Some new results on recursive aggregation rules

Transcription

1 Some ew results o recursive aggregatio rules Daiel Gómez Escuela de Estadística Uiversidad Complutese Madrid, Spai dagomez@estad.ucm.es Javier Motero Facultad de Matemáticas Uiversidad Complutese Madrid, Spai javier motero@mat.ucm.es Abstract As poited by Cutello ad Motero i a previous paper, cosistecy of a aggregatio rule based upo a sequece of biary operators ca be justified from a operatioal argumet, by imposig a recursive calculus. Followig this recursive approach, it was later prove that uder certai regularity coditios, strict icreasigess leads to quasi-additive solutios. I this paper we propose a alterative ad more geeral result, avoidig some of those regularity coditios. Moreover, we poit out that i practice we should evaluate oly those aggregatios beig allowed by the decisio maker. Preferece structures are cosidered as a illustrative example. Keywords: Aggregatio fuctios, Recursive rules, Fuzzy Sets. 1 Itroductio. Aggregatio models play a relevat role i decisio makig. I fact, most decisio makig problems require some aggregatio techique i order to help decisio makers to uderstad the iformatio they are give. If we talk about remote sesig, for example, a typical objective it to classify pixels (uits of the lad surface) withi homogeeous classes (see, e.g., Amo et al. [2, 3, 4, 5]). This particular image classificatio problem implies for each pixel a big amout of data, ad data dimesio eeds to be reduced i order to be maaged by the decisio maker. I additio, sice classificatio of each pixel should also take ito accout behavior i its respective eighborhood, iformatio relative to surroudig pixels should also be aggregated. These aggregatio processes are usually solved by meas of a sequetial procedure, but it is obvious that we ca ot restrict to a uique formula, to be applied agai ad agai o matter the cotext. Moreover, we realize that i some cases data show a particular ad iformative structure (e.g., the eighborhood i a surface is ot the eighborhood i the real space). Aggregatio procedures are defied as rules that tell us how to proceed with the iformatio reachig to us, o matter if its dimesio is previously kow. From this fact, Cutello Motero [9] have claim that cosistecy of such a rule ca be guarateed from a operatioal viewpoit, imposig that aggregatio ca always be decomposed ito a sequece of biary operators. I fact, the key idea of recursiveess, as itroduced i [9], is the existece of a alterative represetatio i terms of a iterative applicatio of biary operators, at each stage takig advatage of the last previous aggregatio. Data are therefore beig assumed to be aggregated oe by oe, ad each particular arragemet of data will tell us the sequece of items to be aggregated. Hece, recursiveess i [9] assumes that data show a liear structure, although the decisio maker ca be al-

2 lowed to re-arrage data, always withi a liear structure, as part of a sometimes eeded preprocessig data 1. The followig defiitios were give i [9]. Defiitio 1.- Let us deote π (a 1, a 2,..., a ) = (a π(1), a π(2),..., a π()) A orderig rule π is a cosistet family of permutatios {π } >1 such that for ay possible fiite collectios of umbers, each extra item a +1 is allocated keepig previous items relative positios, i.e., π +1 (a 1, a 2,..., a, a +1 ) = (a π(1),..., a π(j 1), a π+1 (j), a π (j)..., a π ()) for some j {1,..., + 1}. Defiitio 2.- A left-recursive coective rule is a family of coective operators {φ : [0, 1] [0, 1]} >1 such that there exists a sequece of biary operators verifyig ad {L : [0, 1] 2 [0, 1]} >1 φ 2 (a 1, a 2 ) = L 2 (a π(1), a π(2) ) φ (a 1,..., a ) = L (φ 1 (a π(1),..., a π( 1) ), a π() ) for all > 2 ad some orderig rule π. Notice that i o way we are imposig a uique biary operator for the whole iterative process. This was i fact the mai criticism argued i [17] agaist the restrictive result obtaied by Fug-Fu [13]. Right recursiveess ca be aalogously defied, ad the we ca talk about a recursive 1 Re-arragemet of data, i order to be cosistet implies that, oce the relative positio of two elemets is beig fixed, o extra elemet to be aggregated will chage that relative positio. rule whe both left ad right represetatios hold for the same orderig rule (we talk about stadard recursive rules whe they are based upo the idetity orderig rule, i.e., that rule that keeps the data order). The it follows (see [6]) that a coective rule {φ } >1 is recursive if ad oly if a set of geeral associativity equatios (i the sese of Mak [16]) hold for each, oce the orderig rule π has bee already applied: φ (a 1,..., a ) = R (a π(1), φ 1 (a π(2),..., a π() )) = L (φ 1 (a π(1),..., a π( 1) ), a π() ) must hold for all. 2 Some results o recursiveess. Some relevat results o recursive rules have bee obtaied i [7]. I particular, it was prove that assumig certai regularity coditios, recursive rules were restricted to some relevat families of aggregatio rules (quasiadditive rules amog them). Amog those regularity coditios, the most relevat oe was strict mootoicity. From the followig defiitio give i [6], it was obtaied the ext result (see [7]). Defiitio 3.- A regular recursive coective rule is a family of coective operators {φ : [0, 1] [0, 1]} >1 such that there exists a sequece of biary cotiuous operators ad {L : [0, 1] 2 [0, 1]} >1 {R : [0, 1] 2 [0, 1]} >1 verifyig the followig coditios: If x x ad y y, the L (x, y ) L (x, y ) R (x, y ) R (x, y ) If x < x ad y < y, the L (x, y ) < L (x, y ) R (x, y ) < R (x, y )

3 1. p : [0, 1] R +, cotiuous ad strictly icreasig fuctio, 2. {δ : [0, 1] R + } >1, family of cotiuous ad strictly icreasig fuctios, ad 3. {c } 1, sequece of positive real umbers i such a way that φ (a 1,..., a ) = 2 c j c k 1 1 p(a k ) j=2 k= If x < x, the L (x, y) < L (x, y), y R (x, y) < R (x, y), y If y < y, the L (x, y ) < L (x, y ), x R (x, y ) < R (x, y ), x L (x i, x) L (x i, x), x (0, 1) R (x i, x) R (x i, x), x (0, 1) L ( x, x i ) L ( x, x i), x (0, 1) R ( x, x i ) R ( x, x i), x (0, 1) L (0, y ) = L (0, y ) = 0, y, y L (y, 0) = L (y, 0) = 0, y, y R (0, y ) = R (0, y ) = 0, y, y R (y, 0) = R (y, 0) = 0, y, y L (1, y ) = L (1, y ) y, y L (y, 1) = L (y, 1) y, y R (1, y ) = R (1, y ) y, y R (y, 1) = R (y, 1) y, y for all (a 1,..., a ) [0, 1] ad for all 2, takig l j=2 c j = 1 wheever l 2. Proof: see [7]. But i order to apply the above result we should check the above regularity coditios, which may ot be obvious. I what follows, we provide a alterative result to this oe, which is based o the followig key result due to Aczél [1]. Theorem 2.- Amog the fuctios, cotiuous, ivertible i both variables o a real iterval [α, β], F (x, y) = l[f(x) + g(y)] H(x, y) = l[k(x) + h(y)] G(x, y) = f 1 [k(x) + m(y)] K(x, y) = h 1 [m(x) + g(y)] is the geeral solutio of F (G(x, y), z) = H(x, K(y, z)) where f, g, h, k, l, m are arbitrary cotiuous ad strictly mootoic fuctios. Proof: see [1], page 312. Theorem 1.- Let {φ : [0, 1] [0, 1]} >1 be a regular stadard rule. If φ is strictly icreasig i each coordiate for all > 1, the there exist: Theorem 3.- Let {φ : [0, 1] [0, 1]} >1 be a recursive rule. If L ad R are cotiuous fuctios beig ivertible i both variables for all > 1, the there exist:

4 1. p : [0, 1] R +, cotiuous ad strictly fuctio, 2. {δ : [0, 1] R + } >1, family of cotiuous ad strictly fuctios, ad 3. {c } 1, sequece of positive real umbers i such a way that φ (a 1,..., a ) = 2 c j c k 1 1 p(a k ) j=2 k=1 for all (a 1,..., a ) [0, 1] ad for all 2, takig agai l j=2 c j = 1 wheever l 2. Proof: Obviously from the defiitio of {φ } >1 the followig geeralized associativity equatio holds: L (R 1 (u, v), w) = R (u, L 1 (v, w)) Therefore, havig (x 1..., x ) [0, 1], takig u = x 1,v = φ 2 (x 2,..., x 1 ) ad w = x assures the above equatio. Keepig i mid the above relatio, we kow from [1] that the solutio of the above geeral associativity equatio is basically additive. That is, there exist σ, θ, l, p, q, r cotiuous ad strictly fuctios over the compact iterval [0,1], which verify: R 1 (u, v) = σ 1 (p (u) + q (v)) L 1 (v, w) = θ 1 (q (v) + r (w)) R (u, b) = l (p (u) + θ (b)) L (a, w) = l (σ (a) + r (w)) so l is a strict mootoic fuctio ad L is ivertible i both variables. The, fixed z [0, 1], as L is ivertible the there exists (x, y) [0, 1] 2 such that L (x, y) = z. Hece, l (p (x) + θ (y)) = z ad l is a sobreyective fuctio ad ivertible. If we ow deote δ = l 1, we have the followig equatio: R 1 (u, v) = σ 1 L 1 (v, w) = θ 1 R (u, b) = L (a, w) = (p (u) + q (v)) (q (v) + r (w)) (p (u) + θ (b)) (σ (a) + r (w)) ad proof cotiues as theorem 3.1 i Amo et al [7]. More geeral theorems ca be obtaied takig ito accout alterative results give by Aczél [1]. 3 About the uderlyig structure. Recursiveess does ot impose ay restrictio o the ature of data. It is just a theoretical assumptio. But data set uses to show a particular structure. I this sese, the recursive approach developed i [9] (see also [6, 7]) is i some way assumig that data are orgaized accordig to a liear uderlyig structure, i such a way data ca be alteratively aggregated either from the right or from the left (i.e., from a begiig or from the ed). I case we do ot take care of the order, recursiveess refers oly to a algorithmic property (how to proceed its calculus), because data show a uderlyig complete graph ad ay aggregatio is possible. But liear ad complete structures are ot the oly available structures that allow a iterative calculus, as show i [15]. A iterestig case, for example, is that oe where data ca be represeted accordig to a circular structure. The aggregatio ca still be coceived i a recursive way, but startig aywhere either towards its left or towards its right. Such a circular structure should allow us iterestig results i order to characterize aggregatio operators. 4 Preferece modelig. I this sectio we aalyze the preferece structure from a recursive poit of view. I preferece modelig [12], for each pair of alteratives x, y we have four states: x is worse tha y (x > y), idifferece (x y), x is better tha y (x < y) ad icomparability (x y). But as poited out i [15], the aggregatio of the iformatio betwee these four states ca ot be doe arbitrarily. For example, we ca aggregate the degree of x is worse tha y with

5 the degree of x is idifferet to y. Later we ca aggregate the above aggregated class with the degree to which x is better tha y i order to obtai the egatio of the icomparability. But it is ot so obvious the meaig of a aggregated class betwee class x > y ad class x < y. The decisio maker should defie i advace a uderlyig structure, i which the allowed aggregatios are explicated (with o restrictio i the case of complete graphs). A stadard graph associated to preferece structures is the followig: > < If we cosider aggregatio operators that verify above restrictios assurig quasiadditivity, the we kow that there exist p, δ 2, δ 3 ad c 1 such that ad Φ 2 (a, b) = 2 (p(a) + c 1 p(b)) Φ 3 (a, b, c) = ( ) δ3 1 k(p(a) + c 1 p(b) + c 2 1p(c)) Moreover, Therefore, Φ 2 (a, b) = Φ 2 (b, a) Φ 3 (a, b, c) = Φ 3 (c, b, a) 2 (p(a) + c 1 p(b)) = 2 (p(b) + c 1 p(a)) Ad sice 2 is a ijective fuctio, p(a) + c 1 p(b) = p(b) + c 1 p(a) a, b [0, 1] The, Figure 1: Preferece evaluatio system Some of the equatios associated to this graph are give by: Φ 2 (µ(x < y), µ(x y)) = (Φ 2 (µ(x > y), µ(x y))) Φ 2 (µ(x < y), µ(x y)) = (Φ 2 (µ(x > y), µ(x y))) Φ 3 (µ(x < y), µ(x y), µ(x > y)) = (µ(x y)) for all x, y. There is a lot of aggregatio operators that verify these equatios. Notice that i this case, the four states ca be represeted as a circular structure. So we ca cosider recursive rules as aggregatio operators, sice left ad right are well defied. We ca also observe that Φ 2 (a, b) = Φ 2 (b, a) for all a, b (symmetry). p(a)[1 c 1 ] = p(b)[1 c 1 ] So, c 1 = 1 ad we have that Φ 2 (a, b) = 2 (p(a) + p(b)) Φ 3 (a, b, c) = 3 (k(p(a) + p(b) + p(c))) I other words, whe we have a circular structure we ca assume the recursivity of the operators, ad restrict our model to quasiadditive rules. The above equatios may be helpful i order to obtai membership fuctios, sice the followig equatios must hold: 2 [p (µ(x < y)) + p (µ(x y))] = [p (µ(x > y)) + p (µ(x y))] 2 [p (µ(x < y)) + p (µ(x y))] = [p (µ(x > y)) + p (µ(x y))] 3 [p (µ(x > y)) + p (µ(x y)) + p (µ(x y))] = [p (µ(x < y))] 3 [p (µ(x < y)) + p (µ(x y)) + p (µ(x > y))] = [p (µ(x y))]

6 3 [p (µ(x y)) + p (µ(x < y)) + p (µ(x y))] = [p (µ(x > y))] > I 3 [p (µ(x < y)) + p (µ(x y)) + p (µ(x > y))] = [p (µ(x y))] Figure 2: system < Alterative preferece evaluatio These equatios ca be also take ito accout i desigig appropriate learig procedures for preferece structures. However, the uderlyig structure may ot be uique. The above preferece structure is ideed behid most stadard four-state preferece systems (see [10, 11] ad [20] but also [8]), showig a circular structure (each vertex of the uit square is oly coected with its two adjacet vertices), allowig oly certai aggregatios, to be obtaied by meas of a appropriate disjuctio operator (we could assume a fixed t-coorm [10], alteratively justified i [20], or eve allow disjuctio evolve i time as i [7, 9]). Aggregatio of o coected classes should ot be cosidered (see [15]). Moreover, as poited out i [4], a cojuctio operator will be also eeded i order to evaluate the quality of the classificatio system itself, ad the whole logical structure should give us hits o how our classificatio system could be improved for future classificatios (see [2]). The eed of a learig process for classificatio may also suggest that perhaps a ice preferece structure should iclude, apart from the above four states x < y, x y, x > y ad x y, a cetral state meaig udecisiveess or igorace I, beig this extra state coected with each oe of other four states: with o iformatio, the whole preferece itesity should be associated to such a state, ad as we lear more about our prefereces, itesities trasfer betwee coected states till they are fixed, hopefully assigig o itesity at all to such a udecisiveess state. 5 Fial commets. The examples above show that we ofte fid out that the family of valuatio classes is i may cases structured (a graph is beig associated to it). The family of valuatio classes ca vary, the associated uderlyig structure ca be modified, ad future chages associated to some learig process ca be supported by a arbitrary logical structure (ot ecessarily a stadard De Morga s triple, see [7, 9]). These argumets uderlie i [2, 3, 5], where a fuzzy model was cosidered for the classificatio of lad cover from remotely sesed data: each pixel was classified by meas of the whole family of degrees of membership to every class uder cosideratio. As poited out i [4, 15], a cocept should be uderstood as a structured family of properties, which obviously deped o the cotext (see also [18, 19]). A recursive aggregatio procedure ca be the icorporated to our model, i order to allow the aggregated evaluatio of adjacet classes (aggregatio ca ot be properly defied for o-adjacet classes). The model cosidered i [15] was a particular L-fuzzy set [14], where L = [0, 1] C ad C is a structured family of classes (a graph is beig defied o it). Oce a particular structure has bee fixed, each object will be described by meas of a vector µ(x) [0, 1] C but uderstadig its meaig eeds the associated rules for disjuctio, cojuctio ad

7 egatio, to be applied withi the particular biary relatio defied o C (see [2, 4], where some measures for relevace, overlappig ad redudacy were cosidered). I this cotext, the recursive approach proposed i [7, 9] represets a iterestig possibility for those coectives, allowig a sequetial aggregatio of adjacet classes. A importat specific case of the model cosidered i [15] will be Ruspii s partitio [21]. Although Ruspii s defiitio did ot assume ay particular structure o the family of classes C, we should remid that some kid of uderlyig structure appears i most cases. For example, the stadard 5-valued scale Noe, Poor, Average, Very ad Complete, C = { N, P, A, V, C} assumes a liear order. N P A V C Figure 3: Stadard 5-valued evaluatio system Most decisio makers have i mid some uderlyig structure, which of course may ot be a complete graph. Ackowledgmets This research has bee supported by the Govermet of Spai, grat BFM We also thak oe of the referees for a carefully readig ad suggestios, which ideed improved the paper. Refereces [1] J. Aczél (1966): Lectures o fuctioal equatios ad their applicatios. Academic Press, New York. [2] A. Amo, D. Gómez, J. Motero ad G. Bigig (2001): Relevace ad redudacy i fuzzy classificatio systems. Mathware ad Soft Computig 8, [3] A. Del Amo, J. Motero ad G. Bigig (1999): Classifyig pixels by meas of fuzzy relatios. Iteratioal Joural of Geeral Systems 29: [4] A. Del Amo, J. Motero, G. Bigig ad V. Cutello (2004): Fuzzy classificatio systems. Europea Joural of Operatioal Research, to appear. [5] A. Del Amo, J. Motero, A. Feradez, M. Lopez, J. Tordesillas ad G. Bigig (2002): Spectral fuzzy classificatio: a applicatio. IEEE Tras o Systems, Ma ad Cyberetics (C) 32: [6] A. Del Amo, J. Motero ad E. Molia (2000): Additive recursive rules; i J. Fodor et al., eds.: Prefereces ad decisios uder icomplete kowledge. Physica-Verlag, Heidelberg. [7] A. Del Amo, J. Motero, ad E. Molia (2001): Represetatio of cosistet recursive rules. Europea Joural of Operatioal Research 130: [8] T. Bilgiç (1998): Iterval-valued preferece structures. Europea Joural of Operatioal Research 105: [9] V. Cutello ad J. Motero (1999): Recursive coective rules. Iteratioal Joural of Itelliget Systems 14:3 20. [10] J. Fodor ad M. Roubes (1994): Valued preferece structures. Europea Joural of Operatioal Research 79: [11] J. Fodor ad M. Roubes (1994): Fuzzy Preferece Modellig ad Multicriteria Decisio Support (Kluwer, Dordrecht). [12] J. Fodor ad M. Roubes (1995): Structure of valued biary relatios. Mathematical Social Scieces 30:71 94.

8 [13] L.W. Fug ad K.S. Fu. (1975): A axiomatic approach to ratioal decisio makig i a fuzzy eviromet; i L.A. Zadeh et al., eds.: Fuzzy sets ad their applicatios to Cogitive ad decisio processes (Academic Press, New York); pp [14] J. Gogue (1967): L-fuzzy sets. Joural Mathematical Aals Applicatios 18: [15] D. Gómez ad J. Motero (2003): Prefereces, classificatio ad ituitioistic fuzzy sets. Proceedigs EUSFLAT 03 (Uiversity of Applied Scieces, Zittau); pp [16] K.T. Mak (1987): Coheret cotiuous systems ad the geeralized fuctioal equatio of associativity. Mathematics of Operatios Research 12: [17] J. Motero (1985): A ote o Fug-Fu s theorem. Fuzzy Sets ad Systems 17: [18] J. Motero (1986): Comprehesive fuzziess. Fuzzy Sets ad Systems 20: [19] J. Motero (1987): Extesive fuzziess. Fuzzy Sets ad Systems 21, [20] J. Motero, J. Tejada ad V. Cutello (1997): A geeral model for derivig preferece structures from data. Europea Joural of Operatioal Research 98: [21] E.H. Ruspii (1969): A ew approach to clusterig. Iformatio ad Cotrol 15:22 32.