Fuzzy measures and Choquet integral on discrete spaces Yasuo Narukawa 1 , Vicen¸c Torra
2
1
Toho Gakuen, 3-1-10 Naka, Kunitachi, Tokyo, 186-0004 Japan
[email protected] 2 Institut d’Investigaci´ o en Intel·lig`encia Artificial, Campus de Bellaterra, 08193 Bellaterra, Catalonia, Spain
[email protected]
Abstract. This paper studies some relationships between fuzzy relations, fuzzy graphs and fuzzy measure. It is shown that a fundamental theorem of Discrete Convex Analysis is derived from the theory of fuzzy measures and the Choquet integral. Keywords: Fuzzy relation, Fuzzy graph, Fuzzy measure, Choquet integral, Fuzzy integrals, Matroids
1
Introduction
Graphs are used to represent relations between objects. To include fuzziness in such relations fuzzy graphs were introduced. At present a large number and variety of applications have been developed that use graphs for knowledge representation. For example, they are used in the context of inference systems (e.g. in probabilistic and possibilistic networks [6, 8] or fuzzy cognitive graphs [17]), matching algorithms [16], or in dictionaries (as WordNet [13]), for easing information retrieval and recommendation systems [21]. Such applications have fostered the development of new tools for graphs and fuzzy graphs, as well as the development of other graph based formalisms (as e.g. in [3]). Roughly speaking, fuzzy graphs have been defined adding fuzziness either on the vertexes or on the edges. At present several alternative definitions exists for fuzzy graphs, some of them can be found in [7]. Several theoretical results have been obtained for fuzzy graphs. See e.g. [1, 4, 5, 9]. In this paper we propose a method to define fuzzy measures for the subsets of vertexes in a graph. This measure is based on the fuzzy memberships of the vertexes. In some sense, the proposed measures are to evaluate the connectivity that a subset of vertexes can achieve. Choquet integral [11] (see also [10]) is a tightly related concept with fuzzy measures. In fact, they are defined to integrate functions with respect to the fuzzy measures. In this paper we give some results for Choquet integrals. Namely, we show that they can be used to represent some functions. In this paper, we assume that the universal set N is a finite set, that is, N := {1, 2, . . . , n}.
The structure of this paper is as follows. In Section 2 we give some preliminaries that are needed later on in this paper. In Section 3 the fuzzy measure for fuzzy graphs is proposed and studied. Section 4 is devoted to the results about the representation in terms of Choquet integrals. The paper finishes with some conclusions.
2
Preliminaries
Definition 1. A set function µ : 2N → [0, 1] is a fuzzy measure if it satisfies the following axioms: (i) µ(∅) = 0, µ(N ) = 1 (boundary conditions) (ii) A ⊆ B implies µ(A) ≤ µ(B) (monotonicity) for A, B ∈ 2N Definition 2. [11] (see also [2]) Let µ be a fuzzy measure on (N, 2N ). The Choquet integral Cµ (x) of x : N → R+ with respect to µ is defined by Cµ (x) =
n X
x(as(j) )(µ(As(j) ) − µ(As(j+1) ))
j=1
where xs(i) indicates that the indices have been permuted so that 0 ≤ x(as(1) ) ≤ · · · ≤ x(as(n) ), As(i) = {as(i) , · · · , as(n) }, As(n+1) = ∅. A function x : N → R+ is regarded as |N | = n -dimensional vector, that is, n x ∈ R+ . n Definition 1 Let x, y ∈ R+ . We say that x and y are comonotonic if
xi < x j ⇒ y i ≤ y j for i, j ∈ N . A chain of sets in 2N is a set system M ⊂ 2N which is completely ordered with respect to set inclusion, i. e. A, B ∈ M implies A ⊂ B or B ⊂ A. n Definition 2 Let I be a real valued functional on R+ . We say: n (1) I is comonotonic additive if and only if for comonotonic x, y ∈ R+ ,
I(x + y) = I(x) + I(y), n (2) I is comonotonic monotone if and only if for comonotonic x, y ∈ R+
x ≤ y ⇒ I(x) ≤ I(y).
As the conditions for a functional to be the Choquet integral, we have the next theorem. Theorem 3 [20] Let I : [0, 1]n → R+ be comonotonic monotone and comonotonic additive functional with I(1N ) = 1. There exists a fuzzy measure µ on (N, 2N ) such that I(x) = Cµ µ (x) n for all x ∈ R+ .
Next we define a pseudo-addition. Definition 4 A binary operation ⊕ : [0, ∞)×[0, ∞) → [0, ∞) is called a pseudoaddition if the following properties are satisfied: (1) (2) (3) (4) (5)
(commutativity) a ⊕ b = b ⊕ a, (monotonicity) a ≤ a0 , b ≤ b0 implies a ⊕ b ≤ a0 ⊕ b0 , (associativity) (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) (continuity) an → a and bn → b imply an ⊕ bn → a ⊕ b, (zero element) 0 ⊕ a = a ⊕ 0,
for a, b ∈ [0, ∞). For fixed p > 0, let x ⊕ y := (xp + y p )1/p . Then ⊕ is a pseudo-addition. Using pseudo-addition ⊕, the Choquet integral is generalized [2].
3
Fuzzy relations and fuzzy measures
Let N be a finite set and R be a fuzzy relation on N , that is, R ⊂ N × N , where µR : N × N → [0, 1] is its membership function. (N, R, µR ) is regarded as a fuzzy graph. Thus, the set R is defined by pairs (xi , xj ) and corresponds to the edges of the graph and µR , their membership, is defined from N × N into [0,1] in such a way that for the R0 = N × N − R, we have: muR ((x, y)) = 0 for all (x, y) ∈ R0 . We say that T ⊂ R is a fuzzy tree if there exists no xi ∈ N (2 ≤ i ≤ n) such that (x1 , x2 ), . . . , (xi−1 , xi ) ∈ R and x1 = xi . This is, there is no cicle in the graph. TR denotes the set of all fuzzy trees of fuzzy graph (N, R, µR ). Definition 5 Let (N, R, µR ) be a fuzzy graph. We define a set function m : 2R → [0, ∞) by M m(A) := sup{ µ(x, y)|I ⊂ A, I ∈ TR }. (x,y)∈I
The next proposition follows from the definition. Proposition 1. Let (N, R, µR ) be a fuzzy graph and m be a set function defined in Definition 5.
(1) m(∅) = 0 (2) (monotonicity) A ⊂ B implies m(A) ≤ m(B) (3) (⊕submodularity) m(A) ⊕ m(B) ≥ m(A ∪ B) ⊕ m(A ∩ B). Define a set function ν : 2R → [0, 1] by ν(A) := m(A)/m(N ). Then, conditions (1) and (2) in the proposition above imply that the set function ν on 2R is a fuzzy measure. Example 1 Let (N, R) be the fuzzy graph in Figure 1. That is, N := {A, B, C, D} and a fuzzy relation {a, b, c, b} as indicated in the figure.
Fig. 1. Fuzzy Graph A b:0.5 a:1.0 C c:0.6 B
d:0.4
D
If ⊕ := +, then the fuzzy measure ν generated by the fuzzy graph in Figure 1 is the one in Table 1. Table 1. Fuzzy measure generated by the fuzzy graph (⊕ = +) set ∅ ν 0 set {b, c} ν 0.55
{a} 0.5 {b, d} 0.45
{b} {c} {d} {a, b} {a, c} {a, d} 0.25 0.3 0.2 0.75 0.8 0.7 {c, d} {a, b, c} {a, b, d} {a, c, d} {b, c, d} N 0.5 0.8 0.95 1 0.75 1
If ⊕ := ∨, then the fuzzy measure ν 0 generated by the fuzzy graph in Figure 1 is the one in Table 2. ν 0 is a possibility measure. Let ⊕ = + and µ(x, y) = 1 for all (x, y) ∈ R, Then we have m(A) ≤ |A|, monotonicity and submodularity. Therefore, m is a characteristic function of matroid [14, 18].
Table 2. Fuzzy measure generated by the fuzzy graph (⊕ = ∨) set ν0
4
∅ {a} {b} {c} {d} {a, b} {a, c} {a, d} 0 1 0.5 0.6 0.4 1 1 1 {b, c} {b, d} {c, d} {a, b, c} {a, b, d} {a, c, d} {b, c, d} N 0.6 0.5 0.6 1 1 1 0.6 1
Simplex and Choquet integral
First we define a simplex and a barycentric coordinate. Definition 6 Let P0 , P1 , . . . , Pk ∈ RN . (1) We say that {P0 , P1 , . . . , Pk } is affinely independent if k X
αi = 0,
i=0
k X
αi Pi = 0
i=0
and αi ∈ R imply α0 = α1 = · · · = αk = 0. (2) Let us suppose that {P0 , P1 , . . . , Pk } is affinely independent, and define the subset σ(P0 , P1 , . . . , Pk ) ⊂ RN by σ(P0 , P1 , . . . , Pk ) := {x|x =
k X
αi Pi , αi ≥ 0,
i=0
k X
αi = 1}.
i=0
Then, σ(P0 , P1 , . . . , Pk ) is called a simplex. (3) Let x ∈ σ(P0 , P1 , . . . , Pk ). In this case, if there exists unique non-negative Pk real numbers α0 , α1 , . . . , αk such that x = i=0 αi Pi , we say that (α0 , α1 , . . . , αk ) is the barycentric coordinate of x. Example 2 Consider Fig. 2 illustrated below. Triangles σ(P1 , P2 , P3 ), σ(P1 , P3 , P4 ) and σ(P1 , P4 , P5 ) are simplex. K := {σ(P1 , P2 , P3 ), σ(P1 , P3 , P4 ), σ(P1 , P4 , P5 )} is a complex and |K| = σ(P1 , P2 , P3 ) ∪ σ(P1 , P3 , P4 ) ∪ σ(P1 , P4 , P5 )} We have the next proposition that follows immediately from the definition above. Proposition 7 Let C be a maximal chain, that is, C : ∅ = C0 ⊂ C1 ⊂ . . . Cn = N, and z is a point of integers, that is, z ∈ Z N . Let zi := z + χCi where χCi is a characteristic function of Ci ∈ C (i = 0, 1, . . . , n). Then, the set {z0 , z1 , . . . , zn } is affinely independent. Therefore σ(z0 , z1 , . . . , zn ) is a simplex.
Fig. 2. Complex of {P1 , P2 , P3 , P4 , P5 } P1
P2 P5
P4 P3
Next we define the complex and simplical subdivision. Definition 8 Let σ(P0 , P1 , . . . , Pk ) be a simplex. Then, we say that sub-simplexes σ(Pk0 , Pk1 , . . . , Pki ), (i < k) are faces of σ(P0 , P1 , . . . , Pk ) and that each Pi is a vertex. Let K be a set of simplex in RN . We say that K is a complex if every face of σ ∈ K belongs to K, and if for σ1 , σ2 ∈ K, σ1 ∩ σ2 6= ∅ implies that σ1 ∩ σ2 is a face of both σ1 and σ2 . We define the polyhedra |K| by |K| := ∪σ∈K σ. Let X be a topological space and K be a complex. Then, if there exist a homeomorphic map f : |K| → X, we say that (K, f ) is a simplical subdivision of X. Proposition 9 Let K be a set of simplex σ := σ(z0 , z1 , . . . , zn ) for all z ∈ Z N and every maximal chain C. Then |K| = RN . Let i : |K| → RN be an identity map. Then (|K|, i) is a simplical subdivision of R N . Example 3 Let |K| := R2 and f := id (identity map). We illustrate the simplical subdivision of R2 .
Fig. 3. Simplical subdivision of R2 X2
X1
Proposition 2. For every x ∈ RN , [x] denotes [x](i) := [x(i)](i ∈ N ) where [x(i)] is the maximal integer less than x(i). For x ˜ = x − [x], x˜(l1 ) ≥ x ˜(l2 ) ≥ · · · ≥ x˜(ln ) and the maximal chain C : C0 = ∅, Ci = {l1 , l2 , . . . , li }(i = 1, 2, . . . n), denote zi0 := [x] + χCi , then we have x ∈ σ(z00 , z10 , . . . , zn0 ) and a barycentric coordinate of x is (1 − x ˜(l1 ), x ˜(l1 ) − x˜(l2 ), . . . , x ˜(ln−1 ) − x˜(ln )). Let f : Z N → R, then, we define the piecewise linear extension (PL extension) fˆ of f : Definition 10 Let x ∈ RN . Then, there exists a simplex σ(z0 , z1 , . . . , zn ) such that x ∈ σ(z0 , z1 , . . . , zn ). Let (α0 , α1 , . . . , αn ) be a barycentric coordinate of x, P that is x := ni=0 αi zi . Then we define the piecewise linear extension fˆ of f by P n fˆ(x) := i=0 αi fˆ(zi ). Since x, y ∈ σ(z0 , z1 , . . . , zn ) is comonotonic and fˆ is linear on σ(z0 , z1 , . . . , zn ), we have the next lemma. Lemma 11 fˆ is comonotonically additive. It follows from the previous lemma that we can apply the representation theorem presented in [20]. Proposition 12 The PL extension can be represented by Choquet integrals. Applying subadditivity theorem [11, 12], we have the next corollary, that is a fundamental theorem of discrete convex analysis [19]. Corollary 13 [15] Set function f : 2N → R is submodular if and only if its PL-extension fˆ : [0, 1]N → R is convex. The corollary above can easily be extended to ⊕-submodular set function and extended Choquet integral.
5
Conclusion
In this paper we have studied some aspects related with fuzzy measures and fuzzy integrals. We show that a fuzzy relation induces a fuzzy measure. Conversely we can induce a fuzzy relation by some fuzzy measure. The Choquet integral is regarded as a natural extension of some set function (fuzzy measure). These fact says that theory of discrete convex analysis can be included in theory of theory of fuzzy measures and fuzzy integral.
Acknowledgements The authors acknowledge R. Mesiar and the referees for their comments and suggestions. This work was partly funded by the Spanish MCYT (project TIC2001-0633-C03-02) and the Generalitat de Catalunya (AGAUR, 2002XT 00111).
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