AFFINITY SET AND ITS APPLICATIONS *

Transcription

1 oussa Larbani Yuh-Wn Chn FFINITY SET ND ITS PPLICTIONS * bsrac ffiniy has a long hisory rlad o h social bhavior of human, spcially, h formaion of social groups or social nworks. ffiniy has wo manings. Th firs is a naural liking for or aracion o a prson, hing, ida, c. Th scond dfins affiniy as a clos rlaionship bwn popl or hings ha hav similar apparancs, qualiis, srucurs, propris, or faurs. ffiniy hr is simply dfind as h disanc/closnss bwn any wo objcs: h disanc masurmn could b gomric or absrac, or any yp a dcision makr prfrs. nw forcasing mhod wihou hisorical mmory, basd on gam hory and affiniy s is originally proposd. Th prdicion prformanc of his nw modl is compard wih h simpl rgrssion modl for hir prformancs on dcision of buying in or slling ou socks in a dynamic mark. Inrsingly h qualiaiv modl (affiniy modl) prforms br han h quaniaiv modl (simpl rgrssion modl). Possibl affiniy s applicaions ar providd as wll in ordr o ncourag radrs o dvlop affiniy modls for acual applicaions. Kywords ffiniy, forcasing, dcision, disanc. INTRODUCTION ffiniy forms h basis for many aspcs of social bhavior, spcially, h formaion and voluion of groups or nworks [6, 7, 12]. ffiniy has wo manings. Th firs is a naural liking for or aracion o a prson, hing, ida, c. This kind of affiniy is calld dirc affiniy in his papr. Th scond dfins affiniy as a clos rlaionship bwn popl or hings ha hav similar apparancs, qualiis, srucurs, propris, or faurs. This papr * This rsarch is fundd by h Dparmn of anagmn Scinc of Naional Scinc Council, Taiwan ( H Y2).

2 118 oussa Larbani, Yuh-Wn Chn calls i indirc affiniy. Two difficulis aris whn daling wih affiniy. Firs, affiniy is, by dfiniion, a vagu and imprcis concp. Indd, i is vry difficul o prcisly valua an affiniy lik frindship; i can b approximaly dscribd by linguisic rms lik srong or wak. Th scond is ha affiniy ofn, if no always, varis wih im. For xampl, a sudn may hav srong affiniy wih h collg h is sudying now, bu h affiniy bcoms wak afr h graduas. So far as w know, in liraurs, hr is no hory daling wih affiniy as a vagu and im-dpndn concp, and lil scholarly awarnss ha such a simpl affiniy ida could b dvlopd for valuabl modls in managmn scincs. This papr originally proposs a horical framwork for h affiniy concp, diffrn from fuzzy ss [13] and fuzzy rlaions [3]. Fuzzy s hory is h bs ool for rprsning vagu and imprcis concps so far; howvr, h affiniy s proposd hr is no mrly a fuzzy s bcaus assuming any yp of mmbrship funcion hr is unncssary: h affiniy concp is mor gnral han h fuzzy concp. In h affiniy s hory, allowing a dcision makr uss his subjciv prcpion of disanc o from a s is possibl, inrsing, and innovaiv. Thrfor, his work simply dfins affiniy as h closnss/disanc bwn wo objcs [2]: h disanc masurmn could b gomric or absrac, and affiniy could play various rols in a dcision problm dpnding on dcision makr achivmn. cually, h disanc/closnss concp is mor srongly rlad o Topology [4] rahr han Fuzzy ss [5, 13]; howvr, hs opology absracions ar ranslad/simplifid ino usful modling concps and procdurs hr. Th papr is organizd as follows. Scion 1 inroducs h affiniy s and rlad noions, formalizing indirc affiniy and discussing is applicaion. qualiaiv forcasing mhod basd on affiniy s and gam hory is nwly prsnd, diffrn from h radiional quaniaiv forcasing modls bcaus h hisorical rnd is no longr ncssary. Th prformanc of his nw modl is compard wih h simpl rgrssion modl o show is valu. Scion 2 formalizs dirc affiniy. Las scion concluds h papr. 1. FFINITY SET ND INDIRECT FFINITY This papr rfrs o his yp of affiniy, which could b mdiad by som inrmdiums as indirc affiniy. ahmaically, indirc affiniy can b undrsood as a rlaion bwn lmns of a s, h subjcs, wih an objc or mdium, h rlaion is h affiniy islf. Th radiional crisp

3 FFINITY SET ND ITS PPLICTIONS 119 rlaions canno b usd o modl indirc affiniy for h following wo rasons. Firs, affiniy is, by dfiniion, a vagu and imprcis concp. Th scond is ha affiniy ofn, if no always, varis wih im, for xampl h affiniy bwn a sudn and his sudying collg may bcom srongr or wakr or hav ups and downs ovr im ffiniy s and affiniy W sar by prsning h maning w giv o h primiiv noion of s. Sinc h objciv in his scion is o formaliz affiniy im-dpndnc bwn an lmn and a s, our maning should ncompass h variabiliy of shap or conn of a s. Dfiniion 1 y affiniy s w man any objc (ral or absrac) ha cras affiniy bwn objcs. Exampl 1 rligion is an affiniy s, for i cras affiniy bwn popl ha maks hm liv a crain way of lif. Exampl 2 famous aris or scinis or singr or sporsman or sporswoman is an affiniy s for h or sh cras affiniy bwn popl who apprcia him or hr. From h abov xampls w dduc ha our s noion is widr han h radiional s noion and h fuzzy s noaion. L us now giv a formal dfiniion of affiniy bwn a subjc and an affiniy s. Dfiniion 2 L and b a subjc and an affiniy s, rspcivly. L I b a subs of h im axis [0,+ [. Th affiniy bwn and is rprsnd by a funcion. Th valu () xprsss h dgr of affiniy bwn h subjc and h affiniy s a im. Whn () = 1 his mans ha affiniy of wih affiniy s is compl or a maximum lvl a im ; bu i dosn man ha blongs o, unlss h considrd affiniy is blongingnss. Whn

4 120 oussa Larbani, Yuh-Wn Chn ()= 0 his mans ha has no affiniy wih a im. Whn 0 < () < 1, his mans ha has parial affiniy wih a im. Hr w mphasiz h fac ha h noion of affiniy is mor gnral han h noion of mmbrship or blongingnss. Th lar is jus a paricular cas of h formr. Dfiniion 3 Th univrsal s, dnod by U, is h affiniy s rprsning h fundamnal principl of xisnc. W hav: U (. ): [0,+ ) [0,1] U () and U () = 1, for all xising objcs a im and for all ims. In ohr words h affiniy s dfind by h affiniy xisnc has compl affiniy wih all prviously xising objcs, ha xis in h prsn, and ha will xis in h fuur. In gnral, in ral world siuaions, som radiional rfrnial s V, such as ha whn an objc is no in V, () = 0 for all in I [ 0, + [, can b drmind. In ordr o mak h noion of affiniy s opraional and for pracical rasons, in h rmaindr of h papr, insad of daling wih h univrsal s U, w will dal only wih affiniy ss dfind on a radiional rfrnial s V. Thus, in h rmaindr of h papr whn w rfr o an affiniy s, w assum ha ss V and I ar givn. Dfiniion 4 L b an affiniy s. Thn h funcion dfining is: F (.,.): V I [0,1] (, ) F (, )= () n lmn in ral siuaions ofn blongs o a s a som im and no a ohr ims. Such bhavior can b rprsnd using h affiniy s noion. Th bhavior of affiniy s ovr im can also b invsigad hrough is funcion F (.,.). (1)

5 FFINITY SET ND ITS PPLICTIONS 121 Inrpraion 1 1) For a fixd lmn in V, h funcion (1) dfining h affiniy s rducs o h fuzzy s dscribing dgr variaion of affiniy of h lmn ovr im. 2) For a fixd im, h funcion (1) rducs o a fuzzy s dfind on V ha dscribs h affiniy bwn lmns V and affiniy s a im. Roughly spaking i dscribs h shap or conn of affiniy s a im. 3) In addiion o 1) and 2), w can say/valida affiniy s as a spcial fuzzy s, unlss w can prov ha any affiniy s is includd in a fuzzy s and vic vrsa. 4) ny disanc/closnss could b normalizd o [0, 1], howvr, such a normalizaion procss is no ncssarily fuzzy. Th maximum affiniy ()=1 may no b rachd a any im in ralworld problms. In ordr o considr various siuaions w inroduc h following dfiniion. Dfiniion 5 L b an affiniy s and k [0,1]. W say ha an lmn is in h -k-cor of h affiniy s a im, dnod by -k-cor(), if () k, ha is: ( ) k -k-cor() = { } whn k = 1, -k-cor() is simply calld h cor of a im, dnod by -Cor(). Dfiniion 6 n obsrvaion priod is dfind as h priod (coninuous or discr) analyzing h bhavior of an lmn of V wih rspc o an affiniy s (an illusraion is givn in Figur 1 blow).

6 122 oussa Larbani, Yuh-Wn Chn 1-- () k 0 k-lif Cycl: L Obsrvaion Priod: P Lif cycl Fig. 1. Illusraion of h affiniy bwn an lmn and an affiniy s ovr an obsrvaion priod P Dfiniion 7 L b an affiniy s and k [0,1]. subs T (discr or coninuou of I is said o b h k-lif cycl of an lmn wih rspc o if: () k, for all T and () < k, lswhr in I In ohr words, h priod T is h k-lif cycl of wih rspc o if is in h -k-cor() for all in T. I is h im priod ha lmn kps is affiniy a las qual o k in I. Th priod of im TC = { ( ) > 0, I} is calld lif cycl of h lmn wih rspc o h affiniy s Indirc affiniy Indirc affiniy occurs whn affiniy bwn subjcs aks plac via a mdium. This scion givs a formal dfiniion of indirc affiniy. Th noion of harmony bwn objcs wih rspc o an affiniy s is also formalizd.

7 FFINITY SET ND ITS PPLICTIONS 123 Dfiniion 8 L b an affiniy s and k [0,1]. L D b a subs of V. k-indirc affiniy dgr wih rspc o, a im, bwn h lmns of D xiss, if hy all blong o h -k-cor(), ha is D k Cor (). k-indirc affiniy dgr wih rspc o, during an obsrvaion priod T, bwn h lmns of D xiss, if D k Cor () a any im in T. Dfiniion 9 L and D b an affiniy s and a subs of V, rspcivly. Harmony xiss a im bwn h lmns of D wih rspc o, if hy all blong o -Cor() a im, ha is, D Cor (). In ohr words, harmony bwn h lmns of D wih rspc o is rachd a im whn h maximum indirc affiniy dgr bwn hm is k = 1 a his im. Harmony xiss during h obsrvaion priod of im T, wih rspc o, bwn h lmns of D, if hr is harmony wih rspc o bwn hm a any im in T. This dfiniion xprsss h fac ha harmony is h highs lvl of affiniy Opraions on affiniy ss This scion dfins basic affiniy s opraions. Th following dfiniions 10-14, assum ha and ar wo givn affiniy ss dfind on I and V. Dfiniion 10 W say ha and ar qual a im if () = (), for all in V. Thn w wri = a im. If and ar considrd in an obsrvaion priod T, hn = during his priod if () = (), for all in V and all in T. Dfiniion 11 W say ha is conaind in a im if () (), for all in V. Thn w wri a im. In h cas ha and ar considrd in an obsrvaion priod T, hn during his priod if () (), for all in V and all in T.

8 124 oussa Larbani, Yuh-Wn Chn Dfiniion 12 Th union of and a im, dnod by, is dfind by h funcion F (, ) = () = ax{ (), ()}, for all in V. In h cas ha and ar considrd in an obsrvaion priod T, hn during his priod, is dfind by h funcion F (, ) = () = ax{ (), ()}, for all in V and all in T. Dfiniion 13 Th inrscion of affiniy ss and a im, dnod by, is dfind by h funcion F (, ) = () = in{ (), ()}, for all in V. In h cas ha and ar considrd in an obsrvaion priod T, hn during his priod, is dfind by h funcion F (, ) = () = in{ (), ()}, for all in V and all in T. Dfiniion 14 is said o b h complmn of a im if i is dfind by h following funcion F (, ) = () = 1- (), for all in V. In h cas ha an ar considrd in an obsrvaion priod T, hn during his priod, is dfind by h funcion F (, ) = () = 1- (), for all in V and all in T pplicaion of forcasing Th affiniy s s ponial applicaions ar valuabl in analyzing, valuaing, forcasing (prdicing) h im-dpndn bhaviors: for xampl, volving an uncrain dynamic sysm in a human sociy. In addiion, prdicing h dmand curv wih high flucuaions is also possibl by an affiniy s. W will giv a simpl xampl of how h affiniy s can b applid in forcasing ral-world problms lar. In fac, any im sris mhod can b usd o prdic any lmn bhavior in V wih rspc o an affiniy s basd on pas daa, if i is possibl o dfin affiniy s. This papr proposs a nw forcasing mhod basd on affiniy s and gam hory. ssum ha an affiniy s and a univrs V ar givn and som daa ar availabl a som pas priods 1, 2,..., n on h bhavior of lmns in V wih rspc o affiniy s as dscribd in h following marix [1]:

9 FFINITY SET ND ITS PPLICTIONS 125 D = a a Hr w can follow h similar concp in [1],... a n 1n a 2n,..., 1, 2 n ar rgardd as h mulipl aribus of h dcision problm, and and ar wo alrnaivs of his problm. u w will dfin a nw mhod, which is diffrn from [1] o rsolv his affiniy gam. Whr is h affiniy s complmnary o (s Dfiniion 14), nry a 1 j is h affiniy dgr of lmn wih rspc o affiniy s a h priod j and a2 j = 1 a1 j is h affiniy dgr of lmn wih rspc o affiniy s a h sam priod. Hr a dcision makr wans o forcas lmn bhavior a h nx priod n+ 1. Inrsingly w can look a h siuaion as a gam bwn h dcision makr and Naur. Th dcision makr facs an uncrain siuaion rprsnd by fuur lmn bhavior. On way o handl h siuaion is o adop h maximum dcision making undr uncrainy principl [3] by considring h siuaion as a gam agains Naur [1]. Thus, marix D can b considrd as a marix gam bwn h dcision makr and Naur, whr h dcision makr is h maximizing playr who chooss bwn and and Naur is h minimizing playr who chooss h im priods. Dfiniion 15 pair of sragis ( i 0, j0 ) whr i 0 {1,2} and j0 {1,.., n} ar said o b h Nash quilibrium [9] of h marix gam D if: a ij ai j a i 0 j, for all i {1,2} and j { 1,.., n} (2) ssum ha h gam has a Nash quilibrium ( i 0, j0 ). In rms of affiniy, his quilibrium can b inrprd as follows. If i 0 = 1, h dcision makr will favor lmn affiniy wih affiniy s rahr han affiniy wih, wih affiniy dgr a i 0 j 0. Th dcision makr in cas i 0 = 2 will favor lmn affiniy wih rahr han wih, wih affiniy dgr a i 0 j. I may 0 happn ha marix D has no Nash quilibrium in pur sragis, hn h wo playrs hav o us mixd sragis. mixd sragy for Naur is a probabiliy disribuion ovr h s of is pur sragis, ha is, i is a vcor y = y, y,..., y ) such ha: ( 1 2 n

10 126 oussa Larbani, Yuh-Wn Chn n y j j= 1 = 1 and y 0, j = 1, n Similarly, a mixd sragy for h dcision makr is a vcor x = ( x 1, x2 ) such ha: x + x 1 and x 0, i = 1, = Playr payoffs bcom xpcd payoffs. Dcision makr payoff is x T Dy and ha of Naur is x T D y. ny marix gam always includs a Nash quilibrium in mixd sragis [9]. Nash quilibrium in mixd sragis is dfind by: x T D y j x T D i y x T Dy for all mixd sragis x and y. Th mixd sragy x of h dcision makr can b inrprd as follows. Th dcision makr will favor wih wigh x and wih wigh 1 x. H can also us hs wo valuaions o rank ss 2 and from his poin of viw. Th xpcd affiniy dgr of lmn in h priod n+ 1 wih ach of h affiniy ss can b dfind as follows: ( n+ 1 ) = n 1 a1 j y j and ( n+ 1 ) = n 1 a 2 j y j rspcivly. Th mixd sragy y of Naur can b inrprd as h wighs Naur assigns o h priods in ordr o minimiz xpcd dcision makr affiniy. L us illusra our approach by xampls. Exampl 3. Dcision of buy in/sal ou/hold Today, w ar awar ha h sock pric in a mark is qui unsabl; in ohr words, h sock pric curv is highly flucuaing for a company. Now w collc h acual daa of Taiwan TGV Company for wny-wo priods (from Ocobr 1, 2007 o Ocobr 22, 2007) from Taiwan Sock ark [8]. ssum ha a dcision makr wans o prdic if h can buy in or sll ou his socks in h mark by updaing his informaion and using h affiniy gam. Th firs svnn daa ar usd as h raining bas, hn w prdic h rmaining fiv daa. Plas no ha if w wan o prdic h ighn daa

11 FFINITY SET ND ITS PPLICTIONS 127 by affiniy modl, hn h prvious svnn daa will b all includd in an affiniy gam, and if prdicing h ninn daa hn h prvious ighn daa will b includd, c. ssum, for simpliciy, ha by xprinc h classifis his dcisions ino only uy in, Sll ou and Hold. Ths wo possibl sas can b considrd as wo affiniy complmnary ss (uy in) and (Sll ou), rspcivly. His dcision will b h lmn. nd if h affiniy dgr of o, and ha o ar idnical, hn h chooss h Hold sa. Th pric daa of wny-wo das in Ocobr 2007 ar collcd as in Figur 2. ssum his dcision makr has rcordd h affiniy dgrs of sock pric wih rspc o affiniy s by h following funcion: p c1 = ( ), = 2,3,..., n p 1 and if c 1 < 1 hn a1 = 1 c1 ; if c 1 > 1 hn a 2 = c1 1; if c 1 = 1 hn a 1 = a2 = 0. 5, which is a Hold sa: no buying in and no slling ou. Hr a1 = 1 a2 is also assumd. (3) Sock Pric 9,05 9 8,95 8,9 8,85 8,8 8, Da Fig. 2. cual daa of sock pric (in Taiwans Dollars pr sock) Sourc: [8].

12 128 oussa Larbani, Yuh-Wn Chn Tabl 1 Prformanc comparison of affiniy modl and simpl rgrssion odl\priods ffiniy Gam Hold Hold Hold Hold Hold Simpl Rgrssion cual Daa Sourc: [8]. ccording o h acual daa, h affiniy marix is asily compud (s ppndix). Th suggsd dcision is summarizd in Tabl 1, which is compard wih h simpl rgrssion modl (only using im as h xplainabl variabl). Inrsingly h affiniy modl suggss h Hold sa, which sms o b br han prdicing h dclining rnd by h simpl rgrssion modl. caus if a dclining rnd is forcasd, hn h acion of slling ou socks will b considrd by his dcision makr. Howvr, h Hold sa suggsd by h affiniy modl hins h dcision makr maks mor profis if h kps hs socks from h im priod: 17. ffiniy gam prdics ha h sock pric will rmain almos sabl during h analyzd im. I is clar ha affiniy modl prforms br han h simpl rgrssion modl in his xprimn. Of cours, h funcion (3) could b assumd by various yps, h dcision makr can choos any yp ha h prfrs. Th affiniy spiri is vnually, a dcision makr is ncouragd o ry/dvlop any possibl masurmn o find/xplor/analyz h spcial parn in a im-dpndn daa s or inpu/oupu sysm. nd his spcial parn is arbirarily dfind by a dcision makr, jus lik ha disanc/closnss hav gnral dfiniions in Topology [4]. spcial parn could vary wih im and spac, onc h dcision makr cachs h cor of his spcial parn, h could xplain h obsrvaions or prdic som usful oucoms. cually, hr is an old saying: wha you masurd h world filrs wha you s. Thus, various masurmns for modling ar naural and should b ncouragd ffiniy s dpnding on im and ohr variabls Th affiniy of lmn wih rspc o affiniy s in ral-world siuaions ofn dpnds implicily on ohr variabls han im. Ths variabls gnrally xprss condiion or consrain variabiliy ha affc affiniy valu-

13 FFINITY SET ND ITS PPLICTIONS 129 aion. Sudying lmn bhavior wih rspc o im and ohr variabls may b pracically dsirabl. dcision makr may vn sudy lmn bhavior a a fixd im wih rspc o ohr variabls. This scion xnds h affiniy s dfiniion o h cas whr dsird variabls appar xplicily. This dfiniion maks i possibl o sudy affiniy bhavior ovr im and wih rspc o ohr variabls as wll. Dfiniion 16 L and b an lmn and an affiniy s, rspcivly. ssum ha h affiniy of wih rspc o dpnds on som variabl w ha aks is valus in a radiional s W. In ordr o mak h variabl w appar in h affiniy dfiniion bwn and, w inroduc h following affiniy: (. ): I W [0,1] (,w) (,w) Th valu (,w) xprsss h dgr of affiniy bwn lmn and a im wih rspc o w. Thus, dpnding on h problm a hand, h dcision makr can us Dfiniions 2 or 16 of affiniy bwn an lmn and an affiniy s. Dfiniion 17 L b an affiniy s dpnding on a variabl w W. Thn h funcion dfining is dfind by: F (.,.,.): V I W [0,1] (,,w) F (,,w)= (,w) whr V is h radiional rfrnial as in Scion DIRECT FFINITY Dirc affiniy is a naural liking for or aracion o a prson or a hing, or an ida, c. Dirc affiniy involvs wo lmns: h affiniy subjcs and h affiniy ha aks plac bwn hm. ahmaically, dirc affiniy can b undrsood as a binary rlaion bwn lmns of a s, whr h lmns ar h subjcs and h rlaion is affiniy. Th radiional crisp binary rlaions canno b usd o modl dirc affiniy for h following wo rasons: Firs, affiniy is, by dfiniion, a vagu and imprcis concp. Indd, i is vry difficul o giv a prcis valuaion of affiniy lik frindship; i can b approximaly dscribd by linguisic rms lik srong or wak;

14 130 oussa Larbani, Yuh-Wn Chn h scond is ha affiniy ofn, if no always, varis wih im, for xampl frindship may bcom srongr or wakr or hav ups and downs ovr im. Thus, h adqua way o modl dirc affiniy is o us im-dpndn fuzzy rlaions. ffiniy can b considrd as a paricular cas of h following gnral framwork. Dfiniion 18 L V and I b a rfrnial s and a subs of h im axis [0, + [, rspcivly. im dpndn fuzzy rlaion R such ha: R (.) : I (V V) [0,1] (.,.) (,,(, ) R ( ( ) is calld dirc affiniy on h rfrnial V. Inrpraion 2 1. For any fixd im h rlaion (2) rducs o an ordinary fuzzy rlaion [3]: R ( ) : V V [0,1] (.,.) (,, R ( ( ) ha xprsss h innsiy or h dgr of affiniy bwn any coupl of lmns in V. Hnc affiniy fuzzinss bwn lmns is akn ino accoun in Dfiniion For any fixd coupl of lmns (, V, h rlaion (4) rducs o a fuzzy s dfind on h im-s I: R (.) : I [0,1] (, R ( ( ) ha xprsss affiniy voluion ovr im bwn lmns and s. Thus, h im-dpndn fuzzy rlaion (4) xprsss h mos imporan characrisic of dirc affiniy: Fuzzinss and im-dpndnc. Dfiniion 18 can b xndd o affiniy bwn groups of lmns as follows. Dfiniion 19 L R b a im-dpndn fuzzy rlaion dfind on a subs of im axis I and a rfrnial V. L and b wo subss of V. Thn h affiniy bwn and can b dscribd by h following funcion:, (4)

15 FFINITY SET ND ITS PPLICTIONS 131 R (.) : I [0,1] (5) (, ) R (, ) whr R (, ) (.) can b dfind by many ways, dpnding on h dcision makr. W propos h following four xampls: 1) R (, ) ( ) = max R(, s ) ( ), for all I (,, s ( ) 2) R ( ) = (, ) min R(, s ) ( ), for all I (,, s 3) R ( ) = α ) + ( 1 α ) min ( ), for all (, ) max R(, s ) ( (,, s (,, s R(, s ) I, whr α is a numbr in [0,1] ha xprsss h dgr o which h dcision makr prfrs h maximum of affiniy o is minimum. 4) in h cas and ar fini R ( ) = λ (, R(, ( ), for all (, ) (,, s I, whr λ, 0 is h wigh assignd by h dcision makr o h coupl (, for s and λ (, =1. ( (,, s Hr also for pracical purpos w dfin h -k-affiniy. Dfiniion 20 L R b a im-dpndn fuzzy rlaion dfind on a subs of im axis I and a rfrnial V. L k [0,1], and I. Thn: 1) w say ha a coupl (, has k affiniy dgr a im or -k-affiniy dgr if R(, s ) ( ) k, 2) a subs D of V has -k-affiniy dgr if R( D, D ) ( ) k. Thus, h -k-affiniy dgr of subss dpnds on how affiniy is dfind bwn groups or subss as indicad in Dfiniion 19, 1)-4). Rmark 1 Dpnding on informaion availabl for h im-dpndn fuzzy rlaion dscribing affiniy (2)-(3), dirc affiniy can b usd o sudy nworks (social or nonsocial). Indirc affiniy can also b usd o analyz, dscrib, forcas, and prdic nwork bhavior or is lmns rgarding h considrd affiniy. For xampl, wih knowldg ha nwork voluion ovr im follows a diffrnial quaion or a sochasic procss, ha is, h funcion

16 132 oussa Larbani, Yuh-Wn Chn R (, ( ) is a soluion of a diffrnial quaion or a sochasic procss, hn basd on iniial daa on can prdic nwork bhavior a any im I rgarding h considrd affiniy. Social nwork analysis [6, 12] is on ara for dirc affiniy applicaion. In addiion, h dirc affiniy concp is valuabl in dvloping nwork grouping or nwork conrolling. CONCLUSIONS ND RECOENDTIONS This papr proposs a basic framwork for h affiniy concp, allowing is invsigaion by fuzzy s ools and ohr nonfuzzy mhods. Of cours, fuzzy ools ar no h only way o xplor affiniy. Radrs should raliz ha h affiniy modl proposd in Exampl 3 is qui diffrn from h fuzzy s and rough s [10, 11] bcaus w don nd o assum any yp of fuzzy mmbrship funcion [10] or us h uppr bound and lowr bound o approxima a s [11]. Insad, h closnss or disanc bwn any wo objcs wihin a im sris daa s is dircly assumd, hn i will form h basis of an affiniy s. Numrous masurmns of closnss/disanc could xis in Exampl 3, bu w only propos/assum on way hr. W sudid wo yps of affiniy: Indirc affiniy and dirc affiniy. This work poind ou ha indirc affiniy rquirs a mdium and inroducs h affiniy s for indirc affiniy formalizaion, which acually rprsns h mdium. Th affiniy of lmns wih rspc o an affiniy s is rprsnd by a fuzzy s dfind on h im axis. Thn h affiniy bwn lmns (indirc affiniy) is dfind via hir affiniy o h affiniy s. W hav formalizd dirc affiniy as a im-dpndn fuzzy rlaion and prsn a nw forcasing mhod basd on affiniy s and gam hory. Finally, w indica som ponial aras for possibl applicaion of dirc affiniy and indirc affiniy. any issus ar no fully discussd in his papr. On of hm is h numrical drminaion of funcions () and R (, ( ) ha rprsn affiniy in indirc affiniy and dirc affiniy, rspcivly. nohr issu is xploraion of h affiniy s noion. W bliv ha invsigaing affiniy in social nworks or nginring conrol using our framwork is a worhwhil opic of rsarch. W also hop ha his papr will inspir and arac mor rsarchrs for invsigaing h affiniy concp. Th voluionary algorihms will b bnficial whn w ry o find/xplor h spcial parn hiddn in a larg scal daa s; for xampl, volving h spcial parn ha maximizs a spcifid/prdfind affiniy.

17 FFINITY SET ND ITS PPLICTIONS 133 ppndix cual daa and affiniy dgr Da Tradd Socks vrag Pric c +Up/-Down ( ) 1 = p p 1 a 1 a 2 07/Oc/01 3,252, /Oc/02 1,058, /Oc/03 1,660, /Oc/04 1,018, /Oc/05 1,200, /Oc/06 999, /Oc/07 970, /Oc/08 1,177, /Oc/09 1,696, /Oc/10 1,687, /Oc/11 781, /Oc/12 789, /Oc/13 1,409, /Oc/14 622, /Oc/15 783, /Oc/16 1,702, /Oc/17 859, /Oc/18 1,449, /Oc/19 586, /Oc/20 1,985, /Oc/21 1,166, /Oc/22 1,209,

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