XAO Operators â The interval universe - EUSFLAT
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XAO Operators – The interval universe Tomasa Calvo1,2
Jes´ us L´ azaro 1
Dept. of Computer Science, University of Alcal´ a [email protected]
Dept. of Computer Science, University of Alcal´ a 2 LOBFI Research Group, University of the Balearic Islands 1,2 tomasa.calvo@{uah.es, uib.es}
Summary
why we propose the use of intervals to represent knowledge, as in [16], because in a certain manner this kind of data shows the uncertainty degree. This reason motivated us to define interval–valued aggregation operators on the unit interval, but with the idea of maintaining the possibility of handle totally determined data as standard aggregation operators do.
The aim of this paper is to present the first considerations and examples of interval– valued aggregation operators, which allow to extend the usual concept of aggregation operator. We will define this new kind of operators and present its basic properties, as well as the main requirements needed to maintain the properties of standard aggregation operators. Finally, we present a characterization of various classes of these operators.
This paper is organized as follows. In Section 2 interval–valued aggregation operators, or eXtended Aggregation Operators (XAO operators from here on), are formally defined. Section 3 deals with the basic results obtained from the design of these operators and their relationship with standard aggregation operators and double aggregation operators. The study of the basic properties of these operators is developed in Section 4, where certain classes (families) of XAO are also presented. Finally, Section 5 is devoted to show how to apply these operators over imprecise data, and more concretely on linguistic items.
Keywords: Aggregation operator; interval; interval–valued aggregation; idempotency; duality; symmetry; generator operator; copula; semi–copula; t–norm; ordered weighted averaging operator; double aggregation operator.
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2
Introduction
Interest in information fusion (or aggregation) process has greatly increased in last few years, specially from a theoretical point of view. Standard aggregation operators (this is, those defined in the unit interval) are successfully applied over diverse areas as information and communication science [3], multi–criteria decision making theory [10], fuzzy control [15] or pattern recognition [2]. Standard aggregation operators have been studied in fuzzy sets theory [5]; many of these operators are known to belong to more general classes as Choquet integrals [9], uninorms [8], symmetric summation [14], null–norms [4], etc. Recent studies have focused on fitting aggregation operators to data [1] to find the most suitable operator for a determined problem. But in general, in real life data can’t be obtained in a precise way but in a certain interval of values; this is 189
Basic concepts
Definition 1 An aggregation operators is a mapping [ n A: [0, 1] → [0, 1] such that A(0) = 0, A(1) = 1 n∈N
and it is non–decreasing, i.e., A(x) ≤ A(y) whenever x ≤ y. Each aggregation operator A can be represented by a family of n –ary operators An : [0, 1]n → [0, 1] given by An (x1 , . . . , xn ) = A(x1 , . . . , xn ). This representation allows to define most of the properties of aggregation operators. For purpose of clarity in the paper, and without any loss of generality, we will consider n as being fixed. In order to introduce the interval–valued aggregation operators we need to define both the structure of the data and their relationships (interval–ordering, summation and so on). We will use the following notation. Interval used as data will belong to I[0, 1],
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where x ∈ I[0, 1] means x = [a, b] with a, b ∈ [0, 1], a ≤ b. In this spirit, the domain of the operator will be (I[0, 1])n ; the input will be a list of n intervals x = ([a1 , b1 ], . . . , [an , bn ]) in the previous manner, and the resulting output will also be an interval. Let us denote the interval–ordering on I[0, 1] as ≤s . This (partial) ordering will be the usual interval– ordering on R, i.e., [a, b] ≤s [c, d] if and only if b ≤ c, and will be denominated strong ordering; with this ordering every point from the interval [a, b] is lesser than or equal to any point of [c, d]. We can also define a weak ordering by [a, b] ¹w [c, d] if and only if a ≤ c y b ≤ d. Weak interval–ordering means that for every point in [a, b] there exists a greater than or equal point in [c, d]. In special we will note that the strong ordering is also a weak ordering and that [a, b] ¹w [c, b] if a ≤ c, and [a, b] ¹w [a, d] if b ≤ d; these will represent the left reduction (respectively, right reduction) of an interval. We can extend the ¹w interval–ordering to define the product order on (I[0, 1])n in the following way: x ¹w y if xi ¹w yi for all i ∈ {1, . . . , n}. This order is also a parcial ordering. One point or value j ∈ [0, 1] can be extended to a lower interval [a, j] or to an upper interval [j, b], and obviously [a, j] ≤s [j, b] for any a ≤ j ≤ b. In special, any point j ∈ [0, 1] can be expressed as the interval [j, j] ∈ I[0, 1] and therefore interval–valued aggregation operators must maintain certain properties of standard aggregation operators.
example, [1, 2] ∪ [3, 4] or [1, 2] ∩ [3, 4]); this situation is obtained when the intervals are disjoint, so new operations are needed to extend ∪ and ∩. Given two intervals [a, b], [c, d] we define the operation “closing” ⊕ as [a, b] ⊕ [c, d] = [min{a, c}, max{b, d}]; every point from the intervals is in the covering, but the reciprocal is not true. If two non–disjoint intervals are covered, the resulting intervals coincides with the union, i.e., if x ∩ y 6= ∅ then x ⊕ y = x ∪ y. The covering operation is obviously commutative, transitive and idempotent. In special, note that [a, b] ¹w [c, d] implies [a, b] ¹w [a, b] ⊕ [c, d] ¹w [c, d]. In the case in which b = c we can observe that [a, b] ⊕ [b, d] = [a, d]; it will be called a “continuous closing” and be denoted by ⊕c to show that both intervals have only one point in common. Now, in order to generalize intersection we will define “common part” ¯ as [a, b] ¯ [c, d] = [max{a, c}, min{b, d}] if it is a valid interval, and [min{b, d}, max{a, c}] in other case2 . “Common part” is an interval having at least one point of each interval used; this operation is not transitive, although it is commutative and idempotent, and obviously coincides with intersection if this is not empty. To avoid the former case separation (and possible further ones) we define a new operation “validate” applied over an interval as follows: ½ [a, b] si a ≤ b k[a, b]k = [b, a] en otro caso.
To generalize the addition of real numbers we will consider the usual extension [a, b] + [c, d] = [a + c, b + d], which is the interval in which all possible x + y points can be found, where x and y are points from the first and second interval respectively. It must be observed that the difference of real numbers can’t be directly extended as the immediate interpretation [a, b] − [c, d] = [a−c, b−d] may produce illegal intervals. For example, [5, 6] − [4, 8] = [1, −2]1 .
which allows us to easily redefine “common part” as [a, b] ¯ [c, d] = k[max{a, c}, min{b, d}]k.
To avoid these situations we will consider the interval difference as seen in [13], defined by [a, b] − [c, d] = [a−d, b−c]; this is the interval of all the points obtained as x−y, being x and y points from the first and second interval respectively. However, with this definition, − is no longer the inverse of +, as can be seen in [1, 2] + [2, 4] − [2, 4] = [3, 6] − [2, 4] = [−1, 4].
such that
Furthermore, as closed intervals can be seen as sets, both union and intersection must be applicable. But these operations may produce non–interval results (for
Example 1 The following operators are XAO
1 This results can be obtained even when the intervals are in ≤s ; for example, [1, 4] ≤s [5, 6] but [5, 6] − [1, 4] = [4, 2].
2 This is needed since the result may be a non– valid interval, as can be seen with [2, 3] ¯ [4, 5] = [max{2, 4}, min{3, 5}] = [4, 3].
Once that we have seen how to compare and to operate intervals we are ready to define the interval–valued aggregation operators in the following way: Definition 2 An extended [aggregation operator (XAO) is a mapping A : (I[0, 1])n → I[0, 1] n∈N
• A(x) = [0, 0] if x = ([0, 0], . . . , [0, 0]). • A(x) = [1, 1] if x = ([1, 1], . . . , [1, 1]). • A(x) ¹w A(y) whenever x ¹w y (monotonicity).
• i coordinate projection: Ai (x) = xi .
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• Covering: MA⊕ (x) = [min{ai }, max{bi }], i.e., A⊕ (x) = xi . i
• First load: A← (x) = [min{ai }, min{bi }]. • Second load: A→ (x) = [max{ai }, max{bi }]. • Common interval: Given x, let aM = max{ai } and bm = min{bi } be. Then the common interval is A¯ (x) = k[aM , bm ]k. Every xi has at Jleast one point in A¯ (x). Note that A¯ (x) 6= i xi since ¯ is non–transitive. Observe that these operators can be ordered and that A← (x) ¹w Ai (x)∀i ¹w A→ (x) and also A← (x) ¹w A⊕ (x) ¹w A→ (x) are true. Moreover, A← ⊕c A¯ ⊕c A→ = A⊕ whenever bm ≤ aM . In other case, A¯ = A← ∩ A→ . Nevertheless, A← and A→ are not the smallest and biggest XAO respectively, which are • smallest XAO: Am (x) = [min{ai }, min{ai }], • biggest XAO: AM (x) = [max{bi }, max{bi }]; without taking into account degenerated XAOs (which outputs are equals to [0, 0] or [1, 1] for any input excluding the definition conditions) and XAOs defined via the first (or second) endpoints of their input; for example, A(x) = [0, min{bi }] can be seen as a standard aggregation operator since it ignores all the lower endpoints of the intervals.
3
First results
The first idea to obtain a XAO is to use a pair of standard aggregation operators as generators for the output, i.e., Proposition 1 Let B1 , B2 be two aggregation operators in [0, 1]n . The operator A defined by AB1 ,B2 (x) = k[B1 (a), B2 (b)]k is a XAO in (I[0, 1])n , where x = ([a1 , b1 ], . . . , [an , bn ]) and a = (a1 , . . . , an ), b = (b1 , . . . , bn ). The standard aggregation operators B1 y B2 are called the XAO “generators”, and this will be denoted as AB1 ,B2 . Obviously, the “validate” operation can be eliminated when the generators fulfill B1 (v) ≤ B2 (v) ∀v ∈ [0, 1]n . 191
Following Proposition 1 it is clear that standard aggregation operators can be obtained as particular cases of XAO, just considering the case where the generators B1 = B2 and using as inputs intervals of exactly one point (since a = [a, a]). Moreover, and since in general the operations over intervals can be reduced as operating with their (real) limits, we can express almost any XAO by the former Proposition using the appropriate generators. Remark 1 Observe that the generators determine uniquely the operator A, i.e., if AB1 ,B2 = AB3 ,B4 then B1 = B3 , B2 = B4 : if there exists a0 such that B1 (a0 ) 6= B3 (a0 ) then AB1 ,B2 (a0 , b) 6= AB3 ,B4 (a0 , b) (contradiction). The reciprocal is obvious, so the above condition is necessary and sufficient. Remark 2 The set of XAOs obtained in this manner is a convex one due the convexity of standard aggregation operators. Moreover, it is possible to obtain then XAO generators using the same convex combination. To present this idea we will use the following notation: let B1 , B2 be two standard aggregation operators, and given θ ∈ [0, 1], the linear convex combination θB1 + (1 − θ)B2 will be represented by Cθ (B1 , B2 ), which is obviously a standard aggregation operator. Now, from a certain pair of XAO AB1 ,B2 , AB3 ,B4 and a value θ ∈ [0, 1], the operator C obtained by θAB1 ,B2 + (1 − θ)AB3 ,B4 is a XAO in the sense of Proposition 1, since C = ACθ (B1 ,B3 ),Cθ (B2 ,B4 ) , i.e., a linear convex combination of generators produces the same combination of XAO with its respective generators. Nevertheless, there are XAO that can not be obtained in terms of Proposition 1: all those use input endpoints in the definition of the opposite output endpoint. For example, A¯ , Am o AM . In these cases, the lower endpoint of the output (respectively, the upper endpoint) depends not only on the input lower endpoints, and then its value can not be obtained via an standard aggregation operator. This means that there is no standard aggregation operator that can be chosen as generator. To solve this situation we will generalize the definition of XAO replacing standard aggregation operators with Double Aggregation Operators [6] applied on both inputs a and b. [ [0, 1]n → [0, 1] be two Definition 3 Let F , G : n∈N
aggregation operators and H : [0, 1]2 → [0, 1] a binary aggregation operator. A double aggregation operator (DAO) is a function [ A = AF,G,H : [0, 1]n × [0, 1]m → [0, 1] (n,m)∈N[2]
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with N[2] = N2 ∪ ({0} × N) ∪ (N × {0}), defined by AF,G,H (x(n) , y(m) ) = H(F (x(n) , G(y(m) )) si n, m ∈ N, F (x(n) ) si m = 0, = G(y(m) ) si n = 0, for each x(n) = (x1 , . . . , xn ) ∈ [0, 1]n , y(m) = (y1 , . . . , ym ) ∈ [0, 1]m . Remark 3 In the following, we will only consider the case n 6= 0 and m 6= 0, i.e., there always are two inputs. Proposition 2 Let B1 , B2 be two double aggregation operators with domain [0, 1]2n . The operator A defined by AB1 ,B2 (x) = k[B1 (a, b), B2 (a, b)]k is a XAO operator on (I[0, 1])n . Remark 4 Remarks 1 and 2 can be extended to XAO generated by double aggregation operators. Remark 5 Every XAO can be expressed as in Proposition 2. Apparently, there is no direct relation between the properties of B1 , B2 and the AB1 ,B2 ones. For example, using B1 = B2 = min as generators only produces the minimum XAO Am whenever both generators are DAOs. On the other hand, A← is the “normal ” XAO generated by minimum, considering “normal” as when each output endpoint only depends of that kind of input endpoints, and it is independent of other values or constants3 . For purpose of clarity, we will always assume that the generators will be standard aggregation operators; when this generators need to be DAO it will be specifically indicated. Proposition 3 Every “normal” XAO can be obtained using standard aggregation operators as generators. If the XAO is “normal” is because the output has its endpoints defined as a function of the corresponding inputs endpoints, i.e., if x = ([a1 , b1 ], . . . , [an , bn ]) then C(x) = [f (a), g(b)]. These functions f , g fulfill f (0) = g(0) = 0, f (1) = g(1) = 1, and as f , g are non– decreasing (since C is itself non–decreasing); therefore, f and g are aggregation operators and C = Af,g . 3
This external values or constants are needed because in other case there could exist XAO which could not be generated from aggregation operators. For example, the degenerated maximum XAO: [0, 0] if xi = [0, 0]∀i C(x) = If this XAO were [1, 1] in other case. AB1 ,B2 for certain B1 , B2 aggregation operators, then having xi = [0, bi ] with bi 6= 0 would result in [1, 1] = C(x) = AB1 ,B2 (x) = [B1 (0), B2 (b)] = [0, B2 (b)], which is not possible.
Proposition 4 Every XAO can be obtained using double aggregation operators. Obviously, if A is “normal” it is enough that its double generators take into account only one endpoint from the input (for example, B1 (a, b) = P1 (B11 (a), B21 (b)), i.e., the projection of the first coordinate of two standard aggregation operators). Similarly, in the case of A being not “normal” only two functions f and g which considerate both inputs are needed, although a certain set of constants may also be required4 : f (a, b, k). This functions must fulfill f (0, 0, k) = 0, f (1, 1, k) = 1, and must be non–decreasing; any function in this manner can be represented by a double aggregation operator [6].
4
Study of properties
In this Section we recall some basic properties of standard aggregation operators and adapt them to interval–valued operators; several classes of XAO are presented (XAO t–norms. . . ) and related with the standard case. Along this Section it will be considered that the generators B1 and B2 (independently of being standard or double aggregation operators) fulfill B1 (x) ≤ B2 (x); this allows us to study the properties of AB1 ,B2 without considering the “validate” operation. 4.1
Basic properties
As the XAO are a generalization of the standard aggregation operators, we could study their properties as if they were standard aggregation operators (i.e. applying them on point–intervals as [j, j]). Another possible direction is to observe the generators in order to relate them with the desired properties; this will be the option applied in the paper. For example, the idempotency will be naturally defined as A(x) = [a, b] if xi = [a, b] ∀i. The following Proposition characterizes the idempotent XAO. Proposition 5 A XAO is idempotent on I[0, 1] if and only if its generators are idempotent on [0, 1]. Remark 6 To clarify the notation we will indicate here that the domain of the XAO should be (I[0, 1])n , so strictly speaking it would be “idempotent on I[0, 1]n ”; on the other hand, its generators would be 4 The operator C defined in footnote 3 can be obtained with the following DAOs as generators: B1 (a, b) = 0 if a = b = 0 B2 (a, b) = It is clear that the con1 otherwise. stant (1) is needed and that it is non–input dependent; in this example, B1 (a, b) = f (a, b, (1)).
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idempotent on [0, 1]n or [0, 1]2n (standard or double aggregation operators, respectively). To avoid the excess of notation, we will only indicate the domain of each component of the inputs to avoid confusion between intervals (XAO) and points (generators). Other well–known property is symmetry; a XAO A will be called symmetric when A(x) = A(xσ ) for any permutation σ of {1, . . . , n}, where xσi = xσ(i) . Proposition 6 A XAO is symmetric on I[0, 1] if and only if its generators are symmetric on [0, 1]. Clearly, if both generators are symmetric the XAO is symmetric as well; on the other hand, if there were a non–symmetric generator for a certain permutation aσ , the non–symmetry of A could be deducted from x = (a, a). To talk about continuity we will need a distance. Here we will use the Hausdorff’s metric: D([a, b], [c, d]) = max{|a − c|, |b − d|}. This metric seems not appropriated if it is considered as a distance (since D([1, 2], [1, 10]) = D([1, 2], [9, 10]) but the first two intervals are “closer” than the last ones) but it is very useful when applied to check continuity, as if two intervals have a small value for D then both intervals have similar endpoints. Proposition 7 A XAO is continuous on I[0, 1] if and only if its generators are continuos on [0, 1]. The idea behind Proposition 7 is that if one generator is discontinuous in ai , as the Hausdorff metric takes into account the maximum of the differences it is not important if the other generator is continuos or not. We can combine Proposition 4 and 7 in the following statement: Proposition 8 Any continuous XAO can be obtained by means of B1 , B2 DAO such that B1 ≤ B2 . Another interesting property is “additivity with respect to ∗”, in the sense of A(x ∗ y) = A(x) ∗ A(y). For example, if we consider ∗ = ⊕, then the XAO Ai or A⊕ fulfills it. Proposition 9 A XAO is additive with respect to ∗ on I[0, 1] if and only if its generators are additive for the corresponding endpoint functions on [0, 1]. Let B1 , B2 be the generators of C. Then it is that C(x ∗ y) = C(x) ∗ C(y) can be expressed respect B1 and B2 as [B1 (ax ∗1 ay ), B2 (bx ∗2 by )] = [B1 (ax ), B2 (bx )] ∗ [B1 (ay ), B2 (by )] = [B1 (ax ) ∗1 B1 (ay ), B2 (bx ) ∗2 B2 (by )], i.e., B1 and B2 must be additive with respect to ∗i . 193
4.2
Operating with data
As we defined the + operation it is possible to speak of interval shifting, considering [a+c, b+c] = [a, b]+[c, c]. Analogously to shift invariant aggregation operators [12], a XAO will be said to be shift invariant when for all x + c ∈ I[0, 1], it is A(x + c) = A(x) + c. The following proposition characterizes these operators. Proposition 10 A XAO A is shift invariant if and only if A is idempotent and additive with respect to +. In the same spirit, a XAO will be called scalable if A(k · x) = k · A(x); observe that if k < 0 the endpoints of the interval are reversed and “validate” is needed. When a operator is scalable and shift invariant is said to be invariant for linear transformations. In general, let Φ, ϕ be bijections on I[0, 1] and [0, 1] respectively; let define Φ([a, b]) = k[ϕ(a), ϕ(b)]k. Note that “validate” is only needed if ϕ is decreasing. With this bijections and given a certain XAO A, we can define a XAO B in the following way B(x1 , . . . , xn ) = Φ−1 (A(Φ(x1 ), . . . , Φ(xn ))) B will be called the “Φ–transformed of A”. Special cases of this kind of transformation are the prior linear transformations, where ϕ(x) = kx + c was used for certain values k, c. Proposition 11 A XAO is invariant for Φ in I[0, 1] if and only if its generators are invariant for ϕ in [0, 1]. 4.3
Duality
Many standard aggregation operators can be obtained through a duality transformation. Here we present an analogous transformation for XAO. To apply duality a strong negation n is needed. As this negation will be applied on intervals we have to define this operation first. Let n : [0, 1] → [0, 1] be a negation. Then we define n : I[0, 1] → I[0, 1] by n([a, b]) = [n(b), n(a)]. In this paper we will only focus on strong negations, i.e., such that it is n(n([a, b])) = [a, b]. This situation is another special case of bijections as seen in previous Subsection, but taking into account that n−1 = n and that negations are decreasing functions (this is the reason why negations are not defined as n([a, b]) = k[n(a), n(b)]k, as the interval would be always reordered). With this, given a certain XAO A we will define its dual XAO Ad as Ad (x1 , . . . , xn ) = n(A(n(x1 ), . . . , n(xn ))). Example 2 Let n(x) = 1 − x and A = A⊕ be; then, we have the following chain of equalities:
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d
A⊕ ([a1 , b1 ], . . . , [an , bn ]) = n(A([n(b1 ), n(a1 )], . . . , [n(bn ), n(an )])) = n([min{n(bi )}, max{n(ai )}]) = n([1 − max{bi }, 1 − min{ai }]) = [min{ai }, max{bi }] = A⊕ ([a1 , b1 ], . . . , [an , bn ]) d
i.e., A⊕ = A⊕ . As in the standard situation, a XAO will be called selfdual if Ad = A. Proposition 12 Let C = AB1 ,B2 be a XAO; then Cd = AB2d ,B1d is its dual XAO. The proof is directly derived from the definition of duality: Cd (x) = n(AB1 ,B2 (n(x))) = n([B1 (n(b)), B2 (n(a))] = [n(B2 (n(a))), n(B1 (n(b)))] = [B2d (a), B1d (b)]. Having Proposition 12, selfduality is characterized as follows: Proposition 13 A XAO is selfdual if and only if its generators are dual of each other. 4.4
Families of XAO
Now we will present several families of XAO, which will be defined (if possible) as the families of standard aggregation operators [5] obtained when the XAO is applied to point–interval inputs. For example, a XAO fulfilling symmetry, associativity and with neutral element [1, 1] will be a t–norm. We will also show the relationship between the XAO generators families and the XAO itself; this will solve questions as “Let B1 and B2 be t–norms; is AB1 ,B2 a t–norm?”. Initially, every families of operators based in linear operations on data will be maintained, since these operations are also valid on intervals; for example, [2x1 + 3x2 , 2y1 + 3y2 ] = 2 · [x1 , y1 ] + 3 · [x2 , y2 ]. The following Proposition presents this case when a weighted operator is used. Proposition 14 The XAO AB1 ,B2 has the weighting triangle 4 = (win ) if and only if B1 = B2 are aggregation operators with the same weighting triangle 4. For example, weighted means fulfill Proposition 14. Note that XAO weighted means can also be characterized by their additivity for +: Proposition 9 shows that if a XAO is additive its generators are also additive for the same operation, and (standard) weighted means are the aggregation operators additive for +. Now, to prove that both generators have the same weighting triangle is enough to recall that XAO are non– decreasing operators, and since two different means are incomparable5 the only possibility is that both means have the same weighting triangle. 5
For example, let B1 (x1 , x2 ) = 13 x1 + 23 x2 , B2 (x1 , x2 ) =
The next step would be the Ordered Weighted Averaging operators (OWA operators) but this family needs a total ordering of the data which is not possible when working with intervals, since ¹w is a partial ordering. Moreover, even in the case of B1 , B2 being OWA the operator AB1 ,B2 may have no weighting triangle associated. Example 3 Take the OWA B1 = B2 = min as generators, and then AB1 ,B2 = A← . This XAO is not an OWA since, in first place, it has no weighting triangle as can be seen considering A← ([1, 4], [2, 3]) = [1, 3]: there are no w1 , w2 such that w1 [1, 4] + w2 [2, 3] = [1, 3] with w1 + w2 = 1. 4.5
Family of copula
To extend the standard family of copula we will recall the definition of semi–copula (see e.g. [7]) as it is a generalization of copulas, quasi–copulas, t–norms. . . Definition 4 A function S : [0, 1]2 → [0, 1] is a semi– copula if and only S it fulfills the following two conditions: (a) S(x, 1) = S(1, x) = x ∀x ∈ [0, 1]; (b) S(x, y) is non–decreasing in each argument. From this definition, and in the spirit of maintaining its generality for the whole family, we present the following definition for a XAO semi–copula. Definition 5 A XAO A : (I[0, 1])2 → I[0, 1] is called a semi–copula if A([a, b], [1, 1]) = A([1, 1], [a, b]) = [a, b]. Remark 7 Note that XAO are non–decreasing, and therefore condition (b) is no longer needed. Also, as [1, 1] is a neutral element of A, then it follows that for all [a, b] ∈ I[0, 1], [0, 0] ≤ A([a, b], [0, 0]) ≤ A([1, 1], [0, 0]) ≤ [0, 0]; i.e., [0, 0] is an annihilator of A. Example 4 The XAO 1) Sm (x1 , x2 ) = ½
min{x1 , x2 } if max{x1 , x2 } = [1, 1], [0, 0] otherwise.
2) SM (x1 , x2 ) = 2 x 3 1
+ 13 x2 be weighted means. As (0.2, 0.5) ≤ (0.6, 0.9) provides B1 (0.2, 0.5) ≤ B2 (0.6, 0.9), while (0.1, 0.7) ≤ (0.2, 0.8) results in B1 (0.1, 0.7) > B2 (0.2, 0.8), i.e., B1 and B2 are incomparables. 194
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½
[0, 0] if min{x1 , x2 } = [0, 0], A← (x1 , x2 ) otherwise.
are semi–copulae and they are the smallest and biggest semi–copula, respectively. Now we are going to particularize the semi–copula family to define more particular classes • Copula: given x1 , x2 , y1 , y2 ∈ I[0, 1] such that x1 ¹w y1 and x2 ¹w y2 , if A fulfills A(x2 , y2 ) − A(x1 , y2 ) − A(x2 , y1 ) + A(x1 , y1 ) ºw [0, 0] or “equivalently” A(x1 , y2 ) + A(x2 , y1 ) ¹w A(x1 , y1 ) + A(x2 , y2 ). The first expression is the standard definition of copula but using a XAO; as − is not the inverse of + in I[0, 1] the expression lacks the idea of the standard case. On the other hand, the second expression is much more stable and clear. For example, if we consider x1 = [2, 5], y1 = [3, 7], x2 = [5, 8], y2 = [5, 9] and A = A← (x) = [min{ai }, min{bi }], we would like to expect “good properties” for A since it is generated by the minimum standard aggregation operator (which is a standard copula). It is easily proven that A only fulfills the second expression. • Quasi–copula: let x1 , x2 , y1 , y2 be defined as before; the direct translation of quasi–copula would be |A(x1 , y1 ) − A(x2 , y2 )| ¹w |x1 − x2 | + |y1 − y2 | but the absolute value does nothing related to intervals; anyway, when this expression is applied on standard aggregation operators it is the 1–Lipschitz property, where the absolute value is used as a distance indicator. This leads us to suggest the following expression as a possible interpretation for quasi–copula: D(A(x1 , y1 ), A(x2 , y2 )) ≤ D(A(x1 , [1, 1]), A(x2 , [1, 1])) + D(A([0, 0], y1 ), A([0, 0], y2 )). where D is the Hausdorff metric. • t–norm: given x, y, z ∈ I[0, 1], we define a XAO A as a t–norm if it fulfill the standard properties (1) A(x, y) = A(y, x), i.e., A is commutative, and (2) A(A(x, y), z) = A(x, A(y, z)), i.e., A is associative.
Having these classes defined, we are ready to examine the relationship between the XAO and its generators for operators in these families. One possible method to generate a XAO t–norm is to use standard t–norms as generators (as seen in Proposition 1), i.e., given T1 , T2 : [0, 1]n → [0, 1] t– norms, the XAO defined by A([a1 , b1 ], . . . , [an , bn ]) = [T1 (a1 , . . . , an ), T2 (b1 , . . . , bn )] is a t–norm in I[0, 1]. Q Example 5 Using B1 = min and B2 = as generators, the following XAO is obtained AB1 ,B2 (x1 , x2 ) = [min{ax1 , ax2 }, bx1 · bx2 ] which is clearly a t–norm. On the other hand, if AB1 ,B2 is a t–norm, then B1 and B2 must also be t–norms. To prove this only commutativity (symmetry) and associativity of Bi must be checked. Symmetry is obtained from Proposition 6, and associativity is directly achieved from the XAO associativity. Proposition 15 A XAO is a t–norm in I[0, 1] if and only if its generators are t–norms in [0, 1]. Now, and in a similar way to the standard case, we will obtain the family of t–conorms via duality: S(x1 , . . . , xn ) = n(T(n(x1 ), . . . , n(xn ))); these operators can be also obtained using t–conorms as generators in Proposition 1 or in the same spirit of Proposition 15. The previous ideas are also valid relating copulae when the condition A(x1 , y2 ) + A(x2 , y1 ) ¹w A(x1 , y1 ) + A(x2 , y2 ) is considered. Moreover, this expression allows t–norms to be automatically copulae (as in the standard case)6 . To prove this it is enough to observe that the ¹w ordering implies two standard ≤ orderings, which B1 and B2 will fulfill since they are standard copulae. Proposition 16 A XAO is a copula in I[0, 1] if and only if its generators are copulae in [0, 1].
5
Applications
Since one of the possible uses of XAO is based in the uncertainty of data, this uncertainty (i.e., the data interval) should be quantified somehow; for example, using the length of the interval L([a, b]) = b − a. The lesser the length, the more precise are the data; it may even be completely determined to a certain point [a, a], in which case the length will be zero. 6
Note that it also fulfills the standard conditions imposed in [0, 0] and [1, 1] since it is a semi– copula. 195
With this, now it is clear that A← is a copula since it is generated by the min standard aggregation operator although it failed to fulfill the expression with interval differences.
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Moreover, as XAO can symbolize the data uncertainty a possible application is when data are naturally imprecise or subject to incomplete knowledge, for example, linguistic items. Given a certain range of discrete values (e.g. bad, average and good) which can be associated to determined points in a continuous scale (respectively, 0, 0.5 and 1), it is possible to generate the interval of points corresponding each linguistic label (say [0, 0.25], [0.25, 0.75] and [0.75, 1])7 . Lexical modifier can be used in two ways: considered directly on the base interval, which produces a modification of the linguistic term (Example 6), or applying a determined metric to the input data (Example 7).
6
Conclusions
Extended aggregation operators generalize standard aggregation operators and, under certain circumstances, generalize double aggregation operators as well; its main advantage is the uncertainty treatment, which can be measured both in data as in outputs. We have also shown that extended aggregation operators maintain the families and properties of well– known standard aggregation operators; moreover, extended aggregation operators inherit the basic characteristics of its generators. Acknowledgements
Example 6 Lets consider the bad label, the interval [0, 0.25] and its associated discrete value vb = 0. We can use the very modifier, for example, as an interval reduction through the mean operator. In b this case, very bad would be the interval bad+v , i.e., 2 0.25+0 [ 0+0 , ] = [0, 0.125]. Analogously, very average 2 2 and very good would use va = 0.5 and vg = 1 to obtain [0.375, 0.625] and [0.875, 1] respectively. Here we have used the mean operator to modify the linguistic intervals, but any XAO generated by aggregation operators used in the usual treatment of linguistic items can be applied. Example 7 The ouput of a XAO is [0.1, 0.2]. Which linguistic label defines it more appropriately? A first idea would be to use the Hausdorff metric on bad, average and good; the following results would be obtained: rb = D([0.1, 0.2], [0, 0.25]) = 0.1, ra = D([0.1, 0.2], [0.25, 0.75]) = 0.55 and rg = D([0.1, 0.2], [0.75, 1]) = 0.8, so the “closer” label would be bad. Although this approach is enough to solve simple problems, the “closer” idea can be improved when instead of using the Hausdorff metric, a more complex similarity function is applied; for example, consider L(x∩y) S(x, y) = L(x⊕y) which takes into account both the points in common of the intervals as the full covering. Note that it is needed that all intervals representing linguistic labels have the same length, otherwise there will be a distortion in the similarity measurement8 (this can always be obtained with the transformation of bijections as seen in Subsection 4.2). 7
This is an arbitrary assignment and it is only devoted to provide an example. 8 For example, let l1 = [0, 10] and l2 = [10, 30] be intervals representing two linguistic labels, and let x = [5, 20] be the interval to measure; observe that l1 +[5, 10] = x and that x + [5, 10] = l2 , i.e., apparently x is as similar to l1 as to l2 . But when applying S defined as before, it is that 8 5 and S(l2 , x) = 20 . This situation happens S(l1 , x) = 20 because l2 is longer than l1 .
This work has been partially supported by the projects of Spanish Science and Technology Ministry MTM2004–3175, TIC2002–11942–E, BFM2003-05308, the European project COST 274 TARSKI and the project of the Balearic Islands PRIB–2004–9250.
References [1] G. Beliakov, How to build aggregation operators from data. Int. J. Intell. Syst. 18 (8): 903–923 (2003). [2] J.C. Bezdek y S.K. Pal. Fuzzy models for pattern recognition. IEEE Press, Piscataway. N.Y., 1992. [3] I. Bloch, H. Maˆıtre. Fusion of Image Information under Imprecision. In Aggregation of Evidence under Fuzziness, B. Bouchon-Meunier (ed), Series Studies in Fuzziness, Physica Verlag, Springer, 1997. [4] T. Calvo, B. De Baets y J. Fodor. The functional equations of Alsina and Frank for uninorms and null-norms, Fuzzy Sets and Systems 120 (2001) 15-24. [5] T. Calvo, G. Mayor y R. Mesiar (eds.) Aggregation Operators: New Trends and Applications, Berlin, Germany: Physica-Verlag, 2002. [6] T. Calvo y A. Pradera. Double aggregation operators. Fuzzy Sets and Systems 142 (2004) 15–33. [7] F. Durante y C. Sempi. Semicopulæ. Kybernetika (In press). [8] J. Fodor, R. Yager, A. Rybalov. Structure of uninorms, Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 5 (1997) 411-427. [9] M. Grabisch, T. Murofushi y M. Sugeno (Eds). Fuzzy Measures and Integrals, Heidelberg, Germany: Physica-Verlag, 1999. 196
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[10] F. Herrera, E. Herrera–Viedma y J. L. Verdegay. A rational consensus model in group decision making using linguistic assessments. Fuzzy Sets and Systems 88 (1997) 31–49. [11] S. Jenei. A more efficient method for defining fuzzy connectives. Fuzzy Sets and Systems 90 (1997) 25–35. [12] J. L´azaro, T. R¨ uckschlossov´a, y T. Calvo. Shift invariant binary aggregation operators. Fuzzy Sets and Systems 142, 1 (2004), 51-62. [13] R. Moore, Interval Analysis. Prentice Hall, Englewood Cliffs, NJ, 1966. [14] W. Silvert. Symmetric summation: A class of operations on fuzzy sets, IEEE Trans. Systems, Man Cybernet. 9 (10) (1979) 657-659. [15] R.R. Yager. On the control of hierarchical clustering, Proceedings Eight IPMU International Conference on Information Processing and Management of Uncertainty in Knowledge–based Systems, Madrid, Spain; 2000. pp 1335–1341. [16] L.A. Zadeh. The concept of a linguistic variable and its applications to approximate reasoning - I, Inform. and Control 8 (1975) 199–249.
197
XAO Operators – The interval universe Tomasa Calvo1,2
Jes´ us L´ azaro 1
Dept. of Computer Science, University of Alcal´ a [email protected]
Dept. of Computer Science, University of Alcal´ a 2 LOBFI Research Group, University of the Balearic Islands 1,2 tomasa.calvo@{uah.es, uib.es}
Summary
why we propose the use of intervals to represent knowledge, as in [16], because in a certain manner this kind of data shows the uncertainty degree. This reason motivated us to define interval–valued aggregation operators on the unit interval, but with the idea of maintaining the possibility of handle totally determined data as standard aggregation operators do.
The aim of this paper is to present the first considerations and examples of interval– valued aggregation operators, which allow to extend the usual concept of aggregation operator. We will define this new kind of operators and present its basic properties, as well as the main requirements needed to maintain the properties of standard aggregation operators. Finally, we present a characterization of various classes of these operators.
This paper is organized as follows. In Section 2 interval–valued aggregation operators, or eXtended Aggregation Operators (XAO operators from here on), are formally defined. Section 3 deals with the basic results obtained from the design of these operators and their relationship with standard aggregation operators and double aggregation operators. The study of the basic properties of these operators is developed in Section 4, where certain classes (families) of XAO are also presented. Finally, Section 5 is devoted to show how to apply these operators over imprecise data, and more concretely on linguistic items.
Keywords: Aggregation operator; interval; interval–valued aggregation; idempotency; duality; symmetry; generator operator; copula; semi–copula; t–norm; ordered weighted averaging operator; double aggregation operator.
1
2
Introduction
Interest in information fusion (or aggregation) process has greatly increased in last few years, specially from a theoretical point of view. Standard aggregation operators (this is, those defined in the unit interval) are successfully applied over diverse areas as information and communication science [3], multi–criteria decision making theory [10], fuzzy control [15] or pattern recognition [2]. Standard aggregation operators have been studied in fuzzy sets theory [5]; many of these operators are known to belong to more general classes as Choquet integrals [9], uninorms [8], symmetric summation [14], null–norms [4], etc. Recent studies have focused on fitting aggregation operators to data [1] to find the most suitable operator for a determined problem. But in general, in real life data can’t be obtained in a precise way but in a certain interval of values; this is 189
Basic concepts
Definition 1 An aggregation operators is a mapping [ n A: [0, 1] → [0, 1] such that A(0) = 0, A(1) = 1 n∈N
and it is non–decreasing, i.e., A(x) ≤ A(y) whenever x ≤ y. Each aggregation operator A can be represented by a family of n –ary operators An : [0, 1]n → [0, 1] given by An (x1 , . . . , xn ) = A(x1 , . . . , xn ). This representation allows to define most of the properties of aggregation operators. For purpose of clarity in the paper, and without any loss of generality, we will consider n as being fixed. In order to introduce the interval–valued aggregation operators we need to define both the structure of the data and their relationships (interval–ordering, summation and so on). We will use the following notation. Interval used as data will belong to I[0, 1],
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where x ∈ I[0, 1] means x = [a, b] with a, b ∈ [0, 1], a ≤ b. In this spirit, the domain of the operator will be (I[0, 1])n ; the input will be a list of n intervals x = ([a1 , b1 ], . . . , [an , bn ]) in the previous manner, and the resulting output will also be an interval. Let us denote the interval–ordering on I[0, 1] as ≤s . This (partial) ordering will be the usual interval– ordering on R, i.e., [a, b] ≤s [c, d] if and only if b ≤ c, and will be denominated strong ordering; with this ordering every point from the interval [a, b] is lesser than or equal to any point of [c, d]. We can also define a weak ordering by [a, b] ¹w [c, d] if and only if a ≤ c y b ≤ d. Weak interval–ordering means that for every point in [a, b] there exists a greater than or equal point in [c, d]. In special we will note that the strong ordering is also a weak ordering and that [a, b] ¹w [c, b] if a ≤ c, and [a, b] ¹w [a, d] if b ≤ d; these will represent the left reduction (respectively, right reduction) of an interval. We can extend the ¹w interval–ordering to define the product order on (I[0, 1])n in the following way: x ¹w y if xi ¹w yi for all i ∈ {1, . . . , n}. This order is also a parcial ordering. One point or value j ∈ [0, 1] can be extended to a lower interval [a, j] or to an upper interval [j, b], and obviously [a, j] ≤s [j, b] for any a ≤ j ≤ b. In special, any point j ∈ [0, 1] can be expressed as the interval [j, j] ∈ I[0, 1] and therefore interval–valued aggregation operators must maintain certain properties of standard aggregation operators.
example, [1, 2] ∪ [3, 4] or [1, 2] ∩ [3, 4]); this situation is obtained when the intervals are disjoint, so new operations are needed to extend ∪ and ∩. Given two intervals [a, b], [c, d] we define the operation “closing” ⊕ as [a, b] ⊕ [c, d] = [min{a, c}, max{b, d}]; every point from the intervals is in the covering, but the reciprocal is not true. If two non–disjoint intervals are covered, the resulting intervals coincides with the union, i.e., if x ∩ y 6= ∅ then x ⊕ y = x ∪ y. The covering operation is obviously commutative, transitive and idempotent. In special, note that [a, b] ¹w [c, d] implies [a, b] ¹w [a, b] ⊕ [c, d] ¹w [c, d]. In the case in which b = c we can observe that [a, b] ⊕ [b, d] = [a, d]; it will be called a “continuous closing” and be denoted by ⊕c to show that both intervals have only one point in common. Now, in order to generalize intersection we will define “common part” ¯ as [a, b] ¯ [c, d] = [max{a, c}, min{b, d}] if it is a valid interval, and [min{b, d}, max{a, c}] in other case2 . “Common part” is an interval having at least one point of each interval used; this operation is not transitive, although it is commutative and idempotent, and obviously coincides with intersection if this is not empty. To avoid the former case separation (and possible further ones) we define a new operation “validate” applied over an interval as follows: ½ [a, b] si a ≤ b k[a, b]k = [b, a] en otro caso.
To generalize the addition of real numbers we will consider the usual extension [a, b] + [c, d] = [a + c, b + d], which is the interval in which all possible x + y points can be found, where x and y are points from the first and second interval respectively. It must be observed that the difference of real numbers can’t be directly extended as the immediate interpretation [a, b] − [c, d] = [a−c, b−d] may produce illegal intervals. For example, [5, 6] − [4, 8] = [1, −2]1 .
which allows us to easily redefine “common part” as [a, b] ¯ [c, d] = k[max{a, c}, min{b, d}]k.
To avoid these situations we will consider the interval difference as seen in [13], defined by [a, b] − [c, d] = [a−d, b−c]; this is the interval of all the points obtained as x−y, being x and y points from the first and second interval respectively. However, with this definition, − is no longer the inverse of +, as can be seen in [1, 2] + [2, 4] − [2, 4] = [3, 6] − [2, 4] = [−1, 4].
such that
Furthermore, as closed intervals can be seen as sets, both union and intersection must be applicable. But these operations may produce non–interval results (for
Example 1 The following operators are XAO
1 This results can be obtained even when the intervals are in ≤s ; for example, [1, 4] ≤s [5, 6] but [5, 6] − [1, 4] = [4, 2].
2 This is needed since the result may be a non– valid interval, as can be seen with [2, 3] ¯ [4, 5] = [max{2, 4}, min{3, 5}] = [4, 3].
Once that we have seen how to compare and to operate intervals we are ready to define the interval–valued aggregation operators in the following way: Definition 2 An extended [aggregation operator (XAO) is a mapping A : (I[0, 1])n → I[0, 1] n∈N
• A(x) = [0, 0] if x = ([0, 0], . . . , [0, 0]). • A(x) = [1, 1] if x = ([1, 1], . . . , [1, 1]). • A(x) ¹w A(y) whenever x ¹w y (monotonicity).
• i coordinate projection: Ai (x) = xi .
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• Covering: MA⊕ (x) = [min{ai }, max{bi }], i.e., A⊕ (x) = xi . i
• First load: A← (x) = [min{ai }, min{bi }]. • Second load: A→ (x) = [max{ai }, max{bi }]. • Common interval: Given x, let aM = max{ai } and bm = min{bi } be. Then the common interval is A¯ (x) = k[aM , bm ]k. Every xi has at Jleast one point in A¯ (x). Note that A¯ (x) 6= i xi since ¯ is non–transitive. Observe that these operators can be ordered and that A← (x) ¹w Ai (x)∀i ¹w A→ (x) and also A← (x) ¹w A⊕ (x) ¹w A→ (x) are true. Moreover, A← ⊕c A¯ ⊕c A→ = A⊕ whenever bm ≤ aM . In other case, A¯ = A← ∩ A→ . Nevertheless, A← and A→ are not the smallest and biggest XAO respectively, which are • smallest XAO: Am (x) = [min{ai }, min{ai }], • biggest XAO: AM (x) = [max{bi }, max{bi }]; without taking into account degenerated XAOs (which outputs are equals to [0, 0] or [1, 1] for any input excluding the definition conditions) and XAOs defined via the first (or second) endpoints of their input; for example, A(x) = [0, min{bi }] can be seen as a standard aggregation operator since it ignores all the lower endpoints of the intervals.
3
First results
The first idea to obtain a XAO is to use a pair of standard aggregation operators as generators for the output, i.e., Proposition 1 Let B1 , B2 be two aggregation operators in [0, 1]n . The operator A defined by AB1 ,B2 (x) = k[B1 (a), B2 (b)]k is a XAO in (I[0, 1])n , where x = ([a1 , b1 ], . . . , [an , bn ]) and a = (a1 , . . . , an ), b = (b1 , . . . , bn ). The standard aggregation operators B1 y B2 are called the XAO “generators”, and this will be denoted as AB1 ,B2 . Obviously, the “validate” operation can be eliminated when the generators fulfill B1 (v) ≤ B2 (v) ∀v ∈ [0, 1]n . 191
Following Proposition 1 it is clear that standard aggregation operators can be obtained as particular cases of XAO, just considering the case where the generators B1 = B2 and using as inputs intervals of exactly one point (since a = [a, a]). Moreover, and since in general the operations over intervals can be reduced as operating with their (real) limits, we can express almost any XAO by the former Proposition using the appropriate generators. Remark 1 Observe that the generators determine uniquely the operator A, i.e., if AB1 ,B2 = AB3 ,B4 then B1 = B3 , B2 = B4 : if there exists a0 such that B1 (a0 ) 6= B3 (a0 ) then AB1 ,B2 (a0 , b) 6= AB3 ,B4 (a0 , b) (contradiction). The reciprocal is obvious, so the above condition is necessary and sufficient. Remark 2 The set of XAOs obtained in this manner is a convex one due the convexity of standard aggregation operators. Moreover, it is possible to obtain then XAO generators using the same convex combination. To present this idea we will use the following notation: let B1 , B2 be two standard aggregation operators, and given θ ∈ [0, 1], the linear convex combination θB1 + (1 − θ)B2 will be represented by Cθ (B1 , B2 ), which is obviously a standard aggregation operator. Now, from a certain pair of XAO AB1 ,B2 , AB3 ,B4 and a value θ ∈ [0, 1], the operator C obtained by θAB1 ,B2 + (1 − θ)AB3 ,B4 is a XAO in the sense of Proposition 1, since C = ACθ (B1 ,B3 ),Cθ (B2 ,B4 ) , i.e., a linear convex combination of generators produces the same combination of XAO with its respective generators. Nevertheless, there are XAO that can not be obtained in terms of Proposition 1: all those use input endpoints in the definition of the opposite output endpoint. For example, A¯ , Am o AM . In these cases, the lower endpoint of the output (respectively, the upper endpoint) depends not only on the input lower endpoints, and then its value can not be obtained via an standard aggregation operator. This means that there is no standard aggregation operator that can be chosen as generator. To solve this situation we will generalize the definition of XAO replacing standard aggregation operators with Double Aggregation Operators [6] applied on both inputs a and b. [ [0, 1]n → [0, 1] be two Definition 3 Let F , G : n∈N
aggregation operators and H : [0, 1]2 → [0, 1] a binary aggregation operator. A double aggregation operator (DAO) is a function [ A = AF,G,H : [0, 1]n × [0, 1]m → [0, 1] (n,m)∈N[2]
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with N[2] = N2 ∪ ({0} × N) ∪ (N × {0}), defined by AF,G,H (x(n) , y(m) ) = H(F (x(n) , G(y(m) )) si n, m ∈ N, F (x(n) ) si m = 0, = G(y(m) ) si n = 0, for each x(n) = (x1 , . . . , xn ) ∈ [0, 1]n , y(m) = (y1 , . . . , ym ) ∈ [0, 1]m . Remark 3 In the following, we will only consider the case n 6= 0 and m 6= 0, i.e., there always are two inputs. Proposition 2 Let B1 , B2 be two double aggregation operators with domain [0, 1]2n . The operator A defined by AB1 ,B2 (x) = k[B1 (a, b), B2 (a, b)]k is a XAO operator on (I[0, 1])n . Remark 4 Remarks 1 and 2 can be extended to XAO generated by double aggregation operators. Remark 5 Every XAO can be expressed as in Proposition 2. Apparently, there is no direct relation between the properties of B1 , B2 and the AB1 ,B2 ones. For example, using B1 = B2 = min as generators only produces the minimum XAO Am whenever both generators are DAOs. On the other hand, A← is the “normal ” XAO generated by minimum, considering “normal” as when each output endpoint only depends of that kind of input endpoints, and it is independent of other values or constants3 . For purpose of clarity, we will always assume that the generators will be standard aggregation operators; when this generators need to be DAO it will be specifically indicated. Proposition 3 Every “normal” XAO can be obtained using standard aggregation operators as generators. If the XAO is “normal” is because the output has its endpoints defined as a function of the corresponding inputs endpoints, i.e., if x = ([a1 , b1 ], . . . , [an , bn ]) then C(x) = [f (a), g(b)]. These functions f , g fulfill f (0) = g(0) = 0, f (1) = g(1) = 1, and as f , g are non– decreasing (since C is itself non–decreasing); therefore, f and g are aggregation operators and C = Af,g . 3
This external values or constants are needed because in other case there could exist XAO which could not be generated from aggregation operators. For example, the degenerated maximum XAO: [0, 0] if xi = [0, 0]∀i C(x) = If this XAO were [1, 1] in other case. AB1 ,B2 for certain B1 , B2 aggregation operators, then having xi = [0, bi ] with bi 6= 0 would result in [1, 1] = C(x) = AB1 ,B2 (x) = [B1 (0), B2 (b)] = [0, B2 (b)], which is not possible.
Proposition 4 Every XAO can be obtained using double aggregation operators. Obviously, if A is “normal” it is enough that its double generators take into account only one endpoint from the input (for example, B1 (a, b) = P1 (B11 (a), B21 (b)), i.e., the projection of the first coordinate of two standard aggregation operators). Similarly, in the case of A being not “normal” only two functions f and g which considerate both inputs are needed, although a certain set of constants may also be required4 : f (a, b, k). This functions must fulfill f (0, 0, k) = 0, f (1, 1, k) = 1, and must be non–decreasing; any function in this manner can be represented by a double aggregation operator [6].
4
Study of properties
In this Section we recall some basic properties of standard aggregation operators and adapt them to interval–valued operators; several classes of XAO are presented (XAO t–norms. . . ) and related with the standard case. Along this Section it will be considered that the generators B1 and B2 (independently of being standard or double aggregation operators) fulfill B1 (x) ≤ B2 (x); this allows us to study the properties of AB1 ,B2 without considering the “validate” operation. 4.1
Basic properties
As the XAO are a generalization of the standard aggregation operators, we could study their properties as if they were standard aggregation operators (i.e. applying them on point–intervals as [j, j]). Another possible direction is to observe the generators in order to relate them with the desired properties; this will be the option applied in the paper. For example, the idempotency will be naturally defined as A(x) = [a, b] if xi = [a, b] ∀i. The following Proposition characterizes the idempotent XAO. Proposition 5 A XAO is idempotent on I[0, 1] if and only if its generators are idempotent on [0, 1]. Remark 6 To clarify the notation we will indicate here that the domain of the XAO should be (I[0, 1])n , so strictly speaking it would be “idempotent on I[0, 1]n ”; on the other hand, its generators would be 4 The operator C defined in footnote 3 can be obtained with the following DAOs as generators: B1 (a, b) = 0 if a = b = 0 B2 (a, b) = It is clear that the con1 otherwise. stant (1) is needed and that it is non–input dependent; in this example, B1 (a, b) = f (a, b, (1)).
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idempotent on [0, 1]n or [0, 1]2n (standard or double aggregation operators, respectively). To avoid the excess of notation, we will only indicate the domain of each component of the inputs to avoid confusion between intervals (XAO) and points (generators). Other well–known property is symmetry; a XAO A will be called symmetric when A(x) = A(xσ ) for any permutation σ of {1, . . . , n}, where xσi = xσ(i) . Proposition 6 A XAO is symmetric on I[0, 1] if and only if its generators are symmetric on [0, 1]. Clearly, if both generators are symmetric the XAO is symmetric as well; on the other hand, if there were a non–symmetric generator for a certain permutation aσ , the non–symmetry of A could be deducted from x = (a, a). To talk about continuity we will need a distance. Here we will use the Hausdorff’s metric: D([a, b], [c, d]) = max{|a − c|, |b − d|}. This metric seems not appropriated if it is considered as a distance (since D([1, 2], [1, 10]) = D([1, 2], [9, 10]) but the first two intervals are “closer” than the last ones) but it is very useful when applied to check continuity, as if two intervals have a small value for D then both intervals have similar endpoints. Proposition 7 A XAO is continuous on I[0, 1] if and only if its generators are continuos on [0, 1]. The idea behind Proposition 7 is that if one generator is discontinuous in ai , as the Hausdorff metric takes into account the maximum of the differences it is not important if the other generator is continuos or not. We can combine Proposition 4 and 7 in the following statement: Proposition 8 Any continuous XAO can be obtained by means of B1 , B2 DAO such that B1 ≤ B2 . Another interesting property is “additivity with respect to ∗”, in the sense of A(x ∗ y) = A(x) ∗ A(y). For example, if we consider ∗ = ⊕, then the XAO Ai or A⊕ fulfills it. Proposition 9 A XAO is additive with respect to ∗ on I[0, 1] if and only if its generators are additive for the corresponding endpoint functions on [0, 1]. Let B1 , B2 be the generators of C. Then it is that C(x ∗ y) = C(x) ∗ C(y) can be expressed respect B1 and B2 as [B1 (ax ∗1 ay ), B2 (bx ∗2 by )] = [B1 (ax ), B2 (bx )] ∗ [B1 (ay ), B2 (by )] = [B1 (ax ) ∗1 B1 (ay ), B2 (bx ) ∗2 B2 (by )], i.e., B1 and B2 must be additive with respect to ∗i . 193
4.2
Operating with data
As we defined the + operation it is possible to speak of interval shifting, considering [a+c, b+c] = [a, b]+[c, c]. Analogously to shift invariant aggregation operators [12], a XAO will be said to be shift invariant when for all x + c ∈ I[0, 1], it is A(x + c) = A(x) + c. The following proposition characterizes these operators. Proposition 10 A XAO A is shift invariant if and only if A is idempotent and additive with respect to +. In the same spirit, a XAO will be called scalable if A(k · x) = k · A(x); observe that if k < 0 the endpoints of the interval are reversed and “validate” is needed. When a operator is scalable and shift invariant is said to be invariant for linear transformations. In general, let Φ, ϕ be bijections on I[0, 1] and [0, 1] respectively; let define Φ([a, b]) = k[ϕ(a), ϕ(b)]k. Note that “validate” is only needed if ϕ is decreasing. With this bijections and given a certain XAO A, we can define a XAO B in the following way B(x1 , . . . , xn ) = Φ−1 (A(Φ(x1 ), . . . , Φ(xn ))) B will be called the “Φ–transformed of A”. Special cases of this kind of transformation are the prior linear transformations, where ϕ(x) = kx + c was used for certain values k, c. Proposition 11 A XAO is invariant for Φ in I[0, 1] if and only if its generators are invariant for ϕ in [0, 1]. 4.3
Duality
Many standard aggregation operators can be obtained through a duality transformation. Here we present an analogous transformation for XAO. To apply duality a strong negation n is needed. As this negation will be applied on intervals we have to define this operation first. Let n : [0, 1] → [0, 1] be a negation. Then we define n : I[0, 1] → I[0, 1] by n([a, b]) = [n(b), n(a)]. In this paper we will only focus on strong negations, i.e., such that it is n(n([a, b])) = [a, b]. This situation is another special case of bijections as seen in previous Subsection, but taking into account that n−1 = n and that negations are decreasing functions (this is the reason why negations are not defined as n([a, b]) = k[n(a), n(b)]k, as the interval would be always reordered). With this, given a certain XAO A we will define its dual XAO Ad as Ad (x1 , . . . , xn ) = n(A(n(x1 ), . . . , n(xn ))). Example 2 Let n(x) = 1 − x and A = A⊕ be; then, we have the following chain of equalities:
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d
A⊕ ([a1 , b1 ], . . . , [an , bn ]) = n(A([n(b1 ), n(a1 )], . . . , [n(bn ), n(an )])) = n([min{n(bi )}, max{n(ai )}]) = n([1 − max{bi }, 1 − min{ai }]) = [min{ai }, max{bi }] = A⊕ ([a1 , b1 ], . . . , [an , bn ]) d
i.e., A⊕ = A⊕ . As in the standard situation, a XAO will be called selfdual if Ad = A. Proposition 12 Let C = AB1 ,B2 be a XAO; then Cd = AB2d ,B1d is its dual XAO. The proof is directly derived from the definition of duality: Cd (x) = n(AB1 ,B2 (n(x))) = n([B1 (n(b)), B2 (n(a))] = [n(B2 (n(a))), n(B1 (n(b)))] = [B2d (a), B1d (b)]. Having Proposition 12, selfduality is characterized as follows: Proposition 13 A XAO is selfdual if and only if its generators are dual of each other. 4.4
Families of XAO
Now we will present several families of XAO, which will be defined (if possible) as the families of standard aggregation operators [5] obtained when the XAO is applied to point–interval inputs. For example, a XAO fulfilling symmetry, associativity and with neutral element [1, 1] will be a t–norm. We will also show the relationship between the XAO generators families and the XAO itself; this will solve questions as “Let B1 and B2 be t–norms; is AB1 ,B2 a t–norm?”. Initially, every families of operators based in linear operations on data will be maintained, since these operations are also valid on intervals; for example, [2x1 + 3x2 , 2y1 + 3y2 ] = 2 · [x1 , y1 ] + 3 · [x2 , y2 ]. The following Proposition presents this case when a weighted operator is used. Proposition 14 The XAO AB1 ,B2 has the weighting triangle 4 = (win ) if and only if B1 = B2 are aggregation operators with the same weighting triangle 4. For example, weighted means fulfill Proposition 14. Note that XAO weighted means can also be characterized by their additivity for +: Proposition 9 shows that if a XAO is additive its generators are also additive for the same operation, and (standard) weighted means are the aggregation operators additive for +. Now, to prove that both generators have the same weighting triangle is enough to recall that XAO are non– decreasing operators, and since two different means are incomparable5 the only possibility is that both means have the same weighting triangle. 5
For example, let B1 (x1 , x2 ) = 13 x1 + 23 x2 , B2 (x1 , x2 ) =
The next step would be the Ordered Weighted Averaging operators (OWA operators) but this family needs a total ordering of the data which is not possible when working with intervals, since ¹w is a partial ordering. Moreover, even in the case of B1 , B2 being OWA the operator AB1 ,B2 may have no weighting triangle associated. Example 3 Take the OWA B1 = B2 = min as generators, and then AB1 ,B2 = A← . This XAO is not an OWA since, in first place, it has no weighting triangle as can be seen considering A← ([1, 4], [2, 3]) = [1, 3]: there are no w1 , w2 such that w1 [1, 4] + w2 [2, 3] = [1, 3] with w1 + w2 = 1. 4.5
Family of copula
To extend the standard family of copula we will recall the definition of semi–copula (see e.g. [7]) as it is a generalization of copulas, quasi–copulas, t–norms. . . Definition 4 A function S : [0, 1]2 → [0, 1] is a semi– copula if and only S it fulfills the following two conditions: (a) S(x, 1) = S(1, x) = x ∀x ∈ [0, 1]; (b) S(x, y) is non–decreasing in each argument. From this definition, and in the spirit of maintaining its generality for the whole family, we present the following definition for a XAO semi–copula. Definition 5 A XAO A : (I[0, 1])2 → I[0, 1] is called a semi–copula if A([a, b], [1, 1]) = A([1, 1], [a, b]) = [a, b]. Remark 7 Note that XAO are non–decreasing, and therefore condition (b) is no longer needed. Also, as [1, 1] is a neutral element of A, then it follows that for all [a, b] ∈ I[0, 1], [0, 0] ≤ A([a, b], [0, 0]) ≤ A([1, 1], [0, 0]) ≤ [0, 0]; i.e., [0, 0] is an annihilator of A. Example 4 The XAO 1) Sm (x1 , x2 ) = ½
min{x1 , x2 } if max{x1 , x2 } = [1, 1], [0, 0] otherwise.
2) SM (x1 , x2 ) = 2 x 3 1
+ 13 x2 be weighted means. As (0.2, 0.5) ≤ (0.6, 0.9) provides B1 (0.2, 0.5) ≤ B2 (0.6, 0.9), while (0.1, 0.7) ≤ (0.2, 0.8) results in B1 (0.1, 0.7) > B2 (0.2, 0.8), i.e., B1 and B2 are incomparables. 194
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½
[0, 0] if min{x1 , x2 } = [0, 0], A← (x1 , x2 ) otherwise.
are semi–copulae and they are the smallest and biggest semi–copula, respectively. Now we are going to particularize the semi–copula family to define more particular classes • Copula: given x1 , x2 , y1 , y2 ∈ I[0, 1] such that x1 ¹w y1 and x2 ¹w y2 , if A fulfills A(x2 , y2 ) − A(x1 , y2 ) − A(x2 , y1 ) + A(x1 , y1 ) ºw [0, 0] or “equivalently” A(x1 , y2 ) + A(x2 , y1 ) ¹w A(x1 , y1 ) + A(x2 , y2 ). The first expression is the standard definition of copula but using a XAO; as − is not the inverse of + in I[0, 1] the expression lacks the idea of the standard case. On the other hand, the second expression is much more stable and clear. For example, if we consider x1 = [2, 5], y1 = [3, 7], x2 = [5, 8], y2 = [5, 9] and A = A← (x) = [min{ai }, min{bi }], we would like to expect “good properties” for A since it is generated by the minimum standard aggregation operator (which is a standard copula). It is easily proven that A only fulfills the second expression. • Quasi–copula: let x1 , x2 , y1 , y2 be defined as before; the direct translation of quasi–copula would be |A(x1 , y1 ) − A(x2 , y2 )| ¹w |x1 − x2 | + |y1 − y2 | but the absolute value does nothing related to intervals; anyway, when this expression is applied on standard aggregation operators it is the 1–Lipschitz property, where the absolute value is used as a distance indicator. This leads us to suggest the following expression as a possible interpretation for quasi–copula: D(A(x1 , y1 ), A(x2 , y2 )) ≤ D(A(x1 , [1, 1]), A(x2 , [1, 1])) + D(A([0, 0], y1 ), A([0, 0], y2 )). where D is the Hausdorff metric. • t–norm: given x, y, z ∈ I[0, 1], we define a XAO A as a t–norm if it fulfill the standard properties (1) A(x, y) = A(y, x), i.e., A is commutative, and (2) A(A(x, y), z) = A(x, A(y, z)), i.e., A is associative.
Having these classes defined, we are ready to examine the relationship between the XAO and its generators for operators in these families. One possible method to generate a XAO t–norm is to use standard t–norms as generators (as seen in Proposition 1), i.e., given T1 , T2 : [0, 1]n → [0, 1] t– norms, the XAO defined by A([a1 , b1 ], . . . , [an , bn ]) = [T1 (a1 , . . . , an ), T2 (b1 , . . . , bn )] is a t–norm in I[0, 1]. Q Example 5 Using B1 = min and B2 = as generators, the following XAO is obtained AB1 ,B2 (x1 , x2 ) = [min{ax1 , ax2 }, bx1 · bx2 ] which is clearly a t–norm. On the other hand, if AB1 ,B2 is a t–norm, then B1 and B2 must also be t–norms. To prove this only commutativity (symmetry) and associativity of Bi must be checked. Symmetry is obtained from Proposition 6, and associativity is directly achieved from the XAO associativity. Proposition 15 A XAO is a t–norm in I[0, 1] if and only if its generators are t–norms in [0, 1]. Now, and in a similar way to the standard case, we will obtain the family of t–conorms via duality: S(x1 , . . . , xn ) = n(T(n(x1 ), . . . , n(xn ))); these operators can be also obtained using t–conorms as generators in Proposition 1 or in the same spirit of Proposition 15. The previous ideas are also valid relating copulae when the condition A(x1 , y2 ) + A(x2 , y1 ) ¹w A(x1 , y1 ) + A(x2 , y2 ) is considered. Moreover, this expression allows t–norms to be automatically copulae (as in the standard case)6 . To prove this it is enough to observe that the ¹w ordering implies two standard ≤ orderings, which B1 and B2 will fulfill since they are standard copulae. Proposition 16 A XAO is a copula in I[0, 1] if and only if its generators are copulae in [0, 1].
5
Applications
Since one of the possible uses of XAO is based in the uncertainty of data, this uncertainty (i.e., the data interval) should be quantified somehow; for example, using the length of the interval L([a, b]) = b − a. The lesser the length, the more precise are the data; it may even be completely determined to a certain point [a, a], in which case the length will be zero. 6
Note that it also fulfills the standard conditions imposed in [0, 0] and [1, 1] since it is a semi– copula. 195
With this, now it is clear that A← is a copula since it is generated by the min standard aggregation operator although it failed to fulfill the expression with interval differences.
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Moreover, as XAO can symbolize the data uncertainty a possible application is when data are naturally imprecise or subject to incomplete knowledge, for example, linguistic items. Given a certain range of discrete values (e.g. bad, average and good) which can be associated to determined points in a continuous scale (respectively, 0, 0.5 and 1), it is possible to generate the interval of points corresponding each linguistic label (say [0, 0.25], [0.25, 0.75] and [0.75, 1])7 . Lexical modifier can be used in two ways: considered directly on the base interval, which produces a modification of the linguistic term (Example 6), or applying a determined metric to the input data (Example 7).
6
Conclusions
Extended aggregation operators generalize standard aggregation operators and, under certain circumstances, generalize double aggregation operators as well; its main advantage is the uncertainty treatment, which can be measured both in data as in outputs. We have also shown that extended aggregation operators maintain the families and properties of well– known standard aggregation operators; moreover, extended aggregation operators inherit the basic characteristics of its generators. Acknowledgements
Example 6 Lets consider the bad label, the interval [0, 0.25] and its associated discrete value vb = 0. We can use the very modifier, for example, as an interval reduction through the mean operator. In b this case, very bad would be the interval bad+v , i.e., 2 0.25+0 [ 0+0 , ] = [0, 0.125]. Analogously, very average 2 2 and very good would use va = 0.5 and vg = 1 to obtain [0.375, 0.625] and [0.875, 1] respectively. Here we have used the mean operator to modify the linguistic intervals, but any XAO generated by aggregation operators used in the usual treatment of linguistic items can be applied. Example 7 The ouput of a XAO is [0.1, 0.2]. Which linguistic label defines it more appropriately? A first idea would be to use the Hausdorff metric on bad, average and good; the following results would be obtained: rb = D([0.1, 0.2], [0, 0.25]) = 0.1, ra = D([0.1, 0.2], [0.25, 0.75]) = 0.55 and rg = D([0.1, 0.2], [0.75, 1]) = 0.8, so the “closer” label would be bad. Although this approach is enough to solve simple problems, the “closer” idea can be improved when instead of using the Hausdorff metric, a more complex similarity function is applied; for example, consider L(x∩y) S(x, y) = L(x⊕y) which takes into account both the points in common of the intervals as the full covering. Note that it is needed that all intervals representing linguistic labels have the same length, otherwise there will be a distortion in the similarity measurement8 (this can always be obtained with the transformation of bijections as seen in Subsection 4.2). 7
This is an arbitrary assignment and it is only devoted to provide an example. 8 For example, let l1 = [0, 10] and l2 = [10, 30] be intervals representing two linguistic labels, and let x = [5, 20] be the interval to measure; observe that l1 +[5, 10] = x and that x + [5, 10] = l2 , i.e., apparently x is as similar to l1 as to l2 . But when applying S defined as before, it is that 8 5 and S(l2 , x) = 20 . This situation happens S(l1 , x) = 20 because l2 is longer than l1 .
This work has been partially supported by the projects of Spanish Science and Technology Ministry MTM2004–3175, TIC2002–11942–E, BFM2003-05308, the European project COST 274 TARSKI and the project of the Balearic Islands PRIB–2004–9250.
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