The interpretation of fuzzy integrals and their application to fuzzy systems q

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1 International Jornal of Approximate Reasoning 41 (2006) The interpretation of fzzy integrals and their application to fzzy systems q Vicenç Torra a, *, Yaso Narkawa b a Institt dõinvestigació en IntelÆligència Artificial, Camps de Bellaterra, E Bellaterra, Catalonia, Spain b Toho Gaken, Naka, Knitachi, Tokyo , Japan Available online 29 Agst 2005 Abstract Fzzy integrals, in general, and Sgeno integrals, in particlar, are well known aggregation operators. They can be sed in a great variety of decision making applications. Nevertheless, their se is not easy as their interpretation is not straightforward. In this paper we stdy the interpretation of fzzy integrals, focsing on Sgeno ones, and we show their application to fzzy inference systems when the rles are not independent. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Fzzy integrals; Sgeno integral; Choqet integral; Twofold integral; Fzzy inference system 1. Introdction Within aggregation operators, fzzy integrals are known to be one of the most powerfl and flexible fnctions as they permit the aggregation of information nder different assmptions on the independence of the information sorces. In particlar, q Work partly fnded by the AGAUR Generalitat de Catalnya (2004XT 00004) and the Spanish Ministry of Edcation and Science nder project SEG C04-02 PROPRIETAS. * Corresponding athor. Tel.: ; fax: addresses: vtorra@iiia.csic.es (V. Torra), narkawa@d4.dion.ne.jp (Y. Narkawa) X/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi: /j.ijar

2 44 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) they can be sed to model sitations in which sorces are independent as well as in sitations in which sch independence cannot be assred. The most well-known integrals are the Choqet [4] and the Sgeno [14] integrals. Generalizations of sch integrals inclde e.g. the t-conorm integral [12], the twofold integral [16] or the Composition aggregation operators in [3]. See also [9]. The flexibility of sch operators is also de to the fact that they generalize several of the most widely-known and sed aggregation fnctions. In particlar, they generalize the arithmetic and weighted mean, as well as the median and linear combination of order statistics (or OWA [19]). The flexibility of sch operators is tightly related with the difficlties of sing them in practical applications. Fzzy integrals combine the data spplied by several information sorces according to a fzzy measre. This fzzy measre, that (sing Artificial Intelligence terminology) represents the backgrond knowledge on the information sorces, is a set fnction from the set of information sorces into an appropriate domain (e.g. the [0, 1] interval or an ordered set D). Typically, this fzzy measre represents the importance or relevance of the sorces when compting the aggregation. For bilding a real system, several difficlties arise. One of them is that the set fnction needs to be defined, and this reqires 2 n 1 vales, where n is the nmber of information sorces. Ths, there is a crse of dimensionality. Another difficlty for the se of fzzy integrals in real applications is that their interpretation is not easy. In this respect, an interpretation was introdced in [10] for the Choqet integral. Instead, for the Sgeno integral, althogh there exist some work on the mathematical properties of sch integrals, there is not yet a clear interpretation of their operational principles. Let alone, abot the meaning of the fzzy measres they need to operate. In this paper we stdy the interpretation of fzzy integrals, focsing on Sgeno integrals. We give some examples and describe an application that corresponds to their se in fzzy systems. The strctre of the paper is as follows. In Section 2 we review some reslts that are needed in the rest of the paper. Then, in Section 3 we stdy the interpretation of the Sgeno integral. Section 4 is devoted to the se of Sgeno integrals for fzzy inference systems. Section 5 gives interpretation of other integrals. Finally, the paper finishes with some conclsions. 2. Preliminaries Basic definitions of aggregation operators are described in this section. We focs on fzzy integrals (Choqet, Sgeno and twofold integrals). Weighted minimm and weighted maximm are also reviewed as they will also be sed in the rest of the paper. See [11,1] for details on fzzy measres and fzzy integrals and [2] for a broader view of the field of aggregation operators. Definitions given below are based on a set X that corresponds to the set of information sorces. In this paper, we assme that X is finite.

3 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) Definition 1. A set fnction l:2 X! [0,1] is a fzzy measre if it satisfies the following axioms: (i) l(;) = 0, l(x) = 1 (bondary conditions). (ii) A B implies l(a) 6 l(b) (monotonicity) for A,B 2 2 X. Definition 2 [4]. Let l be a fzzy measre on (X,2 X ). Then, the Choqet integral C l (f) off:x! [0,1] with respect to l is defined by C l ðf Þ :¼ Xn f ðx sðjþ ÞðlðA sðjþ Þ lða sðjþ1þ ÞÞ; j¼1 where f(x s(i) ) indicates that the indices have been permted so that 0 6 f ðx sð1þ Þ 6 6 f ðx sðnþ Þ 6 1; A sðiþ ¼fx sðiþ ;...; x sðnþ g; A sðnþ1þ ¼;: Definition 3 [14]. Let l be a fzzy measre on (X,2 X ). Then, the Sgeno integral S l (f) of a fnction f:x! [0, 1] with respect to l is defined by S l ðf Þ :¼ _n j¼1 f ðx sðjþ Þ^lðA sðjþ Þ; where f(x s(i) ) indicates that the indices have been permted so that 0 6 f ðx sð1þ Þ 6 6 f ðx sðnþ Þ 6 1; A sðiþ ¼fx sðiþ ;...; x sðnþ g; A sðnþ1þ ¼;: Here ^ denotes the minimm and _ denotes the maximm. Definition 4 ([16,13]). Let l C and l S be two fzzy measres on (X,2 X ). Then, the twofold integral of a fnction f:x! [0,1] with respect to the fzzy measres l S and l C is defined by! TI ls ;l C ðf Þ :¼ Xn _ i! f ðx sðjþ Þ^l S ða sðjþ Þ l C ða sðiþ Þ l C ða sðiþ1þ Þ ; i¼1 j¼1 where f(x s(i) ) indicates that the indices have been permted so that 0 6 f ðx sð1þ Þ 6 6 f ðx sðnþ Þ 6 1; A sðiþ ¼fx sðiþ ;...; x sðnþ g; A sðnþ1þ ¼;: Definition 5. Let be a n-dimensional vector, that is, :¼ ( 1,..., n ) 2 R n. (1) is a possibility distribtion or a possibilistic weighting vector of dimension n if and only if i 2 [0,1] for all i 2 { 1,..., n } and max i i =1. (2) [5] Let be a possibilistic weighting vector of dimension n, then a mapping WMin : [0,1] n! [0,1] is a weighted minimm of dimension n if WMinða 1 ;...; a n Þ¼min maxð1 i ; a i Þ: i

4 46 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) (3) [5] Let be a possibilistic weighting vector of dimension n, then a mapping WMax: [0,1] n! [0, 1] is a weighted maximm of dimension n if W Maxða 1 ;...; a n Þ¼max minð i ; a i Þ: i 3. Interpretation of the Sgeno integral Althogh in Definition 3 the Sgeno integral takes the range [0,1] for the inpt vales and also [0,1] for the range of fzzy measres l, the definition wold be valid for other domains. In particlar, any linearly ordered scale D (i.e., D is a linearly ordered set of categories) is sitable for compting S l ðf Þ :¼ _n f ðx sðjþ Þ^lðA sðjþ Þ: j¼1 This is so, becase this expression only involves operations that are consistently defined in an ordinal scale. Note that _(a,b) =a if and only if a P b and that ^(a,b) =a if and only if a 6 b. At the same time, the ordering s in f(x s(j) ) can be compted as long as there is an order on f(x j ). Taking into accont this ordinal setting, it is clear that given a set X, both f(x) (for x 2 X) and l(a) (for A X) shold be into the same codomain D. Otherwise, the integral cannot be applied becase the minimm cannot be applied to combine f(x) and l(a). So, in some sense, both l and f shold denote the same concept. As l denotes some importance, reliability, satisfaction or similar concepts, the same shold apply to f. Accordingly, the Sgeno integral combines a kind of e.g. importance or reliability leading to another vale for importance or reliability. In fact, there are some applications of Sgeno integral in the literatre (e.g. [14,18]) that fit with this perspective. This sitation is illstrated in the following example, where the vales to be aggregated and the measre (the vales in the codomain D) are related with reliability. Example 1. Let X be a set of experts X ={x 1,x 2,...,x n } that evalate the reliability of a given machine. Say, a new Japanese copy machine. Then, we consider f(x i ) as the reliability of sch copy machine according to expert x i. At the same time, we also consider the reliability of sbsets of experts. So, l(a) is the reliability of experts in A all together. For expressing the reliabilities, any ordered set D is appropriate. Nevertheless, for applying the Sgeno integral we need to consider the same set D to express both the reliability of experts and the reliability of copy machines. In other words, we have that for all A X it holds l(a) 2 D and that for all x i 2 X it holds f(x i ) 2 D. Making the example concrete, let X ={x 1,x 2,x 3 } be a set of three experts, then with l({x 1 }) = 0.2, l({x 2 }) = 0.3 and l({x 3 }) = 0.4 we express that the expert x 3 is more reliable than the expert x 2 and that x 2 is more reliable than x 1. Let l({x 1,x 2 }) = 0.3 and l({x 1,x 3 }) = 0.4 represent that joining x 1 to x 2 or to x 3 does not imply a larger reliability than the one of x 2 or x 3 alone. Instead, with l({x 2,x 3 }) = 0.8 we express that joining both x 2 and x 3 their reliability is greatly increased. Finally, according to the

5 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) bondary conditions, we set l(;) = 0 and l(x) = 1. This is, the set of all experts has the maximm reliability (eqal to 1), and that the empty set is not reliable. Then, when experts assign a particlar reliability to the copy machine, we can compte an overall reliability sing the Sgeno integral. For example, let f(x 1 ) = 0.3, f(x 2 ) = 0.7 and f(x 3 ) = 0.6 be expertsõ opinions on the reliability of the copy machine. In this case, the Sgeno integral leads to 0.6, that corresponds in this case to the vale of one of the most relevant experts. Another example follows. In this case, the codomain D stands for satisfaction. We se then this example to show the application of the Sgeno integral to alternative selection. Example 2. Let s consider a traveler in Japan that intends to visit Tokyo, Kyoto and Nagano and considers several alternative places for staying. Then, let X ={x 1,x 2,x 3 } denote the three mentioned cities. That is, x 1 corresponds to Tokyo, x 2 to Kyoto and x 3 to Nagano. Then, we consider the degree of satisfaction of the traveler visiting sch cities. Sch degree is expressed with the fzzy measre l(a) described in Table 1. The measre of satisfaction is monotonic increasing (the more cities are visited, the greater the satisfaction) and is bonded. Here, the bondary conditions mean that no visit implies no satisfaction, and that visiting all cities has maximm satisfaction (eqal to 1). The degree of satisfaction with respect to visiting certain towns, will be combined with the degree of satisfaction of doing the travel itself to the particlar towns. Sch degree of satisfaction will be in proportion to the accessibility of sch towns from the particlar location of the traveler. This degree of satisfaction will be expressed by a fnction f:x! D. Assming that the traveler is located at Tskba, the most accessible town is Tokyo. Then, the second accessible town is Nagano and, finally, Kyoto. Therefore, f(x 1 )>f(x 3 )>f(x 2 ). Table 2 gives measres for sch accessibility from Tskba. The vales for the measre are expressed sing the same terms than the degree of satisfaction. This is, f(x) is comparable with l(a). Using l and f, we can define l f (x i ) :¼ l({x jf(x) P f(x i )}). This expression stands for the degree of satisfaction of visiting x i and all those cities that are at least as Table 1 Fzzy measre for the traveler example: satisfaction degree for visiting cities in X ={x 1,x 2,x 3 } Set {x 1 } {x 2 } {x 3 } {x 1,x 2 } {x 2,x 3 } {x 1,x 3 } X l Here, x 1 corresponds to Tokyo, x 2 to Kyoto and x 3 to Nagano. Table 2 Accessibility degrees from Tskba Set x 1 x 2 x 3 f

6 48 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) Table 3 Satisfaction degree for each city for the traveler example Set x 1 x 2 x 3 l f accessible as x i. Roghly speaking, in what accessibility concerns, if we visit a place x i with a given accessibility f(x i ), then we assme that all places with a greater accessibility than x i will be also visited. So, in fact, we are considering each f(x i )asa threshold for selecting the visits. Formally speaking, if we visit x i, it is possible to visit as well all those cities with a greater degree of possibility than x i. This is, all those cities in the set {xjf(x) P f(x i )}. Accordingly, l f (x i ) is the degree of satisfaction the traveler achieves when decides to visit x i. Table 3 gives the vales for l f for each of sch towns. The next step for compting a degree of satisfaction of being located in a particlar town (in this case Tskba), we need to consider the combination of the degrees of satisfaction of visiting some towns and of the degrees of accessibility of visiting sch towns. To do sch combination, we first consider for each city x i, both degrees of accessibility f(x i ) and satisfaction l f (x i ). Two cases can be considered with respect to sch f(x i ) and l f (x i ): Case f(x i ) P l f (x i ): In this case, as it is relatively easy to access town x i, mch concern is given to the satisfaction l f (x i ). Accordingly, the degree of x i cannot be larger than l f (x i ). Case l f (x i ) P f(x i ): In this case, the traveler mst give special importance to the physical accessibility of x i, and this constraints the degree of x i. Ths, the combination is f(x i ). To take both aspects into accont, the degrees are combined by means of the ^ (the minimm) operator. Ths, f(x i ) ^ l f (x i ) is the evalation of visiting x i. Taking everything into accont, the place x i with the largest evalation f(x i ) ^ l f (x i ) stands for the evalation of staying in Tskba. This largest evalation corresponds to the Sgeno integral of f with respect to l, that in this example is eqal to SI l ðf Þ¼max xi f ðx i Þ^l f ðx i Þ¼0:7. In this latter example, we have shown the se of the Sgeno integral to evalate the satisfaction of the traveler of staying in a particlar town. This interpretation of the Sgeno integral can be sed in an alternative selection problem. For example, to determine which is the most sitable town for the traveler to stay. In this sitation, several towns {t i } i will be considered and for each one a fnction f ti shold be defined. At the same time, the satisfaction of visiting certain towns will be expressed (as in Example 2) in terms of a fzzy measre l. This fzzy measre will be constant throgh the whole comptation. Then, the integration of f ti with respect to l for each t i will give the degree of satisfaction of finally staying in town t i. The town with the best evalation will be the one selected.

7 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) Table 4 Accessibility degrees from Osaka Set x 1 x 2 x 3 g Table 5 Satisfaction degree for each city for the traveler example when staying in Osaka Set x 1 x 2 x 3 l g Example 3. Let s consider two alternatives for the travelerõs staying: Tskba and Osaka. Then, given the conditions of Example 2, the satisfaction degree of staying in Tskba is 0.7 (see Example 2) and the one of staying in Osaka is SI l (g) = 0.6 when the accessibility degrees from Osaka, denoted by g, are the ones displayed in Table 4. Table 5 gives fnction l g. As, SI l (g) <SI l (f), it means that the traveler will chose to stay in Tskba instead of staying in Osaka. From an operational perspective, it can be considered that the Sgeno integral proceeds like by satration. It selects the importance that overcomes (satrates) a certain degree or threshold. In fact, as the threshold is decreasing while the inpts are increasing, it finds a tradeoff (or compromise) between the importance or reliability of the set and the importance that the members of the set have assigned. This follows from the graphical interpretation of the integral (see, e.g., [21]). 4. Sgeno integral for fzzy inference systems In this section we consider a different scenario where the central role is played by certainty degrees. This is, we will describe an application of the Sgeno integral where certainty degrees will be aggregated. The application scenario consists on a set of fzzy rles (althogh any knowledge based system wold lead to a similar scenario) where each rle assigns degrees of satisfaction to particlar otpt vales. Before going into details, we will first review fzzy inference systems Fzzy inference systems Fzzy inference systems are knowledge based systems defined in terms of sets of rles that inclde fzzy sets in their definition. In this section we will only review those elements that are needed latter on in this paper. In particlar, we will focs on standard (flat or one-stage) fzzy inference systems. See e.g. [6,20] for more details and e.g. [17,8] for some recent state-of-the-art applications.

8 50 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) Fzzy rles R i (for i in {1,...,n}) in fzzy inference systems have the following strctre: R i : IF x 1 is A 1 i and... and x m is A m i THEN y is B i : Here, A j i and B i denote fzzy terms, defined by fzzy sets. For example, A j i might correspond to a fzzy set expressing that x j is low, medim or high. For simplicity, we will only consider here systems that have a single inpt variable x. Therefore, the strctre of the rles is as follows: R i : IF x is A i THEN y is B i : Given a set of fzzy rles {R i } i, and given a particlar vale for variable x, say x 0, the system comptes the otpt vale for variable y. At present, there are two main approaches for compting sch distribtion. They correspond to two interpretations of the set of rles: conjnctive and disjnctive rles. We will stdy them below. Nevertheless, in both cases, the otpt is compted throgh the obtention of the so-called possibility distribtion on the range of y, a fnction from the range of y into [0,1]. The possibility distribtion of the system is bilt, in an element-wise manner, from similar distribtions obtained for each rle R i. This is, the final possibility, or certainty, degree that y takes a particlar vale y 0 is defined as the aggregation of the degrees that y eqals y 0 according to each rle R i. Denoting by f i (y 0 ) the certainty degree that rle R i assigns to y 0 and denoting by l(a) the certainty degree of rles R i 2 A, we will show that the otcome of a fzzy inference system for y 0 (for both conjnctive and disjnctive interpretations) can be expressed in terms of a Sgeno integral. This constrction sing Sgeno integral links, as it will be shown below, fzzy inference with Weighted Minimm and Weighted Maximm [22], and the latter operators with the Sgeno integral (one of their generalizations). The next two sections focs on the two types of fzzy systems. First we consider the case of disjnctive rles and, then, the case of conjnctive rles. As we see it, the examples considered validate or interpretation of the Sgeno integral. The se of Sgeno integral for disjnctive rles was previosly sggested in [15] The case of disjnctive rles Let s consider a fzzy inference system with n rles interpreted in a disjnctive manner. Then, the otpt of the system when x = x 0 is compted as follows: (1) Compte the satisfaction degree for the antecedent of all rles R i. This corresponds to compte a i, where a i corresponds to the degree of satisfaction of x 0 is A i. In or case, as there is a single condition in the antecedent, a i = A i (x 0 ). (2) Compte the conclsion of rle R i. For systems defined in terms of disjnctive rles, the otpt of a rle is often compted sing MamdaniÕs approach. This approach is eqivalent to compte the otpt for A 0 ={x 0 } as either [ j (A 0 R j )ora 0 ([ j R j ) with being a max-min composition and R j being

9 the intersection of A j and B j. See e.g. [7] for a proof. We apply MamdaniÕs approach compting [ j (A 0 R i ).From an operational point of view, MamdaniÕs approach is as follows: for each rle R i, its otpt fzzy set B i is clipped according to the degree of satisfaction a i. According to this, the otpt of sch fzzy rle R i is B i ^ A i (x 0 ). (3) Compte the otpt of the set of rles {R i } i. This is, the fzzy otpt ~B (for the whole system) is compted as the nion of the otpts of each rle R i. Using, maximm for the nion (the most sal operator) we obtain ~B ¼ _n ðb i ^ A i ðx 0 ÞÞ. i¼1 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) (4) Finally, the otpt fzzy set ~B is sally defzzified. In what follows, we skip the defzzificatioin stage as it is not relevant for or stdy. Let s now consider the membership of the fzzy otpt ~B for a given element y 0 in Y. This is, let s consider the comptation ~Bðy 0 Þ¼ _n ðb i ðy 0 Þ^A i ðx 0 ÞÞ. i¼1 On the light of the weighted maximm (see Definition 3) this expression can be rewritten as ~Bðy 0 Þ¼W MaxðB 1 ðy 0 Þ;...; B n ðy 0 ÞÞ; ð1þ where the weighting vector is defined as =(A 1 (x 0 ),...,A n (x 0 )) or, in general, =(a 1,...,a n ). Note that the weighting vector is independent of the vale y 0. Ths, for a given x 0, the same aggregation operator with the same weights is applied to all y 0 in Y The case of conjnctive rles Let s now consider the case of conjnctive rles. In this case there are two alternative expressions for compting the otpt of the system for a particlar inpt A 0. Sch expressions are: \ j (A 0 R j )anda 0 (\ j R j ). Althogh these two expressions do not lead, in general, to the same otpt, they are eqal when A 0 is a single vale. As this is the case here, we will se \ j (A 0 R j ) for convenience. In this case, the otpt of the system when x = x 0 is compted as follows: (1) Compte A 0 R j for each rle R j. (2) Compte the intersection of all sch otpts. Using the minimm (denoted ^) to compte the intersection, we have that the otpt is ~B ^n ¼ ða 0 R i Þ. i¼1

10 52 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) This can be rewritten for all y 0 2 Y as follows: ~Bðy 0 Þ¼^n ðða 0 R i Þðy 0 ÞÞ; i¼1 where R i is the relation bilt from A i and B i sing an implication fnction I (see [7] for details on implication fnctions and a description of several of their families). Formally speaking, R i is defined as R i = I(A i,b i ). Now, de to the fact that A 0 ={x 0 }, we have that the expression above of A 0 R i corresponds to I(A i (x 0 ), B i (y)). Considering this latter expression, we can rewrite ~Bðy 0 Þ as follows: ~Bðy 0 Þ¼^n ðiða i ðx 0 Þ; B i ðy 0 ÞÞÞ. i¼1 Selecting an appropriate implication I, the expression above for ~Bðy 0 Þ can be expressed in terms of a weighted minimm (see Definition 3). In particlar, the Kleene Dienes implication I(a,b) = max(1 a,b) is the one that makes this correspondence possible. Note that with this implication, ~Bðy 0 Þ can be rewritten as ~Bðy 0 Þ¼^n ^n ðiða i ðx 0 Þ; B i ðy 0 ÞÞÞ ¼ maxð1 A i ðx 0 Þ; B i ðy 0 ÞÞ. ð2þ i¼1 i¼1 Therefore, when the weighting vector is defined as =(A 1 (x 0 ),...,A n (x 0 )), the following eqality holds: ~Bðy 0 Þ¼WMinðB 1 ðy 0 Þ;...; B n ðy 0 ÞÞ: ð3þ 4.4. Using the Sgeno integral The last two sections have shown that inference in fzzy rle based systems can be formalized in terms of aggregation operators. We have seen that weighted max was sed when rles are interpreted in a disjnctive manner, and that weighted min was sed when rles are interpreted in a conjnctive manner. As the Sgeno integral is known to generalize both WMin and WMax, the otpt of a fzzy inference system can be generally nderstood as the integration of the vales B i (y 0 ) with respect to a fzzy measre constrcted from the vales in the weighting vector =(A 1 (x 0 ),..., A n (x 0 )). In particlar, let l w max and l wmin be fzzy measres with l w max ðzþ ¼max i2z i and l wmin ðzþ ¼1 max i62z i where Z X and X :¼ {1,2,...,n}. Then, since W Max ðf Þ¼SI l w maxðf Þ and W Min ðf Þ¼SI l wminðf Þ, we can rewrite expressions (1) and (3), respectively, as follows: ~Bðy 0 Þ¼SI l w maxðb 1 ðy 0 Þ;...; B n ðy 0 ÞÞ; ð4þ ~Bðy 0 Þ¼SI l wminðb 1 ðy 0 Þ;...; B n ðy 0 ÞÞ; ð5þ where =(A 1 (x 0 ),...,A n (x 0 )).

11 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) The conseqences of this formalization is that both conjnctive and disjnctive fzzy rle based systems are expressed in a nified way in terms of the Sgeno integral. The solely difference between the two approaches is how the fzzy measre is defined from. In the case of disjnctive rles, a possibility measre l w max is sed. Instead, in the case of disjnctive rles, a necessity measre l wmin is sed. Moreover, as it is known (see [7]) that \ j (A 0 R j ) [ j (A 0 R j ), it is easy to see that the two Sgeno integrals defined above (or, more precisely, the two fzzy measres l w max and l wmin in conjnction with the Sgeno integral) define an interval for each y 0. It is clear that the se of other fzzy measres wold lead to other vales for the certainty degree of y 0 (in, or arond, the same interval). An important aspect that cannot be skipped is that the weighting vectors sed above are not, strictly speaking, possibilistic weighting vectors (or possibility distribtions). This is so becase, in general, does not satisfy max i = 1 as it is often the case that there is no i sch that A i (x 0 ) = 1. The practical conseqences of this fact is that the aggregation operator C that se them does not satisfy nanimity (i.e., it does not hold C ða; a;...; aþ 6¼ a). Nevertheless, this property is not a conseqence of sing the Sgeno integral bt it is a property that already held for the weighted minimm and the weighted maximm. The rewriting of the fzzy inference system in terms of Sgeno integrals yields to an important conseqence. While W Min and W Max assme independence between the vales to be aggregated, the Sgeno integral does not reqire sch independence. Therefore, the Sgeno integral is a natral operator to combine the conclsions of several rles in a fzzy rle based system when sch rles are not independent. Fig. 1 illstrates this sitation. Fig. 1 represents (left) the case of a fzzy rle based system with two inpt variables x and y. The rles are assmed to follow a grid-like strctre, a common strctre in real-world fzzy control applications. This is the case of fzzy systems represented in a tablar form. In general, sch systems are defined in terms of fzzy partitions [7], one for the domain of each variable. In or case, let fa x i g i and fa y jg j be the fzzy partitions of the domains of x and y, then for each pair ða x i ; Ay jþ a fzzy rle is defined. Accordingly, for almost any pair of inpt vales (x 0,y 0 ), for rles are applied. This is the case, for example, when firing the rles for x 0 = 2and y 0 =2 Fig. 1. Graphical representation of two different fzzy inference systems with two inpt variables: regions correspond to fzzy rles (the membership fnctions for each of the two variables is also given).

12 54 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) indicated with a circle in Fig. 1 (left). Exceptions to this sitation correspond to the knots or the lines in the figre, when only one or two fzzy rles are applied. Note that the regions in the figre represent a rle for which the sets A x i and A y j are satisfied for a degree of at least 0.5. Instead, in the case represented on the right hand side, not all rles follow the grid-like strctre and there is a region (arond x 0 =2,y 0 = 2) where the nmber of rles to be applied do not have an homogeneos strctre. In fact, the nmber of rles to be applied depends on the vales (x 0,y 0 ). The reglarity of the rles in systems following a grid-like strctre makes the se of Sgeno integral with fzzy measres l w max or l wmin adeqate. In this case there are no major interactions between the otcomes of the rles. Each rle has its own area of inflence. Instead, rles not following this pattern are sally not independent, as there are regions that accmlate several rles. In sch regions, the otcome of the rles bias the otcome of the whole system. In this case, other fzzy measres might be sed to redce the inflence of sch accmlation of rles and, ths, take into accont the interaction among rles. The example given below illstrate sch sitations. Example 4. Let s consider the following set of for rles, each with one inpt variable x and one otpt vale y, to model the relation (x,x 2 ): R 1 : IF x is 1 THEN y is 1, R 2 : IF x is 2 THEN y is 3.7, R 3 : IF x is 1.95 THEN y is 4.04, R 4 : IF x is 1.9 THEN y is 4.4. Let s consider the following trianglar fzzy nmbers to represent the fzzy sets in the conseqent: (0.2, 1.00, 2.3), (1.6, 3.7, 6.3), (1.7, 4.04, 6.6) and (1.8, 4.4, 6.8). Here, a trianglar fzzy nmber (a,b,c) stands for the trianglar fzzy set defined with the normal point b and the spport (a,c). It can be easily observed that rles R 2, R 3 and R 4 are redndant as they try to give information abot the same region on the domain of x (i.e. the region arond the vale 2). Now, let s consider the application of the rles to the inpt vale x 0 = 1.6 considering that the rles are disjnctive. Assme that all rles are fired and that the degree of satisfaction of rles R i (a i = A i (x)) are a = (0.25,0.625,0.6875,0.75). Then, the fzzy set ~B will inclde the otcomes of the for rles. Fig. 2 shows the reslt of firing sch rles. This otpt has been compted following the description in Section 4.2 sing the fzzy measre l wmax defined in Table 6. Recall that this procedre corresponds to the Mamdani approach for applying fzzy rles. This figre shows that as rles R 2, R 3 and R 4 conclde all abot a vale near 4, sch region has a larger inflence in the otpt (a larger dark region) than the one that wold be obtained by a single rle. It shold be nderlined that sch inflence can be positive (i.e., increasing the otpt vale) or negative (i.e., decreasing the

13 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) Fig. 2. Otcome of the rles. Table 6 Fzzy measre l ¼ l w max for Example 4 l({r 1,R 2,R 3,R 4 }) = 0.75 l({r 1,R 2,R 3 }) = l({r 1,R 2,R 4 }) = 0.75 l({r 1,R 3,R 4 }) = 0.75 l({r 2,R 3,R 4 }) = 0.75 l({r 1 }) = 0.25 l({r 2 }) = l({r 4 }) = 0.75 l({r 1,R 2 }) = l({r 1,R 3 }) = l({r 1,R 4 }) = 0.75 l({r 2,R 3 }) = l({r 2,R 4 }) = 0.75 l({r 3,R 4 }) = 0.75 l({r 3 }) = l(;)=0 otpt vale). The sign depends on the shape and position of the membership fnctions. The inflence of the rles in the otpt can be qantified applying a defzzification method to the fzzy set. We have selected the center of gravity as sch defzzification method. The example described above with for rles leads to a defzzified vale eqal to Instead, if only rles R 1 and R 3 were applied (with a 1 and a 2 ), the final otpt wold be Alternatively, if we replace rles R 2, R 3 and R 4 by a rle with an average conseqent fired with the average (a 2 + a 3 + a 4 )/3 we get a defzzified vale of Note that the ideal otpt for x = 1.6 eqals to = Ths, in this example, the redndancy of the rles bias the otpt towards larger vales. A way to solve the problem illstrated in Example 4, is to se the Sgeno integral bt with a fzzy measre different than l w max. In particlar, and as an example, we can consider the fzzy measre l 1 defined in Table 7. This measre redces the effect of the redndant rles. It has been defined so that l 1 (Z) is lower than l when Z

14 56 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) Table 7 Fzzy measre l 1 to solve the inconveniences of Example 4 l 1 ({R 1,R 2,R 3,R 4 }) = 0.75 l 1 ({R 1,R 2,R 3 }) = l 1 ({R 1,R 2,R 4 }) = 0.5 l 1 ({R 1,R 3,R 4 }) = 0.5 l 1 ({R 2,R 3,R 4 }) = 0.75 l 1 ({R 1 }) = 0.25 l 1 ({R 2 }) = l 1 ({R 4 }) = 0.25 l 1 ({R 1,R 2 }) = 0.25 l 1 ({R 1,R 3 }) = 0.25 l 1 ({R 1,R 4 }) = 0.25 l 1 ({R 2,R 3 }) = l 1 ({R 2,R 4 }) = 0.5 l 1 ({R 3,R 4 }) = 0.5 l 1 ({R 3 }) = l 1 (;)=0 Fig. 3. Using Sgeno integral for combining fzzy rles. contains some of the {R 2,R 3,R 4 } bt not all of them (in fact, it is proportional to the nmber of redndant rles in Z). Note e.g. that l({r 1,R 3 }) = in the original fzzy measre bt that l 1 ({R 1,R 3 }) = 0.25 in the new one. Similarly, l({r 1, R 3,R 4 }) = 0.75 while for the new measre l 1 ({R 1,R 3,R 4 }) = 0.5. With sch new fzzy measre l 1, the defzzified vale (for the inpt x = 1.6) eqals to , that is more similar to the goal 2.56 than the original Note that this vale is also similar to the otcome when only one of the redndant rles is sed. Fig. 3 illstrates the reslts of the combination sing Sgeno integral with both the original measre l and the alternative measre l Interpreting the other integrals Another well-known integral is the Choqet integral that generalizes, among other operators, the weighted mean. The main difference between the Choqet

15 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) integral and the weighted mean is that the latter ses a weighting vector, that corresponds to a probability distribtion, while the Choqet integral ses a fzzy measre. Under this consideration, the fzzy measre can be nderstood as a kind of probability distribtion with some ncertainty. On this basis, and taking into accont that the weighted mean can be nderstood as an expected vale, the Choqet integral can be interpreted as a kind of expectation for sch probability with ncertainty. Twofold integrals were defined as a generalization of Choqet and Sgeno integrals. They correspond, in fact, to two-step fzzy integrals: a Choqet integral of Sgeno integrals. Accordingly, sch integrals can be interpreted in terms of the Choqet integral and the Sgeno integral. In particlar, trning into the example of the rle based system, we can se the twofold to define a fzzy inference system with randomness on the rles. This is detailed in the following example: Example 5. Let s consider a rle based fzzy inference system. Let B i (y 0 ) be the certainties that rles R i assign to a particlar vale y 0. Then, l S (A) is the certainty assigned to the set of rles A. Natrally, l S (A) is compted from a i (the degree in which rles x i have been fired) either sing l wmax measre. Additionally, l C ; l wmin or any other composite corresponds to some prior knowledge abot the appropriatedness/accracy of the rles. So, a probability distribtion (or a fzzy measre) is defined over the set of rles. Then, to combine the vales of B i (y 0 ) taking into accont l S (A) andl C the twofold integral of B i (y 0 ) with respect to l S (A) and l C will be sed. 6. Conclsions In this paper we have considered the interpretation of the Sgeno integral. In particlar, we have shown that in the Sgeno integral both the measre and the vales being aggregated are in the same domain. Sch vales can be interpreted as importances, reliabilities or certainties. Several examples are given. In particlar, we have shown that the Sgeno integral can natrally be applied to fzzy inference systems. In fact, we have shown that the Sgeno integral is a natral extension of the operators sed in fzzy inference systems to aggregate the otcomes of the rles. Besides, we have otlined an interpretation for the Choqet and twofold integrals. References [1] P. Benventi, R. Mesiar, D. Vivona, Monotone set fnctions-based integrals, in: E. Pap (Ed.), Handbook of Measre Theory, Elsevier, [2] T. Calvo, G. Mayor, R. Mesiar, Aggregation Operators, Physica-Verlag, [3] T. Calvo, A. Mesiarová, L. Valásková, Composition of aggregation operators one more new constrction method, Kybernetika 39 (2003) [4] G. Choqet, Theory of capacities, Ann. Inst. Forier 5 (1954) [5] D. Dbois, H. Prade, Weighted minimm and maximm operations, Inform. Sci. 39 (1986)

16 58 V. Torra, Y. Narkawa / Internat. J. Approx. Reason. 41 (2006) [6] D. Driankov, H. Hellendoorn, M. Reinfrank, An Introdction to Fzzy Control, Springer-Verlag, USA, [7] G. Klir, B. Yan, Fzzy Sets and Fzzy Logic: Theory and Applications, Prentice-Hall, UK, [8] L. Magdalena, On the role of context in hierarchical fzzy controllers, Int. J. Int. Syst. 17 (5) (2002) [9] R. Mesiar, A. Mesiarová, Fzzy integrals, in: Proceedings of the Modeling Decisions for Artificial Intelligence, LNAI, 3131, 2004, pp [10] T. Mrofshi, M. Sgeno, An interpretation of fzzy measres and the Choqet integral as an integral with respect to a fzzy measre, Fzzy Sets Syst. 29 (1989) [11] T. Mrofshi, M. Sgeno, Fzzy measres and fzzy integrals, in: M. Grabisch, T. Mrofshi, M. Sgeno (Eds.), Fzzy Measres and Integrals: Theory and Applications, Physica-Verlag, 2000, pp [12] T. Mrofshi, M. Sgeno, Fzzy t-conorm integral with respect to fzzy measres: generalization of Sgeno integral and Choqet integral, Fzzy Sets Syst 42(1) (1991) [13] Y. Narkawa, V. Torra, Twofold integral and Mlti-step Choqet integral, Kybernetika 40 (2004) [14] M. Sgeno, Theory of fzzy integrals and its application, Doctoral Thesis, Tokyo Institte of Technology, [15] E. Takahagi, On fzzy integral representation in fzzy switching fnctions, fzzy rles and fzzy control rles, in: IFSA, Proc. 8th IFSA World Congres, 1999, pp [16] V. Torra, Twofold integral: a Choqet integral and Sgeno integral generalization, Btlletí de lõassociació Catalana dõintel.ligència Artificial, vol. 29, pp (in Catalan), Preliminary version: IIIA Research Report TR (in English) [17] E. Tnstel, M.A.A. de Oliveira, S. Berman, Fzzy behavior hierarchies for mlti-robot control, Int. J. Int. Syst. 17 (5) (2002) [18] A. Verkeyn, D. Bottledooren, B. De Baets, G. De Tré, Sgeno integrals for the modelling of noise annoyance aggregation, in: Proc. IFSA 2003, LNAI, vol. 2715, 2003, pp [19] R.R. Yager, On ordered weighted averaging aggregation operators in mlti-criteria decision making, IEEE Trans. SMC 18 (1988) [20] R.R. Yager, D.P. Filev, Essentials of Fzzy Modeling and Control, John Wiley & Sons, New York, [21] M. Yoneda, S. Fkami, M. Grabisch, Hman factor and fzzy science, in: K. Asai (Ed.), Fzzy Science, Kaibndo, 1994, pp (in Japanese). [22] V. Torra, Y. Narkawa, On the interpretation of some fzzy integrals, LNAI 3131 (2004)