Bootstrapping Spurious Regression

By

Peter C.B. Phillips

September 2001

COWLES FOUNDATION DISCUSSION PAPER NO. 1330

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS

YALE UNIVERSITY

Box 208281

New Haven, Connecticut 06520-8281

Shwhu F1 E1 Skloolsv

Frzohv Irxqgdwlrq/ \doh Xqlyhuvlw|/

Xqlyhuvlw| ri Dxfnodqg ) Xqlyhuvlw| ri \run

4: Pd| 5334

Devwudfw

Wkh errwvwuds lv vkrzq wr eh lqfrqvlvwhqw lq vsxulrxv uhjuhvvlrq1 Wkh idloxuh ri wkh errwvwuds lv vshfwdfxodu lq wkdw wkh errwvwuds hhfwlyho| wxuqv d vsxulrxv uhjuhvvlrq lqwr d frlqwhjudwlqj uhjuhvvlrq1 Lq sduwlfxodu/ wkh vhuldo fruuhodwlrq frh!flhqw ri wkh uhvlgxdov lq wkh errwvwuds uhjuhvvlrq grhv qrw frqyhujh wr xqlw|/ vr wkh errwvwuds lv qrw hyhq uvw rughu frqvlvwhqw1 Wkh eorfn errwvwuds vhuldo fruuhodwlrq frh!flhqw grhv frqyhujh wr xqlw| dqg lv wkhuhiruh uvw rughu frqvlvwhqw/ exw kdv d vorzhu udwh ri frqyhujhqfh dqg d glhuhqw olplw glvwulexwlrq iurp wkdw ri wkh vdpsoh gdwd vhuldo fruuhodwlrq frh!flhqw1 Wkh dqdo|vlv fryhuv vsxulrxv uhjuhvvlrqv lqyroylqj erwk ghwhuplqlvwlf wuhqgv dqg vwrfkdvwlf wuhqgv1 Wkh uhvxowv uhlqirufh hduolhu zduqlqjv derxw urxwlqh xvh ri wkh errwvwuds zlwk ghshqghqw gdwd1

MHO Fodvvlfdwlrq= F55

DPV 4<<4 vxemhfw fodvvlfdwlrq= 95P43

Nh| zrugv dqg skudvhv= Dv|pswrwlf wkhru|/ Errwvwuds/ Eurzqldq prwlrq/ Frlqwhjudwlrq/ ON uhsuhvhqwdwlrq/ Qrqvwdwlrqdulw|/ Uhvlgxdo gldjqrvwlfv/ Xqlw urrw1

41 Lqwurgxfwlrq

Wkh errwvwuds lv qrz d srsxodu dqg sudfwlfdo wrro ri lqihuhqfh lq hfrqrphwulf zrun1 Rqh dvshfw ri lwv dsshdo lv lwv zlgh dssolfdelolw|/ hqdeolqj lwv xvh lq pdq| glhuhqw hfrqrphwulf prghov dqg doorzlqj iru erwk furvv vhfwlrq dqg wlph ghshqghqw gdwd1 Lq wlph vhulhv vlwxdwlrqv lw lv nqrzq wr eh lpsruwdqw wkdw wkh errwvwuds fdswxuh dffxudwho| wkh whpsrudo ghshqghqfh surshuwlhv ri wkh ruljlqdo wlph vhulhv li lw lv wr eh d xvhixo dlg wr lqihuhqfh1 Wzr dssurdfkhv duh qrz lq frpprq xvh wr ghdo zlwk whpsrudo ghshqghqfh= wkh vlhyh errwvwuds +Nuhlvv/ 4<<5/ dqg Exkopdqq/ 4<<:/ 4<<;/ dprqj rwkhuv,/ zkhuh d vhtxhqfh ri qlwh glphqvlrqdo sdudphwulf prghov +olnh dxwruhjuhvvlrqv, lv xvhg wr uhpryh whpsrudo ghshqghqfh> dqg eorfn errwvwuds phwkrgv +Fduovwhlq/ 4<;9/ dqg N qvfk/ 4<;<, zkhuh eorfnlqj whfkqltxhv duh xvhg wr ghdo zlwk ghshqghqfh1

W_{Vrph ri wkh uhvxowv uhsruwhg khuh zhuh phqwlrqhg e| wkh dxwkru dw wkh \run Hfrqrphwulfv Phhwlqj lq} Pd|/ 53331 D gudiw yhuvlrq ri wkh sdshu zdv zulwwhq lq Iheuxdu|/ 53341 Wkh sdshu zdv w|shg lq VZ5181 Frpsxwlqj dqg judsklfv zhuh grqh lq JDXVV1

Li whpsrudo ghshqghqfh lv srruo| fdswxuhg ru ljqruhg/ wkhq wkh errwvwuds fdq qrw eh h{shfwhg wr shuirup zhoo hyhq dv|pswrwlfdoo| dqg pd| zhoo surgxfh lqfrqvlvwhqw hvwlpdwhv1 Krurzlw} +4<<<, zduqv ri wkh gl!fxowlhv ri xvlqj wkh errwvwuds zlwk ghshqghqw gdwd/ pdn0 lqj xqlqiruphg dssolfdwlrqv ri wkh errwvwuds zlwk ghshqghqw gdwd rqh klv errwvwudsGrq*wv1 Wkh suhvhqw sdshu kljkoljkwv wkhvh sureohpv lq d frqwh{w ri vxevwdqwldo hfrqrphwulf lqwhu0 hvw1 Zh vkrz wkdw lq vsxulrxv uhjuhvvlrqv xvh ri wkh fuxgh errwvwuds +zkhuh ghshqghqfh lv ljqruhg, surgxfhv d vshfwdfxodu dv|pswrwlf idloxuh1 Wkh fuxgh errwvwuds lv vkrzq wr frq0 yhuw vsxulrxv uhjuhvvlrqv ri ixoo udqn lqwhjudwhg wlph vhulhv lqwr frlqwhjudwlqj uhjuhvvlrqv dqg vsxulrxv uhjuhvvlrqv ri lqwhjudwhg wlphv vhulhv rq ghwhuplqlvwlf wuhqgv lqwr wuhqg vwdwlrqdu| wlph vhulhv1 Wkh sdshu suryhv wkhvh uhvxowv xvlqj doprvw vxuh lqyduldqfh sulqflsohv dqg wkh Orëyh Ndukxqhq +ON, uhsuhvhqwdwlrq ri Eurzqldq prwlrq xvhg uhfhqwo| lq Skloolsv +4<<;, iru vwxg|lqj vsxulrxv uhjuhvvlrqv1 Wkh phwkrgv duh olnho| wr eh xvhixo lq rwkhu dssolfdwlrqv lqyroylqj qrqvwdwlrqdu| wlph vhulhv1

Ilj1 4= Vsxulrxv Uhjuhvvlrq ri L+4, Gdwd rq d Olqhdu Wuhqg

Iljv1 4 dqg 5 looxvwudwh wkhvh hhfwv zlwk vrph vdpsoh gdwd dqg errwvwuds vdpsoh gdwd1 Lq Ilj1 4 wkh vdpsoh gdwd i[| = w @ 4> ===qj frpsulvh q @ 833 revhuydwlrqv ri d vwdqgdug

Jdxvvldq udqgrp zdon zklfk lv vkrzq djdlqvw wkh wwhg ohdvw vtxduhv uhjuhvvlrq olqh [e| @

e_{ew= Wkh udqgrp zdqghulqj fkdudfwhulvwlfv derxw wkh uhjuhvvlrq olqh duh dssduhqw1 Iurp} wudglwlrqdo olplw wkhru| +Gxuodxi dqg Skloolsv/ 4<;;, lw lv nqrzq wkdw wkh uhjuhvvlrq frh!flhqw e_{e$R 3 exw lv vwdwlvwlfdoo| vljqlfdqw zlwk suredelolw| rqh dvq$ 4=Wkh errwvwuds vdpsoh}

i[W

| = w @ 4> ===qj lv jhqhudwhg iurp [|W @ eew.xW|/ zkhuh ixW| = w @ 4> ===> qj duh udqgrp

gudzv zlwk uhsodfhphqw iurp wkh hpslulfdo glvwulexwlrq ri fhqwuhg yhuvlrqv ri wkh uhvlgxdov

ixe|@[|eew=w@ 4> ===> qj>wkdw lv iurp d fuxgh dssolfdwlrq ri wkh errwvwuds1 Ilj1 5 vkrzv d

w|slfdo vhw ri errwvwuds vdpsoh gdwd jhqhudwhg lq wklv zd| iru wkh jlyhq gdwdi[|=w@ 4> ===qj1

Lqfoxghg lq Ilj1 5 lv wkh ohdvw vtxduhv uhjuhvvlrq olqh[eW

uhvxow lv vwulnlqjo| dssduhqw 0 wkh errwvwuds gdwd qrz dsshdu wr eh vwdwlrqdu|/ doehlw zlgho| ydu|lqj/ derxw wkh wuhqg ixqfwlrqeeW₁

Ilj1 5= Vsxulrxv Uhjuhvvlrq ri Errwvwudsshg Vdpsoh rq d Olqhdu Wuhqg

Lq d vlplodu zd|/ Ilj1 6 vkrzv vdpsoh gdwd furvvsorwv iru wzr lqghshqghqw vwdqgdug Jdxvvldq udqgrp zdonv i\|> [| = w @ 4> ===qj wrjhwkhu zlwk wkh +vsxulrxv, uhjuhvvlrq olqh

e

\| @ ee[|= Wkh udqgrp zdqghulqj ehkdylru ri wkh vdpsoh gdwd durxqg wkh olqh lv fohduo|

hylghqw1 Wkh errwvwuds vdpsoh i\W

| = w @ 4> ===qj lv jhqhudwhg iurp \|W @ee[|.xW|/ zkhuh ixW

| = w @ 4> ===> qj duh udqgrp gudzv zlwk uhsodfhphqw iurp wkh hpslulfdo glvwulexwlrq ri

fqwuhg yhuvlrqv ri wkh uhvlgxdovixe|@\|ee[|=w@ 4> ===> qj>djdlq d fuxgh dssolfdwlrq ri wkh

errwvwuds1 Ilj1 7 vkrzv errwvwuds vdpsoh gdwd jhqhudwhg lq wklv zd|/ wrjhwkhu zlwk wkh ohdvw vtxduhv uhjuhvvlrq olqh\eW

| @eeW[|iru wklv errwvwuds vdpsoh1 Wkh uhvxow lv djdlq vwulnlqj1 Wkh

errwvwuds gdwd i\W

| > [| =w @ 4> ===qj qrz dsshdu wr frlqwhjudwh derxw wkh wwhg uhjuhvvlrq

olqh/ doehlw zlwk zlgh yduldwlrq1

Wklv sdshu h{whqgv suhylrxv uhvhdufk sulpdulo| e| ghyhorslqj phwkrgv iru dqdo|}lqj wkh dv|pswrwlf surshuwlhv ri wkh errwvwuds dqg eorfn errwvwuds rq lqwhjudwhg wlph vhulhv1 Wkhuh kdv ehhq iuhtxhqw vshfxodwlrq derxw wkh surshuwlhv ri vxfk xvhv ri wkh errwvwuds +h1j1/ Ol dqg Pdggdod/ 4<<9 ) 4<<:/ dqg Klqnoh|/ 4<<:, exw qr sulru dqdo|vlv1 Xvlqj wkhvh phwkrgv wkh sdshu vkrzv wkdw gluhfw dssolfdwlrq ri wkh errwvwuds frqyhuwv lqwhjudwhg wlph vhulhv lqwr dv|pswrwlfdoo| vwdwlrqdu| vhulhv/ wkhuhe| ixqgdphqwdoo| fkdqjlqj wkh qdwxuh ri wkh vhulhv1 Wkh idloxuh lv d uvw rughu lqfrqvlvwhqf| lq wkdw wkh vhuldo fruuhodwlrq frh!flhqwv ri d errwvwudsshg lqwhjudwhg surfhvv gr qrw whqg wr xqlw|1 Wkh lpsdfw ri wkh lqfrqvlvwhqf| lv wkdw wkh errwvwuds frqyhuwv vsxulrxv uhjuhvvlrqv lqwr frlqwhjudwlqj ru wuhqg vwdwlrqdu| uhjuhvvlrqv1 Xqolnh wkh errwvwuds/ wkh eorfn errwvwuds lv vkrzq wr eh uvw rughu frqvlvwhqw/ pdlqwdlqlqj wkh ihdwxuh ri dq lqwhjudwhg surfhvv wkdw lwv vhuldo fruuhodwlrq frh!flhqwv whqg wr xqlw|1 Krzhyhu/ wkh olplw glvwulexwlrq ri wkh errwvwudsshg vhuldo fruuhodwlrq frh!flhqw glhuv iurp wkdw ri wkh

vdpsoh vhuldo fruuhodwlrq frh!flhqw/ lv qr orqjhu d xqlw urrw glvwulexwlrq dqg frqyhujhv dw d glhuhqw udwh1

Ilj1 6= Vsxulrxv Uhjuhvvlrq ri L+4, Gdwd rq Lqghshqghqw L+4, Gdwd

Ilj1 7= Errwvwudsshg Vsxulrxv Uhjuhvvlrq ri lqghshqghqw L+4, Gdwd

Wkh sdshu lv rujdqlvhg dv iroorzv1 Vhfwlrq 5 jlyhv vrph suholplqdu| wkhru| dqg wkhq ghyhorsv errwvwuds dv|pswrwlfv iru lqwhjudwhg wlph vhulhv/ lqfoxglqj gluhfw errwvwuds dqg

eorfn errwvwuds dssurdfkhv1 Vhfwlrq 6 surylghv uhvxowv iru wkh errwvwuds dqg eorfn errwvwuds lq wkh surwrw|slfdo vsxulrxv uhjuhvvlrq ri dq lqwhjudwhg surfhvv rq d ghwhuplqlvwlf wuhqg dqg Vhfwlrq 7 grhv wkh vdph iru vsxulrxv uhjuhvvlrqv dprqj ixoo udqn lqwhjudwhg wlph vhulhv1 Lpsolfdwlrqv ri wkh uhvxowv duh frqvlghuhg lq Vhfwlrq 8 dqg surriv duh jlyhq lq Vhfwlrq 91 D qrwdwlrqdo wdeoh lv surylghg dw wkh hqg ri wkh sdshu1

51 Errwvwuds Olplw Wkhru| iru Lqwhjudwhg Wlph Vhulhv

+d, Suholplqdulhv

Wkh cyhfwru wlph vhulhv{| lv d ixoo udqn xqlw urrw surfhvv jhqhudwhg e|

{|@{|3.x|> +4,

zlwk {f @R@r+4, dqg zlwkx| d olqhdu surfhvv wkdw vdwlvhv wkh iroorzlqj frqglwlrq/ zkhuh mmdmm@ pd{mdmiru pdwul{d=

Dvvxpswlrq O Iru doo w A3> x| kdv Zrog uhsuhvhqwdwlrq

x|@F+O,%|@ " [ 'f f%|3> " [ 'f mrmmfmm?4> v4> F+4,9@ 3> +5, zlwk %|@llg+3> 0, dqg zlwk H+mm%|mm^,?4> iru vrph t A7=

Wkh vxppdelolw| frqglwlrq lq +5, lv vdwlvhg e| d zlgh fodvv ri sdudphwulf dqg qrqsdud0 phwulf prghov irux| dqg/ lq frqmxqfwlrq zlwk wkh prphqw frqglwlrq/ hqdeohv wkh xvh ri wkh

dv|pswrwlf whfkqltxhv lq Skloolsv dqg Vror +4<<5, dqg dq doprvw vxuh lqyduldqfh sulqflsohv +LS, iru wkh sduwldo vxpv ri x|= Wr shuirup wkh odwwhu/ zklfk duh hvshfldoo| xvhixo khuh/ zh

h{sdqg wkh suredelolw| vsdfh dv qhhghg vr wkdw wkh sduwldo vxp surfhvv V& @S&|'x| ri x|

fdq eh uhsuhvhqwhg xs wr d qhjoljleoh huuru lq whupv ri d Eurzqldq prwlrq ghqhg rq wkh vdph vsdfh1 Dq LS ri wklv w|sh iru sduwldo vxpv ri vfdodu olqhdu surfhvvhv zdv suryhg lq Skloolsv +4<<<, dqg lv vwudljkwiruzdugo| h{whqghg wr wkh fdvh c4=

514 Ohppd Ohw V& @ S&'x iru n 4> dqg Vf @ 3> iru n @ 3> zkhuh x vdwlvhv Dvvxpswlrq O=Wkhq/ wkh suredelolw| vsdfh rq zklfk wkh x dqg V&duh ghqhg fdq eh h{sdqghg lq vxfk d zd| wkdw wkhuh lv d surfhvv glvwulexwlrqdoo| htxlydohqw wr V& dqg d yhfwru Eurzqldq prwlrq E+, zlwk yduldqfh pdwul{ @F+4, 0F+4, rq wkh qhz vsdfh iru zklfk

vxs f$&$? _sV& qE+qn, @r@r+ 4 q23R,= +6, surylghg Hmmx|mm^?4iru vrph t A5s A7=

Vlqfh {| @ V|.R@r+4,> lw iroorzv iurp +6, wkdw/ diwhu fkdqjlqj wkh suredelolw| vsdfh dv

uhtxluhg/ zh kdyh vxs f$|$? _s{| qE+ w q, @r@r+ 4 q23R,= +7,

Zh vkdoo riwhq surfhhg dv li wklv fkdqjh kdv ehhq pdgh zlwkrxw frqwlqxdoo| dgglqj wkh txdolfdwlrq dqg qrwlqj wkdw lq wkh ruljlqdo vsdfh wklv frqyhujhqfh wudqvodwhv lqwr zhdn frqyhujhqfh ri phdvxuhv1

Zh vkdoo dovr pdnh h{whqvlyh xvh ri wkh ON uhsuhvhqwdwlrq ri wkh vwrfkdvwlf surfhvvE+u,= OhwZ+u,ehcyhfwru vwdqgdug Eurzqldq prwlrq1 Ryhu wkh lqwhuydou5^3>4`>wkhcyhfwru surfhvvZ+u, kdv ON uhsuhvhqwdwlrq +htxdwlrq +7, ri Skloolsv/ 4<<;,

Z+u, @s5[" &' vlqn 2 u n 2 &> &llgQ+3> L,>

zklfk krogvd=v1 dqg xqlirupo| lqu= IruE+u, @ 2Z+u, zh kdyh wkh fruuhvsrqglqj uhsuh0

vhqwdwlrq E+u, @s5[" &' vlqn  2 u n 2 &> & llgQ+3>,= +8, Zlwk wklv dssur{lpdwlrq lq kdqg/ zh fdq ghyhors vrph suholplqdu| olplw wkhru| iru dssolfd0 wlrqv ri wkh errwvwuds wr lqwhjudwhg wlph vhulhv1

+e, Gluhfw Errwvwuds Olplwv iru Lqwhjudwhg Surfhvvhv

Zh xvh wkh iroorzlqj vfkhph dqg qrwdwlrq iru errwvwuds vdpsohv1 Vxssrvh zh duh jlyhq gdwd+h|,? ru hvwlpdwhv ri wklv gdwd +h1j1 iurp uhjuhvvlrq uhvlgxdov, ghqrwhg+eh|,? ghqhg rq

d suredelolw| vsdfh +>I> S,= Gluhfw udqgrp uhvdpsolqj iurp wkh hpslulfdo glvwulexwlrq ri

+eh|,? surgxfhv wkh fuxgh errwvwuds vdpsoh +hW|,?= Khuh/ hdfk revhuydwlrq hW| lv gudzq iurp

wkh srsxodwlrq +eh|,? dqg hdfk srlqw kdv htxdo suredelolw|/ ?> ri ehlqj gudzq1 Li zh fhqwuh

wkh ruljlqdo revhuydwlrqv dqg gudz iurp wkh srsxodwlrq+eh|_? S?r'ehr,?>wkhq wkh errwvwuds

vdpsoh lv ghqrwhg +hhW

|,?= Iroorzlqj frqyhqwlrq/ zh ohw SW ghqrwh suredelolw| frqglwlrqdo rq

wkh uhdol}hg wudmhfwru| ri wkh rujlqdo gdwd+eh|," =H{shfwdwlrq zlwk uhvshfw wrSW lv ghqrwhg

HW_{> frqyhujhqfh lq glvwulexwlrq +uhvshfwlyho|/ lq suredelolw|, zlwk uhvshfw wr S}W _{lv ghqrwhg} $_W +uhvshfwlyho|/ $_RW, dqg zkhq frqyhujhqfh rffxuv zlwk S suredelolw| xqlw| +l1h1 iru

doprvw doo uhdol}dwlrqv ri+h|," ,/ zh zulwh$_W d=v=dqg$_RW d=v=

Vxssrvh zh kdyh gdwd+{|,? jhqhudwhg dv lq +4, dqg ohw+{W|,? dqg+{hW|,? eh fuxgh Errwvwuds

vdpsohv frqvwuxfwhg dv deryh1 Zh fdq zulwh{W

| @{ zkhuh m lv d udqgrp lqgh{ gudzq iurp i4>5> ===> qj1 Wkdw lv/ m lv xqlirup ryhu wkh lqwhjhuv i4>5> ===> qj zlwk pdvv suredelolw| _? rq hdfk lqwhjhu1 Wkhq/ lq ylhz ri Ohppd 514 dqg +7,/ wkhuh h{lvwv d suredelolw| vsdfh lq zklfk zh fdq zulwh {W | s_q @ s{_q @E m q .r@r 4 q23R @E+U?, .r@r 4 q23R > +9,

zkhuhU? @ _? lv xqlirupo| glvwulexwhg ryhui_?>_?2> ==>4jiru hdfk m=Qrz/ lq d voljkw dexvh ri

qrwdwlrq/ zh fdq zulwh

U? $_W U> +:,

zkhuhU kdv dq lqghshqghqw frqwlqxrxv xqlirup glvwulexwlrq ryhu ^3>4` iru hdfkm> zulwwhq

dvllgX^3>4`=Wkh frqyhujhqfh +:, hylghqwo| dovr krogv doprvw vxuho| +S, =Lw iroorzv e| wkh frqwlqxrxv pdsslqj wkhruhp wkdw vlq n4₅ U? $_Wvlq n 4₅ U d=v=+S,= +;,

Pruhryhu/ iru m 9@ n> U? dqg U?& duh vwdwlvwlfdoo| lqghshqghqw dqg vr wkh olplw yduldwhv

U dqgU& duh dovr lqghshqghqw=Lw lv wkhuhiruh dssduhqw iurp +8,0+;, wkdw wkh vwdqgdugl}hg

errwvwuds surfhvv q3

2{W_| kdv dq dv|pswrwlf uhsuhvhqwdwlrq lq whupv ri d yhfwru Eurzqldq

prwlrq hydoxdwhg dw d udqgrp dujxphqwU 5^3>4`>yl}1/

{W | s q @ s 5[" &' vlqn 2 U? n₂ &.r@r 4 q23R _Ws5 " [ &' vlqn 2 U n₂ &= +<, Wkxv/ mxvw dv{W

| lv d udqgrp gudz iurp wkh hpslulfdo glvwulexwlrq ri+{|,?>wkh odujh vdpsoh

ehkdylru ri wkh vwdqgdugl}hg surfhvvq3

2{W_| lv dqdorjrxv wr wkdw ri d udqgrp gudz iurp wkh

wudmhfwru| ri wkh Eurzqldq prwlrq olplw riq3

2{_|=Wklv lv irupdol}hg lq wkh iroorzlqj ohppd1

515 Ohppd Ohw +{W

|,? dqg +h{W|,? eh errwvwuds vdpsohv iurp d xqlw urrw surfhvv +{|,? jhqhudwhg dv lq +4, zlwk x vdwlvi|lqj Dvvxpswlrq O1 +d, %_IWd?oo ? $_W E+Uo, d=v=+S,> +e, h%_IWd?oo ? $_WE+Uo, U_ f E+v,gv d=v=+S,>

zkhuh iUo=u5^3>4`j lv idplo| ri lqghshqghqw xqlirup yduldwhv rq ^3>4`=

Wkh errwvwudsshg surfhvv q32{W

d?oo frqyhujhv zhdno| wr d udqgrpo| vdpsohg yhuvlrq ri

wkh Eurzqldq prwlrqE wr zklfk q3

2{_d?oo frqyhujhv/ dqg q32h{W

d?oo frqyhujhv wr d udqgrpo|

vdpsohg yhuvlrq ri wkh fruuhvsrqglqj ghphdqhg Eurzqldq prwlrqE+u, @E+u,U_fE+v,gv= Xqolnh wkh Eurzqldq prwlrq surfhvv iurp zklfk wkhvh gudzv duh pdgh/E+Uo,lv qrw sdwkzlvh

frqwlqxrxv dqg lv qrw d vhsdudeoh surfhvv1

Zh surfhhg wr qg glvwulexwlrqdo olplwv ri errwvwudsshg vwdwlvwlfv lq uhjuhvvlrqv lqyroylqj L+4,yduldeohv olnh{|=Wkhvh olplwv duh xvhixo lq wkh uhjuhvvlrq wkhru| ri wkh iroorzlqj vhfwlrq1

Lq ylhz ri wkh glhuhqw olplw ehkdylru ri {W

| dqg{hW|> zh frqwlqxh wr uhsruw uhvxowv iru erwk

yhuvlrqv ri wkh errwvwuds lq zkdw iroorzv1 Ohppd 516 ehorz jlyhv vrph olplw wkhru| iru vdpsoh prphqwv ri wkh errwvwudsshg gdwd dqg ruljlqdo vdpsoh gdwd1

516 Ohppd Xqghu wkh vdph frqglwlrqv dv Ohppd 515=

+d, q3 2S?_|'{W_| $_{_}W U_fE+v,gv d=v=+S,> +e, q32S?_|'h{W_| $_RWU_fE+v,gv@ 3 d=v=+S,> +f, q32S? |'{W|{W| $_W U_fE+v,E+v,gv d=v=+S,> +g, q32S? |'{hW|h{W| $_W U_fE+v,E+v,gv d=v=+S,> +h, q32S? |'{W|{|$_W U_fE+v,gv U_fE+v,gv d=v=+S,> +i, q32S? |'{hW|{|$_WU_fE+v,gv U_fE+v,gv @ 3 d=v=+S,>

+j, q3D*2S? {W|w$_W U_fE+u,gu U_fugu d=v=+S,> +k, q3D*2S?  {hW|w$_W U_fE+v,gu U_fugu @ 3 d=v=+S,= Hylghqwo|/ wkh errwvwuds vdpsoh prphqwvq3

2S?_|'{W_| dqgq32S?_|'{W_|{W_| kdyh wkh vdph

olplwv dv wkh fruuhvsrqglqj vdpsoh dyhudjhv q32S?_|'{_| dqg q32S?_|'{_|{_|= Fruuhvsrqg0

lqjo|/ vdpsoh prphqwv ri wkh fhqwuhg errwvwuds vdpsoh +{hW

|,? kdyh olplwv wkdw fruuhvsrqg

wr wkrvh ri wkh ghphdqhg vdpsoh gdwd1 Krzhyhu/ vdpsoh fryduldqfhv ehwzhhq wkh errwvwuds gdwd dqg wkh ruljlqdo vdpsoh/ q32S?

|'{W|{| dqg q32S?|'h{W|{|> glhu iurp wkh olplw ri

q32S?

|'{|{|>dqg wkh errwvwuds fryduldqfhvq3D*2S?{W|wdqgq3D*2S?{hW|wgr qrw kdyh wkh

vdph olplwv dvq3D*2S?

{|wdqgq3D*2S? +{|{,w> zklfk duhU_fE+u,ugudqgU_fE+u,ugu=

Qh{w/ frqvlghu wkh vfdodu fdvh dqg vhwc@ 4lq +4,1 Lq wklv fdvh zh vkdoo xvh$2_{wr ghqrwh wkh}

yduldqfh ri wkh Eurzqldq prwlrqE=Ghqh wkhk*wk rughu vhuldo fruuhodwlrq ri wkh gdwde @

S_?

|'{|{|3@S?|'{2|3>dqg wkh errwvwuds vhuldo fruuhodwlrqveW @S?|'{W|{|3W @S?|'{W2|3>

hW @S?|'h{W|{hW|3@S?|'{hW2|3>dqgee @S?|'hW|hW|3@S?|'hW2|3>zkhuh h|@{W| {W= 517 Wkhruhp Vhuldo fruuhodwlrqv ri vfdodu errwvwuds vdpsohv +{W

|,? dqg +h{W|,? iurp d xqlw urrw surfhvv vdwlvi|lqj wkh frqglwlrqv ri Ohppd 515 kdyh wkh iroorzlqj olplwv iru k 9@ 3 dv q$ 4= eW $_W₄ @ kU_ f+E+u,,gu l2 U_ f E+u,2gu d=v=+S,> +43, hW>ee $RW 3 d=v=+S,= +44,

Wkh olplw yduldwh 4 lq +43, hylghqwo| vdwlvhv 3 ? 4 ? 4> d=v=+S,= Wkxv/ wkh wkhruhp

vkrzv wkdw wkh uvw rughu vhuldo fruuhodwlrq frh!flhqw ri d fuxgh errwvwuds vdpsoh ri dqL+4,

surfhvv qr orqjhu kdv d suredelolw| olplw ri xqlw|1 Lqvwhdg/ eW frqyhujhv zhdno| wr d olplw

udqgrp yduldeoh wkdw lv vwulfwo| ohvv wkdq xqlw|/ exw dovr vwulfwo| srvlwlyh1 Vlqfhe $R 4 iru

dook>wkh fuxgh errwvwudseW lv qrw hyhq uvw rughu frqvlvwhqw1 Lw iroorzv wkdw d xqlw urrw whvw

rq wkh errwvwuds vdpsoh+{W

|,? wkdw lv edvhg rqeWzloo uhmhfw wkh qxoo k|srwkhvlv doprvw vxuho|

dv q $ 4=Qrwh wkdw wklv lv wkh fdvh hyhq wkrxjk {W

d?oo lv ri rughu sq dqgq3  2{W

d?oo whqgv

wr d olplwlqj vwrfkdvwlf surfhvv wkdw lv d udqgrpl}hg yhuvlrq ri Eurzqldq prwlrq/ dv vkrzq lq Ohppd 516 +d,1 Ixuwkhu/ doo vhuldo fruuhodwlrqv eW kdyh wkh vdph olplw 4 dv q $ 4 iru

dq|k9@ 3=Wkxv/ wkh errwvwuds surfhvv {W

| lv vwurqjo| ghshqghqw zlwk whpsrudo ghshqghqfh

fkdudfwhulvwlfv dqdorjrxv wr wkrvh ri d wlph vhulhv zlwk d udqgrp {hg hhfw1

Wkh vhuldo fruuhodwlrq frh!flhqwvhWiru wkh fhqwuhg errwvwuds gdwd duh doo }hur lq wkh olplw1

Wkxv/ glhuhqfhv ehwzhhq fhqwuhg dqg xqfhqwuhg errwvwuds vdpsolqj shuvlvw lq wkh olplw ehkdylru ri wkh vhuldo fruuhodwlrqv dqg wkh errwvwuds vdpsoh +{hW

|,? ehkdyhv dv|pswrwlfdoo|

olnh dq xqfruuhodwhg vhtxhqfh1 Djdlq/ xqlw urrw whvwv zloo uhmhfw wkh qxoo k|srwkhvlv zlwk suredelolw| rqh dvq$ 4=Zkhq wkh vhuldo fruuhodwlrq frh!flhqw eW lv uhghqhg lq whupv ri

ghyldwlrqv iurp vdpsoh phdqv/ l1h1 dvee

>wkhq +44, krogv udwkhu wkdq +43,1 Lq doo wkhvh fdvhv/

+f, Eorfn Errwvwuds Olplwv iru Lqwhjudwhg Surfhvvhv

D vlplodu dqdo|vlv fdq eh shuiruphg iru wkh eorfn errwvwuds1 Zh zloo xvh Fduovwhlq +4<;9, eorfnlqj1 Khuh/ wkh gdwd duh vxeglylghg lqwrP vxffhvvlyh eorfnviD =m@ 4> ==> Pjri htxdo

vl}h p> zlwk q@pP= Dvvxph 

6 . 6? $3> vr wkdw  .? $3 dqg wkhq wkh vl}h ri hdfk

eorfn dv zhoo dv wkh qxpehu ri eorfnv whqg wr lqqlw| dw d udwh vorzhu wkdq q= Zlwk wklv vfkhpdwlf/ zh fdq zulwh revhuydwlrqv dv {|@ {Er36n&>vr wkdw wkh w*wk revhuydwlrq lq wkh

vdpsoh dsshduv dv wkhn*wk revhuydwlrq lq wkh v*wk eorfnDr=

Eorfn errwvwuds vdpsohv+{KW

| ,? duh frqvwuxfwhg e| udqgrpo| vdpsolqj wkhP eorfnv +zlwk

uhsodfhphqw, dqg wkhq duudqjlqj wkhp hqg0wr0hqg lq wkh rughu lq zklfk wkh| duh vdpsohg1 Fdoo wkh vdpsohg eorfnviDW

=m@ 4> ==> Pj=D w|slfdo eorfn errwvwuds revhuydwlrq fdq wkhq

eh zulwwhq dv {KW

| @ {E36n& zkhuh m lv d udqgrp lqgh{ gudzq iurp i4>5> ===> Pj1 Wkdw

lv/m lv xqlirup ryhu wkh lqwhjhuv i4>5> ===> Pj zlwk pdvv suredelolw| 

 rq hdfk lqwhjhu1 Zh

fdq frqvwuxfw fhqwuhg eorfn errwvwuds vdpsohv +h{KW

| ,? lq wkh vdph zd|1 Zlwk vrph plqru

fkdqjhv lq wkh surriv/ doo wkh uhvxowv jlyhq ehorz fduu| ryhu wr wkh prylqj eorfn errwvwuds +N qvfk/ 4<;<,1 Lq wklv eorfnlqj vfkhph/ zh ohwQ> ===> Q ehllgxqlirup udqgrp gudzv iurp i3>4>5> ===> qpj=Wkhq d w|slfdo prylqj eorfn errwvwuds revhuydwlrq lv{6KW

E36n&@{n&

iru4mP dqg4np=

Dv lq wkh suhylrxv vhfwlrq/ lw iroorzv iurp Ohppd 514 wkdw wkhuh h{lvwv d suredelolw| vsdfh lq zklfk zh fdq zulwh {KW | s q @ {E36n&_s q @E _E36n& q .r@r 4 q23R @E U._qn .r@r 4 q23R > +45, zkhuh U @ 3_ lv xqlirupo| glvwulexwhg ryhu i3>_ > ==>3_ j iru hdfk m= Dv q> P $ 4>

zlwk d voljkw dexvh ri qrwdwlrq/ zh kdyh

U $_W U> +46,

zkhuh U lv llgX^3>4`1 Vlploduo|/ iru dq| u5 ^3>4` wkhuh h{lvw lqwhjhuv vo dqg no iru zklfk

^qu` @ +vo4,p.no dqg zh fdq zulwh {KW d?oo s q @ {Eao36n&_s o q @E _Ea o36n&o q .r@r 4 q23R @E Uco. n_qo .r@r 4 q23R > +47, zkhuh Mo lv xqlirupo| glvwulexwhg ryhu i4>5> ===> Pj> Uco @ ao_3 lv xqlirupo| glvwulexwhg

ryhui3> 

> ==>3 j dqgUco $_W Uo>zkhuhUo lv xqlirup ryhu^3>4`iru hdfk u=

Eorfn errwvwuds yhuvlrqv ri Ohppdv 515 dqg 516 iroorz lq d vwudljkwiruzdug zd| dqg zh gr qrw uhshdw wkhp khuh1 Lqvwhdg/ zh jlyh wkh pdlq uhvxow/ d eorfn errwvwuds yhuvlrq ri Wkhruhp 5171 Wkh vhuldo fruuhodwlrq frh!flhqwv eKW >hKW> dqg eKe duh ghqhg dv lq Vhfwlrq 5

+e, deryh xvlqj wkh eorfn errwvwuds gdwd1

518 Wkhruhp Vhuldo fruuhodwlrqv ri vfdodu eorfn errwvwuds vdpsohv +{KW

| ,? dqg +{hKW| ,? iurp d xqlw urrw surfhvv vdwlvi|lqj wkh frqglwlrqv ri Ohppd 515 dqg zlwk ₆ . 6_? $ 3 kdyh wkh iroorzlqj olplwv iru doo {hg k9@ 3dv q$ 4=

e

Wkxv/ wkh vhuldo fruuhodwlrq frh!flhqwv ri wkh eorfn errwvwudsshg wlph vhulhv{KW

| frqyhujh

wr xqlw|/ hyhq wkrxjk wkh vwdqgdugl}hg surfhvv q3

2{KW_d?oo frqyhujhv zhdno| wr d udqgrpo|

vdpsohg yhuvlrq ri wkh Eurzqldq prwlrq E mxvw dv lq wkh fuxgh errwvwuds +Ohppd 515,1 Khqfh/ lq frqwudvw wr wkh fuxgh errwvwuds/ wkh eorfn errwvwuds vhuldo fruuhodwlrq frh!flhqwv duh uvw rughu frqvlvwhqw1

Vrph ixuwkhu dqdo|vlv uhyhdov wkdw wkh glvwulexwlrq ri wkh eorfn errwvwuds vhuldo fruuhodwlrq frh!flhqw eKW lv lqfrqvlvwhqw iru wkh xqlw urrw olplw glvwulexwlrq ri wkh uvw rughu vhuldo

fruuhodwlrq frh!flhqw ri dq lqwhjudwhg surfhvv1 Lwv frqyhujhqfh udwh R+q@P, lv dovr vorzhu wkdq wkh frqyhqwlrqdo rughuq udwh iru xqlw urrw glvwulexwlrqv1

519 Wkhruhp Wkh uvw rughu vhuldo fruuhodwlrq frh!flhqw ri d vfdodu eorfn errwvwuds vdpsoh

+{KW

| ,? iurp d xqlw urrw surfhvv vdwlvi|lqj wkh frqglwlrqv ri Ohppd 515 dqg zlwk 6 .6? $3 kdv wkh iroorzlqj olplw glvwulexwlrq dv q$ 4=

q P e KW_ 4$_W U_ f E+u,2gu U_ f E+u,gu 2 U_ f E+u,2gu ?3 d=v=+S,= +49,

Li fhqwuhg errwvwuds vdpsohv +{hKW

| ,? duh xvhg wkhq wkh fruuhvsrqglqj olplw lv

q

P+eKW 4,$RW 4 d=v=+S,= +4:,

Wkh olplw uhvxowv +49, dqg +4:, vkrz wkdw li d frqyhqwlrqdo xqlw urrw urrw whvw zhuh dssolhg wr eorfn errwvwudsshg vdpsoh gdwd wkhq wkh suhvhqfh ri d xqlw urrw zrxog eh uhmhfwhg zlwk d suredelolw| dssurdfklqj xqlw| dv q $ 4= Frqyhuvho|/ li wkh errwvwuds glvwulexwlrq ri eKW

zhuh xvhg wr frqvwuxfw fulwlfdo ydoxhv lq lqihuhqfh edvhg rq e derxw wkh suhvhqfh ri d xqlw

dxwruhjuhvvlyh frh!flhqw lq +4,/ wkhq wkhuh zrxog eh d }hur uhmhfwlrq udwh dv|pswrwlfdoo| iru wkh xqlw urrw k|srwkhvlv1

Wkh frqyhujhqfh udwh ri eKW lv R+p,> uhyhdolqj wkdw lq wkh eorfn errwvwuds lw lv wkh

qxpehu ri frqvhfxwlyh revhuydwlrqvplq hdfk eorfn wkdw ghwhuplqhv wkh udwh ri frqyhujhqfh ri wkh vhuldo fruuhodwlrq frh!flhqw qrw wkh qxpehu ri eorfnv1 Lq hhfw/ hdfk eorfn lv d vpdoo0 lqqlw|0vl}hg wudmhfwru| ri d xqlw urrw surfhvv1 Krzhyhu/ vlqfh wkh h{suhvvlrq irueKW lqyroyhv

dyhudjhv dfurvv eorfnv dv zhoo dv zlwklq eorfnv/ wkh olplw rip+eKW_ 4,lv qrw d frqyhqwlrqdo xqlw urrw glvwulexwlrq1 Lqvwhdg/ dv wkh surri ri wkhruhp 519 uhyhdov/ d eldv whup lqyroylqj dyhudjhv ri glhuhqfhv ri Eurzqldq prwlrq dfurvv udqgrpl}hg eorfnv grplqdwhv wkh vdpsoh fryduldqfh 

?S?|'{W|3{W| dqg jlyhv ulvh wr wkh olplwv lq +49, dqg +4:,1

Ilj1 8 judskv nhuqho hvwlpdwhv ri wkh ghqvlwlhv ri wkh xqlw urrw glvwulexwlrq rie_ djdlqvw wkrvh ri wkh fuxgh errvwuds hvwlpdwh eW> wkh eorfn errwvwuds hvwlpdwh eKW dqg wkh fhqwuhg

errwvwuds hvwlpdwhhW=Wkh vlpxodwlrq hvwlpdwhv xvhg 43/333 uhsolfdwlrqv/ d vdpsoh vl}h ri 533

dqg eorfn vhwwlqjvp@ 43> P @ 53=Dssduhqwo|/ wkh eorfn errwvwuds vdpsolqj glvwulexwlrq lv forvhu wr wkdw ri wkh xqlw urrw glvwulexwlrq wkdq wkh rwkhu hvwlpdwhv exw lv vxevwdqwldoo| grzqzdug eldvhg/ frqvrqdqw zlwk wkh uhvxowv lq Wkhruhp 5191

Ilj1 8= Nhuqho Hvwlpdwhv ri Ghqvlwlhv rie>eW_/eKW_ >hW_=

61 Uhjuhvvlrq rq d Ghwhuplqlvwlf Wuhqg

Zh frqvlghu wkh surwrw|slfdo vsxulrxv uhjuhvlrq ri d vfdodu wlph vhulhv{|jhqhudwhg dv lq +4,

rq d olqhdu wuhqg

{|@ee?w.xe|> +4;,

Uhjuhvvlrqv rq kljkhu rughu sro|qrpldov dqg rwkhu ghwhuplqlvwlf ixqfwlrqv surgxfh dqdorjrxv uhvxowv dqg wkh vlpsoh uhjuhvvlrq +4;, lv fkrvhq vlpso| iru frqyhqlhqfh1 Ohw v_eK? eh wkh vwdqgdug huuru riee?> w+ae?,eh wkh fruuhvsrqglqjwvwdwlvwlf/ dqg vhwe@S?2 ex|xe|3@S?2 ex2|3

dqgGZ @S?₂+xe|xe|3,2@S?₂ ex2|31 Dv lq Gxuodxi dqg Skloolsv +4<;;,/ s qae?@ q 3D 2S?_|'{_|w q3S? |'w2 $_ U_ f_UuE+u,  f u2 > vd|/ +4<, 4 s qw a e? @ _vae? eK? $_ U_fu22 U_ f +E+u,u,2  2> +53, dqg e $R 4> qGZ $_ 2 .2 U_ f +E+u,u,2 > +54,

zkhuh E lv vfdodu Eurzqldq prwlrq zlwk yduldqfh $2_{= E| fkdqjlqj wkh suredelolw| vsdfh dv}

e| d=v= frqyhujhqfh/ dqg zh vkdoo frpprqo| surfhhg ehorz dv li wkhvh fkdqjhv kdyh ehhq pdgh1

+d, Fuxgh Errwvwudsslqj ri Uhvlgxdov

Khuh errwvwuds vdpsohv +xW

|,? dqg +xhW|,? duh iruphg e| udqgrp vdpsolqj iurp +ex|,?/

errwvwuds gdwd+{WW

| ,? duh jhqhudwhg dffruglqj wr

{WW| @ee?w.xW|> +55,

dqg ohdvw vtxduhv surgxfhv wkh wwhg uhjuhvvlrq {WW

| @eeW?w.xWW| > +56,

zklfk ghqhv wkh uhvlgxdo surfhvv xWW

| = Ohw vWW2 @ ? S?|'xWW2| > ohw veKW_? dqg w+eeW?, eh wkh

vwdqgdug huuru dqgw udwlr ri eeW

? lq +56,/ dqg ohw GZWW eh wkh GZ udwlr lq +56,1 Ghqrwh

vhuldo fruuhodwlrqv e| wkh qrwdwlrqeWW @S?_|'xWW

| xWW|3@S?|'xWW2|3=Wkh iroorzlqj uhvxowv jlyh

wkh olplw ehkdylru ri wkhvh errwvwuds vwdwlvwlfv1

614 Wkhruhp Ohw +{WW

| ,? eh errwvwuds gdwd jhqhudwhg iurp +55, xvlqj +xW|,? zkhuh wkh rulj0 lqdo gdwd +{|,? lq wkh wuhqg uhjuhvvlrq +4;, lv d xqlw urrw surfhvv +4, zlwk x| vdwlvi|lqj Dvvxpswlrq O1 +d, sqaeW ?$_W W@  2 U_ fEo_on_U 21  fo2 d=v=+S,> +e,  ?vWW2$_W 21 @UfiEo+v, . +W,vj2gu d=v=+S,> +f, I_?w+e_eW_{?,$_}W 1 W₊U fo2,  2 j1 d=v=+S,>

+g, eWW $_W Koe}₄ > Koe}₄ ?4 d=v=+S,>iru doo k9@ 3>

+h, GZWW_$

_WKoe}_(` A3 d=v=+S,> +i, UWW2_$

_W Koe}_-2 ?4> d=v=+S,>

zkhuh Eo+v, @ E+v, v lv ghwuhqghg Eurzqldq prwlrq/ EoW+v, @ Eo+v, . +W,v @

E+v,Wv> Koe}4 @ U_ f EoW+v,gv 2 U_ f EoW+v,2gv 5+3>4, d=v=+S, Koe}_(` @ 5 U_ f Eo+v,2gv 2 1 k 4K4 l A3 d=v=+S, K4 @ U_ f Eo+v,gv ₂ U_ f Eo+v,2gv 5+3>4, d=v=+S, dqg Koe}_-2 @ 4 2 1 U_ f E+u,2gu =

615 Uhpdunv

+l, Iurp +d,/ aeW

? $RW 3 dqg vr wkh errwvwuds hvwlpdwh aeW_? lv uvw0rughu frqvlvwhqw1 E|

d vwudljkwiruzdug exw ohqjwk| fdofxodwlrq W _{@ Q+3> y}W

K, zlwk yKW @ $2 2f.Sf> zkhuhdv

@Q+3> yK, zlwk yK @ S_D$2=Hylghqwo|/ wkh errwvwuds vwdwlvwlf sqaeW? dqgsqae? duh qrw

dv|pswrwlfdoo| htxlydohqw dowkrxjk wkh olplw glvwulexwlrqQ+3> yW

K,lv forvh wr wkh fruuhfw

olplw +2f.

Sf *4=5<yhuvxv SD @ 4=53,1

+ll, Wkh uhdvrq iru wkh lqfrqvlvwhqf| ri wkh errwvwuds vwdwlvwlfsqaeW

? lv wkdw lq wkh qxphu0

dwru ri wklv vwdwlvwlf q3D*2S?

{W|w @ q3D*2S? {|w= Lq idfw/ iurp Ohppd 516 +j, zh

kdyh q3D*2S?

{W|w $_W +U_fE+u,gu,+U_fugu, d=v=+S,/ zkhuhdv e| vwdqgdug phwkrgv

q3D*2S?

{|w$_W +U_fE+u,ugu, dv lq +4<,1

+lll, Wkh errwvwuds w vwdwlvwlf w+eeW

?, glyhujhv dw wkh udwh sq> mxvw dv wkh uhjuhvvlrq w

vwdwlvwlfw+ee?,>exww+aeW?,@sqkdv d glhuhqw olplw glvwulexwlrq iurpw+ae?,@sq>zklfk e|

vwdqgdug phwkrgv lv+U_fu2_gu,_2@+U

f Eo+v,gv,=

+ly, Prvw lpsruwdqwo|/ wkh errwvwuds vhuldo fruuhodwlrqeWW kdv d udqgrp olplw Koe}4 zklfk

lv vwulfwo| ohvv wkdq xqlw| zlwk suredelolw| rqh1 Lw iroorzv wkdw xqlw urrw whvwv rq wkh uhvlgxdov ri +56, wkdw duh edvhg rq eWW

 zloo uhmhfw wkh qxoo k|srwkhvlv ri d xqlw urrw

dqg khqfh wkh qxoo ri d vsxulrxv uhjuhvvlrq1 Wkxv/ wkh ruljlqdo uhjuhvvlrq +4;, dqg wkh errwvwuds uhjuhvvlrq +56, kdyh ixqgdphqwdoo| glhuhqw fkdudfwhulvwlfv +4;, lv d wwhg wuhqg zlwk d vljqlfdqw frh!flhqw dqg gldjqrvwlfv wkdw uhyhdo wkh vljqlfdqfh lv vsxulrxv/ zkhuhdv +56, lv d wuhqg uhjuhvvlrq zlwk d vlploduo| vljqlfdqw frh!flhqw dqg gldjqrvwlfv wkdw gr qrw lqydolgdwh wkh uhodwlrqvkls dv|pswrwlfdoo|1

616 Wkhruhp Ohw +{WW

| ,? eh errwvwuds gdwd jhqhudwhg iurp +55, xvlqj fhqwuhg errwvwuds uhvlgxdov +xhW

|,? zkhuh wkh ruljlqdo gdwd +{|,? lq wkh wuhqg uhjuhvvlrq +4;, lv d xqlw urrw surfhvv +4, zlwk x| vdwlvi|lqj Dvvxpswlrq O1 +d, sqaeW ?$_W d=v=+S,> +e,  ?vWW2$_W U_ f Eo+v,2gu d=v=+S,> +f, _I ?w+eeW?,$_W 1+ U fo2,  2 +U foEr2_r,  2 d=v=+S,> +g, eWW $RW 3> d=v=+S,>iru dook9@ 3> +h, GZWW_$ RW5 d=v=+S,> +i, UWW2_$ _W 1 2U fo2_o U_ foEr2_rn12Ufo2_o ?4> d=v=+S,> zkhuh Eo+v, @E+v,v@Eo+v,U_fEo+v,gv=

617 Uhpdunv

+l, Wkh errwvwuds hvwlpdwh aeW

? lv frqvlvwhqw dqg sqaeW? kdv wkh vdph olplw glvwulexwlrq dv s_qae

?= Vlploduo|/ wkh errwvwuds w vwdwlvwlf w+eeW?, glyhujhv dw wkh udwh sq> mxvw dv wkh

uhjuhvvlrq w vwdwlvwlfw+ee?,>exw wkh olplw glvwulexwlrq ri w+aeW?,@sqjlyhq lq sduw +f, lv

glhuhqw iurp wkdw riw+ae?,@sq+jlyhq lq Uhpdun 615+lll,,1

+ll, Dffruglqj wr sduwv +g, dqg +h,/ wkh uhvlgxdov lq wkh uhjuhvvlrq +56, duh dv|pswrwlfdoo| xqfruuhodwhg1 Wklv phdqv wkdw errwvwuds revhuydwlrqv +{WW

| ,? jhqhudwhg iurp

{WW

| @ee?w.xhW| +57,

ehkdyh lq wkh uhjuhvvlrq +56, dv wuhqg vwdwlrqdu| gdwd zlwk zklwh qrlvh uhvlgxdov/ zkhuhdv wkh rujlqdo gdwd +{|,? lv L+4,= Djdlq/ wkh ruljlqdo uhjuhvvlrq +4;, dqg wkh

errwvwuds uhjuhvvlrq +56, kdyh ixqgdphqwdoo| glhuhqw fkdudfwhulvwlfv1

+e, Eorfn Errwvwudsslqj ri Uhvlgxdov

Eorfn errwvwuds vdpsohv ri uhvlgxdov +xKW

| ,? duh frqvwuxfwhg e| udqgrpo| vdpsolqj P

eorfnv +zlwk uhsodfhphqw, ri wkh fhqwuhg uhvlgxdov+xe|q3S?r'exr,? iurp +4;,1 D w|slfdo

eorfn errwvwuds uhvlgxdo fdq wkhq eh zulwwhq dvxhKW

| @xeE36n&q3S?r'exr>zkhuhm lv d

udqgrp lqgh{ gudzq iurpi4>5> ===> Pj1 Eorfn errwvwuds gdwd+{KW

| ,? duh jhqhudwhg dffruglqj

wr

{KWW| @ee?w.hxKW| +58,

dqg ohdvw vtxduhv surgxfhv wkh wwhg uhjuhvvlrq

{KWW| @eeKW?w.xKWW| > +59,

zklfk ghqhv wkh uhvlgxdo surfhvv xKWW

| = Ohw vKWW2 @ ?S?|'xKWW2| > ohw vKW dqg wKW eh wkh

vwdqgdug huuru dqgwudwlr rieeKW

? lq +59,/ dqg ohwGZKWWeh wkh GZ udwlr lq +59,1 Djdlq/ zh

ghqrwh vhuldo fruuhodwlrqv e| wkh qrwdwlrq eKWW = Wkh iroorzlqj uhvxowv jlyh wkh olplw ehkdylru

ri wkhvh errwvwuds vwdwlvwlfv1 Dv lq Vhfwlrq 5+f, dqg vhwwlqjw@ +v4,p.n/ wkhuh h{lvwv d suredelolw| vsdfh lq zklfk zh fdq zulwh {KWW | s q @ s qee?_qw .xeEr36n&q 3S? r'xer s q +5:, @ ^.r@r+4,`+v4,_qp.n.Eo _E r36n& q .r@r 4 q23R @ +v4, P .qn .Eo Ur .n_q .r@r+4,> +5;,

zkhuh Ur @ r_3 lv xqlirupo| glvwulexwhg ryhu i3>_ > ==>3_ j iru hdfk v= Dv q> P $ 4>

zlwk d voljkw dexvh ri qrwdwlrq/ zh kdyh

zkhuh U lv llgX^3>4`1 Vlploduo|/ iru dq| u5 ^3>4` wkhuh h{lvw lqwhjhuv vo dqg no iru zklfk ^qu` @ +vo4,p.no dqg zh fdq zulwh {KWW d?oo s q @u.Eo+Uco, .r@r+4,> +5<, zkhuh Uco @ ao_3 lv xqlirupo| glvwulexwhg ryhu i3>_> ==>3_ j dqg Uco $_W U_o> zkhuh

Uo lv xqlirup ryhu^3>4` iru hdfku=

Zh jlyh d eorfn errwvwuds yhuvlrq ri Wkhruhp 6161

618 Wkhruhp Ohw +{KWW

| ,? eh errwvwuds gdwd jhqhudwhg iurp +58, zlwk 6 .6? $3dqg xvlqj fhqwuhg eorfn errwvwuds uhvlgxdov +xhKW

| ,? zkhuh wkh ruljlqdo gdwd +{|,? lq wkh wuhqg uhjuhvvlrq +4;, lv d xqlw urrw surfhvv +4, zlwk x| vdwlvi|lqj Dvvxpswlrq O1 Wkhq=

+d, sqaeKW ? $_W d=v=+S,> +e,  ?vKWW2$_W U_ f Eo+v,2gu d=v=+S,> +f, I_?wKW_{$_}W 1+ U_ fo2,  2 +U foEr2_r,  2 d=v=+S,> +g, eKWW $RW4> d=v=+S,>iru doo k9@ 3> +h, GZKWW_$ RW 3 d=v=+S,> +i, UKWW2_$ _W 1 2U fo2_o U_ foEr2_rn12Ufo2_o ?4> d=v=+S,= 619 Uhpdunv

+l, Wkh eorfn errwvwuds uhvxowv iru wkh uhjuhvvlrq frh!flhqw aeKW

?> w udwlr wKW dqg U2 duh

wkh vdph dv wkrvh iru wkh fhqwuhg errwvwuds jlyhq lq Wkhruhp 6171

+ll, Wkh vhuldo fruuhodwlrq surshuwlhv ri wkh uhvlgxdov lq wkh eorfn errwvwuds uhjuhvvlrq duh glhuhqw iurp wkrvh lq Wkhruhp 6171 Mxvw dv lq eorfn errwvwudsslqjL+4,gdwd +Wkhruhp 518,/ wkh uhvlgxdo vhuldo fruuhodwlrq frh!flhqwveKWW>eKWW $RW4 dqg wkh Gxuelq Zdwvrq

vwdwlvwlf GZKWW _{$RW 3= Wkxv/ wkhvh gldjqrvwlfv duh qrz uvw rughu frqvlvwhqw lq wkh}

errwvwuds uhjuhvvlrq +59,1 Wklv uhjuhvvlrq lv qrz vsxulrxv lq wkh vdph vhqvh dv wkh ruljlqdo wuhqg uhjuhvvlrq +56,1

+lll, Krzhyhu/ wkh olplw glvwulexwlrqv ri qhlwkhueKWW qruGZKWW duh frqvlvwhqw1 Dv lq Wkhr0

uhp 519 zh qg wkdw wkhvh errwvwuds gldjqrvwlf vwdwlvwlfv kdyh glhuhqw udwhv ri frqyhu0 jhqfh dqg glhuhqw olplwv iurp wkrvh ri wkh ruljlqdo vdpsoh gdwd1 Wkhruhp 61: ehorz jlyhv wkh uhvxow iru eKWW = Wkxv/ errwvwudsslqj wkhvh gldjqrvwlf vwdwlvwlfv +ru yhuvlrqv

ri wkhp wkdw duh qxlvdqfh sdudphwhu iuhh, zloo qrw surgxfh dq dv|pswrwlfdoo| ydolg lqihuhqfh surfhgxuh1

61: Wkhruhp Xqghu wkh vdph frqglwlrqv dv Wkhruhp 618>eKWW  kdv wkh iroorzlqj olplw dv q$ 4= _q P eKWW 4 $_W 4 d=v=+S,=

71 Vsxulrxv Uhjuhvvlrq rq Vwrfkdvwlf Wuhqgv

Ohw}|eh dq+c.4,yhfwru lqwhjudwhg wlph vhulhv jhqhudwhg dv lq +4, zlwk}|@x5|vdwlvi|lqj

O dqg}| vdwlvi|lqj Ohppd 514 zlwk olplwlqj Eurzqldq prwlrqE5 zkrvh yduldqfh pdwul{ lv

55 A31 Sduwlwlrq }| @ +||> {|, lqwr wkh vfdodu || dqg c yhfwru{| dqg ohw E5 @ +E+> E%,

eh d frqirupdeoh sduwlwlrq riE5=Uxq wkh +vsxulrxv, uhjuhvvlrq

||@ee?{|.xe|= +63,

Ohwveh wkh vwdqgdug huuru riee?> weh wkh fruuhvsrqglqjwvwdwlvwlf/ dqg vhwe@S?2ex|ex|3@S?2ex2|3

dqgGZ @S?₂+xe|xe|3,2@S?₂ ex2|31 Dv lq Skloolsv +4<;9,/ a_{e?$_D}3 %%d%+ @+%> vd|/ +64, 4 s qw@ a e? v $_ +% d++%D3%%  2 @|> dqg qGZ $_ ₅₅ d++% >

zkhuh/ lq frqirupdeoh sduwlwlrqv/ zh ghqh ] _ f E5+u,E 5+u,gu@ d++ d%+ d%+ D%% > d++% @d++d%+D3%%d%+> @ 4 D3 %%d%+ > dqg 55 @H+x5|x5|,=Wkhvh frqyhujhqfhv fdq eh uhsodfhg e| d=v=frqyhujhqfh diwhu dssur0

suldwh fkdqjhv lq wkh suredelolw| vsdfh dv zh kdyh glvfxvvhg hduolhu1

Dssolfdwlrq ri errwvwuds phwkrgv wr wkh uhvlgxdov lq +63, ohdgv wr uhvxowv wkdw duh vlplodu wr wkrvh ri wkh suhylrxv vhfwlrq1 Zh wkhuhiruh surylgh wkh pdlq uhvxowv khuh lq eulhi1 Errwvwuds vdpsohv+hxW

|,? duh iruphg e| udqgrp vdpsolqj iurp fhqwuhg yhuvlrqv ri wkh uhvlgxdov+xe|,? lq

+63,/ errwvwuds gdwd+|WW

| ,? duh jhqhudwhg dffruglqj wr

||WW@ee?{|.xhW| +65,

dqg ohdvw vtxduhv surgxfhv wkh wwhg uhjuhvvlrq |WW

| @eeW?{|.xWW| > +66,

zklfk ghqhv wkh uhvlgxdo surfhvvxWW

| =OhwvWW2 @ ?S?|'xWW2| >ohwvW dqgwW eh wkh vwdqgdug

huuru dqgwudwlr rieeW

?/ dqg ohwGZWWeh wkh vhuldo fruuhodwlrq frh!flhqw dqg GZ udwlr1 Dv

hduolhu/ ghqrwh vhuldo fruuhodwlrqv e| wkh qrwdwlrq eWW @ S?|'xWW| xWW|3@S?|'xWW2|3= Wkh iro0

orzlqj uhvxowv jlyh wkh olplw ehkdylru ri wkhvh errwvwuds vwdwlvwlfv1 Zh frqqh wkh glvfxvvlrq wr fhqwuhg errwvwuds vdpsohv1

714 Wkhruhp Ohw +|WW

| ,? eh errwvwuds gdwd jhqhudwhg iurp +65, xvlqj +hxW|,? zkhuh wkh gdwd

+||> {|,? lq wkh vdpsoh uhjuhvvlrq +63, lv jhqhudwhg dv lq +4, zlwk lqfuhphqwv }| @ x5| vdwlvi|lqj O1 +d, aeW ?$_W_+% d=v=+S,> +e,  ?vWW2$_W U_ f E5E5 d=v=+S,> +f, _I ?wW $_W 1+% i#+U f55,#^3%%`j  2 @ W | d=v=+S,> +g, eWW $RW 3> d=v=+S,>iru dook9@ 3> +h, GZWW_{$RW5 d=v=+S,>} +i, UWW2_$ _W 1 +%+Uf%%,1+% #+U f55,#n1+%+Uf%%,1+% ?4> d=v=+S,=

Wkh errwvwuds uhjuhvvlrq frh!flhqw lq +66, lv frqvlvwhqw iru+%dqg wkhwudwlrwW glyhujhv

dw wkh udwhsqmxvw dv lq wkh vsxulrxv uhjuhvlrq +63,/ dowkrxjk wkh olplw yduldwhW

|9@|1 Pruh

lpsruwdqwo|/ krzhyhu/ wkh uhvlgxdov lq wkh uhjuhvvlrq +66, duh dv|pswrwlfdoo| xqfruuhodwhg dqg vr wkh errwvwuds uhjuhvvlrq lv dv|pswrwlfdoo| d frlqwhjudwlqj uhjuhvvlrq1

Glhuhqw uhvxowv dsso| zkhq zh xvh wkh eorfn errwvwuds1 Xvlqj wkh vdph qrwdwlrq dv lq Vhfwlrq 6+f, zlwk wkh d!{ewr vljqli| xvh ri wkh eorfn errwvwuds/ zh qg wkh iroorzlqj olplw wkhru|1

715 Wkhruhp Ohw +|KWW

| ,? eh errwvwuds gdwd jhqhudwhg iurp +65, xvlqj fhqwuhg eorfn errwvwuds uhvlgxdov +xhKW

| ,? zlwk 6 .6? $3dqg zkhuh wkh ruljlqdo gdwd wkh gdwd +||> {|,? lq wkh vdpsoh uhjuhvvlrq +63, lv jhqhudwhg dv lq +4, zlwk lqfuhphqwv }|@x5| vdwlvi|lqj O1 Wkhq=

+d, aeKW ? $_W_+% d=v=+S,> +e,  ?vKWW2$_W U_ f E5E5 d=v=+S,> +f, _I ?wKW $_WW_| d=v=+S,> +g, eKWW $RW4> d=v=+S,>iru doo k9@ 3> +h, GZKWW_$ RW 3 d=v=+S,> +i, UKWW2_$ _W 1 +%+Uf%%,1+% #+U f55,#n1+%+Uf%%,1+% ?4> d=v=+S,=

716 Wkhruhp Xqghu wkh vdph frqglwlrqv dv Wkhruhp 715>eKWW

 kdv wkh iroorzlqj olplw dv q$ 4= _q P eKWW 4 $RW 4 d=v=+S,=

81 Frqfoxvlrqv dqg Lpsolfdwlrqv

Wklv sdshu vkrzv wkdw phfkdqlfdo dssolfdwlrq ri erwk wkh errwvwuds dqg eorfn errwvwuds zloo jlyh shuyhuvh uhvxowv li xvhg lq uhvlgxdo edvhg whvwlqj iru xqlw urrwv wr glvwlqjxlvk vsx0 ulrxv dqg frlqwhjudwlqj uhjuhvvlrqv ru wuhqg vwdwlrqdu| iurp lqwhjudwhg gdwd1 Lq sduwlfxodu/ errwvwudsslqj wkh uhvlgxdov ri vxfk uhjuhvvlrqv wxuqv vsxulrxv uhjuhvvlrqv lqwr frlqwhjudwlqj uhjuhvvlrqv dqg wkhuhe| fkdqjhv wkh fkdudfwhu ri wkh uhjuhvvlrqv lq d ixqgdphqwdo zd|1 Zkloh wkh eorfn errwvwuds uhwdlqv wkh L+4, ihdwxuh ri wkh uhjuhvvlrq uhvlgxdov dqg kdv vhuldo fruuh0 odwlrq frh!flhqwv wkdw whqg wr xqlw|/ wkhlu udwh ri frqyhujhqfh lv vorzhu wkdq wkh R+q, udwh ri wkh vdpsoh uhvlgxdo vhuldo fruuhodwlrqv1 Vhuldo fruuhodwlrqv ri wkh eorfn errwvwuds frqyhujh dw wkh udwhR+p,>zkhuhplv wkh eorfn vl}h dqg wkh olplw lv qr orqjhu ri wkh xqlw urrw w|sh1 Wkxv/ wkh errwvwuds dqg wkh eorfn errwvwuds erwk idlo vhulrxvo| lq uhsurgxflqj wkh surshuwlhv ri vsxulrxv uhjuhvvlrqv dqg duh xqvxlwdeoh iru uhvlgxdo edvhg whvwlqj lq wklv frqwh{w1

Wkhvh uhvxowv uhlqirufh wkh zduqlqjv jlyhq lq hduolhu uhvhdufk derxw wkh gl!fxowlhv hq0 frxqwhuhg e| wkh errwvwuds zlwk ghshqghqw gdwd +Krurzlw}/ 4<<</ dprqj pdq| rwkhuv, dqg zlwk fruuhodwlrq frh!flhqwv +Kdoo/ 4<<5/ s1485,1 Lw vhhpv wkdw errwvwudsslqj lqwhjudwhg gdwd h{dfhuedwhv nqrzq sureohpv ri errwvwuds lqfrqvlvwhqf| lq xqlw urrw lqihuhqfh +h1j1 Edvdzd hw1 do1 4<<4,/ surgxflqj glhuhqfhv qrw rqo| lq wkh olplw glvwulexwlrqv ehwzhhq wkh errwvwuds dqg wkh ruljlqdo vwdwlvwlf exw uvw rughu glhuhqfhv lq wkh olplwv ri wkh vwdwlvwlfv lq vrph fdvhv1 Wkh eorfn errwvwuds vkrzv vrph urexvwqhvv wr qrqvwdwlrqdulw| lq wklv uhjdug/ exw dv Klqnoh| +4<<:, uhpdunv/ lw wrr lv xqdeoh wr zlwkvwdqg wkh lqwhjudwhg prgho1

Wkhvh uhvxowv kdyh lpsolfdwlrqv iru wkh xvh ri wkh errwvwuds lq uhvlgxdo edvhg frlqwhjudwlrq whvwlqj/ zkhuh wkh qxoo k|srwkhvlv lv wkdw wkh uhvlgxdov lq wkh uhjuhvvlrq duh lqwhjudwhg1 Wkh uhvxowv khuh vkrz wkdw urxwlqh dssolfdwlrqv ri wkh errwvwuds lq vxfk vlwxdwlrqv vkrxog eh dyrlghg1 Dq dowhuqdwlyh dssurdfk xqghu wkh qxoo lv wr glhuhqfh wkh uhvlgxdov dqg dsso| vlhyh errwvwuds phwkrgv +Nuhlvv/ 4<<5/ Exkopdqq/ 4<<:/ 4<<;, rq wkh glhuhqfhg uhvlgxdov dqg wkhq errwvwuds lqyduldqfh sulqflsohv +Elfnho dqg Exkopdqq/ 4<<</ dqg Sdun/ 5334, wr errwvwuds wkh glvwulexwlrq ri uhvlgxdo edvhg frlqwhjudwlrq whvwv1 Suhwhvwlqj iru wkh suhvhqfh ri d xqlw urrw lq wkh uhvlgxdov sulru wr wkh xvh ri wkh errwvwuds e| frqvlvwhqw phwkrgv ri prgho vhohfwlrq +h1j1 Skloolsv dqg Sorehujhu/ 4<<9, lv dqrwkhu dowhuqdwlyh1 Rwkhu dssurdfkhv olnh k|eulg uhvdpsolqj +Fkxdqj dqg Odl/ 5333, dqg vxevdpsolqj +Srolwlv/ Urpdqr dqg Zroi/ 4<<<, kdyh ehhq irxqg wr rhu lpsuryhphqwv lq vrph fdvhv zkhuh wkh errwvwuds shuirupv srruo| dqg wkhvh dovr vhhp zruwk| ri vwxg| lq wklv frqwh{w1

91 Dsshqgl{

914 Surri ri Ohppd 514 Wklv lv d vwudljkwiruzdug h{whqvlrq ri ohppd 614 ri Skloolsv+4<<<,1 Xqghu O/ wkh EQ ghfrpsrvlwlrq +Skloolsv dqg Vror/ 4<<5, F+O, @ F+4, .Fh+O, +O4, lv ydolg/ zkhuhFh+O, @S"_'fhfO zlwkhf @S"r'nfr dqgS"'fmmhfmm?4=Wkhq/

x|@F+4,%|.h%|3h%|@F+4, 2_|.h%_|3h%_|>

dqg

V|@F+4, 2V_#|.h%_fh%_|>

zkhuh|lv llg+3> L,>h%|@Fh+O,%|lv vwdwlrqdu|/ dqgV#|@S|_'1 D vwurqj dssur{lpdwlrq

Skloolsv +4<<<, ohdglqj wr vxs f$&$? _V# &Z+n,@r@r+q  ^,> l@ 4> ==> p +67,

jlylqj d xqlirup dssur{lpdwlrq wr V#& ryhu 3 n q lq whupv ri wkh yhfwru vwdqgdug

Eurzqldq prwlrqZ @ +Z,=Wkhq/ vhwwlqjE+u, @F+4, 2Z+u,>zh kdyh vxs f$&$? _sV& qE+qn, F+4, 2pd{ _f$&$?vxs V_s#& qZ+nq, . 5 vxs f$&$? mm_sh%&mm q @ r@r 4 q23R > dv lq ohppd 614 ri Skloolsv +4<<<,1

915 Surri ri Ohppd 515 Iru u 5 ^3>4`> zh fdq zulwh {W

d?oo @ {ao zkhuh Mo lv xqlirup

ryhui4>5> ===> qjiru hdfkudqg wkhq dv lq +9, zh kdyh wkh hpehgglqj {W d?oo s q @ {ao s q @E Mo q .r@r 4 q23R @E+U?co, .r@r 4 q23R > +68,

zkhuhU?co @ a_?o lv xqlirupo| glvwulexwhg ryhui_?>_?2> ==>4jiru hdfku=Pruhryhu/ iruu9@vdqg

qodujh hqrxjk vr wkdwqmuvmA4> Mo dqgMr duh lqghshqghqw gudzv1 Khqfh/U?co dqgU?cr

duh vwdwlvwlfdoo| lqghshqghqw iru odujhqiru doo u9@v=

Vxevwlwxwlqj wkh ON uhsuhvhqwdwlrq +8, iru E lq +68, zh jhw {W d?oo s q @ s 5[" &' vlqn 2 U?co n 2 &.r@r 4 q23R = +69,

Qrz U?co $_W U_o> d=v +S,/ zkhuh U_o kdv d frqwlqxrxv xqlirup glvwulexwlrq ryhu ^3>4` iru

hdfku= Lw iroorzv e| wkh frqwlqxrxv pdsslqj wkhruhp wkdw vlq n4₅ U?co $_Wvlq n4₅ Uo > d=v=+S,= +6:, Khqfh/ {W d?oo s q $_W E+Uo, d=v=+S,= +6;,

Vlqfh U?co dqgU?cr duh dv|pswrwlfdoo| lqghshqghqw iru doo u9@v> lw iroorzv wkdwUo dqg Ur

duh lqghshqghqw iruu 9@v= Wkxv/ iUo =u 5^3>4`j lv idplo| ri lqghshqghqw xqlirup yduldwhv

rq^3>4`=

Lq d vlplodu zd| zh zulwh wkh fhqwuhg errwvwuds dv {hW

| @ { {> dqg wkhq lq wkh vdph qrwdwlrq dv +68, {W d?oo s_q @ {_sao q @E Mo q _q4 [? |' E w q .r@r 4 q23R @ E+U?co, ] _ f E+v,gv.r@r 4 q23R > +6<,

916 Surri ri Ohppd 516 Sduw +d, Xvlqj +<, zh kdyh 4 q ? [ |' {W | s q @ s 5$[" &'  ? S_? 'vlq n₂U? n₂ &.r@r 4 q23R > +73, dqg/ lq ylhz ri +;,/ 4 q ? [ ' vlq n4₅ U? @ _q4[? ' vlq n4₅ U .rRW+4, +74, $RW H vlq n4₅ U d=v=+S, @ ]  f vlq n4 5 u gu= +75, Lw iroorzv iurp +73, 0 +75, wkdw 4 q2 ? [ |' {W| $ _Ws5 " [ &' U_ f vlq n 2 ugu n 2 & d=v=+S, @ ]  f E+u,gu>

wkh odvw olqh iroorzlqj e| yluwxh ri wkh xqlirup lqwhjudelolw| ri wkh ON vhulhv lq wklv fdvh1

Sduw +e, Lq d vlplodu idvklrq/ zh qg

4 q2 ? [ |' h {W | $_W ] _ f E+u,gu@ 3> d=v=+S,=

Sduw +f, Kljkhu rughu vdpsoh prphqwv duh ghdow zlwk lq wkh vdph zd|1 Wkxv/ iru sduw +f, zh kdyh 4 q2 ? [ |' {W|{W| @ _q4 ? [ |' {W | s q { W | s q @ 5 " [ &c6'  ?S?'vlqn 2 U?vlqp ₂U? n  2 p2 2 &6> dqg 4 q ? [ ' vlq n4₅ U? vlq p4₅ U? $ RWH vlq n4 5 U vlq p 4 5 U d=v=+S, @ ]  f vlq n4₅ u vlq p4₅ u gu=

Djdlq/ xqlirup frqyhujhqfh ri wkh ON vhulhv lpsolhv wkdw zh fdq lqwhjudwh whup e| whup/ ohdglqj wr wkh uhvxow 4 q2 ? [ |' {W |{W| $_W 5 " [ &c6' U_ f vlq n 2 uvlqp 2 ugu n ₂ p ₂2 &6 @ ] _ f E+u,E+u, _{gu d=v=+S,}

Dowhuqdwlyho|/ e| gluhfw fdofxodwlrq zh kdyh ] _ f vlq n4 5 u vlq p4 5 u gu @ 4 5 ] _ f ifrv ^+np,u`frv ^+n.p4,u`jgu @ 3 n9@p  2 n@p dqg wkhq 4 q2 ? [ |' {W|{W| $_W " [ &' 4 n 2 2& 6>

zklfk lv d vhulhv uhsuhvhqwdwlrq riU_fE+u,E+u,gu=Sduw +g, iroorzv e| d vlplodu fdofxodwlrq1

Sduw +h, Xvlqj +7, dqg +9, zh fdq zulwh 4 q S_?  {W | s q {| s q @ _q4 S?_' E+U?, .rd1v1 4 q*23*R E m q .rd1v1 4 q*23*R @ 4 q ? [ ' E+U?,E m q .rd1v1+4, @ 54_qS?_ # _" [ &' vlqn 2 U? n 2 & $ 3 C[" ' vlqkc_2?l c  2 4 D_.rd1v1+4, @ 5 [" &c' _& n₂ c₂2 4 q ? [ ' vlq n4 5 U? vlq c 4 5 m q .rd1v1+4, +76, Dv lq +74,/ 4 q S_? vlq n4₅ U? vlq c4_{5 q}m @ _q4S?_H vlq n4₅ U vlq c4_{5 q}m . _q4S?_ vlq n 4₅ U? Hvlq n4₅ U vlq c4_{5 q}m @ H vlq n4₅ U 4 q S_?  vlq c4_{5 q}m +77,

. 4 q S_?  vlq n 4₅ U Hvlq n4₅ U vlq c4_{5 q}m .rRW+4, @ H vlq n4₅ U 4 q S_?  vlq c4_{5 q}m ._q4 S|_vlq c4_{5 q}m +78, @ H vlq n4₅ U ] _ f vlq c 4₅ v gv.r+4, .rRW+4, +79, @ ] _ f vlq n4 5 u gu ] _ f vlq c 4 5 v gv .rRW+4, @ 4 n 2 4 c 2 .rRW+4,1 +7:,

Lq olqh +77, deryh/U lv xqlirupo| glvwulexwhg rq^3>4`>lq olqh +78, @ vlqn2

U Hvlqn₂Ullg+3> 2#,>zlwk2# @H+2,>dqg olqh +7:, iroorzv vlqfhUfvlq n₂u gu@ 4@+n 2,= Iurp +76, dqg +7:, zh ghgxfh wkdw 4 q S_|  {W | s q s{|q @ 5 [" &c' & n 2 c 2 2 .rRW+4, $ _W 5 +_" [ &' & n₂2 , + _" [ &' & n₂2 , d=v=+S, @ ] _ f E+u,gu ] _ f E+u,gu > jlylqj uhvxow +h,1 Sduw +i, iroorzv lq d vlplodu zd|1

Sduw +j, 4 q S_?  { W | s q w q @ q4 ? [ 3 E+U?, .rd1v1 4 q*23*R m q.rd1v1+4, @ _q4 S?_E+U?,_qm .rd1v1+4,=

Vhwd&@Hvlqn₂U@ 4@+n₂,>zkhuhUlv xqlirupo| glvwulexwhg rq^3>4`=Wkhq/

zh kdyh 4 q S_? E+U,_qm @ s 5[" &' _& n 2 4 q ? [ ' vlq n4 5 U? m q @ s5[" &' & n 2 4 q S_? d&_qm . 4_qS? vlq n4₅ U?d& m q @ s5[" &' & n 2 4 q S_? d&_qm . 4_qS? vlq n4₅ Ud& m q .rRW+4,

@ s5[" &' & n 2 2 ] _ f ugu.rR W+4, @ 4₅]  f E+u,gu.rR W+4,> +7;, jlylqj wkh vwdwhg uhvxow1 917 Surri ri Wkhruhp 517

Sduw +d, Iluvw frqvlghu eW @ q32S?

|'{W|{W|3@q32S?|'{W2|3= Iurp Ohppd 516 +f,/ wkh

ghqrplqdwru rieW kdv wkh iroorzlqj olplw

4 q2 S_? {W2|3$_W ] _ f E+u, 2_{gu d=v=+S,= +7<,} Xvlqj +9,/ wkh qxphudwru fdq eh zulwwhq dv 4 q2 S_? {W|{W|3 @ 4_qS? {W | s_q {sW|3_q @ 4_q[? ' ^E+U?, .rd1v1+4,` ^E+U?3, .rd1v1+4,`= +83,

Khuh/U? dqgU?3 duh lqghshqghqw gudzv iurp 4@q> ===>4 dqg/ lq ylhz ri +:,/

U? $_W U> U_?3$_{_}W U_3 d=v=+S,>

zkhuhU dqgU3 duh lqghshqghqw gudzv iurpX^3>4`=Wkxv/ +83, lv dv|pswrwlfdoo| htxly0

dohqw wr 4 q ? [ ' E+U,E+V,> +84,

zkhuhU/V duh lqghshqghqwX^3>4`=Xvlqj wkh ON uhsuhvhqwdwlrq riE zh wkhuhiruh qg

4 q ? [ ' E+U,E+V, @ 5$2 " [ &c' & n₂ c₂ 4 q S_? vlq n 2 Uvlqc₂V $ RW 5$2 " [ &c' _& n 2 c 2 H vlqn 2 UHvlqc₂V @ 5$2 [" &c' & n 2 c  2 ]  f vlq n ₂ugu]  f vlq c₂vgv @ ] _ f +E+u,,gu 2 = +85, Lw iroorzv iurp +7<,0+85, wkdw e W$_W ₄ @ kU_ f+E+u,,gu l2 U_ f E+u,2gu d=v=+S,>

dv vwdwhg1

Qh{w frqvlghueW@q32S?|'{|W{W|3@q32S?|'{W2|3=Wkh ghqrplqdwru kdv wkh vdph olplw

dv +7<,1 Wkh qxphudwru/ iroorzlqj +83,/ fdq eh zulwwhq dv

4 q2 S_? {W|{W|3 @ _q4S? {W | s q {W |3 s q @ _q4[? ' ^E+U?, .rd1v1+4,` ^E+U?3, .rd1v1+4,` @ 4 q ? [ ' E+U?,E+U?3, .rd1v1+4,=

Djdlq/U? dqg U?3 duh lqghshqghqw gudzv iurp4@q> ===>4dqgU?$_W U> U_?3$_{_}W

U3 d=v=+S,> zkhuh U dqg U3 duh lqghshqghqw X^3>4`= Wkxv/ +83, lv dv|pswrwlfdoo|

htxlydohqw wr +84,/ zkhuhU/ V duh lqghshqghqw X^3>4` dqg lqghshqghqw ryhum=Wkxv/ wkh

olplw +85, dssolhv dqg zh ghgxfh wkh vwdwhg uhvxow1

Sduw +e, Iluvw frqvlghu hW @ q32S?

|'{hW|{hW|3@q32S?|'h{W2|3= Iurp Ohppd 516 +f,/ wkh

ghqrplqdwru rihW _{kdv wkh iroorzlqj olplw}

4 q2 S_? {hW2|3$_W ] _ f E+u, 2_{gu d=v=+S,= +86,}

Surfhhglqj dv lq +83, dqg xvlqj +6<,/ wkh qxphudwru fdq eh zulwwhq dv

4 q S_?  _h{W | s q h {W |3 s q @ 4 q ? [ ' E+U?, ] _ f E+u,gu.rd1v1+4, E+U?3, ] _ f E+u,gu.rd1v1+4, > +87, zklfk lv dv|pswrwlfdoo| htxlydohqw wr 4 q ? [ ' E+U, ] _ f E+u,gu E+V, ] _ f E+u,gu @ 4_q[? ' E+U,E+V,=

zkhuhU/V duh lqghshqghqw X^3>4`=Fdofxodwlrqv dqdorjrxv wr wkrvh ohdglqj wr +85, vkrz

wkdw 4 q ? [ ' E+U,$RW ] _ f E+u,gu d=v=+S,> zklfk/ lq frpelqdwlrq zlwk +85,/ uhyhdo wkdw 4 q ? [ ' E+U,E+V,$RW 3 d=v=+S,= +88,

Zh ghgxfh wkdwhW $RW 3 d=v=+S, dqg wkh uhvxow iruhW iroorzv lq wkh vdph zd|1

Ilqdoo|/ li wkh vhuldo fruuhodwlrq frh!flhqw lv uhghqhg lq whupv ri ghyldwlrqv iurp phdqv dvee @q32S?

|'hW|hW|3@q32S?|'h|W2>wkhq wkh qxphudwruq32S?hW|hW|3 hylghqwo| kdv wkh

918 Surri ri Wkhruhp 518 Wdnh wkh vhuldo fruuhodwlrqeKW_@q32S?

|'{KW| {KW|3@q32S?|'{KW2|3=

Lq d pdqqhu dqdorjrxv wr wkh surri ri Ohppd 516 +f,/ wkh ghqrplqdwru rieW fdq eh vkrzq wr kdyh wkh iroorzlqj olplw

4 q2 S_? {KW2|3$_W ] _ f E+u, 2_{gu d=v=+S,= +89,} Xvlqj +45,/ wkh qxphudwru fdq eh zulwwhq dv 4 q S_?  {KW | s q #_{KW |3 s q $ @ _pP4  [ ' 6 [ &'2 E+U.n_q, .rd1v1+4, E+U.n_q4, .rd1v1+4, . 4 pP  [ ' E U._q4 .rd1v1+4, k EU3. p_q .rd1v1+4, l +8:, Lq +8:,/U dqgU3 duh lqghshqghqw gudzv iurp 3>_> ===>3_ dqg/ lq ylhz ri +46,/

U $_WU> U_3$_{_}W U_3 d=v=+S,>

zkhuhU dqgU3 duh lqghshqghqw gudzv iurpX^3>4`=Wkxv/ +8:, lv dv|pswrwlfdoo| htxly0

dohqw wr 4 P  [ ' 4 p 6 [ &'2 E+U._qn, .rd1v1+4, E+U.n_q4, .rd1v1+4, .rd1v1 4 p @  [ ' ] _-n6 ? -n_? E+u,2gu.rd1v1+4,= +8;,

E| yluwxh ri wkh frqwlqxlw| ri Eurzqldq prwlrq dqg wkh idfw wkdw 6

? @  $ 3> +8;, lv dv|pswrwlfdoo| htxlydohqw wr 4 P  [ ' E+U,2 Iurp +46,/ zh kdyh vlq n4₅ U $_W vlq n4₅ U d=v=+S,> dv lq +;,/ dqg 4 P  [ ' vlq n4 5 U vlq c4 5 U $RW H vlq n4 5 U vlq c4 5 U =

Wkhq/ dv lq wkh surri ri Ohppd 516 +f, zh qg wkdw 4 P  [ ' E+U,2 $ _W 5$2 " [ &c' & n 2 c 2 ] _ f vlq n 2 uvlqc 2 ugu @ ]  f E+u, 2_{gu= +8<,} Frpelqlqj +89,0+8<, zh ghgxfh wkdw eKW_$ _W U_ f E+u,2gu U_ f E+u,2gu @ 4 d=v=+S,> dv vwdwhg1 Surriv iruhKW

dqgeKe iroorz lq d vlplodu idvklrq1

919 Surri ri Wkhruhp 519 Zulwh

q P+eW4, @ qP4 ? [ |' {W |{W|3@_q4₂ ? [ |' {W2 |3= +93,

Wkh olplw ri wkh ghqrplqdwru lv jlyhq lq +89,1 Iru wkh qxphudwru/ wkh lghqwlw|

[? |' {W2 | @ ? [ |' {W |{W| . ? [ |' {W |3{W| @ 5 ? [ |' {W |3{W| . ? [ |' +{W |,2 ohdgv wr 4 qP ? [ |' {W|3{W| @ 4 5P + {W ? s q ₂ 4 q ? [ |' +{W |,2 , @ _5P4 + {W ? s q 2 _pP4 [ r' 6 [ &'2 {Er36n&{Er36n&3 2 4 q  [ r' {Er36n{Er336n6 2 , @ _5P4 +_{ E36n6_s q 2 _pP4 [ r' 6 [ &'2 x2_Er36n&[ r' _{ Er36n_s{Er336n6 q 2, +94, zkhuhimr=v@ 4> ===> Pjduh lqghshqghqw xqlirup gudzv iurp4> ===> P>dqgmf @ 3=Iurp +47,

dqg d eorfn errwvwuds yhuvlrq ri Ohppd 515/ zh qg wkdw {E_s36n6

q $_W E+U, d=v=+S,> +95,

zkhuhUlv xqlirup ryhu ^3>4`=Qh{w/ iru doov

4 p 6 [ &'2 x2 Er36n&$RW 2 d=v=+S, +96,

zkhuh2 @Hx2|=Ilqdoo| 4 P  [ r' _{ Er36n_s{r36 q 2 @ _P4 [ r' E+Ur._q4,E+Ur3. p q, .rd1v1+4, 2 @ _P4  [ r' E+Ur,E+Ur3, ₂ .rd1v1+4, $ _W ] _ f ] _ f ^E+u,E+v,` 2_{gugv d=v=+S, +97,}

zkhuhUdqgV duh lqghshqghqw xqlirup yduldwhv rq ^3>4`=Frpelqlqj +94, 0 +97,/ zh rewdlq wkh iroorzlqj olplw iru wkh qxphudwru ri +93,

4 qP ? [ |' {W |3{W| $ _W4 5 ] _ f ] _ f ^E+u,E+v,` 2_{gugv d=v=+S,} @ +] _ f E+u, 2_gu]  f E+u,gu 2,

dqg wkh vwdwhg uhvxow iroorzv1 Li fhqwuhg errwvwuds uhvdpsolqj lv xvhg/ wkhq vpdoo prglfdwlrqv wr wkh deryh dujxphqw uhyhdo wkdw

4 qP ? [ |' h {W |3h{W| $ _W 4 5 ] _ f ] _ f ^E+u,E+v,` 2_{gugv d=v=+S,} @ ]  f E+u, 2_gu> dqg wkhq ? +eW4,$RW4= 91: Surri ri Wkhruhp 614 Sduw +d, VlqfheeW ?@ee?.S?_|'wxW|@S?|'w2> s qeeW?ee? @ q3 D 2S?_|'xW_|w q3S? |'w2 Wkh errwvwuds vdpsoh+xW

|,? lv gudzq udqgrpo| iurp wkh uhvlgxdov+xe|@{|ee?w,?=Iurp +7,

dqg +4<, zh fdq zulwh a x| s q @ s{|q s qae?_qw @E w q +.rd1v1+4,,_qw .rd1v1+4,>

zklfk ohdgv wr wkh iroorzlqj uhsuhvhqwdwlrq iru wkh errwvwuds surfhvv xW | a xW | s q @E+U?,U?.rd1v1+4,> +98,

dv lq +9,/ zkhuhU? lv xqlirupo| glvwulexwhg ryhu i_?>_?2> ==>4jiru hdfkm dqg vdwlvhv +:,1 Lq +98, E+U?,U? @E+U?, ] _ f uE+u, ] _ f u 23_U ?@Eo+U?,> +99,

zkhuhEo+v, @E+v,+U_fuE+u,gu,+U_fu2,3vlv ghwuhqghg Eurzqldq prwlrq1 Wkhq s qeeW ?ee? @ q3 D 2S?_|'xW_|w q3S? |'w2 @  ?S?'^E+U?,U?.rd1v1+4,` ? q3S? |'w2 @  ?S?'^E+U?,U?.rd1v1+4,` ? q3S? |'w2 = Dv lq +7;, 4 q ? [ ' E+U?, m q $_W ]  f E+u,gu ]  f ugu@ 4 5 ]  f E+u,gu d=v=+S,> dqg 4 q ? [ ' U? m q $RWH+U, ] _ f ugu@ ] _ f ugu 2 @ 4 7= Lw iroorzv wkdw s qeeW?ee? $_W U_ f Eo+v,gv U_ f vgv U_ f u2gu @ 2 U_ f E+u,gu e  d=v=+S,> dqg vr s qeeW ? $ _W W @. U_ f Eo+v,gv U_ f vgv U_ f u2gu d=v=+S, +9:, @ 6₅]  f E+u,gu. 4 7> dv uhtxluhg1

Sduw +e, Wkh vwdqgdugl}hg uhvlgxdo iurp wkh errwvwuds uhjuhvvlrq +56, lv xWW | s q @ {WW | s q s qeeW ?_qw @ e_e?w.xW | s q s qeeW ?_qw @ sxW| q s qeeW ?ee? _w q= +9;, Iurp +98,/ _IW| ? @E+U?,U?.rd1v1+4,>dqg iurp sduw +d, dqg +9:, s qeeW ?ee? $_W W

> d=v=+S,=Lw iroorzv iurp +98,0+9;, wkdw wkh olplw glvwulexwlrq riq3_vWW2_@q32S? |'xWW2|

lv wkh vdph dv wkdw ri 4 q ? [ ' Eo+U?, . +W,_qm ₂ @ 4 q ? [ ' Eo+U?,2.5 +W,_q4 ? [ ' Eo+U?,_qm.+W,2 _q4 ? [ ' m q ₂ = +9<, Dv lq Ohppd 516 +f, dqg +j, zh kdyh 4 q ? [ ' Eo+U?,2 $ _W ] _ f Eo+v, 2_{gv d=v=+S,> +:3,} 4 q ? [ ' Eo+U?,_qm $ _W 4 5 ] _ f Eo+u,gu@ ] _ f Eo+v,gv ] _ f vgv d=v=+S,= +:4, Lw iroorzv wkdw 4 qvWW2 $ _W ] _ f Eo+v, 2_{gv. +}W_,]  f Eo+v,gv ] _ f vgv. + W_,2]  f vgv d=v=+S, @ ]  f iEo+v, . + W_,vj2_gu@2 1> vd|> +:5, jlylqj wkh vwdwhg uhvxow1

Sduw +f, Wkh errwvwuds uhjuhvvlrq w0udwlr lv w+aeW ?, @ a eW ? v _KW @ a_eW ? ^vWW2_@S? w2`*2 @ s q sqaeW? ^ ?vWW2@?S?<sub></sub>w2`*2 dqg wkhq 4 s qw+aeW?,$_W WU f u2  2 1 @ W s 61 d=v=+S,= Sduw +g, Zulwh eWW @ q32S? |'xWW| xWW|3@q32S?|'xWW2|3= Dv lq +9;,0+9<, deryh/ q3  2xWW_|

ehkdyhv dv|pswrwlfdoo| olnhEo+U?, . +W,_?=Lw iroorzv wkdwq32S?|'xWW| xWW|3lv dv|ps0

wrwlfdoo| htxlydohqw wr 4 q ? [ ' Eo+U?, . +W,_qm Eo+U?3, . +W,m_q4 @ _q4[? ' Eo+U?,Eo+U?3, . +W,_q4 ? [ ' m qEo+U?3, . +W,q4 ? [ ' m4 q Eo+U?, . +W,2 _q4 ? [ ' m qmq4=

Surfhhglqj dv lq +83,0+85, zh qg 4 q ? [ ' Eo+U?,Eo+U?3, $ _W ] _ f Eo+v,gv 2 d=v=+S,> +:6, 4 q ? [ ' m qEo+U?3, $ _W ] _ f vgv ] _ f Eo+v,gv > +:7, dqg wkhq q32 ? [ |' xWW| xWW|3$_W ] _ f Eo+v,gv. + W_,]  f vgv 2 d=v=+S,= Dv lq +:5, q32[? |' xWW2|3$_W ] _ f iEo+v, . + W_,vj2_gv@2 1 d=v=+S,= +:8, Lw iroorzv wkdw eWW$_W Koe}₄ @ U_ f ^Eo+v, . +W,v`gv 2 U_ f iEo+v, . +W,vj2gv >

dv jlyhq lq wkh wkhruhp1 Wkh uhvxow iru eWW iroorzv lq suhflvho| wkh vdph pdqqhu +f1i1 wkh

surri ri sduw +d, ri Wkhruhp 517,1

Sduw +h, Zulwh GZWW _{@ q}32S? 2+axWW| ,2@q32S?xaWW2| zkhuh axWW| @ {WW| aeW? @ ae? a_eW ?. axW| iurp +55, dqg +56,1 Khqfh 4 q2 ? S |'2+x WW | ,2 @ _q4 S xW | s q xW |3 s q .rRW+4, 2 @ _q4 [? ' ^+E+U?,U?,+E+U?3,+U?3,,`2.rRW+4,> zkrvh dv|pswrwlf ehkdylru lv wkh vdph dv 4 q ? [ ' ^iE+U,+U,j iE+V,Vj`2 @ _q4 ? [ ' ^Eo+U,Eo+V,`2 @ _q4 [? ' Eo+U,2._q4 ? [ ' Eo+V,2_q5 ? [ ' Eo+U,Eo+V,>

zkhuhU/V duh lqghshqghqwllgX^3>4`=Dv lq +:3, dqg +:6, deryh

4 q ? [ ' Eo+U,2> _q4 ? [ ' Eo+V,2 $ _W ] _ f Eo+v, 2_{gv d=v=+S,>} 4 q ? [ ' Eo+U,Eo+V, $ _W ] _ f Eo+v,gv 2 d=v=+S,>

vr wkdw 4 q2 ? S |'2+x WW | ,2 $_W5 ] _ f Eo+v, 2_gv5]  f Eo+v,gv 2 d=v=+S,> zklfk fdq eh zulwwhq lq wkh irup 5 ] _ f Eo+v, 2_gv 5 9 74 U_ f Eo+v,gv 2 U_ f Eo+v,2gv 6 : 8@ 5 ] _ f Eo+v, 2_gvk₄K 4 l > zkhuh K4 @ U_ f Eo+v,gv 2 U_ f Eo+v,2gv = Iurp +:8,/ wkh olplw ri wkh ghqrplqdwru riGZWW _lv

q32[? |' xWW2| $_W ] _ f iEo+v, . + W_,vj2_gv@2 1 d=v=+S,> dqg vr GZWW_$ _W 5 U_ f Eo+v,2gv 2 1 k 4K 4 l d=v=+S,=

Sduw +i, Zulwh UWW2 _{@ 4q}32S?

xaWW2| @q32S?{WW2| =Zh kdyh{WW| @ee?w.xW| dqg vr 4 q ? S |' {WW | s q 2 @ _q4 S? |' _s qee?_qw . x W | s q 2 > zkrvh dv|pswrwlf ehkdylru lv wkh vdph dv 4 q ? S ' m q.E+U,U ₂ @ 24_q S? ' m q 2 ._q4 S? 'E+U, 2_.24 q ? S 'U 2 .54_q S? 'E+U, m q 52q4 ? S 'U m q5q4 ? S 'E+U,U $ _W2 ] _ f u 2_gu.]  f E+u, 2_gu.2]  f u 2_{gu d=v=+S,} .5]  f E+u,gu ]  f ugu5 2]  f u 2_gu5]  f E+u,gu ]  f ugu d=v=+S, @ ]  f E+u, 2_gu= Lw iroorzv wkdw U2 $_W 4 2 1 U_ f E+u,2gu @ 4 U_ f iE_U +v,Wvj2gv  f E+u,2gu d=v=+S,=

91; Surri ri Wkhruhp 616 Wkh surriv khuh dqg lq odwhu dujxphqwv duh vlplodu wr zkdw kdv frph ehiruh dqg vr ghulydwlrqv khqfhiruwk duh vlpso| vnhwfkhg1

Sduw +d, VlqfheeW ?@ee?.S?|'whxW|@S?|'w2> s qeeW ?ee? @ q3 D 2S?_|'hxW_|w q3S? |'w2 = Wkh errwvwuds vdpsoh+xhW

|,? lv gudzq udqgrpo| iurp wkh fhqwuhg uhvlgxdov+ex|q3S?_|'xe|@

{|{ee?+ww,,?=Iurp +7, dqg +4<, zh fdq zulwh a x| s q 4 q2 ? [ |' e x|@E w q ]  f E+u,gu+.rd1v1+4,, w q ] _ f ugu .rd1v1+4,>

zklfk ohdgv wr wkh iroorzlqj uhsuhvhqwdwlrq iru wkh errwvwuds surfhvv hxW | h xW | s q @E+U?,U?.rd1v1+4, @Eo+U?, .rd1v1+4,> +:9, zkhuhu@uU_fugu=Wkhq/ mxvw dv lq sduw +d, ri Wkhruhp 614 zh qg wkdw

s qeeW ?ee? $_W U_ f Eo_U+v,gvUfvgv  f u2gu @ 3 d=v=+S,> +::, dqg vr _s qeeW?$_W d=v=+S,=

Sduwv +e, ) +f, Wkh vwdqgdugl}hg uhvlgxdo iurp wkh errwvwuds uhjuhvvlrq +56, lv xWW | s q @ { WW | s q s qeeW?_qw @ e_e?w.hxW | s q s qeeW?_qw @ shxW| q s qeeW?ee? _w q=

Lw iroorzv iurp +:9, dqg +::, wkdw wkh olplw glvwulexwlrq ri q3_vWW2 _{@ q}32S?

|'xWW2| lv wkh vdph dv wkdw ri 4 q ? [ ' E_o+U?,2$_W ] _ f Eo+v, 2_{gv d=v=+S,>}

xvlqj wkh vdph dujxphqwv dv wkrvh lq Ohppd 516 +f,1 Wkh errwvwuds uhjuhvvlrq w0udwlr lv w+aeW ?, @ a eW ? v _KW @ a_eW ? ^vWW2_@S? w2`*2 @ s q sqaeW? ^<sub>?</sub>vWW2<sub>@</sub>  ? S<sub>?</sub> w2`*2 dqg wkhq 4 s_qw+aeW?,$_W U_fu22 U_ f Eo+v,2gv  2 d=v=+S,=

Sduw +g, ZulwheWW@q32S?

|'xWW| xWW|3@q32S?|'xWW2|3=Dv lq +9;,0+9<, deryh/q3 

2xWW_| eh0

kdyhv dv|pswrwlfdoo| olnhEo+U?, @Eo+U?,U_fEo+v,gv=Lw iroorzv wkdw q32S?|'xWW| xWW|3

lv dv|pswrwlfdoo| htxlydohqw wr 4 q ? [ ' E_o+U?,Eo+U?3, @ 4 q ? [ ' Eo+U?,Eo+U?3, ] _ f Eo+v,gv 4 q ? [ ' Eo+U?3, ]  f Eo+v,gv 4 q ? [ ' Eo+U?, . ] _ f Eo+v,gv 2 = Dv lq +:6, zh qg 4 q ? [ ' Eo+U?,Eo+U?3, $ _W ] _ f Eo+v,gv 2 d=v=+S,> 4 q ? [ ' Eo+U?3, $ _W ] _ f Eo+v,gv> dqg wkhq q32[? |' xWW | xWW|3$RW 3 d=v=+S,>

iurp zklfk wkh uhvxow irueWWiroorzv1 Wkh uhvxow irueWW iroorzv lq wkh vdph zd|1

Sduw +h, Zulwh GZWW _{@ q}32S? 2+axWW| ,2@q32S?xaWW2| zkhuh axWW| @ {WW| aeW? @ ae? a_eW ?. hxW| iurp +55, dqg +56,1 Khqfh 4 q2 ? S |'2+x WW | ,2 @ _q4 S h xW | s q h xW |3 s q .rRW+4, 2 @ 4 q ? [ ' ^E_o+U?,Eo+U?3,`2.rRW+4,> zkrvh dv|pswrwlf ehkdylru lv wkh vdph dv 4 q ? [ ' ^Eo+U,Eo+V,`2 @ _q4 ? [ ' Eo+U,2. 4_q ? [ ' Eo+V,2 5_q ? [ ' Eo+U,Eo+V,>

zkhuhU/V duh lqghshqghqwllgX^3>4`=Dv lq +:3, dqg +:6, deryh

4 q ? [ ' E_o+U,2> _q4 ? [ ' E_o+V,2 $ _W ] _ f Eo+v, 2_{gv d=v=+S,>} 4 q ? [ ' Eo+U,Eo+V, $ _W ] _ f Eo+v,gv 2 @ 3 d=v=+S,>

vr wkdw 4 q2 ? S |'2+x WW | ,2 $_W 5 ] _ f Eo+v, 2_{gv d=v=+S,>}

Iurp wklv uhvxow dqg sduw +e, zh ghgxfh wkdwGZWW_{$RW 5 d=v=+S,>dv vwdwhg1} Sduw +i, Zulwh UWW2 _{@ 4q}32S?

xaWW2| @q32S?{WW2| =Zh kdyh{WW| @ee?w.xhW| dqg vr 4 q ? S |' {WW | s q ₂ @ 4 q ? S |' _s qee?_qw . hx W | s q ₂ > zkrvh dv|pswrwlf ehkdylru lv wkh vdph dv 4 q ? S ' _qm .Eo+U, 2 @ 2_q4 S? ' m q 2 ._q4 S? 'Eo+U, 2_{. 5}4 q ? S 'Eo+U, m q $ _W2 ] _ f u 2_gu.]  f Eo+v, 2_{gv. 5}]  f Eo+u,gu ] _ f ugu d=v=+S, @ ]  f Eo+v, 2_gv.2]  f u 2_gu

Lw iroorzv iurp wklv uhvxow dqg sduw +e, wkdw U2 $_W4 U_ f Eo+v,2gv U_ f Eo+v,2gv.2Ufu2gu @ 2 U_ f u2gu U_ f Eo+v,2gv.2Ufu2gu d=v=+S,= 91; Surri ri Wkhruhp 618

Sduw +d, Vwduw zlwk wkh irupxod

s qeeKW ? ee? @ q3 D 2S?_|'xhKW_| w q3S? |'w2 > dqg/ dv lq +5:,05:/ iruw@ +v4,p.nzh fdq zulwh h xKW | s q @ e xEr36n&q3 S_? r'xer s q @Eo Ur._qn .r@r+4, +:;, Wkhq/ dv lq sduw +d, ri Wkhruhpv 614 dqg 616 zh qg wkdw s qeeKW? ee? $_W U f Eo_U+v,gvUfvgv  f u2gu @ 3 d=v=+S,> +:<, dqg vrsqeeW ?$_W d=v=+S,=

Sduwv +e, ) +f, Wkh vwdqgdugl}hg uhvlgxdo iurp wkh uhjuhvvlrq +59, lv xKWW | s q @ {KWW | s q s qeeKW ? _qw @ e_e?w.xhKW | s q s qeeKW ? _qw @ xh KW | s q s qeeKW ? ee? _w q= +;3, Lw iroorzv iurp +:<, wkdw q3

2xKWW_| lv dv|pswrwlfdoo| htxlydohqw wr q32xhKW_| = Wkhq/ iurp +:;,

dqg wkh frqwlqxlw| riEo+v,> wkh olplw glvwulexwlrq ri q3vKWW2 @q32S?|'xKWW2| lv wkh vdph dv wkdw ri 4 P  [ r' Eo+Ur,2$_W ] _ f Eo+v, 2_{gv d=v=+S,> +;4,}

0 frpsduh wkh dujxphqw ohdglqj wr +;7, ehorz1 Sduw +f, iroorzv gluhfwo|1

Sduw +g, Wdnh wkh vhuldo fruuhodwlrqeKWW @q32S?|'x|KWWxKWW|3@q32S?|'xKWW2|3=Wkh ghqrp0

lqdwru rieKWW _{lv fryhuhg lq sduw +e,1 Lq ylhz ri +;3, dqg +:<,/ wkh qxphudwru lv dv|pswrwlfdoo|}

htxlydohqw wr 4 q S_?  h xKW | s q #_xhKW |3 s q $ @ _pP4 [ r' 6 [ &'2 Eo+Ur .n_q, .rd1v1+4, Eo+Ur .n_q4, .rd1v1+4, ._pP4 [ r' Eo Ur ._q4 .rd1v1+4, k Eo Ur3.p_q .rd1v1+4, l +;5, Lq +;5,/Ur dqgUr3 duh lqghshqghqw gudzv iurp3> > ===>3 dqg/ lq ylhz ri +46,/

Ur $_WU> Ur3 $_W U3 d=v=+S,>

zkhuhU dqgU3 duh lqghshqghqw gudzv iurpX^3>4`=Wkxv/ +;5, lv dv|pswrwlfdoo| htxly0

dohqw wr 4 P  [ r' 4 p 6 [ &'2 Eo+Ur._qn, .rd1v1+4, Eo+Ur.n_q4, .rd1v1+4, .rd1v1 4 p @ [ r' ] _-rn6 ? -rn2? Eo+u,2gu.rd1v1+4, @  [ r' ] _-rn6 ? -rn? Eo+u,2gu.rd1v1+4, +;6,

E| yluwxh ri wkh frqwlqxlw| ri Eo+v, dqg wkh idfw wkdw 6? @  $3>+;6, lv dv|pswrwlfdoo|

htxlydohqw wrP3S r'Eo+Ur,2=Wkhq/ mxvw dv lq wkh surri ri Wkhruhp 516/ zh qg wkdw 4 P  [ r' Eo+Ur,2 $_W ] _ f Eo+u, 2_{gu= +;7,}

Frpelqlqj wkhvh uhvxowv iru wkh qxphudwru dqg ghqrplqdwru/ zh ghgxfh wkdw e KWW  $_W U_ f Eo+u,2gu U f Eo+u,2gu @ 4 d=v=+S,> dv vwdwhg1 Wkh surri irueKWW iroorzv lq d vlplodu idvklrq1

Sduw +h, Zulwh GZKWW _{@ q}32S? 2+xKWW| ,2@q32S?xKWW2| zkhuh xKWW| @ {KWW| aeKW? @ a_e?a_eKW ? . xhKW| =Xvlqj sduw +d, zh kdyh 4 q2 ? S |'2+x KWW | ,2 @ _q4 S? |'2 % h xKW | s q h xKW |3 s q .rRW+4, &2 @ 4 q  [ r' 6 [ &'2 ^E_o+Ur. n q,Eo+Ur. n4 q , .rd1v1+4,`2 . 4 pP  [ r' E<sub>o</sub> Ur.<sub>q</sub>4 E<sub>o</sub>Ur3.p<sub>q</sub> .rd1v1+4, <sub>2</sub> .rRW+4,= +;8, Qrz 4 q  [ r' 6 [ &'2 ^Eo+Ur . n q,Eo+Ur . n4 q ,`2@rRW+ 4 q3B,> A3> +;9,

e| yluwxh ri wkh Kroghu frqwlqxlw| riE_o dqg/ dv lq +97, deryh/

4 P  [ r' Eo Ur._q4 Eo Ur3.p_q 2 $_W ] _ f ] _ f ^Eo+u,Eo+v,` 2_{gugv d=v=+S,} +;:, Wkxv/ 4 q2 ? S |'2+x WW | ,2$RW 3 d=v=+S,> Lw iroorzv gluhfwo| wkdwGZKWW_{$RW 3 d=v=+S,>dv vwdwhg1} Sduw +i, ZulwhUKWW2_{@ 4q}32S?

 xKWW2| @q32S?{KWW2| =Zh kdyh{KWW| @ee?w.xhKW| dqg iurp +5<,/q3 2{KWW_d?oo@u.E_o+U_co, .r_@r+4,=Wkhq/ 4 q ? S |' {KWW | s q 2 @ 4 q ? S |' _s qee?_qw . xh KW | s q 2 > zkrvh dv|pswrwlf ehkdylru lv wkh vdph dv 4 q  [ r' 6 [ &' +v4,_qp.n .Eo+Ur.n_q, 2 $_W 2 ]  f u 2_gu.]  f Eo+v, 2_{gv. 5}]  f Eo+u,gu ] _ f ugu d=v=+S, @ ]  f Eo+v, 2_gv.2]  f u 2_gu=

Lw iroorzv iurp wklv uhvxow dqg sduw +e, wkdw U2 $_W4 U_ f Eo+v,2gv U_ f Eo+v,2gv.2Ufu2gu @ 2 U_ f u2gu U_ f Eo+v,2gv.2Ufu2gu d=v=+S,=