Choquet Integral on the Real Line

as a Generalization of the OWA Operator�

Yasuo Narukawa

Toho Gakuen,3-1-10, Naka, Kunitachi, Tokyo, 186-0004, Japan

Department of Computational Intelligence and Systems ScienceTokyo Institute of Technology

4259 Nagatuta, Midori-ku, Yokohama 226-8502, Japannarukawa@d4.dion.ne.jp

Abstract. The Choquet integral is one of the operators that can be usedfor aggregation and synthesis of information. It integrates a function withrespect to a fuzzy measure. In this paper we study the Choquet integralwith respect to a symmetric fuzzy measure, which is a generalization ofthe OWA operator. We present some results about the approximation ofChoquet integral for the calculation. We also present the inequalities forChoquet integral with respect to a symmetric fuzzy measure.

Keywords: Fuzzy measure, Non additive measure, Choquet integral,Aggregation operator, OWA operator.

1 Introduction

Choquet integral [3] is one of the approaches used in aggregation operators [7, 8]to combine information from several sources. Formally speaking, the integralintegrates a function with respect to a fuzzy measure, where a fuzzy measure isa monotone set function. The Choquet integral with respect to a fuzzy measurecan be applied to the decision modeling with uncertainty and risk [2, 23].

Among the aggregation operators, the Choquet integral is well known as ageneralization of other operators as the weighted mean, the Ordered WeightedAveraging (OWA) operator [24–26] as well as the arithmetic mean.

Due to these properties, the Choquet integral is a flexible operator that canbe used in different applications, and this has caused the interest of severalresearchers for its properties.

There are a lot of papers studying the theory and applications of Choquetintegral. Most of them assume the discrete set as universal set [6, 5, 12]. Otherpapers are on the abstract space [22, 13]. There are very few papers for theChoquet integral of a function on real line [15–17].

� Partial support by the Spanish MEC (projects ARES – CONSOLIDER INGENIO2010 CSD2007-00004 –, eAEGIS – TSI2007-65406-C03-02 –, co-Privacy TIN2011-27076-C03-03) is acknowledged.

V. Torra et al. (Eds.): MDAI 2012, LNAI 7647, pp. 56–65, 2012.c© Springer-Verlag Berlin Heidelberg 2012

Choquet Integral on the Real Line as a Generalization of the OWA Operator 57

This paper is one of the first attempts for the calculation of Choquet integralwith respect to a fuzzy measure of a function on real line.

This paper is devoted to the study of the Choquet integral of a function on thereal line. Especially we study the Choquet integral with respect to a symmetricfuzzy measure, which is the geralization of OWA operator. We introduce someapproximation of such integrals for particular types of fuzzy measures. Then,we study the inequalities of the Choquet integral with respect to a special fuzzymeasure, including symmetric fuzzy measure.

The structure of the paper is organized as follows. In Section 2 some prelim-inaries needed in the rest of the paper are given. In Section 3, we present theresults of the Choquet integral with respect to a symmetric fuzzy measure. InSection 4, we present the inequalities of the Choquet integral. The paper finisheswith a conclusion.

2 Preliminaries

In this section, we define fuzzy measures, the Choquet integral and the OWAoperator, and show their basic properties.

Let X be a unit interval or a subset of natural numbers and B be a class of itsBorel sets, that is, the smallest σ−algebra which includes the class of all closedsets. We say that (X,B) is a measurable space.

Definition 1. [21] Let (X,B) be a measurable space. A fuzzy measure (or anon-additive measure) μ is a real valued set function, μ : B −→ [0, 1] with thefollowing properties;

1. μ(∅) = 02. μ(A) ≤ μ(B) whenever A ⊂ B, A,B ∈ B.We say that the triplet (X,B, μ) is a fuzzy measure space if μ is a fuzzy measure.

A fuzzy measure is said to be continuous if An ↑ A implies μ(An) ↑ μ(A) andAn ↓ A implies μ(An) ↓ μ(A).

We assume that μ is continuous if X is a unit interval.

Definition 2. Let (X,B, μ) be a fuzzy measure space.

1. μ is said to be submodular, if

μ(A) + μ(B) ≥ μ(A ∪B) + μ(A ∩B).

2. μ is said to be supermodular if

μ(A) + μ(B) ≤ μ(A ∪B) + μ(A ∩B).

Definition 3. Let (X,B) be a measurable space. A function f : X → R is saidto be measurable if {x|f(x) ≥ α} ∈ B for all α ∈ R.

58 Y. Narukawa

F(X) denotes the class of non-negative measurable functions, that is,

F(X) = {f |f : X → R+, f : measurable}

Definition 4. [3, 18] Let (X,B, μ) be a fuzzy measure space. The Choquetintegral of f ∈ F(X) with respect to μ is defined by

(C)

∫fdμ =

∫ ∞

0

μf (r)dr,

where μf (r) = μ({x|f(x) ≥ r}).Let A ⊂ X . The Choquet integral restricted on A is defined by

(C)

∫A

fdμ := (C)

∫f · 1Adμ.

Let μ be a fuzzy measure on (X,B). Fμ(X) denotes the class of non-negativemeasurable functions with Choquet integrable, that is,

Fμ(X) = {f |f ∈ Fμ(X), (C)

∫fdμ < ∞}.

The next proposition is obvious from the definition of the Choquet integral.

Proposition 1. Let μ and ν be a fuzzy measure on (X,B) and a, b be a realnumber. We have

(C)

∫A

fd(aμ+ bν) = a(C)

∫A

fdμ+ b(C)

∫A

fdν

for f ∈ Fμ(X) ∩ Fν(X).

In relation to Choquet integral with respect to a submodular (a super-modular)non-additive measure, we have the next famous theorem.

Theorem 1. [3, 4, 19] Let μ be a non-additive measure in (X,B)and f, g ∈ M+.

1. If μ is submodular, then

(C)

∫(f + g)dμ ≤ (C)

∫fdμ+ (C)

∫gdμ.

2. If μ is supermodular, then

(C)

∫(f + g)dμ ≥ (C)

∫fdμ+ (C)

∫gdμ.

Next we will introduce the general definition of aggregation operator (aggrega-tion function [8]).

Choquet Integral on the Real Line as a Generalization of the OWA Operator 59

Definition 5. Let I be a non empty real interval. An aggregation operator Agis a function Ag : I(N) → I with the following properties;

1. (Monotonicity)If ai ≤ bi for all i = 1, . . . , N , a = (a1, . . . , aN ),b = (b1, . . . , bN ) a,b ∈ D,then Ag(a) ≤ Ag(b).

2. (boundary conditions)infx∈I(N) Ag(x) = inf I and supx∈I(N) Ag(x) = sup I.

We say that an aggregation operator Ag satisfies an idempotency (or unanimity)if Ag(a, . . . , a) = a if a ∈ I.

In the following we assume that I be a unit interval [0, 1] and Ag is idempo-tent.

Yager introduced the Ordered Weighted Averaging operator in [24], which isone of the most famous aggregation operator with idempotency.

A weighting vectorw with weights (w1, . . . , wN ) is a vectorw ∈ RN satisfying∑Ni=1 wi = 1 and wi ≥ 0 for all i = 1, 2, . . . , N .

Definition 6. [24] Given a weighting vector w with weights (w1, . . . , wN ), theOrdered Weighted Averaging operator is defined as follows:

OWAw(a) =

N∑i=1

wiaσ(i)

where σ defines a permutation of {1, . . . , N} such that aσ(i) ≥ aσ(i+1),a =(a1, . . . , aN ).

Definition 7. Let X := {1, . . . , N}. A fuzzy measure μ on B is said to be sym-metric [9] if μ(A) = μ(B) for |A| = |B|, A,B ∈ B.Symmetric fuzzy measures on (X,B) can be represented in terms of N weights

wi for i = 1, . . . , N so that μ(A) =∑|A|

i=1 wi. Using a symmetric fuzzy measure,we can represent any OWA operator as a Choquet integral.

Proposition 2. Let X := {1, 2, . . . , N}; then, for every OWAw, there exists asymmetric fuzzy measure satisfying μ({1}) := w1

and μ({1, . . . , i}) := w1 + · · ·+ wi for i = 1, 2, . . . , N , such that

OWAw(a) = (C)

∫adμ

for a ∈ RN+ .

3 Choquet Integral with Respect to a Symmetric FuzzyMeasure

In the following we consider the Choquet integral of a monotone increasingfunction on the real line. We assume that X is a unit interval, that is, X = [0, 1].

60 Y. Narukawa

Let λ be a Lebesgue measure on C, that is, a measure generated by λ([a, b]) =b− a for [a, b] ⊂ X . Let Fc(X) be a class of continuous functions on X .

We will define a continuous version of symmetric fuzzy measure.

Definition 8. Let μ be a fuzzy measure on (X,B) and μ(X) = 1. μ is said tobe symmetric, if λ(A) = λ(B) implies μ(A) = μ(B).

Remark. Let X := {1, . . . , N} and m(A) := |A| for A ⊂ X . Then m is anadditive measure on 2X . If μ is symmetric, we havem(A) = m(B) implies μ(A) =μ(B). The symmetry in Definition 8 is essentially same as one in Definition 7.

Let μ be a symmetric fuzzy measure on ([0, 1],B). Define a function ϕ : [0, 1] →[0, 1] by ϕ(x) := μ([0, x]). Suppose that x < y. Since [0, x] ⊂ [0, y], we haveϕ(x) ≤ ϕ(y).

Let λ(A) := x for arbitrary A ∈ B. Then we have λ(A) = λ([0, x]).Since μ is symmetric, we have

μ(A) = μ([0, x]) = ϕ(x) = ϕ(λ(A))

for arbitrary A ∈ B. Therefore we have the next proposition.

Proposition 3. Let μ be a symmetric fuzzy measure on (X,B). There exists amonotone increasing function ϕ : [0, 1] → [0, 1] such that μ = ϕ ◦ λ.Let xn → x for xn ∈ [0, 1], n = 1, 2, . . . . Define {an} and {bn} by

an = infk≥n

xk, bn = supk≥n

xk.

Since an ↑ x, we have ∪n[0, an] = [0, x]. It follows from the continuity of μ that

limn→∞μ([0, an]) = μ([0, x]),

that is,limn→∞ϕ(an) = ϕ(x).

In the same way, we havelimn→∞ϕ(bn) = ϕ(x).

Since ϕ is monotone and an ≤ xn ≤ bn, we have

limn→∞ϕ(xn) = ϕ(x).

Therefore we have the next proposition.

Proposition 4. Let μ be a symmetric fuzzy measure on (X,B). ϕ in Proposition3 is continuous.

Choquet Integral on the Real Line as a Generalization of the OWA Operator 61

We say that a function ϕ in Proposition 1 is a weight function for a symmetricfuzzy measure μ. Since ϕ is continuous, it follows fromWeierstrass approximationtheorem that ϕ can be approximated by the polynomial, that is , for any ε > 0,there exists real numbers a1, · · · , aN such that |ϕ(x)−∑N

k=1 akxk| < ε for x ∈ X .

Therefore every symmetric fuzzy measure μ can be approximated by a fuzzymeasure

∑Nk=1 akλ

k.

Proposition 5. Let μ be a symmetric fuzzy measure on (X,B). For any ε > 0,there exist real numbers a1, . . . , aN such that

|(C)

∫fdμ−

N∑k=1

ak(C)

∫fdλk| < ε.

for f ∈ Fμ(X).

Example 1. Let μ := λ1/2, that is, ϕ(x) = x1/2. We have a sequence ϕn suchthat ϕn → ϕ uniformly. In fact, ϕ1(x) = x,

ϕ2(x) = f(12 )2C1x(1− x) + f(1)2C2x2 =

√2x− (

√2− 1)x2.

· · ·Therefore we have

(C)

∫fdλ1/2 ≈ √

2(C)

∫fdλ− (

√2− 1)(C)

∫fdλ2.

Moreover suppose that ϕ in Proposition 1 be analytic. Then we can express wby

ϕ(x) :=∞∑k=1

akxk.

Then we have

(C)

∫[0,x]

fdμ =

∞∑k=1

ak(C)

∫fdλk

for x ∈ [0, 1].

Example 2. Let μ(A) := log2(λ(A) + 1), that is, ϕ(x) = log2(x + 1). Since wehave

ϕ(x) =1

log 2

∞∑k=1

(−1)k−1

kxk.

Therefore

(C)

∫fd log2(λ+ 1) =

1

log 2

∞∑k=1

(−1)k−1

k

∫fdλk.

62 Y. Narukawa

4 Inequalities

In this section, we will present some basic inequalities for the Choquet integralwith respect to a fuzzy measure generated by a convex function or a concavefunction.

Definition 9. Let ϕ be a real valued function on closed interval [c,d]. ϕ is saidto be convex if

ϕ(λx + (1− λ)y) ≤ λϕ(x) + (1− λ)ϕ(y)

for x, y ∈ [c, d], 0 < λ < 1.ϕ is said to be concave if

ϕ(λx + (1− λ)y) ≥ λϕ(x) + (1− λ)ϕ(y)

for x, y ∈ [c, d], 0 < λ < 1.

We have the next Jensen’s inequality from the definition [4].

Proposition 6. Let μ be a fuzzy measure on (X,B) with μ(X) = 1.

1. If ϕ is convex, then

(C)

∫ϕ(f)dμ ≥ ϕ((C)

∫fdμ).

2. If ϕ is concave, then

(C)

∫ϕ(f)dμ ≤ ϕ((C)

∫fdμ).

Suppose that ϕ is convex or concave and monotone. Applying the theorem aboveto classical Lebesgue integral, we have

∫ 1

0

ϕ(μ({x|f(x) ≥ a}))da ≥ ϕ(

∫ 1

0

(μ({x|f(x) ≥ a}))da).

Therefore we have the next inequalities.

Proposition 7. Let μ be a fuzzy measure on (X,B).1. If ϕ : [0, 1] → [0, 1] is a non-decreasing convex function on closed interval

with ϕ(0) = 0 and ϕ(1) = 1.

(C)

∫fd(ϕ ◦ μ) ≥ ϕ((C)

∫fdμ).

2. If ϕ is a non-decreasing concave function on closed interval with ϕ(0) = 0and ϕ(1) = 1,

(C)

∫fd(ϕ ◦ μ) ≤ ϕ((C)

∫fdμ).

Choquet Integral on the Real Line as a Generalization of the OWA Operator 63

Using subadditivity of Choquet integral with respect to a submodular fuzzymeasure, we have the next proposition [14, 11].

Proposition 8. Let μ be a submodular fuzzy measure on (X,B) and p ≥ 1,q ≥ 1, 1/p+ 1/q = 1.

1.

(C)

∫fgdμ ≤ ((C)

∫fpdμ)1/p((C)

∫gqdμ)1/q

2.

(C)

∫(f + g)pdμ ≤ ((C)

∫fpdμ)1/p + ((C)

∫gpdμ)1/p

Note that if ϕ : [0, 1] → [0, 1] be concave and continuous, and μ := ϕ ◦ λ. Thenμ is submodular [19].

In the following we suppose that ϕ : [0, 1] → [0, 1] be concave and continuous,and μ := ϕ ◦ λ.Definition 10. We say that a continuous function ϕ : [0, 1] → [0, 1] is semiconvex if there exists C > 0 such that for all x, y ∈ [0, 1] and 0 ≤ a ≤ 1

ϕ(ax + (1− a)y) ≤ C{(aϕ(x)) + (1− aϕ(y))}.We say that a continuous function ϕ : [0, 1] → [0, 1] is strongly semi convex if

there exists C > 0 such that for all xi ∈ [0, 1], 0 ≤ ai ≤ 1 and∑

i ai = 1,

ϕ(∑i

aixi) ≤ C∑i

aiϕ(xi).

Suppose that ϕ is continuous and concave with ϕ(0) = 0 and ϕ(1) = 1. Thenfor all x ∈ [0, 1] we have x ≤ ϕ(x). Therefore we have the next proposition.

Proposition 9. Suppose that ϕ is continuous and concave with ϕ(0) = 0 andϕ(1) = 1. ϕ is strongly semi convex if there exists C > 1 such that for all x,ϕ(x) ≤ Cx.

Example 3. Let ϕ(x) = x(2−x). ϕ is concave with ϕ(0) = 0 and ϕ(1) = 1. Thenwe have x ≤ ϕ(x) ≤ 2x.

Next we will define a maximal function with respect to a fuzzy measure.

Definition 11. Let μ be a fuzzy measure on (X, B) and f ∈ Fμ(X).A maximal function Mμf with respect to μ of f is defined by

Mμf(x) := supr

1

μ([x− r, x+ r])(C)

∫[x−r,x+r]

fdμ.

If μ is a classical measure, Mμf is Hardy Littlewood maximal function.If μ is symmetric with some special conditions, we have the next theorem that

is similar to classical one.

64 Y. Narukawa

Theorem 2. Let ϕ be continuous and concave with ϕ(0) = 0 and ϕ(1) = 1 andϕ be strongly semi convex.

Let μ = ϕ ◦ λ and f ∈ Fμ(X).There exists a constant C such that for all α > 0

μ({x|Mμf(x) > α}) ≤ C

α(C)

∫fdμ.

5 Conclusion

In this paper we have studied some properties of the Choquet integral. We havegiven some new expressions to compute the Choquet integral with respect to afunction on the real line, We have given some inequalities.

As future work we will consider the extension of the space to multi dimensionalEuclidean space, and also the application of these results.

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