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 (96-2416-H-212-002-Y2).
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
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. 1.1. 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
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)
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).
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. 1.2. 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.
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. 1.3. 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.
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. 1.4. 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]:
FFINITY SET ND ITS PPLICTIONS 125 D = a a 1 11 21...... 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 0 0 0 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
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, 2 1 2 = 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
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 8,75 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Da Fig. 2. cual daa of sock pric (in Taiwans Dollars pr sock) Sourc: [8].
128 oussa Larbani, Yuh-Wn Chn Tabl 1 Prformanc comparison of affiniy modl and simpl rgrssion odl\priods 18 19 20 21 22 ffiniy Gam Hold Hold Hold Hold Hold Simpl Rgrssion 8.85 8.84 8.84 8.83 8.82 cual Daa 8.95 8.90 8.90 8.90 8.90 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. 1.5. 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-
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 1. 2. 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;
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 18. 2. 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)
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
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.
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 3,252,380 8.98-0.03 07/Oc/02 1,058,000 8.99 +0.01 1.001114 0.998886 0.001114 07/Oc/03 1,660,201 8.98-0.01 0.998888 0.001112 0.998888 07/Oc/04 1,018,000 8.98 +0.00 1 0.50 0.50 07/Oc/05 1,200,500 8.97-0.01 0.998886 0.001114 0.998886 07/Oc/06 999,000 8.95-0.02 0.99777 0.00223 0.99777 07/Oc/07 970,912 8.95 +0.00 1 0.50 0.50 07/Oc/08 1,177,913 8.91-0.04 0.995531 0.004469 0.995531 07/Oc/09 1,696,000 8.87-0.04 0.995511 0.004489 0.995511 07/Oc/10 1,687,000 8.90 +0.03 1.003382 0.996618 0.003382 07/Oc/11 781,000 8.90 +0.00 1 0.50 0.50 07/Oc/12 789,000 8.88-0.02 0.997753 0.002247 0.997753 07/Oc/13 1,409,000 8.91 +0.03 1.003378 0.996622 0.003378 07/Oc/14 622,300 8.92 +0.01 1.001122 0.998878 0.001122 07/Oc/15 783,535 8.83-0.09 0.98991 0.01009 0.98991 07/Oc/16 1,702,000 8.88 +0.05 1.005663 0.994337 0.005663 07/Oc/17 859,000 8.90 +0.02 1.002252 0.997748 0.002252 07/Oc/18 1,449,956 8.95 +0.05 07/Oc/19 586,000 8.90-0.05 07/Oc/20 1,985,956 8.90 +0.00 07/Oc/21 1,166,913 8.90 +0.00 07/Oc/22 1,209,000 8.90 +0.00
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