Discrete Choquet integral and some of its symmetric extensions

Fuzzy Sets and Systems 184 (2011) 148 – 155 www.elsevier.com/locate/fss

Discrete Choquet integral and some of its symmetric extensions R. Mesiara,b , A. Mesiarová-Zemánkovác,d,∗ , K. Ahmadc a Slovak University of Technology, Bratislava, Slovakia b IRAFM, University of Ostrava, Czech Republic c Department of Computer Science, Trinity College, Dublin, Ireland d Mathematical Institute of SAS, Bratislava, Slovakia

Available online 4 December 2010

Abstract Classical extensions of the Choquet integral (defined on [0,1]) to [−1,1] are the asymmetric and the symmetric Choquet integral, the second one being called also the Šipoš integral. Recently, the balancing Choquet integral was introduced as another kind of a symmetric extension of the discrete Choquet integral. We introduce and discuss a new type of such extension, the fusion Choquet integral, and discuss its properties and relationship to the balancing and the symmetric Choquet integral. The symmetric maximum introduced by Grabisch is shown to be a special case of the fusion and the balancing Choquet integral. Several extensions of OWA operators are also discussed. © 2010 Elsevier B.V. All rights reserved. Keywords: Capacity; Choquet integral; Fusion Choquet integral; Balancing Choquet integral; OWA operator; Šipoš integral; Symmetric maximum

1. Introduction For any finite space X (with cardinality n), a function f : X −→ R can be identified with an n-dimensional vector x = (x1 , . . . , xn ). Therefore, throughout this paper we will deal with input vectors x ∈ Rn , not speaking formally about the functions. For a capacity m: 2 X −→ [0, 1], m(∅) = 0, m(X ) = 1 and m(A) ≤ m(B) whenever A ⊆ B ⊆ X , the discrete Choquet integral C(m, x) for x ∈ [0, 1]n (or x ∈ [0, ∞[n ) is defined as C(m, x) =

n 

x(i) (m(A,i ) − m(A,i+1 )),

(1)

i=1

where  is a permutation of (1, . . . , n) such that x(1) ≤ · · · ≤ x(n) , and A,i = {(i), . . . , (n)}, with convention A,n+1 = ∅. Observe that though there may be several different permutations  fitting the above requirements, they always yield the same result and thus the formula (1) is well defined. ∗ Corresponding author.

E-mail address: [email protected] (A. Mesiarová-Zemánková). 0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2010.11.013

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Note that the discrete Choquet integral has numerous applications in multicriteria decision aid (x represents the score vector here), in the game theory, etc. The scale [0, 1] (or [0, ∞[) is a unipolar scale for decision problems. Several decision problems deal, however, with a bipolar scale [−1, 1] (or ]−∞, ∞[= R) and then an appropriate extension of the (discrete) Choquet integral is needed. This is for example the case of sentiment analysis applications discussed in [1]. There are two basic bipolar extensions of the discrete Choquet integral. The first one is described by the same formula as (1) and only the domain of x is extended. Such an integral is called the asymmetric Choquet integral, with notation AC(m, x). Observe that the symmetry AC(m, −x) = −AC(m, x) is not guaranteed, in general. To avoid this lack of symmetry, the symmetric Choquet integral, called also the Šipoš integral [11], was introduced. Using the notation SC(m, x), the Šipoš integral is defined as SC(m, x) = C(m, x+ ) − C(m, x− ),

(2)

where x+ = (max(x1 , 0), . . . , max(xn , 0)) and x− = (max(−x1 , 0), . . . , max(−xn , 0)). More details on the asymmetric and the symmetric (discrete) Choquet integral, including the discussion of their properties, can be found in monographs [3,10,13]. For the simplification of notation, let X = {1, . . . , n}. If the contribution of (criteria) i and j to a capacity m is the same, i.e., if m(A ∪ {i}) = m(A ∪ { j}) for each A ⊆ X \ {i, j}, one could expect some compensation of inputs xi = c and x j = −c. For both AC and SC there is no such compensation. Example 1. Let n = 3, m(A) = (card( A)/3)3 , x = (−c, c, 1), c ∈ [0, 1]. Then AC(m, x) = 1 (1 + 6c). SC(m, x) = 27

1 27 (1

− 12c), while

To ensure (at least a partial) compensation by the extended discrete Choquet integral, recently the balancing Choquet integral, with notation BC(m, x), was introduced [8] and applied in [1]. This integral and some of its properties and representation are discussed in Section 2. In Section 3, we introduce another possible symmetric extension of the discrete Choquet integral, called the fusion Choquet integral, and discuss some of its properties. In Section 4, the relationship of the balancing and the fusion Choquet integrals with the symmetric maximum introduced and discussed by Grabisch in [5,6] is described. Section 5 is devoted to the study of bipolar extensions of OWA operators, originally introduced by Yager [12]. Finally, some concluding remarks are included. 2. The balancing Choquet integral The balancing Choquet integral on X, BC(m, ·): Rn −→ R, was introduced in [8] and applied for sentiment classification in [1]. Definition 1. Let X = {1, . . . , n}, and m: 2 X −→ [0, 1] be a capacity. Assume an input vector x = (x1 , . . . , xn ), with xi ∈ R for i ∈ {1, . . . , n} and let : X −→ X be a permutation related to a non-decreasing sequence of absolute values of x1 , . . . , xn , i.e., |x(1) | ≤ · · · ≤ |x(n) |, such that if 0 < x(i) = −x( j) for some i, j ∈ {1, . . . , n} then i ≤ j. The balancing Choquet integral with respect to m is given by BC(m, x) =

n 

x(i) w(i) ,

i=1

where w(i) = m({(i), (i + 1), . . . , (n)}) − m({(i + 1), (i + 2), . . . , (n)}) if { j ∈ {1, . . . , n}|x( j) = −x(i) } = ∅, and if |x(i−1) | < x(i) = · · · = x(i+k) = −x(i+k+1) = · · · = −x(i+k+ j) < |x(i+k+ j+1) | (where x(0) = −∞ and x(n+1) = ∞) then 1. w(r ) = m({(r ), . . . , (i + k), (i + k + j + 1), . . . , (n)}) − m({(r + 1), . . . , (i + k), (i + k + j + 1), . . . , (n)}) for r ∈ {i, . . . , i + k − 1}, 2. w(i+k) = m({(i + k), (i + k + j + 1), . . . , (n)}),

150

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3. w(r ) = m({(r ), . . . , (n)}) − m({(r + 1), . . . , (n)}) for r ∈ {i + k + 1, . . . , i + k + j − 1}, 4. w(i+k+ j) = m({(i + k + j), . . . , (n)}). i+k+ j  Note that in such a case ri+k =i w(r ) = m({(i), . . . , (i + k), (i + k + j + 1), . . . , (n)}) and r =i+k+1 w(r ) = m({(i + k + 1), . . . , (n)}). Remark 1. For an input vector x = (x1 , . . . , xn ) consider the set C0 = { j ∈ {1, . . . , n}|x j = 0} and the sets Ci ⊆ {1, . . . , n}, Ci  ∅ with i = 1, . . . , p, such that 0 < |xk | = |x j | for all k, j ∈ Ci and |xk | < |x j | for all k ∈ Ci , j ∈ Cr p p with i < r and i=0 Ci = {1, . . . , n}. Denote Ci+ = { j ∈ Ci |x j > 0}, Ci− = Ci \ Ci+ . Further denote Di = k=i Ck p  and |Ci |x = |x j | for j ∈ Ci . The sets Ci will be called value classes of input x. Then BC(m, x) = i=1 j∈Ci x j w j , where for all i ∈ {1, . . . , p} we have  x j w j = |Ci |x (m(Ci+ ∪ Di+1 ) − m(Ci− ∪ Di+1 )), j∈Ci

where D p+1 = ∅. Therefore, we can easily see that if m(Ci+ ∪ Di+1 ) = m(Ci− ∪ Di+1 ), which  means that the subset of negative and the subset of positive inputs from the set Ci have the same weights, then j∈Ci x j w j = 0—what indicates the desired cancellation effect. In the next proposition, an alternative definition of the balancing Choquet integral is shown. Proposition 1. Let BC(m, ·): Rn −→ R be the balancing Choquet integral with respect to a capacity m. Then for any x ∈ Rn it holds   n n  1  x(i) (m(A,i ) − m(A,i+1 )) + x(i) (m(A,i ) − m(A,i+1 )) , (3) BC(m, x) = 2 i=1

i=1

where ,  are permutations of (1, . . . , n) such that |x(1) | ≤ · · · ≤ |x(n) |, |x(1) | ≤ · · · ≤ |x(n) |, and if |x(i) | = |x(i+1) | (|x(i) | = |x(i+1) |) for some i ∈ {1, . . . , n − 1}, then x(i) ≤ x(i+1) (x(i) ≥ x(i+1) ). Proof. Due to Remark 1, the contribution of the inputs xi such that |xi | = |Ck | to the balancing Choquet integral BC(m, x) is |Ck |x · (m(Ck+ ∪ Dk+1 ) − m(Ck− ∪ Dk+1 )) if k ∈ {1, . . . , p}, for k = 0 this contribution vanishing. On the is n x(i) (m(A,i ) − other side, the contribution of the inputs xi such that |xi | = |Ck |x for k ∈ {1, . . . , p} to the value i=1 m(A,i+1 )) is |Ck |x · (−m(Ck ∪ Dk+1 ) − m(Dk+1 ) + 2m(Ck+ ∪ Dk+1 )), while in the case of , this contribution is |Ck |x · (m(Ck ∪ Dk+1 ) + m(Dk+1 ) − 2m(Ck− ∪ Dk+1 )). Thus contribution of these inputs to formula (3) is |Ck |x · (m(Ck+ ∪ Dk+1 ) − m(Ck− ∪ Dk+1 )), i.e., the same as in the case of BC(m, x). Evidently, the contribution of the inputs xi with xi = 0 to formula (3) is vanishing, i.e., formula (3) is an alternative definition of the balancing Choquet integral, independently of the choice of fitting permutations  and .  The following properties of the balancing Choquet integral were shown in [8]. Proposition 2. (i) For the positive input vectors the balancing Choquet integral (as well as the symmetric and the asymmetric Choquet integral) and the Choquet integral coincide. (ii) For the negative inputs, the balancing Choquet integral coincide with the symmetric Choquet integral. (iii) The balancing Choquet integral is homogenous and idempotent. (iv) The balancing Choquet integral with respect to a capacity m is non-decreasing if and only if m is submodular, i.e., m(A) + m(B) ≥ m(A ∪ B) + m(A ∩ B) for all A, B ⊆ X . (v) The balancing Choquet integral with respect to a capacity m is continuous if and only if the measure m is modular. Note that a modular capacity is necessarily additive. Therefore the balancing Choquet integral is continuous if and only if it is equal to a weighted mean.

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Example 2. Using the same notation as in Example 1 and allowing c to attain any value from [0, ∞[, it holds

BC(m, x) =

⎧ ⎪ ⎨ ⎪ ⎩

1 27 7 27 19 27

if |c| < 1, if |c| = 1, if |c| > 1,

i.e., only the relationship of c and 1 matters, but not the actual value of c, illustrating the compensative character of the balancing Choquet integral. 3. The fusion Choquet integral As already mentioned, in the definition of the discrete asymmetric Choquet integral (1) the choice of a fitting permutation  is irrelevant, as all such permutations yield the same result. However, this means that also the formula   n  x(i) (m(A,i ) − m(A,i+1 ))| is a permutation, x(1) ≤ · · · ≤ x(n) , AM i=1

where AM is the arithmetic mean, yields the asymmetric Choquet integral AC(m, x). Similarly, for the balancing Choquet integral it holds  n  BC(m, x) = AM x(i) (m(A,i ) − m(A,i+1 ))| is a permutation, |x(1) | ≤ · · · ≤ |x(n) |, and if i=1

|x(i) | = |x(i+1) | = |x(i+2) | then it does not hold sign x(i) = −signx(i+1) = signx(i+2) ).

(4)

Our next proposal relaxes the requirements on signs of inputs in formula (4). Definition 2. Let m: 2 X −→ [0, 1] be a capacity. Then the mapping FC(m, ·): Rn −→ R given by   n  x(i) (m(A,i ) − m(A,i+1 ))| is a permutation and |x(1) | ≤ · · · ≤ |x(n) | FC(m, x) = AM

(5)

i=1

is called the fusion Choquet integral. Note that one can restrict the domain of FC to [−1, 1]n with no other modifications. Remark 2. The original idea of the Choquet integral on non-negative reals, see [2], is based on a distinguished permutation  making the inputs reordering non-decreasing, x(1) ≤ · · · ≤ x(n) . This permutation  generates the weights from the given capacity m, and then a weighted arithmetic mean yields the value of the corresponding Choquet integral. The same idea is linked to the asymmetric Choquet integral [3] and it is related to the comonotone additivity of this integral. As shown in Proposition 1, the same philosophy can be applied to the recently introduced balancing Choquet integral [8], only the specification of the applied permutations  and  is much more complicated. The idea of the fusion Choquet integral deals with all permutations  resulting into a non-decreasing sequence |x(1) | ≤ · · · ≤ |x(n) |. Evidently, the fusion and the balancing Choquet integrals may differ only for inputs x ∈ Rn such that card{i xi | = c} > 2 for some c > 0, and {c, −c} ⊆ {x1 , . . . , xn }. Proposition 3. For the fusion Choquet integral the following holds: (i) FC(m, k · x) = k · FC(m, x), i.e., the fusion Choquet integral is homogenous. (ii) FC(m, ·) is non-decreasing whenever m is submodular. (iii) The fusion Choquet integral is continuous if and only if m is modular.

152

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(iv) FC(m, x) = BC(m, x) = SC(m, x) whenever x ∈ [0, ∞[n ∪]−∞, 0]n , i.e., if there are no inputs with opposite signs. (v) FC(m, c) = c for any c ∈ R, c = (c, . . . , c). Proofs of (i), (iv) and (v) follow directly from Definition 2, while the properties (ii) and (iii) can be shown using the same arguments as in the proof of similar properties of the balancing Choquet integral ([8]). Example 3. Under the notation of Example 1, in Example 2 we have shown that BC(m, x) =

7 27 ,

if c=1, i.e., if x=(−1,1,1). However, for the fusion Choquet integral it holds FC(m, (−1, 1, 1)) = 13 , i.e., FC(m, (−1, 1, 1))  BC(m, (−1, 1, 1)). 4. Symmetric maximum For the strongest capacity m ∗ : 2 X −→ [0, 1],  0 if A = ∅, m ∗ (A) = 1 else, n ∗ ∗ it is well known that AC(m

n , x) = max(x1 , . . . , xn ) = i=1 xi .∗However, then AC(m , −x) = max(−x1 , . . . , −xn ) = − min(x1 , . . . , xn ) = − i=1 xi , which is not related to AC(m , x), in general. In the case of the symmetric Choquet integral SC(m ∗ , x) = max(0, x1 , . . . , xn )+min(0, x1 , . . . , xn ) = −SC(m ∗ , −x). For n = 2, in the case of the balancing and the fusion Choquet integral, it holds ⎧ ⎨ x if |x| > |y|, BC(m ∗ , (x, y)) = FC(m ∗ , (x, y)) = 0 if x = −y, (6) ⎩ y else. However, formula (6) was already introduced by Grabisch in [5,6] (on the domain [−1, 1]2 ) as the symmetric maximum function, with notation ∨ (see Fig. 1), see also [5–7]. Observe that any symmetrized chain can be considered to introduce the symmetric maximum.

1 0.5 0 -0.5 -1 -1

1 0.5 0 -0.5 -0.5

0

1 0.5 0 -0.5 -1 -1

1 0.5 0 -0.5 -0.5

0 0.5

0.5 1

-1

Fig. 1. Symmetric maximum (left) and symmetric minimum (right).

1

-1

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Proposition 4. On the domain [−1, 1]2 , the symmetric maximum ∨ coincide with BC and FC with respect to the strongest capacity m ∗ . The symmetric maximum ∨ is not associative, and thus there are several possible extensions of this operator for dimension n, n > 2. We recall three such extensions (for more details see [6,7]). All of them coincide if there are no n n inputs with maximal absolute value but different signs, and then ∨ i=1 xi = x j , where |x j | = i=1 |xi |. In the opposite case, some cancellation rule should be applied. n (i) Weak cancellation rule: in such a case all inputs x j satisfying |x j | = i=1 |xi | are omitted and ∨ is applied to remaining inputs. (ii) Strong cancellation rule: in such a case, one positive and one negative input with the maximal absolute value is omitted, and ∨ is applied to remaining inputs. (iii) Splitting rule has the philosophy related to the Šipoš integral,   n     + n xi− . xi

(7) ∨ − ∨ i=1 xi = i=1

It is not difficult to check the following corollary. Corollary 1. For n > 2 and x ∈ [−1, 1]n , splitting method given in (7) corresponds to the balancing Choquet integral, n BC(m ∗ , x) = ∨ i=1 xi . On the other hand, FC(m ∗ , x) = where c =

n

a−b · c, a+b

i=1 |x i |,

(8)

a = card{i|xi = c} and b = card{i|xi = −c}.

Hence the fusion Choquet integral with respect to the strongest capacity yields a new type of n-ary symmetric maximum (for n > 2). Recall that due to the submodularity of m ∗ , FC(m ∗ , ·) is non-decreasing. Similarly, the symmetric minimum can be discussed, based on BC(m ∗ , ·) or FC(m ∗ , ·), where m ∗ : 2 X −→ [0, 1] is the weakest capacity given by  1 if A = X, m ∗ (A) = 0 else. However, m ∗ is not submodular and thus both MC(m ∗ , ·) and FC(m ∗ , ·) are not non-decreasing. For n = 2 it holds ⎧ ⎨ x if |x| < |y|, MC(m ∗ , (x, y)) = FC(m ∗ , (x, y)) = 0 if x = −y, ⎩ y else, see Fig. 1. 5. OWA operators OWA operators were introduced by Yager in 1988 [12]. For a fixed weighting vector n (on [0, 1] or [0, ∞[ scale) w ∈ [0, 1]n , i=1 wi = 1, OWAw : [0, ∞[n −→ [0, ∞[ is given by OWAw (x) =

n 

wi x(i) ,

(9)

i=1

where  is a permutation such that x(1) ≤ · · · ≤ x(n) . Comparing (9) with (1), it is easy to see the result of Grabisch [4] (see also [9]), i.e., that OWAw = C(m, ·), where for any permutation  of (1, . . . , n), m(A,i ) − m(A,i+1 ) = wi , card A wn−i+1 . i.e., m(A) depends on cardinality of A only, m(A) = i=1

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Such capacities for which m(A) = m(B) whenever card A = card B are called the symmetric capacities; and OWA operators on [0, ∞[ (or on [0, 1]) and the symmetric capacities are in one-to-one correspondence. Extensions of the Choquet integral to the bipolar scale [−1, 1] (or R) thus can be seen also as bipolar extensions of OWA operators (see also [8]). Definition 3. Let m: 2 X −→ [0, 1] be a symmetric capacity. Then functionals AOWAw = AC(m, ·), SOWAw = SC(m, ·), BOWAw = BC(m, ·), FOWAw = FC(m, ·): Rn −→ R are called asymmetric, symmetric, balancing and fusion OWA operators, respectively. Recall that the formula for the asymmetric OWA operator AOWAw is the same as for the OWAw operator on [0, ∞[, i.e., formula (9) is applied then. In the case of the symmetric OWA operator SOWAw , SOWAw (x) = OWAw (x+ ) − OWAw (x− ). Concerning BOWAw and FOWAw operators, they are non-decreasing only if the corresponding symmetric capacity is submodular, i.e., if w1 ≤ · · · ≤ wn . Based on Proposition 1 and Definition 2, these two operators are defined as BOWAw (x) =

n 

wi ·

i=1

x(i) + x(i) , 2

(for  and  see Proposition 1) and  n   FOWAw (x) = AM wi x(i) | is a permutation and |x(1) | ≤ · · · ≤ |x(n) | . i=1

The formulas for BOWAw and FOWAw can be simplified as follows: Proposition 5. Let m be a symmetric capacity on X inducing weighting vector w, wi = m(A)−m(B), card A = n−i +1 and card B = n − i. Then FOWAw = FC(m, ·): Rn −→ R and BOWAw = BC(m, ·): Rn −→ R are given by FOWAw (x) =

k  aj − bj · cj · aj + bj j=1

nj 

wi ,

i=n j−1 +1

and BOWAw (x) =

k 

n j −d j

sign(a j − b j ) · c j ·

j=1



wi ,

i=n j−1 +d j +1

where c j = |C j |x for j = 0, . . . , p with C j being value classes of input x defined in Remark 1, j − with a j = card C + i=1 (ai + bi ). j , b j = card C j and d j = min(a j , b j ), n j =

p

j=0 C j

= {1, . . . , n}

6. Concluding remarks We have introduced a new approach to the symmetric aggregation on bipolar scales [−1, 1] and R =]−∞, ∞[. As we have seen, there are several possible symmetric extensions of the Choquet integral and the actual choice of the relevant one depends on the modelled situation and the constraints of decision maker. Based on formulas for FOWAw and BOWAw , see Proposition 5, one can tell that the fusion Choquet integral is stressing more the cardinal information contained in the input vector x, while the balancing Choquet integral is more linked to the ordinal information in x.  Observe that if the underlying capacity m is additive, m(A) = i∈A wi , then n all discussed extensions, i.e., AC, SC, BC and FC turn into the weighted arithmetic mean W : Rn −→ R, W (x) = i=1 wi xi . As a by-product, interesting modification of the symmetric maximum of Grabisch, and new extensions of Yager’s OWA operators were obtained. We expect applications of the fusion Choquet integral and related balancing Choquet integral in all decision and evaluation problems where the bipolar scales for input values are taken into account.

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Acknowledgements The authors are grateful to the anonymous referees for their valuable comments and suggestions. This work was supported by Grant MSM VZ 619889 8701 and Grants VEGA 2/0032/09 and VEGA 1/0080/10. The support of Trinity College Dublin is gratefully acknowledged. References [1] K. Ahmad, A. Mesiarová-Zemánková, R. Mesiar, Fuzzy polarity lexicon and application of the balancing Choquet integral in sentiment analysis, in preparation. [2] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953–1954) 131–295. [3] D. Denneberg, Non-Additive Measure and Integral, Kluwer Academic Publishers, Dordrecht, 1994. [4] M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets Syst. 69 (1995) 279–298. [5] M. Grabisch, The symmetric Sugeno integral, Fuzzy Sets Syst. 139 (2003) 473–490. [6] M. Grabisch, The M˝obius function on symmetric ordered structures and its application to capacities on finite sets, Discrete Math. 287 (1–3) (2004) 17–34. [7] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Cambridge University Press, 2009. [8] A. Mesiarová-Zemánková, R. Mesiar, K. Ahmad, The balancing Choquet integral, Fuzzy Sets Syst. 161 (2010) 2243–2255. [9] T. Murofushi, M. Sugeno, Some quantities represented by the Choquet integral, Fuzzy Sets Syst. 56 (1993) 229–235. [10] E. Pap, Null-Additive Set functions, Kluwer Academic Publishers, Dordrecht, 1995. [11] J. Šipoš, Integral with respect to premeasure, Math. Slovaca 29 (1979) 141–145. [12] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern. 18 (1988) 183–190. [13] Z. Wang, G.J. Klir, Generalized measure theory, IFSR International Series on Systems Science and Engineering, vol. 25, Springer, Boston, 2009.