IFSA-EUSFLAT 009 Some results on Lischitz quasi-arithmetic means Gleb Beliakov, Tomasa Calvo, Simon James 3,3 School of Information Technology, Deakin University Burwood Hwy, Burwood, 35, Australia Deartamento de Ciencias de la Comutación, Universidad de Alcalá Crta Barcelona Km 33,6 887-Alcalá de Henares Madrid, Sain Email: { gleb, 3 sgj}@deakineduau, tomasacalvo@uahes Abstract We resent in this aer some roerties of k-lischitz quasi-arithmetic means The Lischitz aggregation oerations are stable with resect to inut inaccuracies, what is a very imortant roerty for alications Moreover, we rovide sufficient conditions to determine when a quasi arithemetic mean holds the k-lischitz roerty and allow us to calculate the Lischitz constant k Keywords k-lischitz aggregation functions, quasi-arithmetic means, stability, triangular norms Introduction Aggregation of several inut values into a single outut value is an indisensable tool in many discilines and alications such as decision making [], attern recognition, exert and decision suort systems, information retrieval, etc [] There is a wide range of aggregation functions which rovide flexibility to the modeling rocess, including different tyes of aggregation functions There are several recent books that rovide details of many aggregation methods [3, 4, 5, 6, 7] For alications it is imortant to design aggregation functions that are stable with resect to small erturbations of inuts eg, due to inut inaccuracies Such aggregation functions need not only be continuous, but Lischitz continuous [8] Kernel and -Lischitz aggregation functions have been studied in [9, 0, ] It is known, for instance, that -Lischitz triangular norms are coulas [, 3, 7] More recently, k-lischitz t-norms and t-conorms were studied [3, 4, 5] k-lischitz t-norms do not increase the erturbation of inuts by more than a factor of k, which is suitable for many alications There are many other generated functions constructed similarly to the Archimedean triangular norms with the hel of additive generators In this article we examine quasi-arithmetic means and establish conditions under which these functions are Lischitz or not Lischitz Firstly, in Section we recall some basic notions to develo the rest of the work Section 3 contains the main results involving quasi arithmetic means At the end we rovide some conclusions Preinaries and related works We restrict ourselves to aggregation functions defined on [0, ] n Definition A function f :[0, ] n [0, ] is called an aggregation function if it is monotone non-decreasing in each variable and satisfies f0,,0 = 0, f,, = Now, we will ay attention to a secial class of aggregation function, -the class of weighted quasi arithmetic means-, for this we need to consider a continuous strictly monotone function g :[0, ] [, ], which we call a generating function or generator Of course, g is invertible, but it is not necessarily a bijection ie, its range may be Rang [, ] Other two examles of generated functions are Archimedean t-norms and t-conorms Further there exists a class of uninorms, known as reresentable uninorms or generated uninorms, that can also be built by means of additive generators Further, a vector w =w,,w n is called a weighting vector if w i [0, ] and n i= w i = Definition For a given generating function g, and a weighting vector w, the weighted quasi-arithmetic mean is the function n M w,g x =g w i gx i i= From this definition, we have the following articular quasi arithmetic means: n Arithmetic mean M x = n x i i= Geometric mean G x = n x i n Harmonic mean Power mean i= H x =n n i= x i M r x = n n x r i r, if r 0 i= and M 0 x =G x Another class of aggregation oerators is the following Definition 3 Let g : [0, ] [, ] be a continuous strictly monotone function and let w be a weighting vector The function n GenOW A w,g x =g w i gx i i= is called a generalized OWA also known as ordered weighted quasi-arithmetic mean [4] As for OWA, x i denotes the i-th largest value of x Another aggregation oerator that include the revious one for the case of a symmetric fuzzy measure is the generalized discrete Choquet integral which is defined as follows 370
IFSA-EUSFLAT 009 Definition 4 Let g : [0, ] [, ] be a continuous strictly monotone function The generalized Choquet integral with resect to a fuzzy measure v is the function C v,g x =g C v g x, where C v is the discrete Choquet integral with resect to v and g x =gx,,gx n Now, we consider the crucial concet of this work Definition 5 An aggregation function f is called Lischitz continuous if there is a ositive number k, such that for any two vectors x, y in the domain of definition of f: f x f y kd x, y, 3 where d x, y is a distance between x and y The smallest such number k is called the Lischitz constant of f in the distance d We shall call such functions k-lischitz Tyically the distance is chosen as a -norm d x, y = x n / y, with x = x i, for finite, and x = i= max x i In this work we concentrate on the norm i=,,n Definition 6 A function f is called locally Lischitz continuous on Ω if for every x Ω there exists a neighbourhood Dx such that f restricted to Dx is Lischitz Of course, duality wrt standard negation reserves Lischitz roerty and the Lischitz constant It is easy to see that if an aggregation function A is k Lischitz, it is also m Lischitz for any m k Also any convex combination of k Lischitz aggregation functions f = αf + βf, α + β =,α,β 0,isk Lischitz The class of k Lischitz t norms, whenever k >, has been already characterized see [3] Note that Lischitz t norms are coulas, see, eg, [3, 5] A strictly decreasing continuous function g :[0, ] [0, ] with g = 0 is an additive generator of a Lischitz Archimedean t norm if and only if g is convex The k Lischitz roerty imlies continuity of the t norm Recall that a continuous t norm can be reresented by means of an ordinal sum of continuous Archimedean t norms, and that a continuous Archimedean t norm can be reresented by means of a continuous additive generator [3, 6] Characterization of all k Lischitz t norms can be reduced to the roblem of characterization of all Archimedean k Lischitz t norms Definition 7 Let g :[0, ] [0, + ] be a strictly monotone function and let k ]0, + [ be a real constant Then g will be called k convex if gx + kε gx gy + ε gy holds for all x [0, [,y ]0, [, with x y and ε ]0, min y, x k ] Obviously, if k =the function g is convex Observe that, a k convex monotone function is also continuous in ]0, [, as was earlier A decreasing function g can be k convex only for k Moreover, when a decreasing function g is k convex, it is also m convex for all m k In the case of a strictly increasing function g, it can be k- convex only for k Moreover, when g is k convex, it is m convex for all m k Considering k and a strictly decreasing function g, we rovide the following characterization given in [3] Proosition Let T :[0, ] [0, ] be an Archimedean t norm and let g :[0, ] [0, + ],g = 0 be an additive generator of T Then T is k Lischitz if and only if g is k convex Another useful characterizations are the following Corollary Y-H Shyu [7] Let g :[0, ] [0, ] be an additive generator of a t norm T which is differentiable on ]0, [ and let g x < 0 for 0 <x< Then T is k Lischitz if and only if g y kg x whenever 0 <x<y< Corollary Let T :[0, ] [0, ] be an Archimedean t norm and let g :[0, ] [0, ] be an additive generator of T such that g is differentiable on ]0, [\S, where S [0, ] is a discrete set Then T is k Lischitz if and only if kg x g y for all x, y [0, ],x y such that g x and g y exist The following useful results follow from Corollary, with it we can determine whether a given iecewise differentiable t norm is k Lischitz Corollary 3 Let T :[0, ] [0, ] be an Archimedean t norm and let g : [0, ] [0, ] be its additive generator differentiable on ]0, [, and g t < 0 on ]0, [ If inf t ]x,[ g t k su g t t ]0,x[ holds for every x ]0, [ then T is k Lischitz Corollary 4 Let g :[0, ] [0, ] be a strictly decreasing function, differentiable on ]0, a[ ]a, [ If g is k-convex on [0,a[ and on ]a, ], and if then g is k convex on [0, ] inf t ]a,[ g t k su g t, t ]0,a[ Remark Generated uninorms are not k Lischitz since that they are not continuous at 0, and,0 in the binary case Nullnorms are aggregation functions related to t norms and t conorms In this case, it is clear that a nullnorm V is k Lischitz if and only if the underlying t-norm and t conorm are k Lischitz, and the k Lischitz constant of V is the maximum of Lischitz constants of the underlying t norm and t conorm 37
IFSA-EUSFLAT 009 3 Quasi-arithmetic means Consider a univariate continuous strictly monotone function g :[0, ] [, ], called generator For a given g, the quasi-arithmetic means are defined in Definition, and are denoted by M g We start with bivariate quasi-arithmetic means of such oerations Quasi-arithmetic means are continuous if and only if Rang [, ] [8] Moreover, its generator is not defined uniquely, ie, if gt is a generating function of some weighted quasi-arithmetic mean, then agt +b, a, b R, a 0is also a generating function of the same mean rovided Rang [, ] For this reason, one can assume that g is monotone increasing, as otherwise we can simly take g We shall consider two cases: I g0 = 0, g =, and II g0 =,g = 0 Of course, by duality we also cover the case g =,g0 = 0, and by using aroriate linear transformations, all generators can be reduced to the mentioned cases Let us make some reinary remarks on convexity As oosed to the case of convex additive generators of t-norms, where the resulting t-norms are -Lischitz, convexity of the generator g does not lay any role by itself for quasiarithmetic means Since both g and g are generators of the same mean, and obviously when g is convex g is concave, convexity of g by itself does not lead to the Lischitz condition Also note that gx = lnx is a convex generator of the geometric mean Gx, y = xy, which is not Lischitz Further, even if g is convex and increasing or convex and decreasing, this does not imly Lischitz condition either: note that g d x = g x is a generator of a quasi-arithmetic mean dual to the one generated by g, and Lischitz condition is reserved under duality If g is convex increasing, then g d is convex decreasing and vice versa Thus we will look for a different condition 3 Case of finite generators We start with the case I of g finite First, let us show that g must be Lischitz on [0,] For short, we will denote M w,g by M g Lemma Let g be finite locally Lischitz and continuously differentiable excet at a oint a [0; ] Then M g is not k Lischitz for any k Proof Suose that M g is k-lischitz, which means it is differentiable almost everywhere in its domain Rademacher s theorem, and we must have x, y k, x, y a whenever such a derivative exists, and similarly for the other artial derivative Since M g is symmetric, only the derivative with resect to x is needed = g M g x, y g x k Since g is strictly increasing we must have g x kg M g x, y for all x, y [0, ] such that x a and M g x, y a or g x k inf y [0,] g M g x, y 4 Since g is finite, M g does not have an absorbing element Let x a g x = y a : z = M g a, y a such that g z M < because g is locally Lischitz Then inequality 4 fails, because we can always choose such x a that g x > km, which would give us km < g x kg z km, which is false Then >k, hence M g is not Lischitz Now, since g is Lischitz on [0, ], it is differentiable almost everywhere, which means that the left- and right-derivatives exist in [0, ] We start with the case of g differentiable on [0, ], and then adat it to g differentiable almost everywhere by using left- and right-derivatives Let M g be k-lischitz Then we must have x, y k Following the same rocedure as in Lemma, we get the condition g x k inf y [0,] g M g x, y 5 for all x [0, ] Finally, by using left- and right- derivatives g,g + we obtain a general condition for non-smooth increasing generators g x k inf z [Mx,0,Mx,] g z 6 g +x k inf z [Mx,0,Mx,] g +z for all x ]0, [, and only one of the above inequalities for x =0and x = Remark If g is finite and concave increasing, then it is sufficient to check gx g x kg Mx, = kg g +, and similarly for left- and right-derivatives if g is not smooth If g is finite and convex increasing, it is sufficient to check gx g x kg Mx, 0 = kg g Let us rovide some examles of Lischitz and non- Lischitz quasi-arithmetic means Examle If g is linear M g is the arithmetic mean, g x = const, and M g is k-lischitz for k = Examle If gx =x, > M g is a ower mean M [], g x = x, and M g is k-lischitz for k = It follows from x k x = kx Examle 3 If gx =x, 0 << M g is a ower mean M [], M g is not Lischitz by Lemma 37
IFSA-EUSFLAT 009 3 Case of generators infinite at 0 Now we turn to the case II, g increasing with g0 =, which entails that 0 is the absorbing element of M g Wehave an analogue of Lemma The roof is similar, excet that it fails for a =0, hence the modification Let g be finite locally Lischitz and continuously differentiable excet at a oint a [0; ] Then Mg is not klischitz for any k Lemma Let g be locally Lischitz and continuously differentiable excet at a oint a ]0, ] Then M g is not k- Lischitz for any k Let g be finite locally Lischitz and continuously differentiable excet at a oint a [0; ] For x ]0, ] we have condition 6, to which we add condition g +x x 0 + g +Mx, y k 7 for any fixed y ]0, ] This condition may or may not be satisfied deending on the rate at which Mx, y 0 as x 0 The choice of y > 0 is irrelevant as gy is finite and disaears under the it Examle 4 If gx = x, < M g is a ower mean M [], g x = x, and M g is k-lischitz for k = Differentiating M g when x 0 We will exress this rate through the growth of an auxiliary function /g, for which the growth is exressed in traditional terms eg, olynomial when x First, two simle auxiliary results Lemma 3 If two functions f,g are continuous and differentiable at x =0and, f0 = g0 and fx gx for x>0, then f 0 g 0 Proof: Follows directly from the definition of the derivative The next result is a well-known condition for comarability of quasi-arithmetic means, see, eg, [9] Theorem Let g,g be the generators of quasi-arithmetic means M g and M g,and g decreasing Then M g M g if and only if g g is convex Theorem Let g be an increasing decreasing twice continuously differentiable on ]0, ] generator of a quasi-arithmetic mean M g where g = h, and gx = gx = x 0 x 0 + Ifh hh 0 then M g is not Lischitz Proof We will show that M [] M g for any <<0 decreasing, and hence by Lemma 3 is not Lischitz If x g is convex, for <<0by Theorem, with g x =x, M [] M g Let us show that x h 0 = Given <, = k = x +y x x x + y = + x y + x y = x 0 Examle 5 Let M [], <<0be a ower mean with a generator given by gx = x = x q, q > The Lischitz constant will be k = su M =q To see this k = q x q +y q q q q x = q x q q x q + y q q q = + q q x y q } = su { q q + = q = Condition 7 deals with the asymtotic behavior of the additive generators near 0 Its direct verification for a given g may be difficult In the remainder of this section we will establish two sufficient conditions that guarantee that a quasi-arithmetic mean is not Lischitz although it is continuous These conditions are easier to verify, and they rovide a tool for a quick screening of additive generators with resect to their suitability for alications One sufficient condition involves an inequality on the derivatives of the inverse of an additive generator The other condition is that a decreasing additive generator cannot decrease slower than a certain rate /olynomial x y x h = h h x h = h h + h h = h h + hh 0 Given h < 0 for <0,h > 0, convexity will hold if for all <0 h hh 0 8 Therefore h hh 0 imlies x h 0 and M [] M g, and by Lemma 3 the Lischitz constant of M g is greater than that of M [], which is, and 0 Remark 3 The generator g can be either increasing or decreasing Clearly when changing g to g, we change hx to h x Then h changes the sign but h does not, hence the inequality in Theorem is the same for either increasing or decreasing generators Examle 6 Using the geometric mean M g, take gx =lnx with hx =h x =h x =e x Then h hh x = e x e x =0 Therefore M g is not Lischitz For the sake of convenience, we will formulate our next result for decreasing additive generators satisfying g0 = To obtain the resective condition on the increasing generators, we simly invert the sign of g Theorem 3 Let h = g be the inverse of a decreasing generator g of a quasi-arithmetic mean M g If the function h = h grows faster than any ower x q,q > 0, then M g is not Lischitz 373
IFSA-EUSFLAT 009 Proof Fix y so that gy =h y =0, which is always ossible we remind that g is defined u to an arbitrary linear transformation M g x,y x 0 dh = x 0 h x dx, = h x 0 h x = x 0 h h x h x h h x Let z = h x Then x, y = x 0 h z z h z Since h decreases faster than the ower function z = Cz r, l Hôital s rule gives hz 0 = z z = h z z z = h z z z For convenience of notation take such that z = z q Then z = z q x 0 x,y = z h z h z = z h z z q h z z = q Since q can be arbitrarily large, the derivative is unbounded and M g is not Lischitz Examle 7 Let the generator be gx = ln x as in Examle 6 Clearly its inverse is ex x, and the auxiliary function hx = exx, which grows faster than any olynomial, hence the corresonding geometric mean is not Lischitz Further take any ower of the logarithm gx = ln x r,r > The auxiliary function hx = exx r, it grows faster than a olynomial, hence the resulting mean is not Lischitz either Note that this quasi-arithmetic mean is related to the Aczél-Alsina family of t-norms [3, 5] by the equation M g =Tr AA r, which shows directly that M g is not Lischitz fx = M g x, = Tr AA x, r = x r is not Lischitz Examle 8 Consider the generator gx = ln x, hx =e x From the revious examle, r =and we know the resulting mean is not Lischitz, however this would not have been aarent from the alication of Theorem, as h hh = 4x e x 4x e x 4 x 3 e x = 4 x 3 e x < 0 33 Weighted quasi-arithmetic means We adat conditions 6 and 7 for the case of unequal weights For this we take artial derivatives with resect to all arguments The Lischitz constant is the largest, hence we have conditions g x k min g max w i z z 9 g +x k min g max w i z +z where the minimum for z is over [Mx, 0,,0,Mx,,,], and g +x x 0 + g +Mx,c,,c k 0 max w i with c ]0, ] Conditions 9 and 0 can also be used for symmetric means in the multivariate case, where max w i = n It is clearly seen that the higher the number of variables, the smaller is the Lischitz constant, if it exists Remark 4 Similar, results can be obtained for generalized OWAs and generalized Choquet integrals 4 Conclusions k-lischitz aggregation functions are imortant for alications because they can control the changes in the oututs due to inut inaccuracies, to a fixed factor of k k-lischitz triangular norms and conorms have been already characterized by k-convex additive generators, however no analogous results were available for quasi-arithmetic means We have found verifiable conditions which guarantee that an aggregation function is k Lischitz for a given k, or alternatively, not Lischitz We also resented various examles of both Lischitz and non-lischitz aggregation functions Our results will benefit those who design aggregation functions for ractical alications, as they allow one to make an informed choice on suitability of secific functions for these alications Acknowledgements This work was suorted by the Sanish roject MTM006-083, PR007-093 and the Euroean roject 4343-008- LLP-ES-KA3-KA3MP References [] H-J Zimmermann and P Zysno Latent connectives in human decision making Fuzzy Sets and Systems, 4:37 5, 980 [] D Dubois and H Prade On the use of aggregation oerations in information fusion rocesses Fuzzy Sets and Systems, 4:43 6, 004 [3] EP Klement, R Mesiar, and E Pa Triangular Norms Kluwer, Dordrecht, 000 [4] T Calvo, A Kolesárová, M Komorníková, and R Mesiar Aggregation oerators: roerties, classes and construction methods In T Calvo, G Mayor, and R Mesiar, editors, Aggregation Oerators New Trends and Alications, ages 3 04 Physica-Verlag, Heidelberg, New York, 00 [5] G Beliakov, A Pradera, and T Calvo Aggregation Functions: A Guide for Practitioners Sringer, Heidelberg, Berlin, New York, 007 [6] V Torra and Y Narukawa Modeling Decisions Information Fusion and Aggregation Oerators Sringer, Berlin, Heidelberg, 007 374
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