Fuzzy Sets and Systems 160 (2009) 726 – 732 www.elsevier.com/locate/fss
On some construction methods for 1-Lipschitz aggregation functions Jana Kalická∗ Department of Mathematics, SvF, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia Available online 4 July 2008
Abstract Flipping method for constructing copulas from a given copula is extended to aggregation functions. Aggregation functions admitting flipping method are characterized and studied. Moreover, the algebraic structure of several kinds of dualities on aggregation functions, including the flipping method, the reverse method and the standard duality, is classified. Based on composition of construction methods, several new construction methods for 1-Lipschitz aggregation functions are introduced, and their algebraic relationships are studied. © 2008 Elsevier B.V. All rights reserved. Keywords: Aggregation function; Lipschitz property; Duality; Flipping methods
1. Introduction Binary aggregation functions are fundamental in fusion theory, for details see, e.g., [2]. Recall that a mapping A : [0, 1]2 → [0, 1] is called a binary aggregation function whenever it is non-decreasing in both variables and A(0, 0) = 0,
A(1, 1) = 1.
The class of all binary aggregation functions will be denoted by A and we will use simply the name aggregation function for members of A. There are several universal construction methods to build a new aggregation function from a given one. For example, the standard duality d
:A→A
assigns to any A ∈ A its dual Ad ∈ A, given by Ad (x, y) = 1 − A(1 − x, 1 − y). For any fixed automorphism : [0, 1] → [0, 1], for each A ∈ A, A ∈ A is given by A (x, y) = −1 (A((x), (y))). ∗ Tel.: +421 59274417.
E-mail address:
[email protected]. 0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2008.06.017
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Note that the standard duality is involutive, i.e., (Ad )d = A, while (A ) = A for all A ∈ A only if = id|[0,1] . In special subclasses of aggregation functions, there are specific construction methods not applicable to all aggregation functions. For example, for any A ∈ A we can define the mapping A¯ : [0, 1]2 → R by ¯ A(x, y) = x + y − A(x, y).
(1)
However, A¯ is an aggregation function only if A is 1-Lipschitz, i.e., if |A(x, y) − A(x , y )| |x − x | + |y − y | for all x, y, x , y ∈ [0, 1], see [5]. In the class A(1) of all 1-Lipschitz aggregation functions, the construction method ¯ : A(1) → A(1) defined by (1), is again an involutive mapping. Another involutive construction method on A(1) is the reverse : A(1) → A(1) given by y) = x + y − 1 + A(1 − x, 1 − y) A(x, see [8]. of a copula is called a survival Observe that in the class C of copulas (for more details we recommend [7]), the reverse C copula. The aim of this paper is to discuss a construction method known for copulas as flipping method (see [6]) in the broader framework of aggregation functions and its relation to some other construction methods. Moreover, based on consecutive composition of discussed construction methods, new types of construction methods for 1-Lipschitz aggregation functions are introduced and their algebraic structure is studied. The paper is organized as follows. In Section 2, flipping method for aggregation functions is described and aggregation functions admitting flipping are studied. Section 3 deals with several new types of construction methods for 1-Lipschitz aggregation functions. Section 4 is devoted to the study of relationships of several construction methods for aggregation functions from the algebraic point of view. Finally, some conclusions are given. 2. Flipping of aggregation functions Copulas can be understood as probability distributions on Borel subsets of [0, 1]2 with uniformly distributed marginals. Flipping of a copula (more exactly, x-flipping) geometrically means the switching of a copula C describing the dependence structure of a random vector (X, Y ), where FX = FY and FX |[0,1] = id|[0,1] , to a copula C ∗ describing the dependence structure of the random vector (1 − X, Y ). Similarly, y-flipping leads to a copula C∗ describing the dependence structure of the random vector (X, 1 − Y ). That is, for a given copula C : [0, 1]2 → [0, 1] the copulas C ∗ , C∗ : [0, 1]2 → [0, 1] are given by C ∗ (x, y) = y − C(1 − x, y)
(2)
C∗ (x, y) = x − C(x, 1 − y).
(3)
and
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Formally, formulas (2) and (3) can be rewritten into C ∗ (x, y) = y + C(1, 0) − C(1 − x, y)
(4)
C∗ (x, y) = x + C(0, 1) − C(x, 1 − y).
(5)
and
For any aggregation function A ∈ A we can introduce two mappings A∗ , A∗ : [0, 1]2 → R given by A∗ (x, y) = y + A(1, 0) − A(1 − x, y)
(6)
A∗ (x, y) = x + A(0, 1) − A(x, 1 − y).
(7)
and ∈ A given by A(x, y) = A(y, x), Note that if we exchange the role of variables, i.e., when for any A ∈ A we introduce A ∗ ∗ . Therefore, any result introduced for A mappings can straightforwardly be variables transformed into then (A∗ )=( A) the corresponding result for A∗ mappings, and thus we will deal since now in this section with A∗ mappings only. Theorem 1. Let A ∈ A. Then A∗ ∈ A if and only if (i) A is 1-Lipschitz in the second variable and (ii) A(0, 1) = A(1, 0). Proof. For a given A ∈ A, let A∗ ∈ A. Then 1 = A∗ (1, 1) = 1 + A(1, 0) − A(0, 1), i.e., A(1, 0) = A(0, 1). Moreover the non-decreasingness of A∗ in the second variable implies 0 A∗ (x, y) − A∗ (x, y ) = y − y − (A(1 − x, y) − A(1 − x, y )) for all x, y, y ∈ [0, 1], y y , i.e., A(1 − x, y) − A(1 − x, y ) y − y , which implies the 1-Lipschitz property of A in the second variable. To see the opposite implication, (ii) implies A∗ (1, 1) = 1. Moreover, for any A ∈ A, A∗ (0, 0) = 0 and A∗ is non-decreasing in the first variable. Finally, (i) implies the non-decreasingness of A∗ in the second variable. Summarizing, we see that any A ∈ A fulfilling (i) and (ii) yields an aggregation function A∗ . Since (A∗ )∗ = A, conditions (i), (ii) for A∗ are also fulfilled. Note that in [3] a special case of Theorem 1 concerning x-flipping of aggregation functions under the condition A(0, 1) = A(1, 0) = 0 is given. Denote by A∗ the class of all aggregation functions A ∈ A such that also A∗ ∈ A, and for ∈ [0, 1] denote by A∗ the subclass of aggregation functions from A∗ for which A(1, 0) = . Evidently (A∗ )∈[0,1] is a partition of A∗ . Similarly we can introduce the class A∗ of all aggregation functions A ∈ A such that A∗ ∈ A, i.e., A such that A(1, 0) = A(0, 1) and A is 1-Lipschitz in the first variable. Moreover, the class L = A∗ ∩ A∗ is the class of all 1-Lipschitz aggregation functions with the same value at the opposite corner points (0, 1) and (1, 0). Similarly, the classes (A∗ ) and L , ∈ [0, 1], are introduced. Example 1. Define A ∈ A by A(x, y) = x 2 y. Then evidently A ∈ A∗0 and A∗ ∈ A∗0 is given by A∗ (x, y) = (2x −x 2 )y. ∈ ∈ A∗ . Observe that B = (A + A∗ )/2 ∈ A∗ is invariant with respect to However, A ∈ / A∗ . Similarly, A / A∗ but A 0 ∗ x-flipping for any A ∈ A0 , and in our case B = is the standard product. Evidently, ∗ = and ∈ L0 . The following results can be checked straightforwardly.
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Remark 1. The classes A∗ , A∗ , L and the classes A∗ , (A∗ ) , L , ∈ [0, 1] are convex compact (with uniform convergence) subclasses of A. and A are elements of L and Remark 2. Let ∈ [0, 1] be fixed. For any A ∈ L , also A∗ , A, = ( = A. (A∗ )∗ = ( A) A) Moreover Ad , A¯ ∈ L1− . Recall that an aggregation function Q ∈ A is called a quasi-copula whenever it is 1-Lipschitz and 1 is its neutral element. Remark 3. L0 is just the class of all quasi-copulas. Remark 4. For each ∈ [0, 1], A ∈ A∗ and B ∈ A∗ , , ∈ [0, 1], the convex combination C = A + (1 − )B is an element of A∗ , where = + (1 − ). Moreover, C ∗ = A∗ + (1 − )B ∗ . Similar claims hold for aggregation functions from A∗ and from L. Note also that the weakest quasi-copula, i.e., the weakest element of L0 , is the Fréchet–Hoeffding bound W : [0, 1]2 → [0, 1] given by W (x, y) = max{x + y − 1, 0}. The strongest quasi-copula, i.e., the strongest element for L0 , is the upper Fréchet–Hoeffding bound Min : [0, 1]2 → [0, 1] given by Min(x, y) = min{x, y}. By duality (which reverses the order of aggregation functions), due to Remark 2, the weakest element of L1 is the aggregation function Max = Mind Max(x, y) = max{x, y} and the strongest element of L1 is the aggregation function W d : [0, 1]2 → [0, 1] given by W d (x, y) = min{x + y, 1}. Evidently W and W d are extremal elements of L. d Proposition 1. Let ∈]0, 1[ be fixed. Then the weakest element W ∈ L and the strongest element W1− ∈ L are
W (x, y) = max{Max(x, y) + − 1, W (x, y)} and d d W1− (x, y) = min{Min(x, y) + , W (x, y)}.
Proof. Evidently, for any A ∈ L it holds A W . Moreover, suppose that B : [0, 1]2 → R is non-decreasing 1-Lipschitz function and B(0, 1) = B(1, 0) = . Then evidently B(1, y) and B(x, 1) for all x, y ∈ [0, 1]. Moreover, B(x, y) B(x, 1)−(1− y) y +−1 and B(x, y) B(1, y)−(1−x) x +−1, i.e., B(x, y) max{x, y}+ − 1. Now it is clear that F : [0, 1]2 → R given by F (x, y) = Max(x, y) + − 1 is the weakest non-decreasing 1-Lipschitz function satisfying F (1, 0) = = F (0, 1). Combining A W and A F valid for all A ∈ A we see that A Max(F , W ) = W . Evidently, W (1, 0) = W (0, 1) = and 1-Lipschitz property of W follows from the kernel property of Max function (i.e., the Chebyshev norm of Max is just 1), see [5]. Hence W ∈ L . The rest of the
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proof follows from the properties of the standard duality of aggregation functions, taking into account that W1− is the weakest element of L1− . Note that though for any A ∈ A∗ it holds A(1, 0) = A(0, 1), A need not be commutative in general. For example, the mapping C : [0, 1]2 → [0, 1] given by C(x, y) = x y + x y 2 (1 − x)(1 − y) is a non-commutative copula, see [7], and thus C ∈ L0 . Note that C ∗ is also a copula and it is given by C ∗ (x, y) = y − (1 − x)y − x y 2 (1 − x)(1 − y) = x y − x y 2 (1 − x)(1 − y). 3. New construction methods on L Each construction method on L introduced in the previous sections will be denoted by letter m with sign indicating the method. For example, m∗ : L → L is given by m ∗ (A) = A∗ . Evidently, these construction methods can be composed, as indicated in Section 2. Recall, for example, the equality m∗ ◦ m =m ◦ m∗. = m 1 . Then symbol m 1 is a symbol indicating a construction method assigning to any Denote the composition m ∗ ◦ m (A∗ ) i.e., 1-Lipschitz aggregation function A ∈ L another aggregation function A1 ∈ L, A1 = A1 (x, y) = y + A(0, 1) − A(y, 1 − x). Similarly we introduce the next construction methods: • • • • • • • •
m2 = m ◦ m ∗ , and hence A2 (x, y) = x + A(1, 0) − A(1 − y, x). m = m d ◦ m ∗ , and hence A (x, y) = x − A(1, 0) + A(1 − x, y). m = m d ◦ m ∗ , and hence A (x, y) = y − A(0, 1) + A(x, 1 − y). , and hence A3 (x, y) = x + y − A(y, x). m 3 = m¯ ◦ m , and hence A4 (x, y) = 1 − A(1 − y, 1 − x). m4 = md ◦ m m5 = m ◦ m , and hence A5 (x, y) = x − A(0, 1) + A(y, 1 − x). ◦ m , and hence A6 (x, y) = y − A(1, 0) + A(1 − y, x). m6 = m m7 = m ◦m , and hence A7 (x, y) = x + y − 1 + A(1 − y, 1 − x).
Remark 5. All introduced construction methods when applied to a member of L yield a (possibly different) member of L. Some of the original properties may be violated. For example, from a copula we may obtain a 1-Lipschitz aggregation function which is not more a copula. In the next propositions we summarize construction methods preserving some of the distinguished properties of aggregation functions. All proofs are based on the consecutive application of construction methods discussed in Sections 1 and 2 and therefore omitted. Proposition 2. Let m be one of the introduced construction methods. Then m(C) is a copula for any copula if and only if m ∈ {id, m ∗ , m ∗ , m , m , m 1 , m 2 , m 7 } = MC . Moreover, then and only then for arbitrary ∈ [0, 1], 0.5, for all A ∈ L it holds that also m(A) ∈ L . Proposition 3. Let m be one of the introduced construction methods. Then m(A) is a symmetric aggregation function for any symmetric A ∈ L if and only if m ∈ {m 3 , m 4 , m 7 , m, ¯ md, m , m , id} = M B . 4. Algebraic structure of construction methods in L Denote by M the set of all bijective mappings m : L → L, m (A) = A . It is a classical algebraic result that (M, ◦) is a (non-abelian) group with identity element id, id(A) = A, where ◦ is the standard composition of mappings.
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Table 1 m\m
m∗
m∗
m¯
md
m
m
m
m
id
m1
m2
m3
m4
m5
m6
m7
m∗ m∗ m¯ md m m m m id m1 m2 m3 m4 m5 m6 m7
id m m m m¯ md m∗ m2 m∗ m7 m m6 m5 m4 m3 m1
m id m m md m¯ m∗ m1 m∗ m m7 m5 m6 m3 m4 m2
m m id m m∗ m∗ md m3 m¯ m5 m6 m m7 m1 m2 m4
m m m id m∗ m∗ m¯ m4 md m6 m5 m7 m m2 m1 m3
m¯ md m∗ m∗ id m m m6 m m4 m3 m2 m1 m7 m m5
md m¯ m∗ m∗ m id m m5 m m3 m4 m1 m2 m m7 m6
m∗ m∗ md m¯ m m id m7 m m2 m1 m4 m3 m6 m5 m
m1 m2 m3 m4 m5 m6 m7 id m m∗ m∗ m¯ md m m m
m∗ m∗ m¯ md m m m m id m1 m2 m3 m4 m5 m6 m7
m m7 m5 m6 m3 m4 m2 m∗ m1 m id m m md m¯ m∗
m7 m m6 m5 m4 m3 m1 m∗ m2 id m m m m¯ md m∗
m5 m6 m m7 m1 m2 m4 m¯ m3 m m id m m∗ m∗ md
m6 m5 m7 m m2 m1 m3 md m4 m m m id m∗ m∗ m¯
m3 m4 m1 m2 m m7 m6 m m5 m4 m¯ m∗ m∗ m id m
m4 m3 m2 m1 m7 m m5 m m6 m¯ md m∗ m∗ id m m
m2 m1 m4 m3 m6 m5 m m m7 m∗ m∗ md m¯ m m id
Table 2 m\m
id
m∗
m∗
m
m
m1
m2
m7
id m∗ m∗ m m m1 m2 m7
id m∗ m∗ m m m1 m2 m7
m∗ id m m∗ m2 m7 m m1
m∗ m id m∗ m1 m m7 m2
m m∗ m∗ id m7 m2 m1 m
m m1 m2 m7 id m∗ m∗ m
m1 m m7 m2 m∗ m id m∗
m2 m7 m m1 m∗ id m m∗
m7 m2 m1 m m m∗ m∗ id
Therefore any subgroupoid (S, ◦) of (M, ◦), such that for each s ∈ S there are s1 , s2 ∈ S satisfying s ◦s1 = s2 ◦s = id, is also a subgroup of (M, ◦). Theorem 2. The set {id, m ∗ , m ∗ , m , m , m d , m , m , m, ¯ m 1 , m 2 , m 3 , m 4 , m 5 , m 6 , m 7 } = K of all introduced construction methods (together with the identity) equipped with the composition operator form a group (K, ◦) which is described in Table 1. Proof. It is enough to show that K ⊂ M. The bijectivity of each construction method from K follows from the fact that for each element k ∈ K it holds k ◦ k ◦ k ◦ k = id. Observe that up to m 1 , m 2 , m 5 and m 6 , all other construction methods are involutions, i.e., they are dualities on L (and the orbits of mentioned four construction , m 2 }, ◦) and the other is subgroup methods form two subgroups isomorphic to (Z4 , +), one is the subgroup ({id, m 1 , m ({id, m 5 , m , m 6 }, ◦)). Moreover, if we denote , m, ¯ m d , m , m , id} M1 = {m ∗ , m ∗ , m then (M1 , ◦) is a subgroup of (K, ◦) isomorphic to (Z2 , +)3 . There are several subgroups on (K, ◦) isomorphic to , m 3 }, ◦), ({id, m 7 , m , m }, ◦). On the other hand, the subgroup (id, m 1 , m 2 , m , ◦) is (Z2 , +)2 , for example ({id, m 4 , m isomorphic to Z4 . Obviously, all construction methods preserving copulas which are given in Proposition 2 also form a subgroup (MC , ◦) of (K, ◦), which is described in Table 2. Note that the group (MC , ◦) is non-abelian group isomorphic to a subgroup of permutations of (1, 2, 3, 4) generated by (1, 3, 2, 4) (corresponding to m ∗ ) and by (2, 1, 4, 3) (corresponding to m 7 ). Similarly, symmetry preserving constructions form a subgroup (M B , ◦) of (K, ◦) described in Table 3.
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J. Kalická / Fuzzy Sets and Systems 160 (2009) 726 – 732
Table 3 m\m
m3
m4
m¯
md
m7
m
m
id
m3 m4 m¯ md m7 m m id
id m m m7 md m¯ m4 m3
m id m7 m m¯ md m3 m4
m m7 id m m4 m3 md m¯
m7 m m id m3 m4 m¯ md
md m¯ m4 m3 id m m m7
m¯ md m3 m4 m id m7 m
m4 m3 md m¯ m m7 id m
m3 m4 m¯ md m7 m m id
Note that any discussed construction method preserves the class L0.5 and that the only aggregation function stable (invariant) with any of the discussed construction methods is the arithmetic mean M ∈ L, M(x, y) = (x + y)/2. , id} acting on L0 (more precisely, on copulas which are Recall that the stability for members of M2 = {m ∗ , m ∗ , m can be proper subclass of L0 , however, results for copulas concerning the stability with respect to m ∗ , m ∗ , and m straightforwardly extended to all quasi-copulas) was discussed in [4]. As distinguished examples of copulas invariant under all copula-preserving construction methods given in Proposition 2, recall the product copula (x, y) = x y and the copula K averaging the boundary copulas Min and W . 5. Conclusion We have introduced flipping of aggregation functions as a new construction method for 1-Lipschitz aggregation functions fulfilling A(1, 0) = A(0, 1), (1-Lipschitz property in one variable is sufficient). Moreover, we have introduced several new methods for constructing 1-Lipschitz aggregation functions from L, and we have discussed the algebraic structure of these construction methods. Note that several other construction methods for aggregation functions are summarized in a recent handbook [1]. Acknowledgements The support of Grants VEGA 1/3006/06, VEGA 1/3014/06 and APVV-0375-06 is kindly acknowledged. References [1] G. Beliakov, A. Pradera, T. Calvo, Aggregation functions: a guide for practitioners, Studies in Fuzziness and Soft Computing, Vol. 221, Springer, Berlin, 2007. [2] T. Calvo, A. Kolesárová, M. Komorníková, R. Mesiar, Aggregation functions: properties, classes and construction methods, in: Aggregation Operators, New Trends and Applications, Physica-Verlag, Heidelberg, 2002, pp. 3–106. [3] E.P. Klement, A. Kolesárová, Intervals of 1-Lipschitz aggregation functions, quasi-copulas and copulas with given affine section, Monatsh. Math. 152 (2007) 151–167. [4] E.P. Klement, R. Mesiar, E. Pap, Invariant copulas, Kybernetika 38 (2002) 275–286. [5] A. Kolesárová, J. Mordelová, 1-Lipschitz and kernel aggregation operators, in: AGOP2001, Oviedo, 2001, pp. 71–75. [6] P. Mikusinski, H. Sherwood, M.D. Taylor, Shuffles of Min, Stochastica 13 (1992) 61–74. [7] R.B. Nelsen, An Introduction to Copulas, Lecture Notes in Statistics, Vol. 139, Springer, Berlin, 1999. [8] M. Šabo, V. Strežo, On reverses of some binary functions, Kybernetika 41 (4) (2005) 435–450.