{2}+\nu _{A}(x)^{2}\leq 1}, which is reminiscent of the Pythagorean theorem.[10][11][12] Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition ofμA(x)+νA(x)≤1{\displaystyle \mu _{A}(x)+\nu _{A}(x)\leq 1} is not valid. However, the less restrictive condition ofμA(x)2+νA(x)2≤1{\displaystyle \mu _{A}(x)^{2}+\nu _{A}(x)^{2}\leq 1} may be suitable in more domains.[13][14]

Fuzzy logic

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Main article:Fuzzy logic

As an extension of the case ofmulti-valued logic, valuations (μ:Vo→W{\displaystyle \mu :{\mathit {V}}_{o}\to {\mathit {W}}}) ofpropositional variables (Vo{\displaystyle {\mathit {V}}_{o}}) into a set of membership degrees (W{\displaystyle {\mathit {W}}}) can be thought of asmembership functions mappingpredicates into fuzzy sets (or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logic can be extended to allow for fuzzypremises from which graded conclusions may be drawn.[15]

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in theengineering fields ofautomated control andknowledge engineering, and which encompasses many topics involving fuzzy sets and "approximated reasoning."[16]

Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found atfuzzy logic.

Fuzzy number

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Main article:Fuzzy number

Afuzzy number[17] is a fuzzy set that satisfies all the following conditions:

If these conditions are not satisfied, then A is not afuzzy number. The core of this fuzzy number is asingleton; its location is:

C(A)=x∗:μA(x∗)=1{\displaystyle \,C(A)=x^{*}:\mu _{A}(x^{*})=1}

Fuzzy numbers can be likened to thefunfair game "guess your weight," where someone guesses the contestant's weight, with closer guesses being more correct, and where the guesser "wins" if he or she guesses near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function).

The kernelK(A)=Kern⁡(A){\displaystyle K(A)=\operatorname {Kern} (A)} of a fuzzy intervalA{\displaystyle A} is defined as the 'inner' part, without the 'outbound' parts where the membership value is constant ad infinitum. In other words, the smallest subset ofR{\displaystyle \mathbb {R} } whereμA(x){\displaystyle \mu _{A}(x)} is constant outside of it, is defined as the kernel.

However, there are other concepts of fuzzy numbers and intervals as some authors do not insist on convexity.

Fuzzy categories

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The use ofset membership as a key component ofcategory theory can be generalized to fuzzy sets. This approach, which began in 1968 shortly after the introduction of fuzzy set theory,[18] led to the development ofGoguen categories in the 21st century.[19][20] In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as inL-fuzzy sets.[20][21]

There are numerous mathematical extensions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in 1965 by Zadeh, many new mathematical constructions and theories treating imprecision, inaccuracy, vagueness, uncertainty and vulnerability have been developed. Some of these constructions and theories are extensions of fuzzy set theory, while others attempt to mathematically model inaccuracy/vagueness and uncertainty in a different way. The diversity of such constructions and corresponding theories includes:

  • Fuzzy Sets (Zadeh, 1965)
  • interval sets (Moore, 1966),
  • L-fuzzy sets (Goguen, 1967),
  • flou sets (Gentilhomme, 1968),
  • type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975),
  • interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),
  • level fuzzy sets (Radecki, 1977)
  • rough sets (Pawlak, 1982),
  • intuitionistic fuzzy sets (Atanassov, 1983),
  • fuzzy multisets (Yager, 1986),
  • intuitionistic L-fuzzy sets (Atanassov, 1986),
  • rough multisets (Grzymala-Busse, 1987),
  • fuzzy rough sets (Nakamura, 1988),
  • real-valued fuzzy sets (Blizard, 1989),
  • vague sets (Wen-Lung Gau and Buehrer, 1993),
  • α-level sets (Yao, 1997),
  • shadowed sets (Pedrycz, 1998),
  • neutrosophic sets (NSs) (Smarandache, 1998),
  • bipolar fuzzy sets (Wen-Ran Zhang, 1998),
  • genuine sets (Demirci, 1999),
  • soft sets (Molodtsov, 1999),
  • complex fuzzy set (2002),
  • intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003)
  • L-fuzzy rough sets (Radzikowska and Kerre, 2004),
  • multi-fuzzy sets (Sabu Sebastian, 2009),
  • generalized rough fuzzy sets (Feng, 2010)
  • rough intuitionistic fuzzy sets (Thomas and Nair, 2011),
  • soft rough fuzzy sets (Meng, Zhang and Qin, 2011)
  • soft fuzzy rough sets (Meng, Zhang and Qin, 2011)
  • soft multisets (Alkhazaleh, Salleh and Hassan, 2011)
  • fuzzy soft multisets (Alkhazaleh and Salleh, 2012)
  • pythagorean fuzzy set (Yager, 2013),
  • picture fuzzy set (Cuong, 2013),
  • spherical fuzzy set (Mahmood, 2018).

Fuzzy relation equation

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Thefuzzy relation equation is an equation of the formA ·R =B, whereA andB are fuzzy sets,R is a fuzzy relation, andA ·R stands for thecomposition ofA with R[citation needed].

Entropy

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A measured of fuzziness for fuzzy sets of universeU{\displaystyle U} should fulfill the following conditions for allx∈U{\displaystyle x\in U}:

  1. d(A)=0{\displaystyle d(A)=0} ifA{\displaystyle A} is a crisp set:μA(x)∈{0,1}{\displaystyle \mu _{A}(x)\in \{0,\,1\}}
  2. d(A){\displaystyle d(A)} has a unique maximum iff∀x∈U:μA(x)=0.5{\displaystyle \forall x\in U:\mu _{A}(x)=0.5}
  3. ∀x∈U:(μA(x)≤μB(x)≤0.5)∨(μA(x)≥μB(x)≥0.5){\displaystyle \forall x\in U:(\mu _{A}(x)\leq \mu _{B}(x)\leq 0.5)\lor (\mu _{A}(x)\geq \mu _{B}(x)\geq 0.5)}⇒d(A)≤d(B){\displaystyle \Rightarrow d(A)\leq d(B)}, which means thatA is "crisper" thanB.
  4. d(¬A)=d(A){\displaystyle d(\neg {A})=d(A)}

In this cased(A){\displaystyle d(A)} is called theentropy of the fuzzy setA.

ForfiniteU={x1,x2,...xn}{\displaystyle U=\{x_{1},x_{2},...x_{n}\}} the entropy of a fuzzy setA{\displaystyle A} is given by

d(A)=H(A)+H(¬A){\displaystyle d(A)=H(A)+H(\neg {A})},
H(A)=−k∑i=1nμA(xi)ln⁡μA(xi){\displaystyle H(A)=-k\sum _{i=1}^{n}\mu _{A}(x_{i})\ln \mu _{A}(x_{i})}

or just

d(A)=−k∑i=1nS(μA(xi)){\displaystyle d(A)=-k\sum _{i=1}^{n}S(\mu _{A}(x_{i}))}

whereS(x)=He(x){\displaystyle S(x)=H_{e}(x)} isShannon's function (natural entropy function)

S(α)=−αln⁡α−(1−α)ln⁡(1−α), α∈[0,1]{\displaystyle S(\alpha )=-\alpha \ln \alpha -(1-\alpha )\ln(1-\alpha ),\ \alpha \in [0,1]}

andk{\displaystyle k} is a constant depending on the measure unit and the logarithm base used (here we have used the natural basee).The physical interpretation ofk is theBoltzmann constantkB.

LetA{\displaystyle A} be a fuzzy set with acontinuous membership function (fuzzy variable). Then

H(A)=−k∫−∞∞Cr⁡{A≥t}ln⁡Cr⁡{A≥t}dt{\displaystyle H(A)=-k\int _{-\infty }^{\infty }\operatorname {Cr} \lbrace A\geq t\rbrace \ln \operatorname {Cr} \lbrace A\geq t\rbrace \,dt}

and its entropy is

d(A)=−k∫−∞∞S(Cr⁡{A≥t})dt.{\displaystyle d(A)=-k\int _{-\infty }^{\infty }S(\operatorname {Cr} \lbrace A\geq t\rbrace )\,dt.}[22][23]

Extensions

[edit]

There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in 1965, many new mathematical constructions and theories treating imprecision, inexactness, ambiguity, and uncertainty have been developed. Some of these constructions and theories are extensions of fuzzy set theory, while others try to mathematically model imprecision and uncertainty in a different way.[24]

See also

[edit]

References

[edit]
  1. ^L. A. Zadeh (1965)"Fuzzy sets"Archived 2015-08-13 at theWayback Machine.Information and Control 8 (3) 338–353.
  2. ^Klaua, D. (1965) Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876. A recent in-depth analysis of this paper has been provided byGottwald, S. (2010). "An early approach toward graded identity and graded membership in set theory".Fuzzy Sets and Systems.161 (18):2369–2379.doi:10.1016/j.fss.2009.12.005.
  3. ^D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.
  4. ^Liang, Lily R.; Lu, Shiyong; Wang, Xuena; Lu, Yi; Mandal, Vinay; Patacsil, Dorrelyn; Kumar, Deepak (2006)."FM-test: A fuzzy-set-theory-based approach to differential gene expression data analysis".BMC Bioinformatics.7 (Suppl 4): S7.doi:10.1186/1471-2105-7-S4-S7.PMC 1780132.PMID 17217525.
  5. ^"AAAI". Archived fromthe original on August 5, 2008.
  6. ^Bellman, Richard; Giertz, Magnus (1973). "On the analytic formalism of the theory of fuzzy sets".Information Sciences.5:149–156.doi:10.1016/0020-0255(73)90009-1.
  7. ^abcdN.R. Vemuri, A.S. Hareesh, M.S. Srinath:Set Difference and Symmetric Difference of Fuzzy Sets, in: Fuzzy Sets Theory and Applications 2014, Liptovský Ján, Slovak Republic
  8. ^Goguen, J.A (1967)."L-fuzzy sets".Journal of Mathematical Analysis and Applications.18:145–174.doi:10.1016/0022-247X(67)90189-8.
  9. ^Bui Cong Cuong, Vladik Kreinovich, Roan Thi Ngan:A classification of representable t-norm operators for picture fuzzy sets, in: Departmental Technical Reports (CS). Paper 1047, 2016
  10. ^Yager, Ronald R. (June 2013). "Pythagorean fuzzy subsets".2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS). pp. 57–61.doi:10.1109/IFSA-NAFIPS.2013.6608375.ISBN 978-1-4799-0348-1.S2CID 36286152.
  11. ^Yager, Ronald R (2013). "Pythagorean membership grades in multicriteria decision making".IEEE Transactions on Fuzzy Systems.22 (4):958–965.doi:10.1109/TFUZZ.2013.2278989.S2CID 37195356.
  12. ^Yager, Ronald R. (December 2015).Properties and applications of Pythagorean fuzzy sets. Cham: Springer. pp. 119–136.ISBN 978-3-319-26302-1.
  13. ^Yanase J, Triantaphyllou E (2019). "A Systematic Survey of Computer-Aided Diagnosis in Medicine: Past and Present Developments".Expert Systems with Applications.138 112821.doi:10.1016/j.eswa.2019.112821.S2CID 199019309.
  14. ^Yanase J, Triantaphyllou E (2019). "The Seven Key Challenges for the Future of Computer-Aided Diagnosis in Medicine".International Journal of Medical Informatics.129:413–422.doi:10.1016/j.ijmedinf.2019.06.017.PMID 31445285.S2CID 198287435.
  15. ^Siegfried Gottwald, 2001.A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd.,ISBN 978-0-86380-262-1
  16. ^Zadeh, L.A. (1975). "The concept of a linguistic variable and its application to approximate reasoning—I".Information Sciences.8 (3):199–249.doi:10.1016/0020-0255(75)90036-5.
  17. ^Zadeh, L.A. (1999). "Fuzzy sets as a basis for a theory of possibility".Fuzzy Sets and Systems.100:9–34.doi:10.1016/S0165-0114(99)80004-9.
  18. ^J. A. Goguen "Categories of fuzzy sets: applications of non-Cantorian set theory" PhD Thesis University of California, Berkeley, 1968
  19. ^Michael Winter "Goguen Categories:A Categorical Approach to L-fuzzy Relations" 2007SpringerISBN 9781402061639
  20. ^abWinter, Michael (2003). "Representation theory of Goguen categories".Fuzzy Sets and Systems.138:85–126.doi:10.1016/S0165-0114(02)00508-0.
  21. ^Goguen, J.A (1967)."L-fuzzy sets".Journal of Mathematical Analysis and Applications.18:145–174.doi:10.1016/0022-247X(67)90189-8.
  22. ^Xuecheng, Liu (1992). "Entropy, distance measure and similarity measure of fuzzy sets and their relations".Fuzzy Sets and Systems.52 (3):305–318.doi:10.1016/0165-0114(92)90239-Z.
  23. ^Li, Xiang (2015)."Fuzzy cross-entropy".Journal of Uncertainty Analysis and Applications.3.doi:10.1186/s40467-015-0029-5.
  24. ^Burgin & Chunihin 1997;Kerre 2001;Deschrijver & Kerre 2003.

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