Fuzzy measure theory

<h2 id="definitions">Definitions</h2>
<p>Let 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathbf {X} }
  
 be a <a href="/facts/Universe_of_discourse/T1WRQxx6">universe of discourse</a>, 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 be a <a href="/facts/Class_(mathematics)/F5EmDd9b">class</a> of <a href="/facts/Subset/SJGERmbA">subsets</a> of 
  
    
      
        
          X
        
      
    
    {\displaystyle \mathbf {X} }
  
, and 
  
    
      
        E
        ,
        F
        ∈
        
          
            C
          
        
      
    
    {\displaystyle E,F\in {\mathcal {C}}}
  
.  A <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        g
        :
        
          
            C
          
        
        →
        
          R
        
      
    
    {\displaystyle g:{\mathcal {C}}\to \mathbb {R} }
  
 where
</p>
<ol><li>
  
    
      
        ∅
        ∈
        
          
            C
          
        
        ⇒
        g
        (
        ∅
        )
        =
        0
      
    
    {\displaystyle \emptyset \in {\mathcal {C}}\Rightarrow g(\emptyset )=0}
  
</li>
<li>
  
    
      
        E
        ⊆
        F
        ⇒
        g
        (
        E
        )
        ≤
        g
        (
        F
        )
      
    
    {\displaystyle E\subseteq F\Rightarrow g(E)\leq g(F)}
  
</li></ol>
<p>is called a <i>fuzzy measure</i>. 
A fuzzy measure is called <i>normalized</i> or <i>regular</i> if 
  
    
      
        g
        (
        
          X
        
        )
        =
        1
      
    
    {\displaystyle g(\mathbf {X} )=1}
  
.
</p>
<h2 id="properties-of-fuzzy-measures">Properties of fuzzy measures</h2>
<p>A fuzzy measure is:
</p>
<ul><li>additive if for any 
  
    
      
        E
        ,
        F
        ∈
        
          
            C
          
        
      
    
    {\displaystyle E,F\in {\mathcal {C}}}
  
 such that 
  
    
      
        E
        ∩
        F
        =
        ∅
      
    
    {\displaystyle E\cap F=\emptyset }
  
, we have 
  
    
      
        g
        (
        E
        ∪
        F
        )
        =
        g
        (
        E
        )
        +
        g
        (
        F
        )
        .
      
    
    {\displaystyle g(E\cup F)=g(E)+g(F).}
  
;</li>
<li>supermodular if for any 
  
    
      
        E
        ,
        F
        ∈
        
          
            C
          
        
      
    
    {\displaystyle E,F\in {\mathcal {C}}}
  
, we have 
  
    
      
        g
        (
        E
        ∪
        F
        )
        +
        g
        (
        E
        ∩
        F
        )
        ≥
        g
        (
        E
        )
        +
        g
        (
        F
        )
      
    
    {\displaystyle g(E\cup F)+g(E\cap F)\geq g(E)+g(F)}
  
;</li>
<li><a href="/facts/Submodular/YNnlKb4U">submodular</a> if for any 
  
    
      
        E
        ,
        F
        ∈
        
          
            C
          
        
      
    
    {\displaystyle E,F\in {\mathcal {C}}}
  
, we have 
  
    
      
        g
        (
        E
        ∪
        F
        )
        +
        g
        (
        E
        ∩
        F
        )
        ≤
        g
        (
        E
        )
        +
        g
        (
        F
        )
      
    
    {\displaystyle g(E\cup F)+g(E\cap F)\leq g(E)+g(F)}
  
;</li>
<li>superadditive if for any 
  
    
      
        E
        ,
        F
        ∈
        
          
            C
          
        
      
    
    {\displaystyle E,F\in {\mathcal {C}}}
  
 such that 
  
    
      
        E
        ∩
        F
        =
        ∅
      
    
    {\displaystyle E\cap F=\emptyset }
  
, we have 
  
    
      
        g
        (
        E
        ∪
        F
        )
        ≥
        g
        (
        E
        )
        +
        g
        (
        F
        )
      
    
    {\displaystyle g(E\cup F)\geq g(E)+g(F)}
  
;</li>
<li>subadditive if for any 
  
    
      
        E
        ,
        F
        ∈
        
          
            C
          
        
      
    
    {\displaystyle E,F\in {\mathcal {C}}}
  
 such that 
  
    
      
        E
        ∩
        F
        =
        ∅
      
    
    {\displaystyle E\cap F=\emptyset }
  
, we have 
  
    
      
        g
        (
        E
        ∪
        F
        )
        ≤
        g
        (
        E
        )
        +
        g
        (
        F
        )
      
    
    {\displaystyle g(E\cup F)\leq g(E)+g(F)}
  
;</li>
<li>symmetric if for any 
  
    
      
        E
        ,
        F
        ∈
        
          
            C
          
        
      
    
    {\displaystyle E,F\in {\mathcal {C}}}
  
, we have 
  
    
      
        
          |
        
        E
        
          |
        
        =
        
          |
        
        F
        
          |
        
      
    
    {\displaystyle |E|=|F|}
  
 implies 
  
    
      
        g
        (
        E
        )
        =
        g
        (
        F
        )
      
    
    {\displaystyle g(E)=g(F)}
  
;</li>
<li>Boolean if for any 
  
    
      
        E
        ∈
        
          
            C
          
        
      
    
    {\displaystyle E\in {\mathcal {C}}}
  
, we have 
  
    
      
        g
        (
        E
        )
        =
        0
      
    
    {\displaystyle g(E)=0}
  
 or 
  
    
      
        g
        (
        E
        )
        =
        1
      
    
    {\displaystyle g(E)=1}
  
.</li></ul>
<p>Understanding the properties of fuzzy measures is useful in application.  When a fuzzy measure is used to define a function such as the <a href="/facts/Sugeno_integral/DCHRRBBI">Sugeno integral</a> or <a href="/facts/Choquet_integral/qApIGbVb">Choquet integral</a>, 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 <a href="/facts/Lebesgue_integral/f4AgvC6x">Lebesgue integral</a>.  In discrete cases, a symmetric fuzzy measure will result in the <a href="/facts/Ordered_weighted_averaging_aggregation_operator/x5bHdbQz">ordered weighted averaging</a> (OWA) operator.  Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
</p>
<h2 id="mbius-representation">Möbius representation</h2>
<p>Let <i>g</i> be a fuzzy measure. The Möbius representation of <i>g</i> is given by the set function <i>M</i>, where for every 
  
    
      
        E
        ,
        F
        ⊆
        X
      
    
    {\displaystyle E,F\subseteq X}
  
, 
</p>

M
        (
        E
        )
        =
        
          ∑
          
            F
            ⊆
            E
          
        
        (
        −
        1
        
          )
          
            
              |
            
            E
            ∖
            F
            
              |
            
          
        
        g
        (
        F
        )
        .
      
    
    {\displaystyle M(E)=\sum _{F\subseteq E}(-1)^{|E\backslash F|}g(F).}

<p>The equivalent axioms in Möbius representation are:
</p>
<ol><li>
  
    
      
        M
        (
        ∅
        )
        =
        0
      
    
    {\displaystyle M(\emptyset )=0}
  
.</li>
<li>
  
    
      
        
          ∑
          
            F
            ⊆
            E
            
              |
            
            i
            ∈
            F
          
        
        M
        (
        F
        )
        ≥
        0
      
    
    {\displaystyle \sum _{F\subseteq E|i\in F}M(F)\geq 0}
  
, for all 
  
    
      
        E
        ⊆
        
          X
        
      
    
    {\displaystyle E\subseteq \mathbf {X} }
  
 and all 
  
    
      
        i
        ∈
        E
      
    
    {\displaystyle i\in E}
  
</li></ol>
<p>A fuzzy measure in Möbius representation  <i>M</i> is called <i>normalized</i>
if 
  
    
      
        
          ∑
          
            E
            ⊆
            
              X
            
          
        
        M
        (
        E
        )
        =
        1.
      
    
    {\displaystyle \sum _{E\subseteq \mathbf {X} }M(E)=1.}

</p><p>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 <i>g</i> in standard representation can be recovered from the Möbius form using the Zeta transform:
</p>

g
        (
        E
        )
        =
        
          ∑
          
            F
            ⊆
            E
          
        
        M
        (
        F
        )
        ,
        ∀
        E
        ⊆
        
          X
        
        .
      
    
    {\displaystyle g(E)=\sum _{F\subseteq E}M(F),\forall E\subseteq \mathbf {X} .}

<h2 id="simplification-assumptions-for-fuzzy-measures">Simplification assumptions for fuzzy measures</h2>
<p>Fuzzy measures are defined on a <a href="/facts/Semiring/7cSvWXVQ">semiring of sets</a> or <a href="/facts/Monotone_class/7kk5VbNj">monotone class</a>, which may be as granular as the <a href="/facts/Power_set/FVuf8PDD">power set</a> 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 <a href="/facts/Multi-criteria_decision_analysis/IXQLJgBx">multi-criteria decision analysis</a> 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 <i>additive</i>, it will hold that 
  
    
      
        g
        (
        E
        )
        =
        
          ∑
          
            i
            ∈
            E
          
        
        g
        (
        {
        i
        }
        )
      
    
    {\displaystyle g(E)=\sum _{i\in E}g(\{i\})}
  
 and the values of the fuzzy measure can be evaluated from the values on X.  Similarly, a <i>symmetric</i> fuzzy measure is defined uniquely by |X| values.  Two important fuzzy measures that can be used are the Sugeno- or 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
-fuzzy measure and <i>k</i>-additive measures, introduced by Sugeno<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> and Grabisch<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> respectively.
</p>
<h3>Sugeno <i>λ</i>-measure</h3>
<p>The Sugeno 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
-measure is a special case of fuzzy measures defined iteratively. It has the following definition:
</p>
<h4>Definition</h4>
<p>Let 
  
    
      
        
          X
        
        =
        
          {
          
            
              x
              
                1
              
            
            ,
            …
            ,
            
              x
              
                n
              
            
          
          }
        
      
    
    {\displaystyle \mathbf {X} =\left\lbrace x_{1},\dots ,x_{n}\right\rbrace }
  
  be a finite set and let 
  
    
      
        λ
        ∈
        (
        −
        1
        ,
        +
        ∞
        )
      
    
    {\displaystyle \lambda \in (-1,+\infty )}
  
. A Sugeno 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
-measure is a function 
  
    
      
        g
        :
        
          2
          
            X
          
        
        →
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle g:2^{X}\to [0,1]}
  
 such that
</p>
<ol><li>
  
    
      
        g
        (
        X
        )
        =
        1
      
    
    {\displaystyle g(X)=1}
  
.</li>
<li>if 
  
    
      
        A
        ,
        B
        ⊆
        
          X
        
      
    
    {\displaystyle A,B\subseteq \mathbf {X} }
  
 (alternatively 
  
    
      
        A
        ,
        B
        ∈
        
          2
          
            
              X
            
          
        
      
    
    {\displaystyle A,B\in 2^{\mathbf {X} }}
  
) with 
  
    
      
        A
        ∩
        B
        =
        ∅
      
    
    {\displaystyle A\cap B=\emptyset }
  
 then 
  
    
      
        g
        (
        A
        ∪
        B
        )
        =
        g
        (
        A
        )
        +
        g
        (
        B
        )
        +
        λ
        g
        (
        A
        )
        g
        (
        B
        )
      
    
    {\displaystyle g(A\cup B)=g(A)+g(B)+\lambda g(A)g(B)}
  
.</li></ol>
<p>As a convention, the value of g at a singleton set 
  
    
      
        
          {
          
            x
            
              i
            
          
          }
        
      
    
    {\displaystyle \left\lbrace x_{i}\right\rbrace }

is called a density and is denoted by 
  
    
      
        
          g
          
            i
          
        
        =
        g
        (
        
          {
          
            x
            
              i
            
          
          }
        
        )
      
    
    {\displaystyle g_{i}=g(\left\lbrace x_{i}\right\rbrace )}
  
. In addition, we have that 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 satisfies the property
</p>

λ
        +
        1
        =
        
          ∏
          
            i
            =
            1
          
          
            n
          
        
        (
        1
        +
        λ
        
          g
          
            i
          
        
        )
      
    
    {\displaystyle \lambda +1=\prod _{i=1}^{n}(1+\lambda g_{i})}
  
.
<p>Tahani and Keller <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> as well as Wang and Klir have shown that once the densities are known, it is possible to use the previous <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> to obtain the values of 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 uniquely.
</p>
<h3><i>k</i>-additive fuzzy measure</h3>
<p>The <i>k</i>-additive fuzzy measure limits the interaction between the subsets 
  
    
      
        E
        ⊆
        X
      
    
    {\displaystyle E\subseteq X}
  
 to size 
  
    
      
        
          |
        
        E
        
          |
        
        =
        k
      
    
    {\displaystyle |E|=k}
  
.  This drastically reduces the number of variables needed to define the fuzzy measure, and as <i>k</i> 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.
</p>
<h4>Definition</h4>
<p>A discrete fuzzy measure <i>g</i> on a set X is called <i>k-additive</i> (
  
    
      
        1
        ≤
        k
        ≤
        
          |
        
        
          X
        
        
          |
        
      
    
    {\displaystyle 1\leq k\leq |\mathbf {X} |}
  
) if its Möbius representation verifies 
  
    
      
        M
        (
        E
        )
        =
        0
      
    
    {\displaystyle M(E)=0}
  
, whenever 
  
    
      
        
          |
        
        E
        
          |
        
        >
        k
      
    
    {\displaystyle |E|>k}
  
 for any 
  
    
      
        E
        ⊆
        
          X
        
      
    
    {\displaystyle E\subseteq \mathbf {X} }
  
, and there exists a subset <i>F</i> with <i>k</i> elements such that 
  
    
      
        M
        (
        F
        )
        ≠
        0
      
    
    {\displaystyle M(F)\neq 0}
  
.
</p>
<h2 id="shapley-and-interaction-indices">Shapley and interaction indices</h2>
<p>In <a href="/facts/Game_theory/zkEIj1ya">game theory</a>, the <a href="/facts/Shapley_value/ElL2j4e6">Shapley value</a> 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.
</p><p>For a given fuzzy measure <i>g</i>, and 
  
    
      
        
          |
        
        
          X
        
        
          |
        
        =
        n
      
    
    {\displaystyle |\mathbf {X} |=n}
  
, the Shapley index for every 
  
    
      
        i
        ,
        …
        ,
        n
        ∈
        X
      
    
    {\displaystyle i,\dots ,n\in X}
  
 is:
</p>

ϕ
        (
        i
        )
        =
        
          ∑
          
            E
            ⊆
            
              X
            
            ∖
            {
            i
            }
          
        
        
          
            
              (
              n
              −
              
                |
              
              E
              
                |
              
              −
              1
              )
              !
              
                |
              
              E
              
                |
              
              !
            
            
              n
              !
            
          
        
        [
        g
        (
        E
        ∪
        {
        i
        }
        )
        −
        g
        (
        E
        )
        ]
        .
      
    
    {\displaystyle \phi (i)=\sum _{E\subseteq \mathbf {X} \backslash \{i\}}{\frac {(n-|E|-1)!|E|!}{n!}}[g(E\cup \{i\})-g(E)].}

<p>The Shapley value is the vector 
  
    
      
        
          ϕ
        
        (
        g
        )
        =
        (
        ψ
        (
        1
        )
        ,
        …
        ,
        ψ
        (
        n
        )
        )
        .
      
    
    {\displaystyle \mathbf {\phi } (g)=(\psi (1),\dots ,\psi (n)).}

</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Probability_theory/mFBn51rE">Probability theory</a></li>
<li><a href="/facts/Possibility_theory/XKHDEpbJ">Possibility theory</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Beliakov, Pradera and Calvo, <i>Aggregation Functions: A Guide for Practitioners</i>, Springer, New York 2007.</li>
<li>Wang, Zhenyuan, and, <a href="/facts/George_J._Klir/R7aAM6C8">George J. Klir</a>, <i>Fuzzy Measure Theory</i>, Plenum Press, New York, 1991.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://pami.uwaterloo.ca/tizhoosh/measure.htm">Fuzzy Measure Theory at Fuzzy Image Processing</a> <a href="https://web.archive.org/web/20190630034036/http://pami.uwaterloo.ca/tizhoosh/measure.htm">Archived</a> 2019-06-30 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Gustave Choquet (1953). "Theory of Capacities". Annales de l'Institut Fourier. 5: 131–295. <a href="/wiki/Gustave_Choquet" target="_blank">/wiki/Gustave_Choquet</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>M. Sugeno (1974). "Theory of fuzzy integrals and its applications. Ph.D. thesis". Tokyo Institute of Technology, Tokyo, Japan. <a href="/wiki/Tokyo_Institute_of_Technology" target="_blank">/wiki/Tokyo_Institute_of_Technology</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>M. Grabisch (1997). "k-order additive discrete fuzzy measures and their representation". Fuzzy Sets and Systems. 92 (2): 167–189. doi:10.1016/S0165-0114(97)00168-1. <a href="/wiki/Fuzzy_Sets_and_Systems" target="_blank">/wiki/Fuzzy_Sets_and_Systems</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>H. Tahani & J. Keller (1990). "Information Fusion in Computer Vision Using the Fuzzy Integral". IEEE Transactions on Systems, Man, and Cybernetics. 20 (3): 733–741. doi:10.1109/21.57289. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Fuzzy measure theory open-in-new

Fuzzy measure theory