An Extension of Classical Functional Dependency: Dynamic Fuzzy Functional Dependency
Descripción
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Ghqlwlrq 4 D ix}}| vhw I lq d xqlyhuvh ri glvfrxuvh X lv fkdudfwhul}hg e| d phpehuvkls ixqfwlrq I = X $ ^3> 4`/ zkhuh I +x, ghqrwhv wkh judgh ri php0 ehuvkls ri x lq wkh ix}}| vhw I 1 Khqfh/ wkh ix}}| vhw I lv ghqrwhg e| wkh iroorzlqj +zkhq X lv frxqwdeoh vhw ri q hohphqwv,= I @ i+x4 ,@x4 . +x5 ,@x5 . = = = . +xq ,@xq j 1 H{dpsoh 4 Ohw X @ ^3> 433`> eh wkh xqlyhuvh ri glvfrxuvh uhsuhvhqwlqj d shu0 vrq*v djh1 Ohw \ dqg R wzr ix}}| vxevhwv ri X / uhsuhvhqwlqj wkh frqfhswv \rxqj dqg Rog/ uhvshfwlyho|1 Wkh phpehuvkls ixqfwlrqv \ dqg R duh jlyhq ehorz= 4 li 3 x 58 \ +x, @ ^4 . + x 58,5 `4 li 58 ? x 433 3 li 3 x 83 R +x, @ +4 . +x 83,5 ,4 li 83 ? x 433 Ohw U+D4 > D5 > = = = > Dq , eh d uhodwlrq vfkhph1 Pdwkhpdwlfdoo|/ dq q0du| ix}}| uhodwlrq u ryhu U lv d ix}}| vxevhw ri G @ grp+D4 , grp+D5 , = = = grp+Dq ,/ dqg lv fkdudfwhul}hg e| d phpehuvkls ixqfwlrq u =G $ ^3> 4`1 Wkxv/ d ix}}| uhodwlrq u ryhu U fdq eh ylhzhg dv d wzr0glphqvlrqdo wdeoh lq zklfk d urz fruuhvsrqgv wr dq remhfw dqg d froxpq wr dq dwwulexwh1 Dq dgglwlrqdo froxpq/ ghqrwhg u +w,/ vwdqgv iru wkh phpehuvkls ydoxh ri wkh wxsoh w lq u1 Ghshqglqj rq wkh frpsoh{lw| ri grp+Dl ,/ l @ 4/= = =/q/ Udmx hw do/ ^54`/ kdyh lqwurgxfhg d fodvvlfdwlrq ri ix}}| uhodwlrqv lqwr wzr fdwhjrulhv/ qdpho| W|sh04 dqg W|sh05 ix}}| uhodwlrq1 Lq w|sh04 ix}}| uhodwlrq/ grp+Dl ,/ pd| eh d fodvvlfdo ru d ix}}| vxevhw ri X1 Lq w|sh05 ix}}| uhodwlrq/ grp+Dl ,/ pd| eh d vhw ri ix}}| vxevhwv lq Xl / l @ 4> = = = > q/ xqlyhuvh ri glvfrxuvh dvvrfldwhg zlwk hdfk dwwulexwh Dl1
H{dpsoh 5 Ohw xv frqvlghu wkh uhodwlrq vfkhph HPS4+Q/ G/ S/ H/ V, uhsuh0 vhqwlqj h{shulphqwhg dqg zhoo sdlg hpsor|hhv/ zkhuh Q @ qdph/ G@ ghsduw0 phqw/ S @ surihvvlrq/ H@ h{shulhqfh dqg V @ vdodu|1 D srvvleoh lqvwdqfh ri u+HPS4, lv jlyhq lq wdeoh 41 Zh ghqh wkh glhuhqw grpdlqv ri dwwulexwhv dv ehorz= 6
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Uhpdun 4 Lq wkh fodvvlfdo uhodwlrqdo gdwd prgho/ wkh uhvwulfwlrq ri d wxsoh rq d vhw ri dwwulexwh lv hqwluho| nqrzq dqg vlqjoh ydoxhg1 Exfnohv hw do ^9/ :` kdyh doorzhg/ dq dwwulexwh ydoxh wr eh d vxevhw ri wkh dvvrfldwhg grpdlq1 Wkh htxly0 dohqfh uhodwlrq ryhu wkh grpdlq lv uhsodfhg e| d ix}}| vlplodulw| uhodwlrq wr lghqwli| vlplodu wxsohv h{fhhglqj d jlyhq wkuhvkrog ri vlplodulw| +Lghd xvhg dovr e| Lfklndzd hw do ^47`,1
2 633) ,^@*|) @t 633) hi*@|L? Dv zh h{whqg wkh uhodwlrqdo gdwd prgho wr kdqgoh ix}}| gdwd/ lw lv qhfhvvdu| wr vxssruw lqwhjulw| frqvwudlqwv h{suhvvlqj dssur{lpdwh htxdolw| vxfk wkdw %ds0 sur{lpdwho| htxdo% ru %pruh ru ohvv htxdo%1 Wkh htxdwlrq +4, vkrzv wkh qhhg wr frpsduh/ zlwk dq htxlydohqfh uhodwlrq/ ydoxhv ri dwwulexwhv ri glhuhqw wxsohv1 Krzhyhu/ zkhq zh ghdo zlwk ix}}| gdwd/ zh qhhg d ghylfh wr frpsduh lpsuhflvho| ru loo nqrzq ydoxhv ri dwwulexwhv1 Iru wklv sxusrvh/ zh xvh d ix}}| uhvhpeodqfh uhodwlrq ghqrwhg khuh HT1 Wkh uhodwlrq HT lv uhh{lyh +l1h1/ HT +d> d, @ 4, dqg v|pphwulf +l1h1/ HT +d> e, @ HT +e> d,,1 Wkh wudqvlwlylw| surshuw| grhv qrw krog dv lw lv dgyrfdwhg e| Uxqghqvwhlqhu hw do ^55` iru sur{lplw| phdvxuhv xvhg iru frpsdulqj grpdlq ydoxhv1 8
Wkh uhodwlrq HT fdq eh h{whqghg ryhu frpsrvlwh grpdlqv dv iroorzv1 Ohw
G @ grp+D4 , grp+D5 , = = = grp+Dq ,/ dqg w4 dqg w5 eh wzr wxsohv lq G1
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Uhpdun 5 Wkh vhohfwlrq ri wkh phpehuvkls HT iru hyhu| grpdlq lv rq wkh ehdu ri wkh gdwd edvh ghvljqhu dv lwv rzq shufhswlrq ri wkh dssur{lpdwh htxdolw| ri grpdlq ydoxhv1 H{dpsoh 7 Ohw xv frqvlghu wkh uhodwlrq vfkhph HPS5+Q/ G/ S/ H/ V/ L,1 Lq wklv fdvh/ wkh grpdlqv ri Q/ G dqg S duh fodvvlfdo vhwv/ zkhuh wkrvh ri H/ V dqg L duh ix}}| vhwv1 Zh ghqh wkh phpehuvkls ixqfwlrq HT +d> e,> ghshqglqj rq wkh qdwxuh ri d dqg e/ iru hyhu| grpdlq dv iroorzv= HT +d> e, @ 3 iru d 9@ e zkhuh d dqg e 5 grp+D,/ D 5 iQ> G> S j/ Iru d dqg e 5 grp+D,/ D 5 iH> V> L j/ l, Li d dqg e duh suhflvh/ wkhq HT +d> e, @ + +4.4mdem , zkhuh @ 4 ; d/ e 5 grp+H ,/ @ 4@5333 ; d/ e 5 grp+V ,/ @ 4@833 ; d/ e 5 grp+L ,1 ll, Li d lv suhflvh dqg e lv d ix}}| vhw/ wkhq= HT +d> e, @ e+d,1 Li e lv suhflvh dqg d lv d ix}}| vhw/ wkhq= HT +d> e, @ d +e,1 +d_ e, fdug+d_ e, lll, Li d dqg e duh ix}}| vhwv/ wkhq HT +d> e, @ pd{i fdug fdug+d, > fdug+e, j1 Zkhuh fdug ghqrwhv fduglqdolw| ri ix}}| vhwv ^4 4, li d 9@ 3 Jrjxhq ,J 4 rwkhuzlvh Nohhqh0Glhqhv ,NG pd{+4 d> e, Oxndvlhzlf} plq+4> 4 d . e, ,O Qdph
Wdeoh 6= Wkh prvw xvhg ix}}| lpsolfdwlrqv1 eh lpsuhflvho| nqrzq dqg duh ghvfulehg e| phdqv ri srvvlelolw| glvwulexwlrqv1 Lq wklv frqwh{w/ wkh dxwkruv lqwurgxfhg wkh iroorzlqj ghqlwlrq =
Ghqlwlrq 5 [ A \ lv ydolg rq u+X , / ;w4 > w5 5 u w4 =[ @ w5 =[ , UHV +w4 =\> w5 =\ ,
+5,
Zkhuh lv d {hg wkuhvkrog dqg UHV lv d sur{lplw| uhodwlrq xvhg wr frpsduh hlwkhu d sdlu ri fodvvlfdo ydoxhv ru srvvlelolw| glvwulexwlrqv1 Zkhq UHV lv dssolhg wr srvvlelolw| glvwulexwlrqv/ lw frpsxwhv wkh htxdolw| srvvlelolw| ghjuhh ri wkh loo0 nqrzq ydoxhv { dqg | xvlqj wkh iroorzlqj irupxod +zkhuh { dqg | duh uhsuhvhqwhg e| wkh srvvlelolw| glvwulexwlrqv { / | / uhvshfwlyho|,= UHV +{> |, @ vxs plq+{ +x,> | +x,,1
x5grp+{, Fkhq hw do ^;/ 4`, / ;w4 / w5 5 u Li ^w4 =[ @ w5 =[ $ w4 =\ @ w5 =\ hovhUHV +w4 =[> w5 =[ , ,Jr UHV +w4 =\> w5 =\ , `= +6,
Wkh ghqlwlrq 6 lv vwurqjhu wkdq wkh fodvvlfdo ghqlwlrq ri IG/ vlqfh wkh iruphu rqh lv lqfoxghg dv dq lqvwdqfh +wkh +li 1 1 1 , sduw, lq ghqlwlrq 61 Wkh vhfrqg sduw ri ghqlwlrq 6 lpsrvhv d frqvwudlqw zkhq wkh uhsuhvhqwdwlyh ydoxhv ri wkh ohiw0kdqg vlgh vkduh frpprq ydoxhv1 Wkh iroorzlqj uhvxowv kdyh ehhq hvwdeolvkhg= l, Dq h{whqghg vrxqg dqg frp0 sohwh vhw ri lqihuhqfh d{lrpv/ ll, Wkh orvvohvv mrlq ghfrpsrvlwlrq wkhruhp lv :
ydolg/ lll, Dq dojrulwkp wr whvw wkh suhvhuydwlrq ri IIGv zkhq ghfrpsrvlwlrq lv dssolhg1 D1 Nlvv ^49`= Wklv sursrvdo frqfhuqv W|sh04 ix}}| uhodwlrqv= wkh kljkhu wkh zhljkw dvvrfldwhg zlwk wxsoh w/ wkh pruh frqghqw wkh lqirupdwlrq vwruhg lq w1 Wkh dxwkru lqwurgxfhg wkh iroorzlqj ghqlwlrq=
Ghqlwlrq 7 [ A \ lv ydolg rq u+X , dqg lwv wuxwk ydoxh WU +[ A \ , ^3> 4` lv htxdo wr =
5
WU +[ A \ , @ plq ^plq^u +w4 ,> u +w5 ,> w4 =[ @ w5 =[ ,NG w4 =\ @ w5 =\ `= +7, w >w 4 5
Wkh h{suhvvlrq 7 fdq eh lqwhusuhwhg dv iroorzv1 Wkh pruh d sdlu ri wxsohv/ zlwk d kljk phpehuvkls ghjuhh/ ylrodwhv wkh fodvvlfdo IG/ wkh ohvv wkh GII [ A \ lv vdwlvhg1 Wkh dlp ri wklv sursrvdo lv qrw wkh uhgxfwlrq +ru holplqd0 wlrq, ri uhgxqgdqf|/ exw udwkhu wkh vdyh ri vwrudjh vsdfh1 Wkh dxwkru pdlqo| lqwurgxfhv d % zhdn % ghfrpsrvlwlrq wkhruhp1 Lq idfw/ li WU +[ A \ , > wkhq wkh vxevhw ri u+[> \> ] ,/ u+[> \> ] , zkrvh wxsoh phpehuvkls lv juhdwhu wkdq / pd| eh ghfrpsrvhg lqwr lwv wzr surmhfwlrqv u ^[> \ ` dqg u ^[> ] ` zkhuh A 4 1 Wkh iroorzlqj sursrvdov dgguhvv W|sh04 dqg W|sh05 ix}}| uhodwlrqv1 Udmx hw do ^54`= Xvlqj Uhvfkhu0Jdlqhv lpsolfdwlrq/ wkh dxwkruv sursrvhg wkh iroorzlqj ghqlwlrq=
Ghqlwlrq 8 [ A \ lv ydolg rq u+X , / ;w4 / w5 5 u ^HT +w4 =[> w5 =[ , ,UJ HT +w4 =\> w5 =\ ,`=
+8,
Lqwxlwlyho|/ wkh frqglwlrq 8 phdqv wkdw wkh dqwhfhghqwv* htxdolw| ghjuhh +l1h1/
HT +w4 =[> w5 =[ ,, pxvw eh ohvv wkdq ru htxdo wr wkh htxdolw| ghjuhh ri wkhlu fruuhvsrqglqj lpdjhv +l1h1/ HT +w4 =\> w5 =\ ,,1 Wkh iroorzlqj uhvxowv kdyh ehhq hvwdeolvkhg= l, d vrxqg exw qrw frpsohwh
d{lrpdwlf v|vwhp +lghqwlfdo wr Dupvwurqj*v v|vwhp ^4`,1 Wr dvvxuh wkh frp0 sohwhqhvv surshuw|/ d ixuwkhu uhvwulfwlrq ri wkh uhodwlrq HT vkrxog eh lpsrvhg1 ll, Wkh orvvohvv mrlq ghfrpsrvlwlrq wkhruhp lv ydolg1 lll, Dq dojrulwkp wr whvw wkh suhvhuydwlrq ri IIGv zkhq d ghfrpsrvlwlrq lv dssolhg1 Ekxlq|d hw do ^7`= Lq rughu wr shuplw wkh dffhswdqfh ri vrph lqvwdqfhv ylrodwlqj wkh frqglwlrq +8,/ wkh dxwkruv vxjjhvwhg wkh iroorzlqj ghqlwlrq=
Ghqlwlrq 9 [ A \ lv ydolg rq u+X , / ;w4 / w5 5 u HT +w4 =[> w5 =[ , ,UJ HT +w4 =\> w5 =\ , ;
+9,
ru +:, HT +w4 =[> w5 =[ , HT +w4 =\> w5 =\ , 4 zkhuh HT +w4 =[> w5 =[ , dqg HT +w4 =\> w5 =\ , duh 0htxdo +l1h1= HT +w4 =[> w5 =[ , dqg HT +w4 =\> w5 =\ , , dqg ? ? 41 Wkh iroorzlqj uhvxowv kdyh ehhq hvwdeolvkhg= l, dq h{whqghg vrxqg dqg frp0 sohwh d{lrpdwlf/ ll, wkh orvvohvv mrlq ghfrpsrvlwlrq wkhruhp lv ydolg dqg dq dojrulwkp wr whvw wkh suhvhuydwlrq ri IIGv zkhq d ghfrpsrvlwlrq lv dssolhg1 Vd{hqd hw do ^56`= Wklv sursrvdo lv dq h{whqvlrq ri Udmx hw do*v sursrvdo1 Lq idfw/ wkh dxwkruv lqwurgxfhg ix}}lfdwlrq ri wkh frqfhsw ri qxoo ydoxhv/ ghqrwhg khuh >/ dqg sursrvhg wkh iroorzlqj ghqlwlrq=
Ghqlwlrq : [ A \ lv ydolg rq u+X , 9 w5 =[ ,= w4 =[ 9@ > @
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