Exps Kxqwlqj lq Kljk�Glphqvlrqdo Gdwd

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Nh|zrugv= Gdwd Plqlqj/ qrlv| ixqfwlrq rswlpl}dwlrq/ fodvvl�fdwlrq/ dvvrfldwlrq/ uxoh lqgxfwlrq1

41 LqwurgxfwlrqWkh sxusrvh ri pdq| gdwd dqdo|vhv fdq eh ylhzhg lq wkh frqwh{w ri �suhglfwlrq�1 Wkh gdwd edvhfrqwdlqv uhshdwhg revhuydwlrqv ri d ghvljqdwhg �rxwsxw� yduldeoh | dorqj zlwk vlpxowdqhrxv ydoxhv ridgglwlrqdo �lqsxw� yduldeohv { @ +{ � > { � > � � �> { � ,1 Wkh jrdo lv wr xvh wkhvh gdwd

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s+| m{, @ s+|>{,U s+|>{,g|

zklfk lv prvw vlpso| +li lqfrpsohwho|, ghvfulehg e| lwv �uvw prphqw

i+{, @ H^| m{` @]

| s+| m{,g|= +415,

Wklv txdqwlw| +415, lv zhoo nqrzq wr plqlpl}h phdq0vtxduhg suhglfwlrq huuru dw hdfk {

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H{suhvvlqj wkh rxwsxw yduldeoh dv| @ H^| m{` . +| �H^| m{`, @ i+{, . % +416,

rqh vhhv wkdw wklv suhglfwlrq sureohp fdq eh fdvw dv rqh ri ixqfwlrq hvwlpdwlrq zkhuh wkh jrdo lv wrdssur{lpdwh +�ohduq�, d ghwhuplqlvwlf �wdujhw� ixqfwlrq i+{, iurp d vhw ri revhuydwlrqv zkhuh lwv ydoxh+dw hdfk {, lv fruuxswhg e| udqgrp qrlvh %1 Wklv qrlvh uhsuhvhqwv wkh udqgrp glvwulexwlrq ri wkh rxwsxw| derxw lwv phdq ydoxh i+{,/ dw hdfk {/ dqg fkdudfwhul}hv wkh idfw wkdw vshfli|lqj d vlpxowdqhrxv vhwri lqsxw ydoxhv grhv qrw vshfli| d xqltxh |0ydoxh> rwkhu idfwruv/ qrw fdswxuhg e| wkh vhw ri phdvxuhglqsxwv/ lq xhqfh wkh rxwsxw ydoxh1

Dowkrxjk uhvwulfwlqj dwwhqwlrq wr wkh �uvw prphqw +415, juhdwo| vlpsol�hv wkh sureohp/ lw lv vwlood iruplgdeoh rqh1 Wkh sureohp ri dffxudwho| dssur{lpdwlqj d jhqhudo ixqfwlrq ri pdq| dujxphqwv/hyhu|zkhuh zlwklq vrph grpdlq ri lqsxw ydoxhv/ edvhg rq vdpsohg gdwd +zlwk ru zlwkrxw qrlvh, uhpdlqvd gl!fxow rqh1 Dv vxfk/ lw kdv ehhq wkh irfxv ri pxfk uhvhdufk lq wkh �hogv ri pdwkhpdwlfv/ qhxudoqhwzrunv/ pdfklqh ohduqlqj/ dqg vwdwlvwlfv ^ vhh Oruhqw} +4<;9,/ Elvkrs +4<<8,/ Ulsoh| +4<<9,/ Plwfkhoo+4<<:,/ dqg Zdked +4<<3,`1 Wkh vhw ri ixqfwlrqv dphqdeoh wr dffxudwh dssur{lpdwlrq zlwk fxuuhqwphwkrgrorj| lv vwloo uhodwlyho| vpdoo dqg wkhuh pd| zhoo eh pdq| wdujhw ixqfwlrqv wr eh hqfrxqwhuhglq sudfwlfh wkdw uhpdlq hoxvlyh1

Riwhq/ ixqfwlrq dssur{lpdwlrq lv dssolhg lq vlwxdwlrqv iru zklfk wkh dfwxdo gdwd dqdo|wlf jrdo lvidu pruh prghvw> wkh lqwhuhvw lv lq vrph surshuw| ri wkh wdujhw ixqfwlrq1 D frpprq surfhgxuh lqvxfk vlwxdwlrqv lv wr dwwhpsw wr hvwlpdwh wkh wdujhw i+{, hyhu|zkhuh lq wkh lqsxw vsdfh dqg dvfhuwdlqwkh surshuw| ri lqwhuhvw iurp wkh uhvxowlqj hvwlpdwh ai+{,1 Iuhtxhqwo| krzhyhu/ wklv vwudwhj| fdq ehfrxqwhu�surgxfwlyh lq wkdw dq dowhuqdwlyh rqh irfxvhg gluhfwo| rq hvwlpdwlqj wkh surshuw| ri lqwhuhvwpd| jlyh ulvh wr kljkhu dffxudf|1

Rqh h{dpsoh ri wklv skhqrphqrq lv +5 0 fodvv, fodvvl�fdwlrq1 Khuh wkh rxwsxw yduldeoh | dvvxphvwzr ydoxhv/ | @ 3 lqglfdwlqj wkh �uvw fodvv dqg | @ 4 lqglfdwlqj wkh vhfrqg1 Wkh wdujhw ixqfwlrq lv

i+{, @ H^| m{` @ Su+| @ 4 m{, +417,dqg wkh rswlpdo +plqlpxp huuru, fodvvl�fdwlrq ghflvlrq lv

| @ 4+i+{, A 4@5,> +418,zkhuh 4+�, wdnhv wkh ydoxh rqh zkhq lwv dujxphqw lv wuxh dqg }hur rwkhuzlvh1 Lw lv frpprq wr dsso|ixqfwlrq dssur{lpdwlrq phwkrgrorj| lq wklv vlwxdwlrq/ lqvhuwlqj wkh hvwlpdwh ai+{, lq +418, wr pdnhsuhglfwlrqv a|1 Ydsqln +4<<8, dqg Iulhgpdq +4<<:, kdyh vkrzq wkdw wklv fdq surylgh srru shuirupdqfhuhodwlyh wr surfhgxuhv wkdw wu| wr gluhfwo| hvwlpdwh wkh ghflvlrq erxqgdu| i+{, @ 4@5/ dqg vrphwlphvohdgv wr frxqwhu�lqwxlwlyh uhvxowv1 Iru h{dpsoh lpsurylqj wkh txdolw| ri wkh ixqfwlrq hvwlpdwh fdqdfwxdoo| ghjudgh fodvvl�fdwlrq shuirupdqfh1

Dqrwkhu h{dpsoh lv +xqlyduldwh, ghqvlw| hvwlpdwlrq zkhuh wkh wdujhw i+{, lv wkh uhodwlyh suredelolw|ri dq revhuydwlrq dw {1 Wkh surshuw| ri lqwhuhvw lv wkh fxpxodwlyh glvwulexwlrq ixqfwlrq

I +{, @] �

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aI +{, @ 4Q

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wkh hqwluh lqsxw vsdfh�i @

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5

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| � +41;,

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51 Ixqfwlrq rswlpl}dwlrqIxqfwlrq dssur{lpdwlrq riwhq lv dssolhg lq frqwh{wv zkhuh wkh surshuw| ri lqwhuhvw lv wkh rswlpdo+pd{lpxp ru plqlpxp, ydoxhv ri wkh wdujhw i+{, +415,1 Vshfl�fdoo|/ rqh vhhnv d vxeuhjlrq ri wkhvsdfh ri lqsxw ydoxhv zlwklq zklfk wkh dyhudjh ydoxh ri wdujhw lv pxfk odujhu +ru vpdoohu, wkdw lwvdyhudjh ryhu wkh hqwluh lqsxw vsdfh +41:,1 Vlqfh plqlpl}lqj d ixqfwlrq fdq eh dfklhyhg e| pd{lpl}lqjlwv qhjdwlyh/ rqo| pd{lpl}dwlrq lv frqvlghuhg khuh zlwkrxw orvv ri jhqhudolw|1 Ohw V� eh wkh vhw ri doosrvvleoh ydoxhv iru wkh lqsxw yduldeoh { �

i{ � 5 V � j �� � � = +514,Wkh lqglylgxdo V � frxog uhsuhvhqw uhdo +shukdsv glvfuhwh, ydoxhv/ ru fdwhjrulfdo +xqrughudeoh, ydoxhv1Wkh hqwluh lqsxw grpdlq V fdq wkhq eh uhsuhvhqwhg e| wkh q0glphqvlrqdo +rxwhu, surgxfw vsdfh

V @ V � � V � � � � � � V � = +515,Wkh jrdo lv wr �qg d vxeuhjlrq U ri wkh lqsxw grpdlq V/ U � V/ iru zklfk

�i � @ dyh ����� i+{, @]

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zkhuh �i lv wkh dyhudjh ryhu wkh hqwluh lqsxw vsdfh +41:,1 Dq lpsruwdqw surshuw| ri dq| vxfk vxeuhjlrqlv lwv vl}h +�vxssruw�,

� � @]

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61 Gluhfw dssolfdwlrqvWkhuh duh pdq| flufxpvwdqfhv lq zklfk dwwhqwlrq lv gluhfwo| irfxvhg rq wkh h{wuhphv ri wkh wdujhwixqfwlrq i+{, +416,1 Lq iruhfdvwlqj ixwxuh uhwxuqv ri �qdqfldo vhfxulwlhv rqh lv jhqhudoo| lqwhuhvwhg lqlghqwli|lqj wkrvh wkdw surylgh wkh kljkhvw uhwxuq1 Khuh wkh rxwsxw | lv wkh uhwxuq dqg wkh lqsxwv {

pljkw eh sdvw uhwxuqv dqg ydulrxv hfrqrplf idfwruv1 Wkh vxssruw � � +517, uhsuhvhqwv wkh iudfwlrq risrwhqwldo ehwv1 Lq pdunhw uhvhdufk | pljkw uhsuhvhqw vrph fxvwrphu ehkdylru dqg wkh lqsxwv { zrxogeh ydulrxv ghprjudsklf yduldeohv1 Wkh vxssruw � � lv wkh vl}h ri wkh lghqwl�hg pdunhw vhjphqw1 Lqphglfdo dssolfdwlrqv wkh rxwsxw frxog eh d phdvxuh ri vhyhulw| ri dq looqhvv dqg wkh lqsxwv ydulrxvv|pswrpv dqg phglfdo phdvxuhphqwv1 Wkh jrdo zrxog eh wr lghqwli| wkh fkdudfwhulvwlfv ri wkh prvwvhyhuho| loo sdwlhqwv/ shukdsv iru ixuwkhu gldjqrvwlf whvwlqj ru hvshfldoo| djjuhvvlyh wuhdwphqw1 Khuhwkh vxssruw lv wkh iudfwlrq ri sdwlhqwv iru zklfk vxfk whvwlqj ru wuhdwphqw lv ihdvleoh1

6

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Lq wkhvh surfhvv frqwuro dssolfdwlrqv wkh vxssruw � � lv jhqhudoo| ri qr gluhfw lqwhuhvw1 Lw fdq eh ylhzhgdv d �phwd�0sdudphwhu ri wkh surfhgxuh zkrvh ydoxh lv wdnhq wr eh dv vpdoo dv srvvleoh frqvlvwhqw zlwkpd{lpl}lqj �i � +516, dv rssrvhg wr �| � +518,1

Lq doo ri wkhvh dssolfdwlrqv/ dv zhoo dv prvw rwkhuv lq gdwd dqdo|vlv/ wkh rxwsxw yduldeoh | uhsuhvhqwvqrlv| phdvxuhphqwv ri wkh wdujhw i+{,1 Wkh vhw ri revhuyhg lqsxwv vhogrp frpsohwho| fkdudfwhul}h doosrvvleoh idfwruv wkdw lq xhqfh ydoxhv ri wkh rxwsxw1

71 Lqgluhfw dssolfdwlrqvEhvlghv wkrvh wkdw gluhfwo| irfxv rq rswlpl}dwlrq/ wkhuh duh pdq| rwkhu frpprqo| dssolhg gdwd dqdo|wlfsurfhgxuhv wkdw fdq eh fdvw zlwklq wkdw iudphzrun1 Rqh vxfk h{dpsoh lv fodvvl�fdwlrq1 Khuh wkh dfwxdorxwsxw txdqwlw| dvvxphv +xqrughudeoh, fdwhjrulfdo ydoxhv if � j � � 1 Lq wklv fdvh wkhuh duh N +�gxpp|�,rxwsxw yduldeohv i| � @ 4+fodvv @ f � ,j �

� 1 Wkh lqsxwv { uhsuhvhqw wkh suhglfwru yduldeohv1 Wkh jrdo lvwr lghqwli| wkrvh uhjlrqv ri wkh lqsxw vsdfh zlwklq zklfk dq revhuydwlrq lv prvw olnho| wr eh iurp rqhri wkh lqglylgxdo fodvvhv1 Wkhvh duh wkh uhjlrqv zkhuh wkh fruuhvsrqglqj wdujhw

i � +{, @ H^| � m{` @ Su+| � @ 4 m{,lv odujhu wkdq wkdw ri dq| rwkhu fodvv1 Wkxv/ fodvvl�fdwlrq fdq eh ylhzhg dv �qglqj uhjlrqv lq zklfkhdfk i � +{, lv uhodwlyho| odujh1 Wkh vxssruw +517 ,+518, ri hdfk ri wkh uhjlrqv fdq eh uhjdughg dv phwd0sdudphwhuv mrlqwo| rswlpl}hg wr pd{lpl}h fodvvl�fdwlrq dffxudf| li suhglfwlrq lv wkh jrdo/ ru fkrvhq wrdlg lqwhusuhwdelolw| lq pruh gdwd dqdo|wlf vlwxdwlrqv1

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i+{, @ H^| m{` @ u+{,!+4 . u+{,,+ryhu wkh srrohg gdwd, lv prqrwrqlf lq u+{, vr wkdw lwv pd{lpd2plqlpd uhsuhvhqw foxvwhuv2krohv1

D sureohp forvho| uhodwhg wr foxvwhulqj lv dvvrfldwlrq1 Khuh rqh vhhnv uhjlrqv ri wkh gdwd vsdfhwkdw duh pruh2ohvv ghqvho| srsxodwhg wkdq zrxog eh wkh fdvh li wkh gdwd yduldeohv zhuh lqghshqghqwri hdfk rwkhu1 Lq wklv fdvh s � +{, lv wdnhq wr eh wkh ghqvlw| forvhvw wr wkdw ri wkh gdwd s+{, xqghu wkhfrqvwudlqw ri lqghshqghqfh

s � +{, @�\

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7

zkhuh hdfk idfwru lq wkh surgxfw lv wkh pdujlqdo glvwulexwlrq ri wkh uhvshfwlyh gdwd yduldeohs � +{ � , @

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Dv wkh deryh glvfxvvlrq looxvwudwhv pdq| gdwd dqdo|vlv sureohpv fdq eh fdvw lq dq rswlpl}dwlrqiudphzrun zkhuh wkh remhfwlyh ixqfwlrq lv revhuyhg zlwk vxshulpsrvhg qrlvh1 Surfhgxuhv wkdw orfdwhuhjlrqv ri wkh lqsxw vsdfh zkhuh wklv wdujhw dvvxphv uhodwlyho| odujh ydoxhv uhsuhvhqw +lq sulqflsdo,srwhqwldo vroxwlrqv wr wkhvh sureohpv1

81 Lqwhusuhwdelolw|Lq ghvfulswlyh gdwd dqdo|vlv lqwhusuhwdelolw| ehfrphv dq lpsruwdqw lvvxh1 Rqh zrxog olnh wr uhvwulfwvroxwlrqv wr wkrvh wkdw fdq eh ghvfulehg dqg lqwhusuhwhg lq whupv ri lpsruwdqw fkdudfwhulvwlfv ri wkhsureohp/ hyhq li wklv pd| vdful�fh vrph srzhu1 Iru wkh sureohp frqvlghuhg khuh/ wklv lpsolhv wkdw wkhvroxwlrq uhjlrq U +516, eh vshfl�hg e| vlpsoh vwdwhphqwv +orjlfdo frqglwlrqv, lqyroylqj wkh ydoxhv riwkh lqglylgxdo lqsxw yduldeohv i{� j � � 1 Vxfk �uxohv� wdnh wkh irup

U @�^

� � �E � = +814,

Wkdw lv/ wkh vroxwlrq uhjlrq lv wdnhq wr eh wkh xqlrq ri d vhw ri vlpso| gh�qhg vxeuhjlrqv iE � j �� 1 Ohw

v � � uhsuhvhqw d vxevhw ri wkh srvvleoh ydoxhv ri lqsxw yduldeoh {� > wkdw lv iv � � � V � j � � zkhuh hdfk V �+514, uhsuhvhqwv doo srvvleoh {� 0ydoxhv1 Wkhq hdfk E � +814, lv wdnhq wr eh d �er{�

E � @ v � � � v � � � � � � � v � � +815,zlwklq wkh hqwluh lqsxw vsdfh +515,1 Hdfk er{ +815, lv ghvfulehg e| wkh lqwhuvhfwlrq ri vxevhwv ri ydoxhvri hdfk lqsxw

{ 5 E � @�_

� � �+{� 5 v � � ,= +816,

Iru lqsxwv wkdw duh uhdo +shukdsv glvfuhwho|, ydoxhg/ wkh vxevhwv duh uhsuhvhqwhg e| frqwljxrxvvxelqwhuydov

v � � @ ^w �� � > w �� � `= +817,Wkxv/ wkh surmhfwlrq ri d er{ E � rq wkh vxevsdfh ri uhdo ydoxhg lqsxwv lv d k|shu0uhfwdqjoh1 Irufdwhjrulfdo lqsxwv wkh lqglylgxdo vxevhwv ri ydoxhv v � � duh h{solflwo| gholqhdwhg1 Qrwh wkdw iru wkh fdvhlq zklfk wkh vxevhw ri ydoxhv +uhdo ru fdwhjrulfdo, lv lq idfw wkh hqwluh vhw v � � @ V � / wkh fruuhvsrqglqjidfwru {� 5 V � fdq eh rplwwhg iurp wkh er{ gh�qlwlrq +816,1 Lq wklv fdvh lw wdnhv wkh vlpsohu irup

{ 5 E � @ _

����� �� �+{� 5 v � � ,= +818,

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gh�qhv rqh ri wkh er{hv frpsulvlqj wkh vxeuhjlrq dvvrfldwhg zlwk kljk iuhtxhqf| ri ylvlwlqj eduv dqgqljkw foxev1

8

91 FryhulqjIurp +516, +814, rqh vhhv wkdw wkh jrdo ri wkh rswlpl}dwlrq surfhgxuh lv wr lqgxfh d vhw ri er{hv+818, iurp wkh gdwd +414, wkdw froohfwlyho| fryhu wkh uhjlrq ri wkh lqsxw vsdfh zkhuh wkh wdujhw i+{,dvvxphv odujh ydoxhv1 Jlyhq dq dojrulwkp iru frqvwuxfwlqj er{hv iurp gdwd +vhh Vhfwlrq :, wklv fdq ehdffrpsolvkhg e| �fryhulqj� ^vhh Plwfkhoo+4<<:,`1 Wkh vdph er{ frqvwuxfwlrq dojrulwkp lv uhshdwhgo|dssolhg lq d vhtxhqwldo pdqqhu wr vxevhwv ri wkh gdwd1 Wkh �uvw er{ E � lv lqgxfhg xvlqj wkh hqwluhgdwd vhw +414,1 Wkh vhfrqg E � lv frqvwuxfwhg xvlqj wkh ruljlqdo gdwd zlwk wkrvh revhuydwlrqv fryhuhge| wkh �uvw er{ uhpryhg= E � � i| � >{ � m{ � @5 E � j1 Dw wkh Nwk lwhudwlrq d er{ E � lv lqgxfhg xvlqj wkhgdwd uhpdlqlqj diwhu wkh uhprydo ri doo revhuydwlrqv fryhuhg e| wkh N � 4 suhylrxvo| lqgxfhg er{hv=E � � i| � >{ � m{ � @5 V

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� � � E � j1 Wklv frqwlqxhv xqwlo hlwkhu wkh hvwlpdwhg wdujhw phdqv zlwklq wkher{hv

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� � �E � ` +914,

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wr wkh vshfl�f gdwd dqdo|wlf jrdo1 Rqh pljkw vhohfw wkrvh zkrvh phdq +914, lv juhdwhu wkdq vrphwkuhvkrog �| �

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9

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e � @ duj pd{� ����� ��� dyh^| � m{ � 5 E � e`= +:15,

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e � � @ i{ m{� ? {� � � � j +:18,e � � @ i{ m{� A {� � � � � � j=

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:

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Lq rughu wr glvfxvv vwdwlvwlfdo shuirupdqfh d irupdo +lghdol}hg, jrdo pxvw eh vshfl�hg dv zhoo dv dphdvxuh ri krz zhoo wkdw jrdo lv dfklhyhg1 Iru d jlyhq ydoxh ri � � wkh irupdo jrdo ri er{ lqgxfwlrqfdq eh h{suhvvhg dv pd{lpl}lqj wkh phdq ri wkh wdujhw ixqfwlrq i+{, zlwklq d er{ +:14, zlwk uhvshfwwr wkh er{ sdudphwhuv v � � +816,/ xqghu d frqvwudlqw rq er{ vxssruw/ � @ � � 1 Wkdw lv/

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dyh^i+{, m{ 5 E`> � � @ � � = +;14,

Ohw ai eh wkh vroxwlrq +hvwlpdwh, lqgxfhg e| wkh shholqj dojrulwkp zkhq dssolhg wr gdwd1 Wkhq rqhphdvxuh ri shuirupdqfh lv dyhudjh +h{shfwhg, phdq0vtxduhg huuru

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;

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Iurp wkh shuvshfwlyh ri lqwhusuhwdwlrq lw lv lpsruwdqw wr eh dzduh ri srvvleoh dowhuqdwlyh gh�qlwlrqviru hdfk lqgxfhg er{ E � 1 Wkh uxoh +818, gh�qhv wkh gdwd +4914, zlwklq wkh er{1 Wklv gdwd fdq ehxvhg wr frpsduh wkh uhodwlyh iuhtxhqf| glvwulexwlrq ri ydoxhv ri hdfk lqsxw yduldeoh { � zlwklq wkh er{s � +{ � m{ 5 E � , wr wkdw ryhu wkh hqwluh gdwd vdpsoh s � +{ � ,1 Lqsxw yduldeohv +rwkhu wkdq wkrvh xvhgwr gh�qh wkh er{, iru zklfk wkh iruphu glvwulexwlrq lv pruh kljko| shdnhg wkdq wkh odwwhu/ uhsuhvhqwfdqglgdwhv iru dowhuqdwlyh gh�qlwlrqv1 Olnh wkh lqsxw yduldeohv wkdw h{solflwo| gh�qh E � +818, wkh udqjhri ydoxhv ri wkhvh rwkhu yduldeohv lv dovr uhvwulfwhg zlwklq wkh er{1

Wkhuh duh d ydulhw| ri zd|v wr frpsduh wzr glvwulexwlrqv1 Rqh srvvlelolw| lv wkurxjk wkhlu udwlru � � +{� , @ s � +{ � m{ 5 E � ,!s � +{� ,= +4917,

Qrwh wkdw wkh vxssruw ri wkh ghqrplqdwru lv dozd|v juhdwhu wkdq wkdw ri wkh qxphudwru1 D xqlirupglvwulexwlrq iru u � � +{� , lpsolhv wkdw {� lv wrwdoo| luuhohydqw wr wkh gh�qlwlrq ri E � > wkh uhodwlyh iuhtxhqf|ri lwv ydoxhv lv wkh vdph lqvlgh ru rxwvlgh ri wkh er{1 Ghsduwxuhv iurp xqliruplw| lqglfdwh dvvrfldwlrqzlwk wkh er{ gh�qlwlrq1 D kljko| shdnhg glvwulexwlrq iru u � � +{� , lpsolhv wkdw wkh lqsxw {� lv kljko|uhohydqw wr wkh gh�qlwlrq ri E � zkhwkhu ru qrw lw lv rqh ri wkh gh�qlqj yduldeohv +818,1

Iljxuh 4915 vkrzv uhodwlyh iuhtxhqf| udwlr sorwv iu ��� +{ � ,j � �� +4917, iru wkh �uvw er{E � ri wkh iuhtxhqw |hu h{dpsoh +| @ qxpehu ri ljkwv shu |hdu, ri Vhfwlrq 481 Wkh odvw elq ri hdfk sorw uhsuhvhqwv wkhydoxh { � @ plvvlqj1 Wkh rwkhu elqv vkrz wkh uhvshfwlyh u ��� +{� , iru wkh qrq0plvvlqj ydoxhv1 Wkh �uvwiudph u �� +{ � , vkrzv wkdw jhqghu lv qrw kljko| uhohydqw wr wkh gh�qlwlrq ri E � > phq +{ � @ 4, duh voljkwo|ryhu�uhsuhvhqwhg dprqj wkhvh iuhtxhqw lhuv1 Wkh vhfrqg sorw vkrzv wkdw zlgrzhg shrsoh +{ � @ 7,duh pxfk ohvv olnho| wr eh dprqj wkhvh iuhtxhqw |huv wkdw lq wkh wrwdo vdpsoh1 Wkh sorw iru { � +djh,lqglfdwhv wkdw |rxqj ^{ � � 5 +57 |hduv rog,` dqg roghu ^{ � � : +88 |hduv rog,` shrsoh duh kljko| xqghu�uhsuhvhqwhg lq E � 1 Wkh glvwulexwlrqv ri { � / { � / dqg { � +hgxfdwlrq/ rffxsdwlrq/ dqg lqfrph, uh hfw

55

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edu

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occup

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45

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Iljxuh 4915= Uhodwlyh iuhtxhqf| udwlr glvwulexwlrqv iru wkh �uvw er{ ri wkh iuhtxhqw |hu h{dpsoh1Vkrzq duh klvwrjudpv ri wkh iuhtxhqf| ri ydoxhv iru hdfk yduldeoh iru revhuydwlrqv lq wkh er{/ uhodwlyhwr wkrvh lq wkh hqwluh gdwd vhw1 Wkhvh fdq eh xvhg wr dvvhvv wkh orfdwlrq ri wkh er{ zlwk uhvshfw wr wkhydoxhv ri doo lqsxw yduldeohv/ lqfoxglqj wkrvh wkdw gr qrw dsshdu h{solflwo| lq wkh er{ gh�qlwlrq1

56

wkh er{ gh�qlwlrq +Vhfwlrq 48,1 Wkh sorw ri u ��� +{ � , vkrzv wkdw { � @ 4 +rffxsdwlrq @ surihvvlrqdo2 pdqdjhuldo, duh doprvw wkuhh wlphv pruh olnho| wr eh dprqj wkh iuhtxhqw |huv ri E � wkdq lq wkhwrwdo vdpsoh1 Wkh glvwulexwlrq ri u � � +{ � , +qxpehu lq krxvhkrog, uh hfwv wkh er{ gh�qlwlrq rq qxpehuri fkloguhq lq wkh krxvh ohvv wkdq 4; |hduv rog1 Wkh sorw ri u ���� � +{ �� , +krxvhkroghu vwdwxv, uh hfwvwkdw ri u ��� +{ � , +djh,1 Wkrvh shrsoh zkr olyh zlwk sduhqwv ru idplo| +{ �� @ 6, whqg wr eh |rxqjhu+{ � � 57 |hduv rog,1 Wkh glvwulexwlrq ri u �W��� � +{ �W� , +w|sh ri krph, lqglfdwhv dq ryhu�uhsuhvhqwdwlrq rifrqgrplqlxp rzqhuv +{ �W� @ 5, dqg d gudpdwlf xqghu�uhsuhvhqwdwlrq ri preloh krph rzqhuv +{ �W� @ 7,lq E � 1 Wkh hwkqlf fodvvl�fdwlrq sorw u � ��� � +{ � � , vkrzv dq ryhu uhsuhvhqwdwlrq ri Hdvw Lqgldqv +{ � � @ 7,dqg zklwhv +{ � � @ :,/ dqg dq xqghu�uhsuhvhqwdwlrq ri Diulfdq Dphulfdqv +{ � � @ 6, dqg hvshfldoo|Klvsdqlfv +{ � � @ 8,1 Wklv lv dovr uh hfwhg lq u � ��� � +{ � � ,> shrsoh zkr vshdn Vsdqlvk dw krph +{ � � @ 5,duh ohvv olnho| wr eh dprqj wkh iuhtxhqw |huv lq E � 1

Wkh odvw iudph ri Ilj1 4915 vkrzv wkh glvwulexwlrq fruuhvsrqglqj wr +4917, iru wkh rxwsxw yduldeoh/u � � +|,1 Rqh vhhv wkdw doo ydoxhv ri | duh uhsuhvhqwhg lq E � 1 Shrsoh zkr | qlqh ru pruh wlphv shu |hdu+| @ <, duh urxjko| irxu wlphv pruh olnho| wr eh lq E � wkdq lq wkh jhqhudo srsxodwlrq/ zkhuhdv wkrvhrggv duh uhyhuvhg iru wkrvh zkr grq*w | dw doo +| @ 3,1

Wkh kljko| qrq0xqlirup glvwulexwlrqv ri u ��� +{ � , +4917, iru pdq| ri wkh lqsxwv qrw gluhfwo| gh�qlqjE � lqglfdwh wkh suhvhqfh ri vwurqj dvvrfldwlrqv dprqj pdq| ri wkh lqsxwv zlwklq wkh er{1 Wkh fkrlfhkhuh ri { � / { � / { � / dqg { � � wr h{solflwo| gh�qh E � lv edvhg rq wkh gdwd> wkh| dsshdu wr eh wkh prvwkljko| uhohydqw dv hvwlpdwhg e| wkh uhgxqgdqw yduldeoh holplqdwlrq vwudwhj| ri Vhfwlrq 431 Dv zlwk dq|hvwlpdwlrq surfhgxuh/ yduldqfh fdq fdxvh huudqw fkrlfhv dprqj forvh dowhuqdwlyhv1 Dovr lw lv srvvleoh +lixqolnho|, wkdw d kljko| uhohydqw lqsxw yduldeoh fdq eh frpsohwho| pdvnhg e| rwkhuv kljko| fruuhodwhgzlwk lw/ dqg qhyhu eh fkrvhq iru shholqj1 Dowkrxjk wkh sdwlhqw vwudwhj| +Vhfwlrq ;, dv zhoo dv pxowlsohwudmhfwrulhv +Vhfwlrq 46, khos plwljdwh wklv srvvlelolw|/ lw fdqqrw eh frpsohwho| holplqdwhg1 H{dplqlqjwkh glvwulexwlrqv ri iu � � +{� ,j � � +4917, fdq khos gldjqrvh vxfk sureohpv dv zhoo dv jlylqj dq ryhudoolpsuhvvlrq ri wkh fkdudfwhulvwlfv ri wkh lqgxfhg er{1 Ri frxuvh/ li wzr ru pruh kljko| fruuhodwhgyduldeohv dsshdu iurp wkh gdwd wr eh qhduo| htxdoo| uhohydqw/ rqo| xvhu mxgjhphqw fdq dvfhuwdlq zklfkrqh +ru erwk, dfwxdoo| duh uhodwhg wr wkh rxwsxw |1

Iru wkh pdunhwlqj gdwd +Wdeoh 4814, wkhuh duh dw prvw qlqh gl�huhqw qrq�plvvlqj ydoxhv iru dq|lqsxw yduldeoh1 Zlwk d vdpsoh vl}h ri Q @ <73< wkh ydoxh ri u� � +{ � , +4917, fdq eh uholdeo| hvwlpdwhgiru hdfk lqglylgxdo ydoxh ri {� 1 Lq pdq| gdwd vhwv uhdo�ydoxhg yduldeohv fdq dvvxph d odujh qxpehuri glvwlqfw ydoxhv1 Wkh jhrorj| gdwd glvfxvvhg lq Vhfwlrq 44 lv dq h{dpsoh1 Iru vxfk uhdo�ydoxhglqsxwv { � / sorwv ri +4917, dqdorjrxv wr Ilj1 4915 duh pdgh e| klvwrjudpplqj wkh { � 0ydoxhv xvlqj dvelq erxqgdulhv wkh �0txdqwlohv ri wkh glvwulexwlrq ri {� ryhu wkh hqwluh gdwd vhw s � +{� ,1 Wkh qxpehuri elqv +4@�, lv fkrvhq e| wkh xvhu edvhg rq vdpsoh vl}h Q dqg wkh ghjuhh ri ghwdlo uhtxluhg1

Ilqdoo|/ lw lv lpsruwdqw wr uhphpehu wkdw wkh sorwv lq Ilj1 4915 gr qrw uhsuhvhqw wkh iuhtxhqf|glvwulexwlrq ri wkh { � 0ydoxhv s � +{ � m{ 5 E � , zlwklq wkh er{ E � 1 Udwkhu/ wkh| uhsuhvhqw wkrvh iuhtxhq0flhv uhodwlyh wr wkh fruuhvsrqglqj iuhtxhqflhv s� +{� , ryhu wkh hqwluh gdwd vhw +4917,1 Iru h{dpsoh/ wkhydoxh | @ < lv qrw wkh prvw iuhtxhqw rxwsxw ydoxh lq E � +odvw iudph,1 Wklv ydoxh lv yhu| lqiuhtxhqw lqwkh hqwluh gdwd vhw1 Wkh ydoxh s � � +<, @ 7=5 lqglfdwhv wkdw wkh rxwsxw ydoxh | @ < lv 7=5 wlphv pruholnho| wr dsshdu lq E � wkdq lq wkh gdwd vhw dv d zkroh1 Sorwv ri s � +{ � m{ 5 E � , fdq dovr vhuyh dv xvhixodgglwlrqdo gldjqrvwlfv dv fdq vwdwlvwlfv rwkhu wkdq +4917, iru frpsdulqj lw wr s � +{ � ,1

49161 Lqwhuer{ glvvlplodulw|Wkh fryhulqj surfhgxuh +Vhfwlrq 9, surgxfhv d vhtxhqfh ri er{hv iE � j �

� wkdw fryhu d vxeuhjlrq U+814, ri wkh lqsxw yduldeoh vsdfh1 Qr frqvwudlqwv duh sodfhg rq wkh uhodwlrqvklsv dprqj wkh er{hv1Wkh| fdq ryhuods ru eh glvmrlqw/ eh forvh ru idu dsduw/ dv glfwdwhg e| wkh qdwxuh ri wkh wdujhw ixqf0wlrq1 Iru h{dpsoh/ li wkh wdujhw ixqfwlrq lv fkdudfwhul}hg e| d vlqjoh surplqhqw prgh/ wkh fryhulqjsurfhgxuh pljkw surgxfh d vhulhv ri qhvwhg er{hv/ hdfk vxffhvvlyho| lqgxfhg er{ frpsohwho| fryhulqjdoo ri wkrvh lqgxfhg ehiruh lw1 Dowhuqdwlyho|/ vxffhvvlyh er{hv pljkw fryhu gl�huhqw �vkrxoghuv� riwkdw prgh surgxflqj d �foxvwhu� ri forvho| uhodwhg er{hv1 Li wkhuh zhuh vhyhudo gl�huhqw surplqhqwprghv/ wkh er{hv pljkw glylgh lqwr fruuhvsrqglqj jurxsv ri qhvwhg2foxvwhuhg vhtxhqfhv1 Nqrzohgjhfrqfhuqlqj uhodwlrqvklsv dprqj wkh er{hv fdq surylgh lqirupdwlrq derxw wkh prgdo vwuxfwxuh ri wkh

57

wdujhw ixqfwlrq1Uhodwlrqvklsv dprqj wkh er{hv fdq eh ghgxfhg iurp wkhlu sdluzlvh glvvlplodulwlhv1 Wkh �glvvlp0

lodulw|� G+E � > E � , ehwzhhq er{hv E � dqg E � lv gh�qhg dv wkh gl�huhqfh ehwzhhq wkh vxssruw ri wkhvpdoohvw er{ E � � wkdw fryhuv erwk ri wkhp/ dqg wkh vxssruw ri wkhlu xqlrq

G+E � > E � , @ �+E � � ,� �+E �VE � ,= +4918,

Khuh wkh vxssruw �+E,> ri d er{ E/ lv gh�qhg dv wkh iudfwlrq ri revhuydwlrqv lq wkh hqwluh gdwd vhw+414, wkdw lw fryhuv1 Wkh plqlpdo fryhulqj er{ lv gh�qhg dv

{ 5 E � � @�_

� � �^{ � 5 v � +n> o,` +4919,

zkhuh wkh ydoxh vxevhwv v � +n> o, 5 V � duh ghulyhg iurp wkrvh ri wkh uhvshfwlyh wzr er{hv iv � � j �� � �dqg iv � � j �� � � +816,1 Iru uhdo ydoxhg yduldeohv {� / v � +n> o, lv wkh vpdoohvw lqwhuydo ^w �� +n> o,> w �� +n> o,` wkdwfryhuv wkh uhvshfwlyh wzr lqwhuydov iru E � dqg E � rq {� +817,

w �� +n> o, @ plq+w �� � > w �� � ,/ w �� +n> o, @ pd{+w �� � > w �� � ,= +491:,Iru fdwhjrulfdo yduldeohv {� / wkh vxevhw v � +n> o, lv wkh xqlrq ri wkh uhvshfwlyh E � dqg E � vxevhwv

v � +n> o, @ v � �V v � � = +491;,

Wkh glvvlplodulw| phdvxuh +4918, dvvxphv ydoxhv lq wkh vhpl�rshq lqwhuydo 3 � G+E � > E � , ? 41Dowkrxjk G+E � > E � , lv qrw vwulfwo| d glvwdqfh/ lw uhwdlqv wkh hvvhqwldo surshuw| wkdw lwv ydoxh lqfuhdvhvdv wkh ydoxh vxevhwv iv � � j �� � � / iv � � j �� � � +uhdo ru fdwhjrulfdo,/ gh�qlqj wkh wzr er{hv E � > E � / lqfuhdvlqjo|gl�hu rq hdfk ri wkh lqsxw yduldeohv {� 1 Qhvwhg er{hv zloo kdyh }hur glvvlplodulw|/ dv zloo �dgmdfhqw�er{hv wkdw kdyh frqwljxrxv lqwhuydov rq rqh uhdo yduldeoh {� / dqg lghqwlfdo vxevhwv rq doo rwkhu ydul0deohv iv � � � @ v � � � j � � �� � 1 Rwkhu frq�jxudwlrqv zloo surgxfh odujhu glvvlplodulw| zlwk wkh kljkhvw ydoxhvrffxuulqj zkhq wkh wzr er{hv duh gh�qhg e| kljko| glvsdudwh ydoxh vxevhwv +818, rq rqh/ ru hvshfldoo|vhyhudo/ lqsxw yduldeohv1 Qrwh wkdw wzr er{hv fdq kdyh qrq�}hur lqwhuvhfwlrq ^�+E �

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