New Concepts and Methods in Information Aggregation

Transcription

1 Nw Concpts and Mthods in Information Aggrgation János Fodor 1, Imr J. Rudas John von Numann Faculty of Informatics, Budapst Tch Bécsi út 96/B, H-1034 Budapst, Hungary {Fodor, Abstract: This papr summarizs som rsults of th authors rsarch that hav bn carrid out in rcnt yars on gnralization of convntional aggrgation oprators. Aggrgation of information rprsntd by mmbrship functions is a cntral mattr in intllignt systms whr fuzzy rul bas and rasoning mchanism ar applid. Typical xampls of such systms consist of, but not limitd to, fuzzy control, dcision support and xprt systms. Sinc th advnt of fuzzy sts a grat numbr of fuzzy connctivs, aggrgation oprators hav bn introducd. Som familis of such oprators hav bcom standard in th fild. Nvrthlss, it also bcam clar that ths oprators do not always follow th ral phnomna. Thrfor, th suggstd nw oprators satisfy natural nds to dvlop mor sophisticatd intllignt systms. Kywords: t-norm and t-conorm, uninorm, nullnorm, distanc-basd conjunctions and disjunctions 1 Introduction Aggrgation of svral inputs into a singl output is an indispnsabl stp in divrs procdurs of mathmatics, physics, nginring, conomical, social and othr scincs. Gnrally spaking, th problms of aggrgation ar vry broad and htrognous. Thrfor, in this contribution w rstrict ourslvs to information aggrgation in intllignt systms. Th problm of aggrgating information rprsntd by mmbrship functions (i.., by fuzzy sts) in a maningful way has bn of cntral intrst sinc th lat 1970s. In most cass, th aggrgation oprators ar dfind on a pur axiomatic basis and ar intrprtd ithr as logical connctivs (such as t-norms and t- conorms) or as avraging oprators allowing a compnsation ffct (such as th arithmtic man). 1 Supportd in part by OTKA T04676

2 On th othr hand, it can b rcognizd by som mpirical tsts that th abovmntiond classs of oprators diffr from thos ons that popl us in practic (s [0]). Thrfor, it is important to find oprators that ar, in a sns, mixturs of th prvious ons, and allow som dgr of compnsation. On can also discrn that popl ar inclind to us standard classs of aggrgation oprators also as a mattr of routin. For xampl, whn on works with binary conjunctions and thr is no nd to xtnd thm for thr or mor argumnts, as it happns.g. in th infrnc pattrn calld gnralizd modus ponns, associativity of th conjunction is an unncssarily rstrictiv condition. Th sam is valid for th commutativity proprty if th two argumnts hav diffrnt smantical backgrounds and it has no sns to intrchang on with th othr. Ths obsrvations advocat th study of nlargd classs of oprations for information aggrgation and hav urgd us to rvis thir dfinitions and study furthr proprtis. Traditional Associativ and Commutativ Oprations Th original fuzzy st thory was formulatd in trms of Zadh's standard oprations of intrsction, union and complmnt. Th axiomatic sklton usd for charactrizing fuzzy intrsction and fuzzy union ar known as triangular norms (t-norms) and triangular conorms (t-conorms), rspctivly. For mor dtails w rfr to th book [9]..1 Triangular Norms and Conorms Dfinition 1 A triangular norm (shortly: a t-norm) is a function T :[ 0,1] [ 0,1] which is associativ, incrasing and commutativ, and satisfis th boundary condition T(1, x) = x for all x [0,1]. Dfinition A triangular conorm (shortly: a t-conorm) is an associativ, commutativ, incrasing S :[ 0,1] [ 0,1] function, with boundary condition S(0, x) = x for all x [0,1]. Notic that continuity of a t-norm and a t-conorm is not takn for grantd. Th following ar th four basic t-norms, namly, th minimum T M th product T P, th Łukasiwicz t-norm T L, and th drastic product T D, which ar givn by, rspctivly:

3 TM ( x, y) = min( x, y), TP ( x, y) = x y, TL ( x, y) = max( x+ y 1,0), 0 if ( xy, ) [0,1[, TD ( x, y) = min( xy, ) othrwis. Ths four basic t-norms hav som rmarkabl proprtis. Th drastic product T D and th minimum T M ar th smallst and th largst t-norm, rspctivly. Th minimum T M is th only t-norm whr ach x [0,1] is an idmpotnt lmnt. Th product T P and th Łukasiwicz t-norm T L ar prototypical xampls of two important subclasss of t-norms (of strict and nilpotnt t-norms, rspctivly). Dfinition 3 A non-incrasing function :[ 0,1] [ 0,1] N satisfying N (0) = 1, N (1) = 0 is calld a ngation. A ngation N is calld strict if N is strictly dcrasing and continuous. A strict ngation N is said to b a strong ngation if N is also involutiv: N( N( x))= x for all x [0,1]. Th standard ngation is simply Ns ( x) = 1 x, x [0,1]. Clarly, this ngation is strong. It plays a ky rol in th rprsntation of strong ngations. W call a continuous, strictly incrasing function :[ 0,1] [ 0,1] ϕ (1) = 1 an automorphism of th unit intrval. Not that :[ 0,1] [ 0,1] ϕ with ϕ (0) = 0, N is a strong ngation if and only if thr is an automorphism ϕ of th unit intrval such that for all x [0,1] w hav N x N x 1 = ϕ ( s ( ϕ )). In what follows w assum that T is a t-norm, S is a t-conorm and N is a strong ngation. Clarly, for vry t-norm T and strong ngation N, th opration S dfind by S( x, y) = N( T( N( x), N( y))), x, y [0,1] (1) is a t-conorm. In addition, T( x, y) = N( S( N( x), N( y ))) ( xy, [0,1] ). In this cas S and T ar calld N -duals. In cas of th standard ngation (i.., whn N( x) = 1 x for x [0,1] ) w simply spak about duals. Obviously, quation (1) xprsss th D Morgan's law in th fuzzy cas. Gnrally, for any t-norm T and t-conorm S w hav T xy, T xy, T xy, and S xy, S xy, S xy,, D M M D whr S M is th dual of T M, and S D is th dual of T D.

4 Ths inqualitis ar important from practical point of viw as thy stablish th boundaris of th possibl rang of mappings T and S. Btwn th four basic t- norms w hav ths strict inqualitis: T < T < T < T. D P L M 3 Nw Associativ and Commutativ Oprations 3.1 Uninorms Uninorms wr introducd by Yagr and Rybalov [19] as a gnralization of t- norms and t-conorms. For uninorms, th nutral lmnt is not forcd to b ithr 0 or 1, but can b any valu in th unit intrval. Dfinition 4 A uninorm U is a commutativ, associativ and incrasing binary oprator with a nutral lmnt [0,1], i.., for all x [0,1] w hav U( x, ) = x. T-norms do not allow low valus to b compnsatd by high valus, whil t- conorms do not allow high valus to b compnsatd by low valus. Uninorms may allow valus sparatd by thir nutral lmnt to b aggrgatd in a compnsating way. Th structur of uninorms was studid by Fodor t al. [11]. For a uninorm U with nutral lmnt ]0,1], th binary oprator T U dfind by Uxy (, ) TU ( x, y)= is a t-norm; for a uninorm U with nutral lmnt [0,1[, th binary oprator S U dfind by U ( + (1 x ), + (1 y ) ) SU ( x, y) = 1 is a t-conorm. Th structur of a uninorm with nutral lmnt ]0,1[ on th squars [0, ] and [,1] is thrfor closly rlatd to t-norms and t-conorms. For ]0,1[, w dnot by φ and ψ th linar transformations dfind by = x φ x and ψ = x x. To any uninorm U with nutral lmnt ]0,1[, 1 thr corrsponds a t-norm T and a t-conorm S such that: for any for any (, ) [0, ] x y : (, ) [,1] xy : U x y T x y 1 (, ) = φ ( ( φ, φ )); U x y S x y 1 (, ) = ψ ( ( ψ, ψ )).

5 On th rmaining part of th unit squar, i.. on E = [0, [ ],1] ],1] [0, [, it satisfis min( x, y) U( x, y) max( x, y ), and could thrfor partially show a compnsating bhaviour, i.. tak valus strictly btwn minimum and maximum. Not that any uninorm U is ithr conjunctiv, i.. U(0,1) = U (1,0) = 0, or disjunctiv, i.. U(0,1) = U (1,0) = Rprsntation of Uninorms In analogy to th rprsntation of continuous Archimdan t-norms and t- conorms in trms of additiv gnrators, Fodor t al. [11] hav invstigatd th xistnc of uninorms with a similar rprsntation in trms of a singl-variabl function. This sarch lads back to Dombi's class of aggrgativ oprators [7]. This work is also closly rlatd to that of Klmnt t al. on associativ compnsatory oprators [15]. Considr ]0,1[ and a strictly incrasing continuous [,1] R 0 mapping h with h (0) =, h () = 0 and h (1) = +. Th binary oprator U dfind by for any U x y h h x h y (, ) [0,1] \{(0,1),(1,0)}, 1 (, ) = ( + ) xy and ithr U(0,1) = U (1,0) = 0 or U(0,1) = U (1,0) = 1, is a uninorm with nutral lmnt. Th class of uninorms that can b constructd in this way has bn charactrizd [11]. Considr a uninorm U with nutral lmnt ]0,1[, thn thr xists a strictly incrasing continuous [,1] R h (1) =+ such that 0 mapping h with h (0) =, h () = 0 and U x y h h x h y 1 (, ) = ( + ) for any ( xy, ) [0,1] \{(0,1),(1,0)} if and only if U is strictly incrasing and continuous on ]0,1[ ; thr xists an involutiv ngator N with fixpoint such that U( x, y) = N( U( N( x), N( y )))) for any ( xy, ) [0,1] \{(0,1),(1,0)}. Th uninorms charactrizd abov ar calld rprsntabl uninorms. Th mapping h is calld an additiv gnrator of U. Th involutiv ngator corrsponding to a rprsntabl uninorm U with additiv gnrator h, as mntiond in condition (ii) abov, is dnotd N and is givn by U N x h h x 1 U = ( ).

6 Clarly, any rprsntabl uninorm coms in a conjunctiv and a disjunctiv vrsion, i.. thr always xist two rprsntabl uninorms that only diffr in th points (0,1) and (1, 0). Rprsntabl uninorms ar almost continuous, i.. continuous xcpt in (0,1) and (1, 0), and Archimdan, in th sns that ( x ]0, [)( U( x, x) < x ) and ( x ],1[)( U( x, x) > x ). Clarly, rprsntabl uninorms ar not idmpotnt. Th classs U min and U max do not contain rprsntabl uninorms. A vry important fact is that th undrlying t-norm and t- conorm of a rprsntabl uninorm must b strict and cannot b nilpotnt. Morovr, givn a strict t-norm T with dcrasing additiv gnrator f and a strict t-conorm S with incrasing additiv gnrator g, w can always construct a rprsntabl uninorm U with dsird nutral lmnt ]0,1[ that has T and S as undrlying t-norm and t-conorm. It suffics to considr as additiv gnrator th mapping h dfind by x f, if x hx =. x g ( 1 ), if x On th othr hand, th following proprty indicats that rprsntabl uninorms ar in som sns also gnralizations of nilpotnt t-norms and nilpotnt t- conorms: ( x [0,1])( U( x, NU( x)) = NU( )). This claim is furthr supportd by studying th rsidual oprators of rprsntabl uninorms in [6]. As an xampl of th rprsntabl cas, considr th additiv gnrator h x dfind by hx = log, thn th corrsponding conjunctiv rprsntabl 1 x uninorm U is givn by U( x, y ) = 0 if ( xy, ) {(1,0),(0,1)}, and xy U( x, y) = (1 x)(1 y) + xy othrwis, and has as nutral lmnt 1. Not that U N ( x) = 1 x. U N is th standard ngator: Th class of rprsntabl uninorms contains famous oprators, such as th functions for combining crtainty factors in th xprt systms MYCIN (s [5, 18]) and PROSPECTOR [5]. Th MYCIN xprt systm was on of th first systms capabl of rasoning undr uncrtainty []. To that nd, crtainty factors wr introducd as numbrs in th intrval [ 1,1]. Essntial in th procssing of ths crtainty factors is th modifid combining function C proposd by van Mll []. Th [ 1,1 ] [ 1,1 ] mapping C is dfind by x+ y(1 x), if min( x, y) 0 Cxy (, ) = x+ y(1 + x), if max( xy, ) 0. x+ y, othrwis 1 min( x, y )

7 Th dfinition of C is not clar in th points ( 1,1) and (1, 1), though it is undrstood that C( 1,1) = C (1, 1) = 1. Rscaling th function C to a binary oprator on [0,1], w obtain a rprsntabl uninorm with nutral lmnt 1 and as undrlying t-norm and t-conorm th product and th probabilistic sum. Implicitly, ths rsults ar containd in th book of Hajk t al. [14], in th contxt of ordrd Ablian groups. 3. Nullnorms Dfinition 5 [3] A nullnorm V is a commutativ, associativ and incrasing binary oprator with an absorbing lmnt a [0,1], i.. ( x [0,1])( V( x, a) = a ), and that satisfis ( x [0, a])( V( x,0) = x) (4) ( x [ a,1])( V( x,1) = x) Th absorbing lmnt a corrsponding to a nullnorm V is clarly uniqu. By dfinition, th cas a = 0 lads back to t-norms, whil th cas a = 1 lads back to t-conorms. In th following proposition, w show that th structur of a nullnorm is similar to that of a uninorm. In particular, it can b shown that it is built up from a t-norm, a t-conorm and th absorbing lmnt [3]. Thorm 1 Considr a [0,1]. A binary oprator V is a nullnorm with absorbing lmnt a if and only if if a = 0 : V is a t-norm; if 0< a < 1: thr xists a t-norm T V and a t-conorm S V such that V( x, y ) is 1 φa ( SV( φa( x), φa( y))), if ( x, y) [0, a] 1 givn by ψa ( TV( ψa( x), ψ a( y))), if ( x, y) [ a,1] ; a, lswhr if a = 1 : V is a t-conorm. Rcall that for any t-norm T and t-conorm S it holds that T( x, y) min( x, y) max( x, y) S( x, y ), for any ( xy, ) [0,1]. Hnc, for a nullnorm V with absorbing lmnt a it holds that ( ( x, y) [0, a ] ) ( V( x, y) max( x, y )) and ( ( xy, ) [ a,1] ) ( V( x, y) min( x, y )). Clarly, for any nullnorm V with absorbing lmnt a it holds for all x [0,1] that V( x,0) = min( x, a) and V( x,1) = max( x, a ). Notic that, without th additional conditions (4), it cannot b shown that a

8 commutativ, associativ and incrasing binary oprator V with absorbing lmnt a bhavs as a t-conorm and t-norm on th squars [0, a ] and [ a,1]. Nullnorms ar a gnralization of th wll-known mdian studid by Fung and Fu [13], which corrsponds to th cas T = min and S = max. For a mor gnral tratmnt of this oprator, w rfr to [10]. W rcall hr th charactrization of that mdian as givn by Czogała and Drwniak [4]. Firstly, thy obsrv that an idmpotnt, associativ and incrasing binary oprator O has as absorbing lmnt a [0,1] if and only if O(0,1) = O(1,0) = a. Thn th following thorm can b provn. Thorm Considr a [0,1]. A continuous, idmpotnt, associativ and incrasing binary oprator O satisfis O(0,1) = O(1,0) = a if and only if it is givn by max( x, y), if ( x, y) [0, a] Oxy (, ) = min( xy, ), if ( xy, ) [ a,1]. a, lswhr Nullnorms ar also a spcial cas of th class of T - S aggrgation functions introducd and studid by Fodor and Calvo [1]. Dfinition 6 Considr a continuous t-norm T and a continuous t-conorm S. A binary oprator F is calld a T - S aggrgation function if it is incrasing and commutativ, and satisfis th boundary conditions ( x [ 0,1] )( F( x,0) = T ( F( 1,0), x) ) ( x [ 0,1] )( F( x,1) = S( F( 1,0 ), x) ). Whn T is th algbraic product and S is th probabilistic sum, w rcovr th class of aggrgation functions studid by Mayor and Trillas [17]. Rphrasing a rsult of Fodor and Calvo, w can stat that th class of associativ T - S aggrgation functions coincids with th class of nullnorms with undrlying t- norm T and t-conorm S. 4 Gnralizd Conjunctions and Disjunctions 4.1 Th Rol of Commutativity and Associativity On possibl way of simplification of axiom skltons of t-norms and t-conorms may b not rquiring that ths oprations to hav th commutativ and th associativ proprtis. Non-commutativ and non-associativ oprations ar

9 widly usd in mathmatics, so, why do w rstrict our invstigations by kping ths axioms? What ar th rquirmnts of th most typical applications? From thortical point of viw th commutativ law is not rquird, whil th associativ law is ncssary to xtnd th opration to mor than two variabls. In applications, lik fuzzy logic control, fuzzy xprt systms and fuzzy systms modling fuzzy rul bas and fuzzy infrnc mchanism ar usd, whr th information aggrgation is prformd by oprations. Th infrnc procdurs do not always rquir commutativ and associativ laws of th oprations usd in ths procdurs. Ths proprtis ar not ncssary for conjunction oprations usd in th simplst fuzzy controllrs with two inputs and on output. For ruls with gratr amount of inputs and outputs ths proprtis ar also not rquird if th squnc of variabls in th ruls ar fixd. Morovr, th non-commutativity of conjunction may in fact b dsirabl for ruls bcaus it can rflct diffrnt influncs of th input variabls on th output of th systm. For xampl, in fuzzy control, th positions of th input variabls th ''rror'' and th ''chang in rror'' in ruls ar usually fixd and ths variabls hav diffrnt influncs on th output of th systm. In th application aras of fuzzy modls whn th squnc of oprands is not fixd, th proprty of noncommutativity may not b dsirabl. Latr som xampls will b givn for paramtric non-commutativ and non-associativ oprations. Th axiom systms of t-norms and t-conorms ar vry similar to ach othr xcpt th nutral lmnt, i.. th typ is charactrizd by th nutral lmnt. If th nutral lmnt is qual to 1 thn th opration is a conjunction typ, whil if th nutral lmnt is zro th disjunction opration is obtaind. By using ths proprtis w introduc th concpts of conjunction and disjunction oprations [1]. Dfinition 7 Lt T b a mapping T :[ 0,1] [ 0,1] [ 0,1] opration if T( x,1) = x for all x [0,1]. Dfinition 8 Lt S b a mapping S : [ 0,1] [ 0,1] [ 0,1] opration if S( x,0)= x for all x [0,1].. T is a conjunction. S is a conjunction Conjunction and disjunction oprations may also b obtaind on from anothr by S x, y = N T N x, N y, and mans of an involutiv ngation N : ( ) T( x, y) = N S( N( x), N( y )). It can b sn asily that conjunction and disjunction oprations satisfy th following boundary conditions: T (1,1) = 1, T(0, x) = T( x,0) = 0, S (0,0) = 0, S(1, x) = S( x,1) = 1. By fixing ths conditions, nw typs of gnralizd oprations ar introducd.

10 Dfinition 9 Lt T b a mapping T :[ 0,1] [ 0,1] [ 0,1]. T is a quasiconjunction opration if T(0,0) = T(0,1) = T (1,0) = 0, and T (1,1) = 1. Dfinition 10 Lt S b a mapping S : [ 0,1] [ 0,1] [ 0,1]. S is a quasidisjunction opration if S(0,1) = S(1,0) = S (1,1) = 1, and S (0,0) = 0. It is asy to s that conjunction and disjunction oprations ar quasi-conjunctions and quasi-disjunctions, rspctivly, but th convrs is not tru. Omitting T (1,1) = 1 and S (0,0) = 0 from th dfinitions furthr gnralization can b obtaind. Dfinition 11 Lt T b a mapping T :[ 0,1] [ 0,1] [ 0,1]. T is a psudoconjunction opration if T(0,0) = T(0,1) = T (1,0) = 0. Dfinition 1 Lt S b a mapping S : [ 0,1] [ 0,1] [ 0,1]. S is a psudodisjunction opration if S(0,1) = S(1,0) = S (1,1) = 1. Thorm 3 Assum that T and S ar non-dcrasing psudo-conjunctions and psudo-disjunctions, rspctivly. Thn thr xist th absorbing lmnts 0 and 1,0 = 0, = 0 S x,1 = S 1, x = 1. such as T( x ) T( x ) and 4. A Paramtric Family of Quasi-Conjunctions Lt us cit th following rsult, which is th bas of th forthcoming paramtric construction, from [1]. Thorm 4 Suppos T 1, T ar quasi-conjunctions, S 1 and S ar psudo disjunctions and ar non-dcrasing functions such g 1 = g 1 = 1. Thn th following functions ar quasi-conjunctions: that 1 (, ) = 1(, ), 1( 1, ) T x y T T x y S g x g y (, ) = ( 1(, ), 1 1(, )) T x y T T x y g S x y (, ) = 1(, ), (, 1(, )) T x y T T x y S h x S x y. By th us of this Thorm th simplst paramtric quasi-conjunction oprations can b obtaind as follows ( pq, 0): T x, y = x p y q, p q = T x, y min x, y, (, ) = p ( + ) T x y xy x y xy. q

11 5 Distanc-basd Oprations Lt b an arbitrary lmnt of th closd unit intrval [0,1] and dnot by d( x, y ) th distanc of two lmnts x and y of [0,1]. Th ida of dfinitions of distanc-basd oprators is gnratd from th rformulation of th dfinition of th min and max oprators as follows x, if d( x,0 ) d( y,0 ) x, if d( x,0 ) d( y,0) min( xy, ) =, max( xy, ) = y, if d( x,0 ) > d( y,0 ) y, if d( x,0 ) < d( y,0) Basd on this obsrvation th following dfinitions can b introducd, s [1]. Dfinition 13 Th maximum distanc minimum oprator with rspct to 0,1 is dfind as [ ] x, if d( x, ) > d( y, ) ( x y) d( x ) d( y ) min max( x, y) = y, if d x, < d y,. min,, if, =, Dfinition 14 Th maximum distanc maximum oprator with rspct to 0,1 is dfind as [ ] x, if d( x, ) > d( y, ) ( x y) d( x ) d( y ) max max( x, y) = y, if d x, < d y,. max,, if, =, Dfinition 15 Th minimum distanc minimum oprator with rspct to [ 0,1] is dfind as x, if d( x, ) < d( y, ) ( x y) d( x ) d( y ) min min( x, y) = y, if d x, > d y,. min,, if, =, Dfinition 16 Th minimum distanc maximum oprator with rspct to 0,1 is dfind as [ ] x, if d( x, ) < d( y, ) ( x y) d( x ) d( y ) max min( x, y) = y, if d x, > d y,. max,, if, =,

12 5.1 Th Structur of Distanc-basd Oprators It can b provd by simpl computation that if th distanc of x and y is dfind as d( x, y) = x y thn th distanc-basd oprators can b xprssd by mans of th min and max oprators as follows. max x, y, if y > x min x, y, if y > x min min max = min x, y, if y < x, min = max x, y, if y < x min,, if = x y y x min x, y, if y = x max x, y, if y > x min x, y, if y > x max max max = min x, y, if y < x, min = max x, y, if y < x max,, if = x y y x max x, y, if y = x Th structurs of th min max and th min min oprators ar illustratd in Figur 1. Figur 1 Maximum distanc minimum oprator (lft) and minimum distanc minimum oprator (right) Conclusion In this papr w summarizd som of our contributions to th thory of nonconvntional aggrgation oprators. Furthr dtails and anothr classs of aggrgation oprators can b found in [1]. Rfrncs [1] Batyrshin, O. Kaynak, and I. Rudas, Fuzzy Modling Basd on Gnralizd Conjunction Oprations, IEEE Transactions on Fuzzy Systms, Vol. 10, No. 5 (00), pp [] B. Buchanan and E. Shortliff, Rul-Basd Exprt Systms, Th MYCIN Exprimnts of th Stanford Huristic Programming Projct, Addison- Wsly, Rading, MA, 1984

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