Idvw Iudfwdo Lpdjh Frpsuhvvlrq zlwk Wuldqjxodwlrq Zdyhohwv
G1 M1 Khehuw dqg H}hnlho Vrxqgdududmdq
DEVWUDFW
Zh dgguhvv wkh sureohp ri lpsurylqj wkh shuirupdqfh ri zdyhohw edvhg iudfwdo lpdjh frpsuhvvlrq
e| dsso|lqj h!flhqw wuldqjxodwlrq phwkrgv1 Zh frqvwuxfw lwhudwlyh ixqfwlrq v|vwhpv +LIV, lq wkh
wudglwlrq ri Eduqvoh| dqg Mdftxlq/ xvlqj qrq0xqlirup wuldqjxodu udqjh dqg grpdlq eorfnv lqvwhdg ri
xqlirup uhfwdqjxodu rqhv1 Zh vhdufk iru pdwfklqj grpdlq eorfnv lq wkh pdqqhu ri ]kdqj dqg Fkhq/
shuiruplqj d idvw zdyhohw wudqvirup rq wkh eorfnv dqg holplqdwlqj orz uhvroxwlrq plvpdwfkhv wr jdlq
vshhg1 Zh rewdlq ixuwkhu lpsuryhphqwv e| wkh h!flhqflhv ri elqdu| wuldqjxodwlrqv +lqfoxglqj wkh
holplqdwlrq ri d!qh dqg v|pphwu| fdofxodwlrqv dqg uhgxfhg sdudphwhu vwrudjh,/ dqg e| suxqlqj wkh
elqdu| wuhh ehiruh frqvwuxfwlrq ri wkh LIV1 Rxu zdyhohwv duh wuldqjxodu Kddu zdyhohwv dqg vhfrqg
jhqhudwlrq lqwhusrodwlrq zdyhohwv dv vxjjhvwhg e| Vzhoghqv* uhfhqw zrun1
4
Lqwurgxfwlrq
Zh suhvhqw d dssurdfk wr wkh iudfwdo lpdjh frpsuhvvlrq lq wkh wudglwlrq ri Eduqvoh|45 dqg Mdftxlq143
Rxu sduwlwlrqlqj lv wuldqjxodu udwkhu wkdq uhfwdqjxodu/ exw lq frqwudvw wr wkh Ghodxqd| wuldqjxodwlrqv
xvhg e| Gdyrlqh hw do/5 zh xvh d elqdu| wuldqjxodwlrq zkrvh frpsxwdwlrqv duh vlpsolhg e| txdgwuhh
orfdwlrqdo frghv/ elqdu| wuhh lqgh{lqj/ dqg wkh dxwrpdwlf uhgxfwlrq ri doo frpsxwdwlrq dqg vwrudjh
wr wkh rqh glphqvlrqdo shulrglf fdvh+fi19 > : > ; ,1 Iru eorfn pdwfklqj zh xvh d phwkrg vlplodu wr wkh
uhfwdqjxodu Kddu0Zdyhohw s|udplg eorfn pdwfklqj ri ]kdqj dqg Fkhq49 zkr shuirup d idvw zdyhohw
wudqvirup rq hdfk grpdlq dqg udqjh eorfn dqg pdwfk wkh sdwwhuq ri zdyhohw frh!flhqwv lq wkh eorfnv1
Zh frqvwuxfw zdyhohwv lq wkh pdqqhu ri Grqrkr*v lqwhusrodwlrq zdyhohwv dqg Vzhoghq*v oliwlqj vfkhph48
lq zklfk wkh zdyhohw frh!flhqw lv wkh glhuhqfh ehwzhhq d qhu ohyho lqwhqvlw| dqg d suhglfwlrq edvhg
rq lqwhusrodwlrq ru vprrwklqj1
Iluvw zh uhgxfh frpsxwdwlrq dqg vwrudjh wr wkh rqh glphqvlrqdo fdvh e| vfdqqlqj wkh lpdjh dorqj d
wuldqjxodu Vlhuslqvnl vfdq sdwk zkrvh g|dglf sduwlwlrqlqjv irup wuldqjxodwlrqv ri wkh lpdjh1 Wklv rqh0
glphqvlrqdo uhsuhvhqwdwlrq lv qdwxudoo| sduwlwlrqhg lqwr udqjh eorfnv frqvlvwlqj ri frqwljxrxv lqwhqvlwlhv
hdfk ri zklfk vfdqv d wuldqjoh lq wkh wzr glphqvlrqdo jhrphwu| ri wkh lpdjh1 Rxu frpsxwhu dojrulwkp
wkhq shuirupv wkh rqh glphqvlrqdo idvw zdyhohw wudqvirup rq wkh lqglylgxdo udqjh dqg grpdlq eorfnv
fkhfnlqj iru pdwfklqj ri zdyhohw frh!flhqwv/ ehjlqqlqj dw orz uhvroxwlrq1 Wkhq lw vdyhv sdudphwhuv iru
wkh frqvwuxfwlrq ri dq lwhudwlyh ixqfwlrq v|vwhp +LIV, zkrvh {hg srlqw zloo dssur{lpdwh wkh suxqhg
lpdjh1 Lq wkh pdqqhu ri ]kdqj dqg Fkhq zh rewdlq d skhqrphqdo vshhg0xs e| uhmhfwlqj dq| pdwfk
zklfk idlov dw d orz0uhvroxwlrq ohyho1 Rxu pdwfklqj ri wuldqjxodu sduwlwlrq eorfnv xvlqj wkh wuldqjxodwlrq
s|udplg dojrulwkp kdv wkh dgglwlrqdo dgydqwdjh ri dxwrpdwlf frpsdulvrq ri urwdwhg/ uhyhuvhg/ dqg
txlqfxq{ pdwul{ wudqvodwhg/ wuldqjxodu eorfnv zlwk qr jhrphwulf frpsxwdwlrqv1 Lqvwhdg ri vwrulqj
wkh sdudphwhuv ri wzr0glphqvlrqdo wudqvodwlrq/ vfdolqj/ lqwhqvlw| rvhwv/ dqg pdwul{ urwdwlrqv/ wkh
wuldqjxodwlrq wuhh vwuxfwxuh doorzv xv wr vwruh rqo| g|dglf olqhdu rvhwv/ elqdu| wuhh ohyho/ dqg frduvhvw
vfdolqj frh!flhqw wkh urwdwlrqv duh dxwrpdwlf dqg uhtxluh qr frpsxwdwlrq ru vwrudjh1 Wkh vwrudjh ri
wkhvh pdsslqj sdudphwhuv uhtxluhv rqo| 53 elwv shu udqjh eorfn1 Dowkrxjk rxu s|udplg dojrulwkpv
duh frpsxwdwlrqdoo| rqh glphqvlrqdo/ wkh zdyhohwv duh qrq0vhsdudeoh iurp wkh wzr0glphqvlrqdo srlqw
ri ylhz1
Ixuwkhu lpsuryhphqwv duh pdgh e| suxqlqj wkh wkh elqdu| wuldqjxodwlrq wuhh ehiruh frqvwuxfwlqj
wkh LIV pdsslqj1 Zh xvh wzr phwkrgv ri suxqlqj= Iru wuldqjxodu Kddu zdyhohwv zh frpelqh dgmdfhqw
wuldqjohv zkrvh lqwhqvlwlhv glhu e| ohvv wkdq d wkuhvkrog ydoxh1 Iru kljkhu rughu lqwhusrodwlrq zdyhohwv
zh frpelqh dgmdfhqw wuldqjohv zkrvh frpprq yhuwh{ ydoxh lv zlwklq d wkuhvkrog ydoxh ri lwv suhglfwlrq1
D frpsdwlelolw| frqvwudlqw rq wkh wuldqjohv/ vlplodu wr wkdw xvhg lq qlwh0hohphqw dqdo|vlv/ lqvxuhv wkdw
wuhh ohyhov ri dgmdfhqw wuldqjohv glhu e| qr pruh wkdq rqh1 Wkh suxqhg elqdu| wuhh lv wkhq vwruhg
dv d olqhduo| rughuhg phpru| vhtxhqfh zkrvh vwuxfwxuh lv frpsohwho| uhsuhvhqwhg e| dq lqwhjhu ydoxh
fdoohg d ohdi ohyho glhuhqfh nh|1 Wklv suxqhg wuhh surgxfhv d suholplqdu| frpsuhvvhg rqh0glphqvlrqdo
uhsuhvhqwdwlrq ri wkh lpdjh1 D zdyhohw0edvhg iudfwdo lpdjh frpsuhvvlrq dojrulwkp vlplodu wr wkh rqh
ghvfulehg deryh lv wkhq dssolhg wr wkh suxqhg lpdjh1
5
Vlhuslqvnl Vfdq dqg Wuldqjxodu Eorfnv
Wkh uvw vwhs lq wkh iudfwdo lpdjh frpsuhvvlrq surjudp lv wr vfdq wkh lpdjh dorqj d wuldqjxodu
+Vlhuslqvnl, sdwk wr uhgxfh wkh LIV frglqj wr wkh rqh glphqvlrqdo fdvh1 Wkh wuldqjxodu vfdqqlqj
dojrulwkp lv edvhg rq lghqwli|lqj wkh sl{hov zlwk wkh ohdyhv ri d ixoo elqdu| wuhh ghulyhg iurp d elqdu|
wuldqjxodwlrq ri wkh xqlw vtxduh L 5 @ ^3> 4`5 +fi19 ,1 Iru vlpsolflw| zh uhvwulfw rxu glvfxvvlrq wr d vtxduh
lpdjh ri vl}h q q zkhuh q @ 5p 1
D elqdu| wuldqjxodwlrq ri wkh xqlw vtxduh lv frqvwuxfwhg e| wkh iroorzlqj julg uhqhphqw surfhvv=
Wkh xqlw vtxduh lv wdnhq wr eh wkh urrw ri d elqdu| wuhh1 Elvhfwlrq ri wkh vtxduh dorqj wkh gldjrqdo
zklfk frqwdlqv wkh ruljlq |lhogv wzr fkloguhq fdoohg wkh ohyho 3 wuldqjohv1 Wkh fkloguhq ri hdfk wuldqjoh
lq wkh elqdu| wuhh duh rewdlqhg e| elvhfwlrq dw wkh orqjhvw hgjh1 Li wkh ohyho ri wkh sduhqw lv o> wkhq
wkh fkloguhq duh dw ohyho o . 41 Dw ohyho o @ 5p 4> irxu wuldqjohv zlwk d frpprq yhuwh{ +fdoohg d for
srlqw ri ghswk p 4, irup d vtxduh ri ohqjwk ri 4@5p zklfk lv d eorfn ri ohyho p 4 lq d txdgwuhh
sduwlwlrqlqj ri wkh xqlw vtxduh11 Wkh irxu wuldqjohv lq d eorfn ri ohyho p 4 duh lghqwlhg zlwk txdgwuhh
eorfnv ri ohyho p dqg zlwk wkh sl{ho fhqwhuv ri wkh q q lpdjh +sl{ho fhqwhuv duh for srlqwv ri ghswk
p,1 Dq rughulqj ri wkh sl{hov wr irup dq lpdjhyhfwru > lv rewdlqhg e| wkh qdwxudo rughulqj wkh ohdyhv
ri wkh elqdu| wuldqjxodwlrq wuhh vxfk wkdw vleolqjv duh rughuhg dffruglqj wr d frxqwhuforfnzlvh urwdwlrq
derxw wkh frpprq for srlqw1 Iljxuh 4 vkrzv dq hqodujhg sruwlrq ri d vl}h +7 7, lpdjh glvsod|lqj wkh
wuldqjxodwlrq dqg wkh gluhfwhg sdwk ri lpdjh yhfwru >1 Lw dovr vkrzv wkh f|folf sdwk dw ghswk 7 dqg
wkh elqdu| wudqjxodwlrq dw ohyho :1
Iljxuh 4=
Dq dojrulwkp iru wkh wuldqjxodu vfdq sdwk lv rewdlqhg e| lghqwli|lqj wkh sl{hov zlwk d nlqg ri
txdgwuhh orfdwlrqdo frgh fdoohg d for frgh/ d olvw ri 70du| gljlwv1 D uhgxqgdqw sdwk wkurxjk wkh for
frghv lv rewdlqhg e| dq lqfuhphqwdwlrq dffruglqj wr d wxuq dj zklfk ghwhuplqhv zkhwkhu wkh sdwk
wxuqv wr wkh ohiw ru uljkw1 Uhgxqgdqf| lv holplqdwhg e| d vlpsoh orjlfdo ghflvlrq frqfhuqlqj d vhtxhqfh ri
wzr frqvhfxwlyh uljkw wxuqv1 Vhh9 iru ghwdlov1 Wkh qrq0ryhuodsslqj gluhfwhg sdwk wkurxjk wkh wuldqjxodu
fhoov dsshdu dv dssur{lpdwlrqv wr wkh vsdfh oolqj fxuyh nqrzq dv wkh Vlhuslqvnl Fxuyh 146 Zh qrwh
wkdw wkh sduhqw ri wzr vleolqj sl{hov uhsuhvhqwv d wuldqjoh ri ohyho 5p 5= E| uhfxuvlrq zh qg wkdw d
wuldqjoh ri dq| ohyho o frqvlvwv ri d olvw ri dgdfhqw sl{hov lq wkh lpdjhyhfwru olvw zlwk g|dglf vwduwlqj
srlqw dqg g|dglf ohqjwk1
Elqdu| wuldqjxodwlrqv duh srsxoduo| uhihuuhg wr/ lq frpsxwhu judsklfv olwhudwxuh/ dv wuldqjxodwlrqv
ri txdgwuhh eorfnv147 Wkh| kdyh dovr dsshduhg dv wkh qdwxudo vhwwlqj iru d vshfldo fodvv ri er{ vsolqhv/
lqfoxglqj wkh txdgudwlf ]zduw vsolqh/4: jhqhudwhg e| irxu0gluhfwlrqdo yhfwruv dqg vxssruwhg e| d wul0
dqjxodwhg rfwdkhgurq18 Wkh elqdu| wuhh dqg txdgwuhh vwuxfwxuh pdnhv srvvleoh wkh irupdwlrq ri dojr0
ulwkpv xvlqj rqo| lqwhjhuv/ orjlfdo frpsxwdwlrqv/ dqg orrnxs wdeohv/ iru vxfk wdvnv dv julg uhqhphqw/
suxqlqj/ dqg qhljkeru qglqj 1 Elqdu| wuldqjxodwlrq julgv kdyh ehhq xvhg iru wkh h!flhqw frpsxwdwlrqv
ri wkh vroxwlrqv ri SGHv e| IHP1 Pdxedfk44 kdv vkrzq wkdw wuldqjxodwlrqv pd| eh jhqhudol}hg wr
qglphqvlrqv1
6
Wuhh Suxqlqj iru Suholplqdu| Frpsuhvvlrq
D suholplqdu| frpsuhvvlrq ri wkh lpdjh fdq eh dfklhyhg zlwk suxqlqj1 Zkhq wzr vleolqj wuldqjx0
odwlrq fhoov frqvlvw ri sl{hov ri rqh lqwhqvlw| ru li wkhuh lv d olqhdu yduldwlrq ri lqwhqvlw| ryhu wkh sduhqw
wuldqjoh/ wkhq wkh wuldqjxodwlrq wuhh pd| eh suxqhg e| holplqdwlqj wkh vleolqj ohdyhv ri wkh wuhh/ ohdylqj
wkh sduhqw dv d ohdi dw wkh qh{w ohyho1 Suxqlqj uhvxowv lq d qrq0xqlirup wuldqjxodwlrq ri wkh xqlw vtxduh1
Li hdfk wuldqjoh ru hdfk yhuwh{ uhsuhvhqwv d sl{ho gdwd srlqw/ wkhq d suxqhg wuhh uhsuhvhqwv d vpdoohu
vhw ri gdwd/ l1h1 d frpsuhvvhg lpdjh1 Li hdfk wuldqjoh lv iruphg e| d vhw ri sl{hov ri wkh vdph lqwhqvlw|/
sduwldo frpsuhvvlrq ri wkh lpdjh rffxuv zlwkrxw dq| orvv1 Wkh lpdjh fdq wkxv eh uhqghuhg dv d vkdghg
wuldqjxodwlrq1 Qrq0}hur wkuhvkroglqj jlyhv orvv| frpsuhvvlrq1
Zh xvh wzr phwkrgv ri suxqlqj= Iru wuldqjxodu Kddu zdyhohwv zh frpelqh dgmdfhqw wuldqjohv zkrvh
lqwhqvlwlhv glhu e| ohvv wkdq d wkuhvkrog ydoxh1 Iru kljkhu rughu lqwhusrodwlrq zdyhohwv/ zh frpelqh
dgmdfhqw wuldqjohv zkrvh frpprq yhuwh{ ydoxh lv zlwklq d wkuhvkrog ydoxh ri lwv suhglfwlrq1 D frp0
sdwdelolw| frqvwudlqw rq wkh wuldqjohv/ vlplodu wr wkdw xvhg lq qlwh0hohphqw dqdo|vlv/ lv xvhg wr hqvxuh
wkdw wuhh ohyhov ri dgmdfhqw wuldqjohv glhu e| qr pruh wkdq rqh1 +D wuldqjoh vkduhv dq hgjh zlwk dw
prvw rqh rwkhu wuldqjoh,1 Wr suhvhuyh hgjh frpsdwlelolw|/ wzr qhljkerulqj wuldqjohv fdq eh phujhg li
dqg rqo| li wkh rwkhu sdlu ri wuldqjohv wkdw vkduh wkh vdph yhuwh{ duh dovr fdsdeoh ri ehlqj phujhg1
Sp
Wuhh suxqlqj suhvhuyhv rughulqj ri wkh ohdyhv ri wkh wuldqjxodwlrq wuhh= Wkh hohphqwv ri d elqdu|
wuhh pd| eh uhsuhvhqwhg dv vljq srlqwv lq wkh xqlw lqwhuydo L> l1h srlqwv ri wkh irup
l
5 l l >
zkhuh l
5 i4> 4j= Wkh urrw lv uhsuhvhqwhg dv 45 1 Wkh fkloguhq ri wkh urrw duh 45 . 47 dqg 45 47 1 Wkh fkloguhq ri
Sp 5ll duh p4.4 . Sp 5ll dqg p4.4 Sp 5ll1 Wkh rughulqj ri wkh uhsuhvhqwdwlyh vljq srlqwv/ lq wkh
5
5
l
l
l
xqlw lqwhuydo/ lqgxfhv d qdwxudo rughulqj ri wkh ohdyhv/ zklfk lv uhwdlqhg gxulqj suxqlqj ru uhqhphqw1
Vlqfh wkh ohdyhv glhu lq ohyho e| dw prvw rqh/ wkh ixoo wuhh vwuxfwxuh pd| eh uhsuhvhqwhg dv iroorzv=
Uhfrug wkh ohyho ri wkh uvw ohdi dqg wkhq uhfrug lq rughu wkh ohdi ohyho glhuhqfhv1 Vlqfh wkh
glhuhqfhv fdq rqo| wdnh wkh ydoxhv 4> 3> 4 wkh olvw ri glhuhqfhv pd| eh ylhzhg dv d olvw ri 6du|
gljlwv ru htxdoo| dv d vlqjoh lqwhjhu zkrvh 6du| gljlwv duh wkh +prg6, ohdi ohyho glhuhqfhv1
Wkh suxqhg elqdu| wuhh lv wkhq vwruhg dv d olqhduo| rughuhg phpru| vhtxhqfh zkrvh vwuxfwxuh
lv frpsohwho| uhsuhvhqwhg e| dq lqwhjhu ydoxh fdoohg d ohdi ohyho glhuhqfh nh| +oogn,1 Suxqlqj dqg
uhqhphqw dojrulwkpv fdq eh irupxodwhg dv pdqlsxodwlrqv ri wkh oogn1 Ixuwkhu frpsuhvvlrq pd| eh
dfklhyhg e| vwdqgdug phwkrgv vxfk dv Uxq Ohqjwk Hqfrglqj+UOH,1 Ghfrglqj xs wr wkh ohyho ri wkh
suxqhg wuhh lv vx!flhqw wr uhvwruh wkh lpdjh1 Wuhh suxqlqj iru elqdu| wuldqjxodwlrq wuhhv zdv lqwurgxfhg
lq:
Dojrulwkp =
Wuhh Suxqlqj
Sl{hol}dwlrq=0
Lqlwldol}dwlrq=0
Suhpdun=0
Frpsdwlelolw|=0
Suxqlqj=0
Sdvv lpdjh wkurxjk wuldqjxodwlrq vfkhph wr rxwsxw yhfwru >1
Vhw Wkuhvkrog ydoxh %1 Vhw Ohyho @ 5p . 4 1 Fuhdwh ohdi ohyho glhuhqfh
olvw ri wkh irup i5p 4> 3> 3> ===> 3j1
Iru l @ Ohyho wr 4 Vwhs 4 > s @ 4 wr Ohyho> n =@ 4>
Iru m @ 4 wr 5l.4 Vwhs 5s >
li +m >^m ` >^m .5Ohyhol ` m? % , wkhq S uhpdun^n` =@ m >
hovh S uhpdun^n` @ 3>
n . .>
Fkhfn pdwfklqj lqglfhv iru hgjh frpsdwlelolw| dqg vhw xqsuxqdeoh lqglfhv
wr 3 lq Suhpdun1 Frpsdfw Suhpdun e| uhprylqj }hurv1
l =@ Ohyho >
Gr =
ohqjwk =@Suhpdun l =ohqjwk>
l @ 4 wr ohqjwk =
Vhw wkh vleolqj ri ^suhpdun^n`` =@ 3>
Xsgdwh ohdi ohyho glhuhqfh olvw> l
>
Z kloh +ohqjwk @ 3,>
Iru
Frpsdfw
7
>
9
> e| uhprylqj }hurv1
LIV Frpsuhvvlrq Xvlqj Elqdu| Wuldqjxodwlrq dqg
Pxowluhvroxwlrq Dqdo|vlv1
Wkh rqh0glphqvlrqdo lpdjhyhfwru lv sduwlwlrqhg lqwr qrq0ryhuodsslqj udqjh eorfnv ri frqwljxrxv
lqwhqvlwlhv1 Ryhuodsslqj grpdlq eorfnv ri odujhu vl}h irup d grpdlq srro1 Iru hdfk udqjh eorfn wkh
iudfwdo lpdjh frpsuhvvlrq dojrulwkp vhdufkhv iru d pdwfklqj grpdlq eorfn iru wkh frqvwuxfwlrq ri
dq lwhudwlyh ixqfwlrq v|vwhp zkrvh {hg srlqw dssur{lpdwhv wkh ruljlqdo lpdjh1 Zh xvh d phwkrg
vlplodu wr wkh uhfwdqjxodu Kddu0Zdyhohw s|udplg eorfn pdwfklqj ri ]kdqj Gdq dqg Fkhq Jdqj/ l1h1/
e| shuiruplqj d idvw zdyhohw wudqvirup rq hdfk grpdlq dqg udqjh eorfn dqg pdwfklqj wkh sdwwhuq ri
zdyhohw frh!flhqwv lq wkh eorfnv1 Frpsxwdwlrq wlph lv vdyhg e| uhmhfwlqj dq| pdwfk zklfk idlov dw d
orz0uhvroxwlrq ohyho dqg fkrrvlqj wkh uvw pdwfk zklfk phhwv d ohdvw0vtxduhv wkuhvkrog fulwhulrq1 Vlqfh
zh xvh wuldqjxodu sduwlwlrq eorfnv frqvlvwlqj ri rughuhg olvwv ri sl{hov/ wkh idvw zdyhohw wudqvirup lv rqh0
glphqvlrqdo1 Wkh elqdu| wuldqjxodwlrq kdv wkh dgglwlrqdo dgydqwdjh ri wkh dxwrpdwlf frpsdulvrq ri 7
urwdwhg/ uhyhuvhg/ dqg txlqfxq{ pdwul{ wudqvodwhg/ wuldqjxodu eorfnv zlwk qr jhrphwulf frpsxwdwlrqv1
Lqvwhdg ri vwrulqj wkh sdudphwhuv ri wzr0glphqvlrqdo wudqvodwlrq/ vfdolqj/ lqwhqvlw| rvhwv/ dqg pdwul{
urwdwlrqv/ wkh wuldqjodwlrq wuhh vwuxfwxuh doorzv xv wr vwruh rqo| g|dglf olqhdu rvhwv/ elqdu| wuhh ohyho/
dqg frduvhvw vfdolqj frh!flhqw1 Wkh urwdwlrqv duh dxwrpdwlf dqg uhtxluh qr frpsxwdwlrq ru vwrudjh1
Dowkrxjk rxu s|udplg dojrulwkpv duh frpsxwdwlrqdoo| rqh glphqvlrqdo/ wkh zdyhohwv duh qrq0vhsdudeoh
iurp wkh wzr0glphqvlrqdo srlqw ri ylhz1 Wkh vxffhvv ri wklv LIV frpsuhvvlrq whfkqltxh ghshqgv rq
qglqj hhfwlyh pdwfklqj fulwhuld/ l1h1 sdlulqj wkh grpdlq0eorfn dqg udqjh0eorfn vxfk wkdw wkh O5
glvwdqfh ehwzhhq wkhp lv plqlpl}hg1
714
LIV Hqfrglqj Vfkhph
Wkh rqh0glphqvlrqdo lpdjhyhfwru
ohqjwk E/ vxfk wkdw
Ul +m , @ >+lE . m ,>
> lv glylghg lqwr QU
l @ 3> 4> 5> ===> QU
iUl jQl@3U 4
duh fdoohg wkh udqjh0eorfnv ri
udqjh0eorfnv duh wlolqj wkh yhfwru >1 Qh{w
ri ohqjwk G vxfk wkdw
Gl +m , @ >+lGk . m ,>
Q qrq0ryhuodsslqj vxe0yhfwruv
=@ E
4>
m @ 3> 4> 5> ===> E
4
Ul / ri
+4,
>/ zlwk l ghqrwlqj wkh udqjh0eorfn lqgh{1 Qrwh wkdw wkh
> lv glylghg lqwr QG =@ + QGkG . 4, ryhuodsslqj vxe0yhfwruv
l @ 3> 4> 5> ===> QG
4>
m @ 3> 4> 5> ===> G
4
+5,
iGl jQl@3G 4
duh fdoohg wkh grpdlq0eorfnv ri >/ dqg wkhlu hqvhpeoh lv fdoohg wkh grpdlq0srro G 1 Wkh
sdudphwhu Gk lv wkh vkliw ehwzhhq wzr dgmdfhqw grpdlq0eorfnv1 Iru hyhu| udqjh0eorfn Ul> zh vhdufk
wkh grpdlq0srro wr qg d grpdlq0eorfn Gpl 5 G dqg d wudqvirupdwlrq $l vxfk wkdw $l +Gpl , lv wkh
ehvw pdwfk iru Ul / l=h=>
gupv +Ul> $l +Gpl ,,
+6,
lv plqlpxp1 Wkh lghd lv wr qg wkh sduw ri wkh lpdjh zklfk uhvhpeohv wkh lpdjh Ul 1 Wklv grpdlq0eorfn
lv vdlg wr fryhu wkh udqjh0eorfn1 Wkh wudqvirupdwlrq $l lv frpsrvhg iurp d vsdwldo frqwudfwlqj pds
iroorzhg e| d pdvvlf pds1 Wkh frqwudfwlqj pds vkulqnv wkh grpdlq0eorfn wr kdoi ri lwv vl}h/ uhsodflqj
d sl{ho sdlu e| lwv dyhudjh1 Wkh pdvvlf pds fkdqjhv wkh frqwudvw dqg euljkwqhvv e| d vfdoh idfwru vl /
m vl m? 4> dqg dq rvhw rl= Plqlpl}lqj h{suhvvlrq +6, phdqv wzr wklqjv1 Iluvw zh qg d jrrg fkrlfh
ri Gpl 1 Qh{w zh qg wkh ehvw frqwudvw dqg euljkwqhvv vhwwlqjv vl dqg rl = Iru hdfk Gpl 5 G/ zh fdq
frpsxwh vl dqg rl xvlqj ohdvw vtxduh uhjuhvvlrq zklfk dovr jlyhv upv glhuhqfh1 D fkrlfh ri Gpl dorqj
zlwk d fruuhvsrqglqj vl dqg rl> wkxv ghwhuplqlqj d pds $l ri wkh irup
$l
Gpl
=
$
U5E $ UE
l=h=
Ul @ $l +Gpl , @ vl #+Gpl , . rl 4E
+7,
zkhuh 4E 0 lv d yhfwru ri vl}h E ri doo 4 v > pl 0 lv wkh lqgh{ ri wkh grpdlq0eorfn zklfk ehvw pdwfkhv
wkh lwk udqjh0eorfn/ dqg # lv d vsdwldo frqwudfwlrq ixqfwlrq/ ghqhg dv iroorzv=
3
#+Gpl ,+m , =@
4
5
+ Gp l +5m , . Gpl +5m . 4,,> m @ 3> 4> ===> E
4
+8,
Zh ixuwkhu dvvxph wkdw E lv dovr dq lqwhjhu srzhu ri 5/ E @ 5o / dqg khqfh G @ 5o.4 1 Iru vlpsolflw| ri
frpsxwdwlrq zh fdq dovr fkrrvh wkh vkliw Gk @ E 1 Wkh sdudphwhuv +pl > vl > rl, duh fdoohg wudqvirupdwlrq
sdudphwhuv1 Wkh pdsslqj $ lv frqwudfwlqj li v ? 41 Krzhyhu/ Mdfrev hw do1< vkrz wkdw lw lv qrw
qhfhvvdu| wr hqirufh wklv frqglwlrq iru doo wkh pds1 Wkh rvhw ydoxh lv d frqvwdqw lq wkh ruljlqdo sdshu
e| Mdftxlq43 1 Odwhu uhvhdufkhuv xvhg pruh jhqhudo irupv ri rvhw ydoxhv1 Iru lqvwdqfh/ J1H1 Rlhq/
V1Ohsvr|/ W1D1Udpvwdg7 vxjjhvw d olqhdu whup lq wkh lqglfhv +l> m ,1 Rxu phwkrg zrunv zlwk hlwkhu
irup ri rvhw = Prqur dqg Gxgeulgjh6 vhw wkh rvhw ydoxh wr d wklug ghjuhh sro|qrpldo/ xvlqj d yhu|
uhvwulfwhg vhw ri grpdlq0eorfnv1 Wkh LIV frgh ri wkh lpdjh lv wkh hqvhpeoh ri wkhvh QU wudqvirupdwlrqv/
zklfk uhodwh grpdlq0eorfn wr udqjh0eorfn/ dqg wkxv frqvlvwv ri 6QU wudqvirupdwlrq sdudphwhuv1
Wkh pdlq gudzedfn ri wkh wudglwlrqdo LIV vfkhph lv lwv kljk frpsxwdwlrqdo frvw1 Wklv kljk frvw
lv d uhvxow ri wkh h{whqvlyh wlph uhtxluhg iru vhdufklqj wkh grpdlq0eorfnv dqg wkh qxphurxv lwhudwlrqv
grqh rq d ixoo vl}h lpdjh1 Wkh iroorzlqj lv rxu hqfrglqj dojrulwkp1
Dojrulwkp =
Hqfrglqj
Sdvv lpdjh wkurxjk wuldqjxodwlrq vfkhph wr rxwsxw yhfwru >=
Q =@ ohqjwk ri +>,> E =@ udqjh0eorfn vl}h> G =@ 5E grpdlq0eorfn vl}h>
Gk =@ E krul}rqwdo vkliw> QU =@ Q
E wrwdo qxpehu ri udqjh0eorfnv>
QG @ QGG . 4 =@ wrwdo qxpehu ri grpdlq0eorfnv1
k
Sl{hol}dwlrq=0
Lqlwldol}dwlrq=0
> 4,E . m ,>
> 4,Gk . m ,>
I ru l @ 4 wr QU = I ru m @ 4 wr E = Ul +m , =@ ++l
I ru l @ 4 wr QG = I ru m @ 4 wr G = Gl +m , =@ ++l
Vhw wkuhvkrog ydoxh %>
H{wudfwlqj=0
Pdwfklqj=0
Ilqg ehvw srvvleoh grpdlq0eorfn iru udqjh0eorfn e| xvlqj PUD
Frpsxwh vl dqg rl e| xvlqj ohdvw vtxduh phwkrg=
Vwruh wudqvirupdwlrq sdudphwhuv +pl > vl > rl ,=
Frpsxwdwlrq=0
Vwrulqj=0
715
Pdwfklqj Fulwhuld iru PUD
Wkh lpdjhyhfwru > lv glylghg lqwr udqjh0eorfnv U ri vl}h E dqg grpdlq0eorfnv G ri vl}h G @
Diwhu wkh dssolfdwlrq ri wkh s|udplg dojrulwkp wkh grpdlq dqg udqjh eorfnv wudqvirup lqwr
5E 1
iOq +U,> Kq +U,> Kq4 +U,> Kq5 +U, > ===> K3 +U,j> iOq +G,> Kq +G,> Kq4 +G,> Kq5 +G, > ===> K3 +G,j
zkhuh Oq +U, dqg Oq +G, duh hdfk dvvrfldwhg zlwk vlqjoh ydoxhv zklfk duh wkh ryhudoo dyhudjhv ri
wkh udqjh0eorfn dqg wkh grpdlq0eorfn uhvshfwlyho|1Qrwh wkdw Kl+U, dqg Kl +G, uhsuhvhqw wkh vhwv ri
zdyhohw frh!flhqwv iru wkh udqjh dqg grpdlq eorfnv/ uhvshfwlyho|1 Ohw H +>, uhsuhvhqw wkh hqhuj| ri
wkh lpdjhyhfwru
> / l1h1
4
H+ , @ Q
>
H +U
S>
Q 4
l@3
^l`
5 dqg ghqh
G, @ H +Oq +U, Oq +G,, .
[3
l@q
H +Kl+U,
Kl +G,,=
Lq dq lwhudwlyh idvklrq zh surfhvv H +Kl +U, Kl +G,, 1 Zkloh lw lv ohvv wkdq d fhuwdlq wkuhvkrog
ydoxh zh dvvxph d srvvleoh pdwfk dqg pryh rq wr wkh qh{w ohyho xqwlo zh uhdfk wkh wrsprvw ohyho/ dw
zklfk srlqw zh frqvlghu wkh sdlu pdwfkhg1 Rwkhuzlvh U dqg G duh qrw pdwfkhg dqg zh pryh rq wr wkh
qh{w srvvleoh sdlulqj1 Wklv pd| surgxfh vhyhudo srvvleoh pdwfkhv ri zklfk zh vhohfw wkh ehvw pdwfk dv
wkh rqh zkhuh wkh hqhuj| ehwzhhq wkh sdlu lv plqlpxp1
Dojrulwkp =
Pdwfklqj
Lqlwldol}dwlrq=0
Ghfrpsrvlwlrq=0
Pdwfklqj=0
U
dqg
G
Vsolw > lqwr udqjh0eorfnv dqg grpdlq0eorfnv1
Vhw Wkuhvkrog ydoxh %1
Ghfrpsrvh wkh eorfnv lqwr or0uhv dqg kl0uhv lpdjhv1
Iru l @ 4 wr QU i
Iru m @ 4 wr QG i
pdwfk =@|hv>
Iru n @ q wr 3 Vwhs 4i
li +H +Kn +Ul Gm ,, A %, wkhq
pdwfk =@qr>
euhdn> jj
ehvwpdwfk=@ plqi H +K +Ul Gpm ,j zkhuh pm 5 ipdwfk @|hv j> j
pm
8
Uhfrqvwuxfwlqj Wkh Lpdjh
Jlyhq wkh iudfwdo frgh ri dq lpdjh/ wkh uhfrqvwuxfwlrq ri wklv lpdjh lv grqh e| lwhudwlqj wkh wudqv0
irupdwlrqv rq dq duelwudu| lqlwldo lpdjh1 Dv wkh ghfrglqj surfhhgv/ wkh lpdjh frqyhujhv wr wkh ruljlqdo
lpdjh1 Wkh frqyhujhqfh wr d ruljlqdo lpdjh/ glvuhjduglqj wkh lqlwldo lpdjh >3 / fdq eh vwdwhg e|
gupv +>> Z q +>3 ,, 4
4
v gupv +>> Z +>,, . v
qg
upv+>> >3 ,
+9,
Wklv uhvxow fdq eh suryhq e| xvh ri wkh wuldqjxodu lqhtxdolw|14 Wkh frqyhujhqfh ri frxuvh uhtxluhv
wkdw wkh frqwudfwlylw| v lv ohvv wkdq rqh1 Krzhyhu/ lw lv qrw uhtxluhg wkdw hyhu| eorfn wudqvirupdwlrq lv
frqwudfwlyh exw wkh vhw ri wudqvirupdwlrqv pxvw eh hyhqwxdoo| frqwudfwlyh4 wr hqvxuh wkh frqyhujhqfh
ri wkh lpdjh1 Wkdw lv/ iru hdfk Ul wkh grpdlq0eorfn Gl wkdw pdsv wr lw/ vkuxqnhq e| wzr/ duh wkhq
pxowlsolhg e| vl > dqg dgghg wr rl> dqg sodfhg lq Ul= Wklv frqvwlwxwhv d vlqjoh ghfrglqj lwhudwlrq1 Wkh
ghfrglqj vwhs lv lwhudwhg xqwlo wkh {hg srlqw lv dssur{lpdwhg> wkdw lv xqwlo ixuwkhu lwhudwlrqv vkrz qr
fkdqjh wr wkh lpdjh ru xqwlo wkh fkdqjh lv ehorz vrph vpdoo wkuhvkrog ydoxh1 Wkh ghfrglqj dojrulwkp=
Dojrulwkp =
Ghfrglqj
Vwduw= 0
Grpdlq Srro= 0
Frpsxwh= 0
Irupdwlrq=0
Fkhfn= 0
Uhvxow= 0
>l3 = Vhw lwhudwlrq l @ 3>fkrrvh A 3
>l3 lqwr QG grpdlq0eorfnv ri ohqjwk G> hdfk xvlqj d vkliw Gk=
Vwduw iurp dq| yhfwru ri ohqjwk Q/
Glylgh
Xvlqj wkh LIV dqg grpdlq0srro
Irup d qhz yhfwru ri ohqjwk Q/
gupv +>l3.4 > >l3 , wkhq
l
> .4 lv wkh ghfrghg yhfwru1
Li
3
9
914
G/ frpsxwh wkh udqjh0eorfnv/ dffruglqj wr +6,
>l3.4 / e| frqfdwhqdwlqj wkh udqjh0eorfnv1
vwrs/ hovh l
# l . 4
Uhvxowv
Shuirupdqfh fulwhuld
Zh ghqh frpsuhvvlrq udwlr dv
wkh qxpehu ri elwv lq wkh ruljlqdo lpdjh
wkh qxpehu ri elwv lq wkh frpsuhvvhg lpdjh
+:,
Rqh zd| wr frpsduh wzr lpdjhv lv wr ghwhuplqh wkh qrupdol}hg phdq0vtxduh huuru1
S
l@ Q
> > ^l`,5
g5qupv +>> > , @ l@4 l@Q
3
3
+ ^l`
S>
l@4
5
^l`
+;,
Wkh Vljqdo0wr0Qrlvh0Udwlr +VQU, lq ghflehov+gE, fruuhvsrqglqj wr wkh deryh huuru lv=
VQ UgE @ 43
orj43 g5
4
qupv +>> > ,
3
gE=
+<,
Zh frqqh rxu phdvxuhphqwv wr ; elwv shu sl{ho +ess, juh|vfdoh lpdjhv/ vr wkh shdn vljqdo0wr0qrlvh0
udwlr +SVQU, lv frpsxwhg dv
S VQU @ 53orj43
915
588
+43,
g5qupv +>> > ,
3
Frpsxwdwlrq ri Elw Udwhv
Vlqfh zh duh xvlqj wuldqjxodu sduwlwlrqv/ qr vwrudjh lv qhfhvvdu|1 Vlqfh udqjh0eorfnv duh uhihuuhg wr
e| dq lqgh{/ qr udqjh0eorfn lqirupdwlrq qhhg eh vwruhg1 Grpdlq0eorfnv duh dovr lqgh{hg dqg uhihuuhg
e| wklv lqgh{ vr rqh e|wh lv xvhg wr vwruh wkh grpdlq0eorfn lqgh{/ yh elwv duh xvhg wr vwruh wkh vfdolqj/
dqg vhyhq elwv iru rvhw ydoxhv1 Wkhuhiruh/ +;.8.:@53, wzhqw| elwv shu udqjh eorfn duh uhtxluhg iru
vwrudjh ri wkh wudqvirupdwlrq sdudphwhuv1
916
H{dpsoh zlwkrxw Suxqlqj
Ohw U3 / U4 /111/ U588 eh wkh 589 sl{ho qrq0ryhuodsslqj udqjh0eorfnv dqg G eh wkh froohfwlrq ri 845
sl{ho +ryhuodsslqj,grpdlq0eorfnv1 Wkh froohfwlrq G frqwdlqv 588 eorfnv1 D fkrlfh ri Gl dorqj zlwk wkh
fruuhvsrqglqj vl dqg rl > ghwhuplqhg d pds $l lq htxdwlrq +6,1 Rqfh zh kdyh wkh froohfwlrq $3 > $4 /111/
$588 zh fdq ghfrgh wkh lpdjh e| hvwlpdwlqj {Z = Iljxuh 5 vkrzv wkh lqlwldo lpdjh dqg wkh ghfrghg
lpdjh1 Wkh uhvxow lv vxusulvlqjo| yhu| jrrg1 Wkh ruljlqdo lpdjh uhtxluhg 98> 869 e|whv ri vwrudjh/
zkhuhdv/ wkh wudqvirupdwlrqv uhtxluhg 973 e|whv/ jlylqj d frpsuhvvlrq udwlr ri 435=7 = 41
Ydoxhv ri wkh SVQU ehwzhhq wkh ruljlqdo Ohqd lpdjh dqg wkh vxffhvvlyh whupv ri wkh uhfrqvwuxfwhg
vhtxhqfh mxvw ghvfulehg duh olvwhg lq wdeoh 41
Wdeoh 4
1
lwhu1&
SVQU
917
4
5
6
7
8
9
:
;
54148<
5414<5
5414<<
541538
541538
541539
541539
541539
H{dpsoh zlwk Suxqlqj
Iljxuh 6+Ohiw, vkrzv wkh uhfrqvwuxfwhg lpdjh rewdlqhg iurp d vlplodu frpsuhvvlrq phwkrg zlwk
suxqlqj1 Wkh wkuhvkrog ydoxh iru Kddu zdyhohw suxqlqj lv 63 sl{ho lqwhqvlw| ohyhov dqg wkh udqjh eorfn
vl}h lv 971 Wkh frpsuhvvlrq udwlr lv 4;9=41 Wkh vhyhuh wuldqjxodu eorfnlqj lq wkh vprrwk duhdv lv gxh wr
wkh odujh wkuhvkrog ydoxh/ exw wkh hgjhv duh ehwwhu zlwk suxqlqj wkdq zlwkrxw1 Iljxuh 6+Uljkw, vkrzv
uhvxowv iurp wkh vdph dojrulwkp zlwk d wkuhvkrog ri 53 dqg d udqjh eorfn vl}h ri 45;1 Wkh frpsuhvvlrq
udwlr lv 653=41
918
Rwkhu whvwv
Suholplqdu| uhvxowv zlwk lqwhusrodwlrq zdyhohwv vkrz vlplodu frpsuhvvlrq uhvxowv zlwk pxfk ohvv
ylvxdo hylghqfh ri wuldqjxodu eorfnlqj1 Surwrw|sh surjudpv zulwwhq lq Mdyd/ uxqqlqj rq 4330533PK}
Shqwlxp surfhvvruv/ zlwkrxw rswlpl}dwlrq/ frpsuhvv wkh lpdjh zlwklq d ihz plqxwhv/ lqfoxglqj doo
suhsurfhvvlqj/ dqg uhfrqvwuxfw xvlqj 60: lwhudwlrqv ri wkh LIV pdsslqj zlwklq d ihz vhfrqgv1 Zh
duh suhvhqwo| h{shulphqwlqj zlwk kljkhu rughu lqwhusrodwlrq zdyhohwv wr phdvxuh wkh wudgh0r ehwzhhq
vprrwkqhvv ri odujh0duhd wuldqjoh lqwhsrodwlrq dqg frpsxwdwlrqdo vshhg1
Iljxuh 5= Ruljlqdo Ohqqd dqg wkh frpsuhvvhg lpdjh zlwk wuldqjoh eorfn vl}h 5891
Iljxuh 6= Frpsuhvvlrq zlwk suxqlqj wkuhvkrog 63 dqg eorfn vl}h 97+Ohiw, Frpsuhvvlrq zlwk suxqlqj
wkuhvkrog 53 dqg eorfn vl}h 45;1 Udwlr 669=41+Uljkw,
:
UHIHUHQFHV
^4` P1 I1 Eduqvoh|1 Iudfwdov Hyhu|zkhuh1 Dfdghplf Suhvv/ 4<<61
^5` F1 M1 Gdyrlqh/ I1 Dgdswlyh ghodxqd| wuldqjxodwlrq iru dwwudfwru lpdjh frglqj1 45 wk Lqwhuqdwlrqdo
Frqihuhqfh rq Sdwwhuq Uhfrjqlwlrq +LFSU,/ Mhuxvdohp/ Rfw1 4<<71
^6` G1P1Pxqur dqg I1Gxgeulgjh1 Iudfwdo eorfn frglqj ri lpdjhv1 Hohfwurqlfv Ohwwhuv/ 5; Qr 44/ Pd|
4<<51
^7` J1H1Rlhq/ V1Ohsrv|/ dqg W1D1Udpvwdg1 Dq lqqhu surgxfw vsdfh dssurdfk wr lpdjh frglqj e|
frqwudfwlyh wudqvirupdwlrqv1 lq Surf1 ri LFDVVS0<4/ sdjhv 5::55::9/ 4<<41
^8` G1 M1 Khehuw1 D er{ vsolqh vxeglylvlrq s|udplg dojrulwkp1 Dssolhg Pdwk Ohwwhuv/ wr dsshdu1
^9` G1 M1 Khehuw1 F|folf lqwhuodfhg txdgwuhh dojrulwkpv iru txlqfxq{ pxowluhvroxwlrq1 Mrxuqdo ri
Dojrulwkpv/ 5:/<:045;1
^:` G1 M1 Khehuw dqg K1 Nlp1 Lpdjh hqfrglqj zlwk wuldqjxodwlrq zdyhohwv1 Surf1 VSLH/ Zdyhohw
Dssolfdwlrqv lq Vljqdo dqg Lpdjh Surfhvvlqj/ 589<=6;46<5/ 4<<81
^;` G1 M1 Khehuw dqg K1 Nlp1 D idvw zdyhohw frpsdvv hgjh ghwhfwru1 Surf1 VSLH/ Zdyhohw Dssolfdwlrqv
lq Vljqdo dqg Lpdjh Surfhvvlqj/ 5;58=765775/ 4<<91
^<` H1 Z1 Mdfrev/ \1 Ilvkhu/ dqg U1G1Ervv1 Lpdjh frpsuhvvlrq= D vwxg| ri wkh lwhudwhg wudqvirup
phwkrg1 Vljqdo Surfhvvlqj/ 5<=584596/ 4<<51
^43` D1 H1 Mdftxlq1 Lpdjh frglqj edvhg rq d iudfwdo wkhru| ri lwhudwhg frqwudfwlyh lpdjh wudqvirupdwlrq1
LHHH Wudqvdfwlrq rq Lpdjh surfhvvlqj/ 4=Qr 4/ 4<<41
^44` M1 P1 Pdxedfk1 Orfdo elvhfwlrq uhqhphqw iru q0vlpsolfldo julgv jhqhudwhg e| uh hfwlrqv1 VLDP
M1 Vfl1 Frpsxw1/ 49=54355:/ 4<<81
^45` P1I1Eduqvoh|1 Iudfwdo ixqfwlrqv dqg lqwhusrodwlrq1 Frqvwu1 Dssur{1// 5=63665</ 4<;91
^46` K1 Vdjdq1 Vsdfh Iloolqj Fxuyhv1 Vsulqjhu0Yhuodj/ 4<<71
^47` K1 Vdphw1 Dssolfdwlrqv ri Vsdwldo Gdwd Vwuxfwxuhv= Frpsxwhu Judsklfv/ Lpdjh Surfhvvlqj/ dqg
JLV1 Dgglvrq0Zhvoh|/ 4<;<1
^48` Z1 Vzhoghqv dqg S1 Vkurhghu1 Exloglqj |rxu rzq zdyhohwv dw krph1 Lq Zdyhohwv lq Frpsxwhu
Judsklfv/ DFP VLJJUDSK Frxuvh Qrwhv/ sdjhv 48;:/ 4<<91
^49` G1 ]kdqj dqg J1 Fkhq1 D idvw iudfwdo lpdjh frglqj edvhg rq pxowluhvroxwlrq dqdo|vlv1 suhsulqw/
4<<:1
^4:` S1 ]zduw1 Pxowlyduldwh vsolqhv zlwk qrqghjhqhudwh sduwlwlrqv1 VLDP M1 Qxphu1 Dqdo1/ 43=9989:6/
4<:61