Hvwlpdwlqj G|qdplf Prghov iurp Uhshdwhg
Furvv0Vhfwlrqv
Pduqr Yhuehhn
Hudvpxv Xqlyhuvlw| Urwwhugdp
Iudqflv Yhood
Hxurshdq Xqlyhuvlw| Lqvwlwxwh
Uhylvlrq Mdqxdu| 64/ 5335
Hfrqrphwulf Lqvwlwxwh Uhsruw HL 5335038
Devwudfw
Dq lpsruwdqw ihdwxuh ri sdqho gdwd lv wkdw lw doorzv wkh hvwlpd0
wlrq ri sdudphwhuv fkdudfwhul}lqj g|qdplfv iurp lqglylgxdo ohyho gdwd1
Vhyhudo dxwkruv dujxh wkdw vxfk sdudphwhuv fdq dovr eh lghqwlhg iurp
uhshdwhg furvv0vhfwlrq gdwd dqg suhvhqw hvwlpdwruv wr gr vr1 Wklv sd0
shu uhylhzv wkh lghqwlfdwlrq frqglwlrqv xqghuo|lqj wkhvh hvwlpdwruv1
Dv jurxslqj gdwd wr rewdlq d svhxgr0sdqho lv dq dssolfdwlrq ri lqvwux0
phqwdo yduldeohv +LY,/ lghqwlfdwlrq uhtxluhv wkdw vwdqgdug LY frqgl0
wlrqv duh phw1 Wklv sdshu h{solflwo| glvfxvv wkh lpsolfdwlrqv ri wkhvh
frqglwlrqv iru hpslulfdo dqdo|vhv1 Zh dovr sursrvh d frpsxwdwlrqdoo|
dwwudfwlyh lqvwuxphqwdo yduldeohv hvwlpdwru wkdw lv frqvlvwhqw xqghu d
uhodwlyho| zhdn vhw ri frqglwlrqv1 D Prqwh Fduor vwxg| lqglfdwhv wkdw
wklv hvwlpdwru pd| zrun zhoo lq sudfwlfh1
Fruuhvsrqghqfh=
Pduqr Yhuehhn/ Hudvpxv Xqlyhuvlw| Urwwhugdp/ Ghsw1 ri Ilqdq0
fldo Pdqdjhphqw dqg Hfrqrphwulf Lqvwlwxwh/ S1R1Er{ 4:6;/ 6333 GU Urwwhugdp/ Wkh
Qhwkhuodqgv1 H0pdlo= P1YhuehhnCien1hxu1qo1 Khosixo frpphqwv dqg vxjjhvwlrqv iurp wzr
dqrq|prxv uhihuhhv/ dqg iurp vhplqdu sduwlflsdqwv dw Wloexuj Xqlyhuvlw|/ FRUH/ dqg wkh
Lqwhuqdwlrqdo Frqihuhqfh rq wkh Dqdo|vlv ri Uhshdwhg Furvv0Vhfwlrqdo Vxuyh|v +Qlmphjhq/
5333, duh judwhixoo| dfnqrzohgjhg1 MHO0frghv= F54/ F56/ F;41
4
4
Lqwurgxfwlrq
Dq lpsruwdqw ihdwxuh ri sdqho gdwd lv lwv idflolwdwlrq ri wkh hvwlpdwlrq ri sd0
udphwhuv fdswxulqj wkh g|qdplf uhodwlrqvklsv lq lqglylgxdo gdwd1 Uhfhqwo|/
Pr!ww +4<<6,/ Froodgr +4<<;,/ Jlupd +5333, dqg PfNhq}lh +5334, kdyh
dujxhg wkdw wkh vdph sdudphwhuv fdq dovr eh hvwlpdwhg yld uhshdwhg furvv0
vhfwlrq +UFV, gdwd1 Krzhyhu/ dowkrxjk wkhvh sdshuv duh frqfhuqhg zlwk
hvvhqwldoo| wkh vdph prgho/ qdpho| wkh uvw rughu dxwruhjuhvvlyh prgho zlwk
h{rjhqrxv yduldeohv/ wkh sursrvhg hvwlpdwruv dqg wkhlu suhvhqwdwlrqv duh yhu|
glhuhqw14 Wklv pdnhv lw gl!fxow wr frpsduh wkhvh surfhgxuhv dqg wkhlu
xqghuo|lqj dvvxpswlrqv1 Zkloh wkhuh duh glhuhqfhv dfurvv hvwlpdwruv wkh
dssurdfk hpsor|hg lv hlwkhu h{solflwo|/ ru lpsolflwo|/ lqvwuxphqwdo yduldeohv
+LY,1 Wkh hvwlpdwruv uvw djjuhjdwh wkh lqglylgxdo gdwd lqwr frkruwv frp0
sulvlqj lqglylgxdov zlwk vlplodu revhuyhg fkdudfwhulvwlfv +h1j1 |hdu ri eluwk,1
Xvlqj wklv svhxgr sdqho wkh odjjhg ghshqghqw yduldeoh lv wkhq uhsodfhg e|
d suhglfwhg ydoxh iurp dq dx{loldu| uhjuhvvlrq dqg wkh g|qdplf prgho lv
vxevhtxhqwo| hvwlpdwhg yld LY1
Dv wkh delolw| wr hvwlpdwh g|qdplfv iurp UFV gdwd lv xvhixo/ dqg wkh
sursrvhg hvwlpdwruv duh doo txlwh glhuhqw/ lw lv lpsruwdqw wr xqghuvwdqg
krz wkh| duh uhodwhg dqg xqghu zklfk frqglwlrqv rqh pljkw eh hpsor|hg
lq idyru ri wkh rwkhuv1 Dffruglqjo|/ wklv sdshu uhylhzv wkh lghqwlfdwlrq
frqglwlrqv iru frqvlvwhqw hvwlpdwlrq ri d olqhdu g|qdplf prgho iurp UFV
gdwd dqg sxwv wkh uhvshfwlyh dvvxpswlrqv ri wkhvh glhuhqw surfhgxuhv lqwr
shuvshfwlyh1 Rqh ri rxu frqfoxvlrqv iurp grlqj vr lv wkdw lw dsshduv wkdw
wkhvh hvwlpdwruv duh hlwkhu wrr vlpsolvwlf ru wrr hoderudwh wr eh xvhixo lq
uhdolvwlf vhwwlqjv1 Pruhryhu/ zh duh deoh wr sursrvh idluo| vlpsoh dowhuqdwlyhv
wkdw duh suhihudeoh xqghu frpsdudeoh frqglwlrqv1
Zh ehjlq rxu dqdo|vlv zlwk dq h{dplqdwlrq ri wkh hvwlpdwru lq Pr!ww
+4<<6,1 Lq wklv surfhgxuh wkh g|qdplf uhodwlrqvkls lv hvwlpdwhg yld ruglqdu|
ohdvw vtxduhv +ROV, zkhuh wkh odjjhg ghshqghqw yduldeoh lv uhsodfhg e| d
suhglfwhg ydoxh1 Zh vkrz wkdw wkh uhvxowlqj hvwlpdwru lv lqfrqvlvwhqw xq0
ohvv vwurqj/ dqg riwhq xquhdolvwlf/ frqglwlrqv duh lpsrvhg rq wkh h{rjhqrxv
yduldeohv1 Wklv lv wuxh hyhq li wkh lqvwuxphqwv xvhg wr suhglfw wkh odjjhg
ghshqghqw yduldeoh vdwlvi| wkh vwdqgdug LY uhtxluhphqwv1 Zh sursrvh wkdw
wklv lqfrqvlvwhqf| fdq eh ryhufrph e| lqvwuxphqwlqj wkh rwkhu h{sodqdwru|
4
Wkh prghov frqvlghuhg e| Jlupd +5333, dqg PfNhq}lh +5334, glhu dv wkh| doorz iru
khwhurjhqhlw| ri wkh sdudphwhuv dfurvv frkruwv1
5
yduldeohv1
Zkhq wkh lqvwuxphqwv duh fruuhodwhg zlwk wkh xqrevhuydeohv wkhvh hv0
wlpdwruv duh lqfrqvlvwhqw1 Krzhyhu/ li rqh lv zloolqj wr dvvxph wkdw wkh
fruuhodwlrq ehwzhhq wkh xqrevhuydeohv dqg wkh +wlph0lqyduldqw, lqvwuxphqwv
lv wlph0lqyduldqw/ rqh fdq fruuhfw iru wklv e| lqfoxglqj wkh lqvwuxphqwv/ zlwk
wlph0lqyduldqw frh!flhqwv/ lq wkh prgho1 Zkhq wkh lqvwuxphqwv frpsulvh d
jurxs ri gxpplhv fruuhvsrqglqj wr pxwxdoo| h{foxvlyh frkruwv/ wkh uhvxow0
lqj hvwlpdwru lv vlpso| wkh vwdqgdug zlwklq hvwlpdwru dssolhg wr d g|qdplf
prgho lq whupv ri frkruw dyhudjhv udwkhu wkdq lqglylgxdo revhuydwlrqv1 Zkloh
wklv hvwlpdwru lv riwhq xvhg lq hpslulfdo zrun iru erwk wkh g|qdplf dqg vwd0
wlf prgho +vhh/ iru h{dpsoh/ Gdujd| dqg Y|wkrxondv/ 4<<<,/ lw kdv uhfhlyhg
vxusulvlqjo| olwwoh dwwhqwlrq lq wkh wkhruhwlfdo olwhudwxuh1
Dowkrxjk wkh frqglwlrqv uhtxluhg iru frqvlvwhqf| ri wkh sursrvhg hvwl0
pdwruv duh uhodwlyho| zhdn/ wkh| duh qrw wulyldoo| vdwlvhg1 Hvwlpdwlrq yld
jurxslqj lv dq LY whfkqltxh dqg/ dffruglqjo|/ wkh jurxslqj yduldeohv vkrxog
vdwlvi| wkh dssursuldwh frqglwlrqv/ lqfoxglqj d udqn frqglwlrq/ iru dq LY hvwl0
pdwru wr eh frqvlvwhqw1 Wklv uhtxluhv wkdw wkh fkrvhq vhw ri frkruw gxpplhv lv
dssursuldwho| lqvwuxphqwlqj hdfk ri wkh h{sodqdwru| yduldeohv lq wkh prgho1
Iru wkh zlwklq hvwlpdwru dw wkh frkruw ohyho/ wklv uhtxluhv wkdw doo h{sodqd0
wru| yduldeohv kdyh wlph0ydu|lqj frkruw hhfwv zklfk/ lq dgglwlrq/ duh qrw
froolqhdu1
Wkh olwhudwxuh rq hvwlpdwlrq iurp UFV gdwd lv fkdudfwhul}hg e| d udqjh
ri dowhuqdwlyh dv|pswrwlfv +frpsduh Yhuehhn/ 4<<9/ PfNhq}lh/ 5334,1 Frq0
vlvwhqf| lq wklv sdshu zloo uhihu wr odujh dqg vpdoo A Wkdw lv/ wkh qxpehu
ri lqglylgxdov lq hdfk furvv0vhfwlrq lv dvvxphg wr eh odujh/ zkloh wkh qxpehu
ri furvv0vhfwlrqv lv {hg1 Dq dgglwlrqdo lpsruwdqw lvvxh lv krz wkh qxpehu ri
frkruwv lv wuhdwhg dv wkh qxpehu ri lqglylgxdov lqfuhdvhv1 Lq pdq| dssolfd0
wlrqv/ frkruwv duh ghqhg rq wkh edvlv ri |hdu ri eluwk/ lq zklfk fdvh lw vhhpv
qdwxudo wr wuhdw wkhlu qxpehu dv {hg +frpsduh Pr!ww/ 4<<6/ Jlupd/ 5333/
dqg PfNhq}lh/ 5334,1 Vxfk dq dssurdfk kdv wzr dgydqwdjhv1 Iluvw/ lw vlp0
solhv wkh dqdo|vlv1 Vhfrqg/ dqg pruh lpsruwdqwo|/ lw lv gl!fxow wr wklqn ri
dssursuldwh vhtxhqfhv ri lqvwuxphqwv wkdw jurz zlwk wkh qxpehu ri lqglylgx0
dov lq d vdpsoh +frpsduh PfNhq}lh/ 5334,/ xqohvv wkh gdwd jhqhudwlqj surfhvv
lv irupxodwhg dw wkh frkruw ohyho1 Wklv/ krzhyhu/ vhhpv xqqdwxudo1 Iru h{dp0
soh/ lw vhhpv lqdssursuldwh wr dvvxph wkdw wkh glvwulexwlrq ri frkruw0vshflf
phdqv lv xqdhfwhg e| wkh ohyho ri djjuhjdwlrq wkdw d uhvhdufkhu hpsor|v lq
frqvwuxfwlqj frkruwv1
Wkh uhpdlqghu ri wklv sdshu lv rujdql}hg dv iroorzv1 Vhfwlrq 5 suhvhqwv
6
wkh prgho ri lqwhuhvw dqg vkrzv wkdw ROV xvlqj wkh svhxgr sdqho zlwk d suh0
glfwhg odjjhg ghshqghqw yduldeoh zloo jhqhudoo| ohdg wr lqfrqvlvwhqw hvwlpd0
wruv1 Vhfwlrq 6 glvfxvvhv vrph srwhqwldoo| vxlwdeoh LY hvwlpdwruv/ lqfoxglqj
wkrvh vxjjhvwhg e| Pr!ww/ Froodgr dqg Jlupd/ dqg fulwlfdoo| uhylhzv wkh
qhfhvvdu| frqglwlrqv iru frqvlvwhqf|1 Vhfwlrqv 7 dqg 8 surylgh dq looxvwud0
wlyh h{dpsoh dqg d Prqwh Fduor h{shulphqw/ uhvshfwlyho|/ zkloh Vhfwlrq 9
frqfoxghv1
5
Prgho dqg Gdwd
Wkh revhuyhg gdwd frqvlvw ri A lqghshqghqw furvv0vhfwlrqv dw glhuhqw srlqwv
lq wlph zlwk hdfk ehlqj d udqgrp vdpsoh ri vrph xqghuo|lqj srsxodwlrq1
Dv lqglylgxdov lq glhuhqw shulrgv duh qrw wkh vdph shrsoh/ zh iroorz wkh
frqyhqwlrq lq wklv olwhudwxuh dqg lqgh{ wkh yduldeohv e| d grxeoh vxevfulsw1
Wkh | ghqrwhv wkh furvv0vhfwlrq/ zkloh E|c c c | lqgh{hv wkh lqglylgxdov
lq furvv0vhfwlrq |1 Wkh prgho lv=
+E|c| ' k+E|c|

3
n %E|c| q n 0E|c| c | ' c c A ( E| ' c c | c
+4,
zkhuh wkh g0glphqvlrqdo yhfwru %E|c| pd| lqfoxgh dq lqwhufhsw/ dqg wlph0
lqyduldqw dqg wlph0ydu|lqj yduldeohv1 Khuh/ +E|c|  uhihuv wr wkh ydoxh ri + dw
|  iru dq lqglylgxdo revhuyhg lq furvv0vhfwlrq | qrwlqj wkdw lw lv xqrevhuyhg
lq wkh dydlodeoh gdwd1 Zh dvvxph wkh huuru whup kdv wkh iroorzlqj surshuwlhv=
3
.i0E|c| %E|c| j ' f/ | ' c c A
+5,
Dv wkh odjjhg ydoxhv iru + duh qrw revhuyhg iru dq| lqglylgxdo/ zh frq0
vwuxfw dq hvwlpdwh e| xvlqj lqirupdwlrq rq wkh +0ydoxhv ri rwkhu lqglylgxdov
revhuyhg dw |  Wr gr vr/ ohw 5E| ghqrwh d vhw ri wlph0lqyduldqw yduldeohv/
lqfoxglqj dq lqwhufhsw whup15 Qrz/ frqvlghu wkh ruwkrjrqdo surmhfwlrq lq
furvv0vhfwlrq | ri +E|c| xsrq 5E| c
. i+E|c| m5E| j ' 5E| B | c | ' c c Ac
W
+6,
zkhuh . ghqrwhv wkh ruwkrjrqdo surmhfwlrq +iru d jlyhq | Wklv uhgxfhg
irup fdq eh hvwlpdwhg frqvlvwhqwo| +iru | $ 4 e| uhjuhvvlqj +E|c| rq 5E|
W
5
Dfwxdoo|/ wkh uhtxluhphqw wkdw }l+w, lv wlph0lqyduldqw lv xqqhfhvvdu|1 Wlph0ydu|lqj
yduldeohv wkdw fdq eh edfnfdvwhg zlwk uhdvrqdeoh dffxudf| pd| dovr eh lqfoxghg +iru h{0
dpsoh/ qxpehu ri fkloguhq,1 Iru wkh vdnh ri qrwdwlrq/ zh devwudfw iurp wklv1
7
wr jlyh B | Iroorzlqj Pr!ww +4<<6,/ rqh rewdlqv dq hvwlpdwh ri +E|c|  dv wkh
suhglfwhg ydoxh iurp wklv uhjuhvvlrq/ vxevwlwxwlqj wkh dssursuldwh 5 ydoxhv
iru wkh lqglylgxdov lq furvv0vhfwlrq | Wkdw lv/
3
+E|c|
3

' 5E| B |  c
+7,
3
qrwlqj wkdw B |  lv hvwlpdwhg iurp gdwd rq glhuhqw lqglylgxdov wkdq wkrvh
lqgh{hg e| E| Iru odwhu uhihuhqfh/ zh dovr ghqh wkh srsxodwlrq htxlydohqw ri
wklv dv +E|c|  ' 5E| B |  Lq wkh vhtxho/ zh vkdoo dvvxph wkdw dv|pswrwlfv duh
vxfk wkdw T*4 B |  ' B |  Lq pdq| flufxpvwdqfhv lw lv frqyhqlhqw wr wklqn
ri 5E| dv d yhfwru ri gxpp| yduldeohv/ fruuhvsrqglqj wr pxwxdoo| h{foxvlyh
frkruwv +vhh Ghdwrq/ 4<;8/ Froodgr/ 4<<;/ Jlupd/ 5333,1 Iru h{dpsoh/ zlwk
UFV gdwd/ lw lv frpprq wr xvh |hdu0ri0eluwk frkruwv +vhh/ h1j1/ Eoxqghoo/
Eurzqlqj/ Phjklu/ 4<<7/ ru Dohvvlh/ Ghyhuhx{ dqg Zhehu/ 4<<:,1 Lq wklv fdvh/
wkh ruwkrjrqdo surmhfwlrq lq +6, fruuhvsrqgv wr wkh frqglwlrqdo h{shfwdwlrq
dqg +7, fruuhvsrqgv wr wdnlqj vdpsoh dyhudjhv zlwklq shuvrq *v frkruw1 Qrwh
wkdw d glhuhqfh ehwzhhq wkh dssurdfkhv ri Pr!ww +4<<6, dqg Jlupd +5333,
lv wkdw udwkhu wkdq xvlqj wkh dyhudjh zlwklq d frkruw/ Jlupd +5333, vxjjhvw
xvlqj wkh ydoxhv ri rqh ru pruh duelwudulo| fkrvhq shuvrqv zlwklq wkh frkruw1
Zh glvfxvv wkh vljqlfdqfh/ dqg phulwv/ ri wklv yduldwlrq ehorz1
Qrz/ lqvhuw wkhvh suhglfwhg ydoxhv lqwr wkh ruljlqdo prgho wr jhw=
3
W
3
3
3
3
+E|c| ' k
+E|c|
zkhuh

3
n %E|c| q n 0E|c| c | ' c c A ( E| ' c c | c
+8,
W
0E|c| ' 0E|c| n kE+E|c|
W
3

+E|c|  +9,
3
Qr pdwwhu krz +E|c|  lv jhqhudwhg/ lwv lqfoxvlrq lpsolhv wkdw rqh ri wkh
h{sodqdwru| yduldeohv lv phdvxuhg zlwk huuru/ dowkrxjk wkh phdvxuhphqw hu0
uru zloo eh +dv|pswrwlfdoo|, xqfruuhodwhg zlwk wkh suhglfwhg ydoxh16 Jlyhq
dvvxpswlrq +5, dqg wkh uhvxow wkdw suhglfwlrq huuru dqg suhglfwru duh ru0
wkrjrqdo/ frqvlvwhqf| ri ROV dssolhg wr wklv htxdwlrq uhtxluhv wkdw 0E|c| lv
xqfruuhodwhg7 zlwk +E|c|  dv zhoo dv wkdw wkh suhglfwlrq huuru +E|c|  +E|c| 
lv xqfruuhodwhg zlwk %E|c| Krzhyhu/ erwk dvvxpswlrqv pd| eh sureohpdwlf1
Zkloh wkh dvvxpswlrq wkdw 0E|c| lv xqfruuhodwhg zlwk +E|c|  fdq eh ghihqghg
e| wkh xvxdo LY dvvxpswlrq wkdw
3
W
W
3
3
3
W
3
.i0E|c| 5E| j ' fc | ' c c Ac
6
7
Xqolnh wkh vwdqgdug wh{werrn phdvxuhphqw huuru h{dpsohv1
Uhfdoo wkdw |al+w,>w4 lv wkh vdpsoh frxqwhusduw ri |l+w,>w4 =
8
+:,
lw h{foxghv wkh srvvlelolw| wkdw wkhuh duh frkruw hhfwv lq wkh xqrevhuydeohv1
Zkloh wklv pd| dsshdu xquhdvrqdeoh/ dqg wrr uhvwulfwlyh iru hpslulfdo dqdo|0
vhv/ wklv dvvxpswlrq lv pdgh lq Pr!ww +4<<6, dqg Jlupd +5333,18 Vhfrqg/
lw lv lqdssursuldwh wr dujxh wkdw %E|c| lv xqfruuhodwhg zlwk wkh suhglfwlrq
huuru +E|c|  +E|c|  1 Frqvlghu/ iru h{dpsoh/ zkhuh kljk %0ydoxhv lq rqh
shulrg rq dyhudjh fruuhvsrqg zlwk kljk %0ydoxhv lq wkh qh{w shulrg1 Li wkh
q frh!flhqwv duh srvlwlyh wklv zloo jhqhudoo| lpso| wkdw d kljk ydoxh iru
%E|c|  / zklfk lv xqrevhuydeoh/ zloo uhvxow lq dq xqghusuhglfwlrq ri +E|c|  1
Rq wkh rwkhu kdqg/ %E|c|  lv srvlwlyho| fruuhodwhg zlwk %E|c| 1 Frqvhtxhqwo|/
wklv zloo surgxfh d srvlwlyh fruuhodwlrq ehwzhhq 0E|c| dqg %E|c| / uhvxowlqj lq
dq lqfrqvlvwhqw hvwlpdwru iru q1 Wklv lqfrqvlvwhqf| fduulhv ryhu wr k xqohvv
+E|c|  lv xqfruuhodwhg zlwk %E|c| 1 Dv d uhvxow/ wkh hvwlpdwru vxjjhvwhg e|
Pr!ww +4<<6,/ edvhg rq dsso|lqj ROV wr +8,/ lv w|slfdoo| lqfrqvlvwhqw xqohvv
wkhuh duh hlwkhu qr wlph0ydu|lqj h{rjhqrxv uhjuhvvruv ru wkh wlph0ydu|lqj
h{rjhqrxv yduldeohv gr qrw h{klelw dq| dxwrfruuhodwlrq1
W
3
3
3
3
3
W
W
3
6
Lqvwuxphqwdo yduldeohv hvwlpdwruv
Wr ryhufrph wkh sureohp ri fruuhodwlrq ehwzhhq wkh uhjuhvvruv dqg wkh huuru
whup lq +8, rqh pd| hpsor| LY1 Qrwh wkdw qrz zh qhhg lqvwuxphqwv iru %E|c|
hyhq wkrxjk wkhvh yduldeohv duh h{rjhqrxv lq wkh ruljlqdo prgho1 Ohw E|c|
ghqrwh dq -0glphqvlrqdo yhfwru ri srwhqwldo lqvwuxphqwv wkdw fdq eh xvhg
wr hvwlpdwh +8,1 Vwdqgdug LY frqglwlrqv uhtxluh wkdw wkh lqvwuxphqwv duh
xqfruuhodwhg zlwk wkh htxdwlrq*v huuru whup1 Wklv uhtxluhv wkdw9
.iE+E|c|
3
.i0E|c| E|c| j ' fc | ' c c A
 +E|c|  E|c| j ' fc | ' c c A
W
3
+;,
+<,
Wkh uvw frqglwlrq vd|v wkdw wkh lqvwuxphqwv vkrxog eh h{rjhqrxv/ zklfk lv
vlplodu wr frqglwlrq +:,1 Wkh vhfrqg frqglwlrq/ +<,/ lpsolhv wkdw wkh suhglf0
wlrq huuru lv ruwkrjrqdo wr wkh lqvwuxphqwv1 Wkdw lv/ wkh lqvwuxphqwv E|c|
vkrxog qrw eh deoh wr suhglfw dq| ri wkh yduldwlrq lq +E|c|  ohiw xqh{sodlqhg
e| 5E| 1 Frqvhtxhqwo|/ d qdwxudo fkrlfh lv E|c| ' 5E| / vxfk wkdw erwk +;,
3
8
Wkh pdmrulw| ri wkh uhvxowv lq PfNhq}lh +5334, lv dovr edvhg rq wklv dvvxpswlrq1
Zh lqwhusuhw wkhvh frqglwlrqv dv ehlqj uhtxluhg iru hdfk shulrg w= Rq wkh srrohg
vdpsoh/ wklv lpsolhv wkdw wkh lqvwuxphqwv lq zl+w,>w pd| eh lqwhudfwhg zlwk wlph gxpplhv1
Dowhuqdwlyho|/ wkh frqglwlrqv fdq eh lpsrvhg wr krog ryhu wkh srrohg vdpsoh rqo|1 Lq wklv
fdvh/ frqvlvwhqf| lv wr eh lqwhusuhwhg iru W +ru erwk W dqg Qw , jrlqj wr lqqlw|1
9
9
dqg +<, duh dxwrpdwlfdoo| vdwlvhg xqghu frqglwlrq +:,1 Jlyhq wklv uhvxow/
dqg wr vlpsoli| wkh glvfxvvlrq/ zh dvvxph wkdw E|c| ' 5E| 1 Wkxv/ +<, lv
dxwrpdwlfdoo| vdwlvhg dqg frqglwlrqv +:, dqg +;, duh lghqwlfdo1
Lw lv zhoo nqrzq wkdw wdnlqj jurxs dyhudjhv lv htxlydohqw wr lqvwuxphqwdo
yduldeohv hvwlpdwlrq zlwk wkh jurxs gxpplhv dv lqvwuxphqwv +Dqjulvw/ 4<<4/
Pr!ww/ 4<<6,1 Wkxv/ zkhq wkh lqvwuxphqwv 5E| duh d vhw ri frkruw gxpplhv/
hvwlpdwlrq ri +8, e| lqvwuxphqwdo yduldeohv lv lghqwlfdo wr dsso|lqj ROV wr
wkh ruljlqdo prgho zkhuh doo yduldeohv duh uhsodfhg e| wkhlu +wlph0vshflf,
frkruw vdpsoh dyhudjhv1
Krzhyhu/ lpsrvlqj +:, pd| qrw eh vx!flhqw wr lghqwli| wkh prgho sdud0
phwhuv1 Wr vhh wklv/ frqvlghu wkh ruwkrjrqdo surmhfwlrq ri +4, xsrq wkh vhw
ri lqvwuxphqwv=
. i+E|c| m5E| j ' k. i+E|c|  m5E| jn. i%E|c| m5E| j qn. i0E|c| m5E| j +43,
W
W
W
W
3
Zkloh +:, lpsolhv wkdw wkh odvw whup lq wklv h{suhvvlrq lv }hur/ lw lv fohdu wkdw
lq rughu wr lghqwli| k dqg qc wkh uvw wzr whupv vkrxog qrw h{klelw shuihfw
froolqhdulw|1 Frqvhtxhqwo|/ d udqn frqglwlrq qhhgv wr eh dovr vdwlvhg1 Wklv
fdq eh irupxodwhg dv=
3
E
E
E
.E
E
C
5E %Ec
5E2 %E2c2
11
1
4
3
4
P
F
E P F
F
E 2 F
F
F
F'E
E 11 F ' P kdv udqn g n 
F
C
D
1
D
5E +Ecf
5E2 +E2c
11
1
5EA %EA cA 5EA +EA cA
3
+44,
PA

Wklv frqglwlrq vd|v wkdw A - Eg n  furvv0prphqw pdwul{ P kdv ixoo
froxpq udqn1 Lw uhtxluhv wkdw wkh lqvwuxphqwv fdswxuh yduldwlrq lq +E|c| 
lqghshqghqw ri wkh yduldwlrq lq %E|c| Lw lv lpsruwdqw wr qrwh wkdw wkh udqn
frqglwlrq iru lghqwlfdwlrq lq +44, lv lq whupv ri wkh srsxodwlrq prphqwv1 Dv
d uhvxow/ wklv rughu frqglwlrq fdqqrw eh gluhfwo| yhulhg iurp wkh gdwd dqg
hvwlpdwlrq huuru pd| wkhuhe| surylgh vsxulrxv lghqwlfdwlrq1:
Wkh sdluzlvh txdvl0glhuhqflqj dssurdfk ri Jlupd +5333, ghyldwhv iurp
wkh deryh hvwlpdwlrq vwudwhj| lq wzr uhvshfwv/ dowkrxjk lw hvvhqwldoo| pdnhv
wkh vdph dvvxpswlrqv1 Iluvw/ wkh odjjhg ydoxh ri + lv qrw dssur{lpdwhg e|
3
:
Iru h{dpsoh/ li wkh lqvwuxphqwv duh frkruw gxpplhv/ lw lv uhtxluhg wkdw wkh srsxodwlrq
frkruw phdqv h{klelw vx!flhqw yduldwlrq1 Lq sudfwlfh/ wkh frkruw vdpsoh dyhudjhv zloo ydu|
dfurvv frkruwv gxh wr vdpsolqj huuru hyhq li doo frkruwv kdyh wkh vdph phdq dqg wkh prgho
lv qrw lghqwlhg1
:
wkh odjjhg frkruw dyhudjh exw e| dq duelwudulo| vhohfwhg revhuydwlrq iurp wkh
frkruw1 Vhfrqg/ wkh lqvwuxphqwv duh qrw wkh frkruw gxpplhv/ exw lqglylgxdo/
ru dyhudjhg/ revhuydwlrqv iurp wkh frkruw1 Dv d uhvxow/ Jlupd*v dssurdfk
hpsor|v d qrlv| dssur{lpdwlrq wr wkh xqrevhuyhg odjjhg ydoxhv dv zhoo dv
qrlv| lqvwuxphqwv1 Dowkrxjk/ xqghu dssursuldwh dvvxpswlrqv/ wklv qrlvh zloo
fdqfho rxw dv|pswrwlfdoo|/ wkhuh grhv qrw vhhp wr eh dq| jdlq lq grlqj vr1
Rqh fdq hdvlo| vhh wklv e| qrwlqj wkdw iru dq duelwudulo| lqglylgxdo wkdw lv
lq wkh vdph frkruw dv lqglylgxdo c
. i%E|c| m% E|c| c 5E| j ' . i%E|c| m5E| j
W
W
+45,
Wkh rqo| uhdvrq zk| lqglylgxdo *v revhuydwlrq surylghv lqirupdwlrq derxw
lqglylgxdo lv ehfdxvh wkh| duh lq wkh vdph frkruw dqg wklv lv douhdg| fds0
wxuhg e| wkh frkruw gxpplhv1 D vlplodu dujxphqw krogv iru wkh dssur{lpd0
wlrq ri +E|c|  Wkh dydlodelolw| ri dssursuldwh lqvwuxphqwv vdwlvi|lqj frqglwlrq +:, pd|
eh udwkhu olplwhg1 Wkxv lw lv qdwxudo wr dvn zkhwkhu zh fdq vrphzkdw uhod{
wkh uhtxluhphqwv rq wkh lqvwuxphqwv1 Rqh srvvlelolw| lv wr h{solflwo| fdswxuh
wkh frkruw hhfwv e| lqfoxglqj frkruw {hg hhfwv/ dv lq Ghdwrq +4<;8, dqg
Froodgr +4<<;,1 Wklv lv grqh e| lqfoxglqj 5E| dv dgglwlrqdo uhjuhvvruv lq +8,
exw zlwk wlph0lqyduldqw frh!flhqwv1; Wkxv/ zh doorz iru frkruw hhfwv e|
uhsodflqj dvvxpswlrq +:, zlwk
3
. i0E|c| m5E| j ' 5E| b
W
+46,
vr wkdw zh fdq zulwh<
+E|c| ' k
+E|c|
3
zkhuh

n %E|c| q n 5E| b n # E|c| c
.i# E|c| 5E| j ' fc | ' c c A
+47,
+48,
uhsodfhv frqglwlrq +:,1 Wklv doorzv rqh wr uhod{ dvvxpswlrq +5, wr
.i# E|c| %E|c| j ' fc | ' c c A
;
+49,
Wkh yhfwru }l+w, qrw qhfhvvdulo| frqvlvwv ri frkruw gxpplhv1 Dowhuqdwlyh ixqfwlrqdo
irupv/ wkdw duh hlwkhu pruh h{leoh ru pruh sduvlprqlrxv/ pd| eh hpsor|hg +vhh Pr!ww/
4<<6,1 Lq wkh glvfxvvlrq zh vkdoo devwudfw iurp wklv srvvlelolw|1
<
Wkh prgho suhvhqwhg khuh lv d vshfldo fdvh ri PfNhq}lh +5334,*v jhqhudo prgho1
;
Xqghu wkhvh frqglwlrqv/ rqh zrxog hvwlpdwh +47, e| LY xvlqj 5E| / lq0
whudfwhg zlwk wlph gxpplhv/ dv lqvwuxphqwv1 Zh vkdoo uhihu wr wklv dv wkh
dxjphqwhg LY hvwlpdwru qrwlqj wkdw d wlph0ydu|lqj b zrxog pdnh wkh prgho
xqlghqwlhg1 Wr dfklhyh lghqwlfdwlrq/ zh qhhg wr dvvxph wkdw . i+E|c|  m5E| j
dqg . i%E|c| m5E| j h{klelw wlph0yduldwlrq dqg duh qrw froolqhdu1 Wklv lpsrvhv
dgglwlrqdo uhvwulfwlrqv xsrq wkh prphqw pdwulfhv P| lq +44,1 Lq sduwlfxodu/
zh qhhg wr vwuhqjwkhq +44, wr=
W
3
W
3
4
P2 P
E
F
11
E
F kdv udqn g n 
1
C
D
PA P
+4:,
Li P| grhv qrw ydu| vx!flhqwo| ryhu wkh furvv0vhfwlrqv/ wkh yduldwlrq ri wkh
lqvwuxphqwv lv froolqhdu zlwk 5E| dqg hvwlpdwlrq zloo euhdn grzq1 D gluhfw
lpsolfdwlrq ri wklv lv wkdw rqh qhhgv dw ohdvw wkuhh furvv0vhfwlrqv +| ' fc c 2
wr lghqwli| wkh prgho xqghu wkhvh dvvxpswlrqv1
Frpsxwdwlrq ri wklv dxjphqwhg LY hvwlpdwru lv uhpdundeo| vlpsoh li 5E|
lv d vhw ri frkruw gxpplhv1 Iluvw/ rqh vlpso| djjuhjdwhv wkh gdwd lqwr frkruw
dyhudjhv/ zklfk jlyhv
+E|c|
+E|c| ' k
3

n%
E|c| q n 5E| b n #E|c| c
+4;,
zkhuh wkh kdwv ghqrwh suhglfwhg ydoxhv iurp d shulrg0e|0shulrg uhjuhvvlrq rq
5E| c wkdw lv/ wkh| ghqrwh +vdpsoh, frkruw dyhudjhv1 Vhfrqg/ dsso|lqj ROV
wr +4;, fruuhvsrqgv wr wkh vwdqgdug zlwklq hvwlpdwru iru Ekc q edvhg xsrq
wuhdwlqj wkh frkruw0ohyho gdwd dv d sdqho1 Wkh xvxdo sureohp zlwk hvwlpdw0
lqj g|qdplf sdqho gdwd prghov +vhh Qlfnhoo/ 4<;4,43 / grhv qrw dulvh ehfdxvh
xqghu dvvxpswlrq +48, wkh huuru whup/ zklfk lv d zlwklq frkruw dyhudjh ri lq0
glylgxdo huuru whupv wkdw duh xqfruuhodwhg zlwk E| c lv dv|pswrwlfdoo| }hur144
Krzhyhu/ lw uhpdlqv zkhwkhu vxlwdeoh lqvwuxphqwv fdq eh irxqg wkdw vdwlvi|
wkh deryh frqglwlrqv1
Wkh hvwlpdwruv suhvhqwhg lq Pr!ww +4<<6,/ Jlupd +5333, ru PfNhq}lh
+5334, gr qrw doorz iru frqglwlrq +48, wr eh uhod{hg1 Wklv phdqv wkdw wlph0
ydu|lqj frkruw hhfwv lq wkh xqrevhuydeohv duh qrw doorzhg1 Vwduwlqj iurp wkh
fdvh zkhuh 5E| lv d vhw ri frkruw gxpplhv/ Froodgr +4<<;, suhvhqwv hvwlpdwruv
43
Zlwk jhqxlqh sdqho gdwd/ wkh zlwklq hvwlpdwru lq wkh g|qdplf prgho kdv d vxevwdqwldo
eldv iru vpdoo dqg prghudwh ydoxhv ri W 1
44
Uhfdoo wkdw/ dv|pswrwlfdoo|/ wkh qxpehu ri frkruwv lv {hg dqg wkh qxpehu ri lqglylg0
xdov jrhv wr lqqlw|1
<
wkdw doorz iru d wlph0ydu|lqj fruuhodwlrq ehwzhhq +E|c|  dqg # E|c| 45 Iroorz0
lqj vwdqgdug g|qdplf sdqho gdwd LY surfhgxuhv/ wkh frkruw {hg hhfwv b
duh uvw holplqdwhg e| uvw0glhuhqflqj/ wr jlyh
W
3
+E|c| +E|
3
' kE
+E|c|  +E| c| 2 nE
%E|c| %E| c|  q n E# E|c| #E|
c|3
3
3
+4<,
3
3
3
3
c|3 c
zkhuh/ ehfdxvh ri wkh ghqlwlrq ri 5E|c| dv frkruw gxpplhv/ +E| c|  '
+E|c|  dqg vlploduo| iru wkh rwkhu yduldeohv dqg odjv146 Lq rwkhu zrugv/ lq
+4<, zh kdyh d vwdqgdug uvw0glhuhqfhg g|qdplf sdqho gdwd prgho zlwk wkh
xqlw ri revhuydwlrq ehlqj d frkruw1 Wkh srsxodwlrq frxqwhusduw lv jlyhq e|
3
3
3
+E|c| +E|
W
W
3
c|3
' kE+E|c|  +E| c| 2 nE%E|c| %E|c|  q n E# E|c| # E|
W
+53,
W
3
3
W
W
3
W
W
3
3
c|3 Li frqglwlrq +48, lv qrw ydolg/ wkh frkruw phdqv # E|c| duh dv|pswrwlfdoo|
qrq}hur dqg fruuhodwhg zlwk 5E| lq d wlph0ydu|lqj zd|1 Ixuwkhupruh/ +E|c| 
dqg # E| c|  duh fruuhodwhg e| frqvwuxfwlrq/ dqg ROV dssolhg wr +4<, lv lq0
frqvlvwhqw1 Frqvhtxhqwo|/ htxdwlrq +4<, pxvw dovr eh hvwlpdwhg e| lqvwux0
phqwdo yduldeohv whfkqltxhv1 Iru wkh jhqxlqh sdqho gdwd prgho/ Duhoodqr
dqg Erqg +4<<4, surylgh d olvw ri lqvwuxphqwv zklfk lv h{sorlwhg e| Froodgr
+4<<;,1 Wkh hvvhqwldo frqglwlrq iru lghqwlfdwlrq/ krzhyhu/ lv wkdw +E| c| 2 c
ru rqh ri lwv odjv/ lv d ydolg lqvwuxphqw iru +E|c|  +E| c| 2 Wklv/ dv ehiruh/
uhtxluhv wlph yduldwlrq lq +E|c|  c exw dovr uhtxluhv wkdw47
W
W
3
W
3
3
W
3
W
3
W
3
3
3
W
3
.iE# E|c| # E|
W
W
c|3 +E|3c|32 j
W
3
' fc
+54,
zklfk lv wulyldoo| vdwlvhg li wkhuh duh qr frkruw hhfwv lq # E|c| +dv zlwk wkh
dxjphqwhg LY hvwlpdwru,148 Lq jhqhudo/ krzhyhu/ lw lv ohvv reylrxv wkdw wklv
frqglwlrq lv vdwlvhg149 Li
. i#E|c| m5E| j ' 5E| b| c | ' c c A
W
45
W=
+55,
Zh vkdoo uhvwulfw dwwhqwlrq wr wkh hvwlpdwru wkdw lv fodlphg wr eh frqvlvwhqw iru {hg
46
Froodgr +4<<;, zulwhv |f>w > zkhuh f lqgh{hv frkruwv1
Wkh htxdolw| vljq lv wr eh lqwhusuhwhg dv dv|pswrwlfdoo| htxdo1
48
Qrwh wkdw dq| wlph0lqyduldqw frkruw hhfwv zrxog kdyh ehhq holplqdwhg e| wkh uvw0
glhuhqfh wudqvirupdwlrq1
49
Uhfdoo wkdw zh ljqruh wkh phdvxuhphqw huuru sureohp lq Froodgr +4<<;, e| dvvxplqj
wkdw a w > dqg wkh rwkhu uhgxfhg irup sdudphwhuv/ dv|pswrwlfdoo| frqyhujh wr wkhlu wuxh
ydoxhv1 Lq Froodgr*v dssurdfk +iroorzlqj Ghdwrq/ 4<;8, wkh qxpehu ri frkruwv lv dvvxphg
47
43
zh wkxv qhhg wkdw/ iru qlwh Ac
.E5E| Eb| b|  5E| B | 2 j ' fc
+56,
3
3
zklfk/ jlyhq wkh ghqlwlrq ri 5E| / uhtxluhv wkdw wkh wzr yhfwruv b| b|  dqg
B | 2 duh ruwkrjrqdo1 Froodgr +4<<;, lpsrvhv wklv frqglwlrq zkloh dvvxplqj
wkdw wkh glphqvlrq ri 5c dqg wkxv wkh glphqvlrqv ri b| dqg B | c lqfuhdvh zlwk
vdpsoh vl}h wkurxjk lpsrvlqj wkdw wkhuh lv qr dxwrfruuhodwlrq lq wkh frkruw
hhfwv iru dq| jlyhq frkruw1 Wkxv/ wklv doorzv iru wlph0ydu|lqj frkruw hhfwv
lq wkh xqrevhuydeohv surylghg wkh| duh xqfruuhodwhg ryhu wlph1
Lw lv kdug wr wklqn ri fdvhv zkhuh +54, lv vdwlvhg zkloh +46, lv qrw/ xq0
ohvv rqh lv zloolqj wr pdnh vdpsolqj dvvxpswlrqv dw wkh frkruw ohyho14: Li
lqvwuxphqwv fdq eh fkrvhq vxfk wkdw +46, lv vdwlvhg +dv lv grqh lq Jlupd/
5333/ dqg PfNhq}lh/ 5334, wkhuh duh qr wlph0ydu|lqj frkruw hhfwv/ dqg wkh
dxjphqwhg lqvwuxphqwdo yduldeohv hvwlpdwru sursrvhg deryh lv qrw rqo| frp0
sxwdwlrqdoo| dwwudfwlyh exw dovr pruh h!flhqw wkdq Froodgr +4<<;,*v frkruw0
ohyho JPP hvwlpdwru1 Wkh uhdvrq lv vlpsoh= xqqhfhvvdulo| lqvwuxphqwlqj
+E|c|  +E| c| 2 zloo ohdg wr d orvv lq h!flhqf| frpsduhg wr wkh dxj0
phqwhg LY hvwlpdwru1 Dv d uhvxow/ hvwlpdwlqj d g|qdplf prgho iurp frkruw
ohyho gdwd fdq eh frpsxwdwlrqdoo| pxfk vlpsohu wkdq iurp jhqxlqh sdqho
gdwd1
3
3
3
7
3
3
D vlpsoh h{dpsoh
Wr looxvwudwh dqg foduli| wkh frqglwlrqv iru frqvlvwhqf| ri wkh uhvshfwlyh hvwl0
pdwruv/ dv glvfxvvhg deryh/ zh frqvlghu d vlpsoh h{dpsoh/ zkhuh wkh prgho
ri lqwhuhvw frqwdlqv rqo| rqh h{rjhqrxv yduldeoh1 Wkh qh{w vhfwlrq suhvhqwv d
Prqwh Fduor vwxg| edvhg xsrq wklv h{dpsoh1 Jlyhq wkdw wkh ghqlwlrq ri d
frkruw lv ohiw wr wkh uhvhdufkhu/ lw lv lpsruwdqw wr vwduw iurp d gdwd jhqhudwlqj
surfhvv dw wkh lqglylgxdo ohyho udwkhu wkdq wkh frkruw ohyho1 Lw lv frqyhqlhqw
wr jurz zlwk vdpsoh vl}h1 Wklv lpsolhv wkdw wkh qxpehu ri lqvwuxphqwv lv lqfuhdvlqj zlwk
Qw dqg lw lv qr orqjhu reylrxv wkdw a w w lv dv|pswrwlfdoo| }hur +qrwh wkdw wkh ghqlwlrq
ri w fkdqjhv zlwk vdpsoh vl}h,1 Nhhslqj wkh qxpehu ri lqvwuxphqwv {hg/ dv zh vkdoo
gr/ wkh hvwlpdwlrq huuru lq a w uhvxowv lq d vpdoo vdpsoh eldv/ zklfk pd| ru pd| qrw eh
qhjoljleoh/ ghshqglqj xsrq wkh qxpehu ri revhuydwlrqv zlwklq d frkruw1 Wkh Prqwh Fduor
vwxg| lq Vhfwlrq 8 zloo dgguhvv wklv lvvxh1 Vhh dovr Yhuehhn dqg Qlmpdq +4<<5/ 4<<6, iru d
glvfxvvlrq ri wklv lvvxh lq wkh vwdwlf prgho1
4:
Lq zklfk fdvh w dqg w duh wuhdwhg dv udqgrp yduldeohv1
44
wr wklqn ri wkh srsxodwlrq ehlqj d odujh sdqho gdwd vhw/ iurp zklfk glhu0
hqw lqglylgxdov duh vdpsohg hdfk shulrg1 Dvvxph wkdw wkh gdwd jhqhudwlqj
surfhvv lv jlyhq e|4;
+| ' k+c|
3

n q%| n 0| c
f k c
+57,
zkhuh wkh huuru whup kdv wkh xvxdo huuru frpsrqhqwv vwuxfwxuh
0| ' w n | +58,
Lw lv dvvxphg wkdw | lv xqfruuhodwhg ryhu wlph1 Ohw 5 ghqrwh d vhw ri wlph0
lqyduldqw yduldeohv wkdw duh xvhg dv lqvwuxphqwv1 Wkh pdlq txhvwlrq lv zkdw
frqglwlrqv qhhg wr eh lpsrvhg xsrq wkh uhodwlrqvklsv ehwzhhq 5 dqg wkh
rwkhu yduldeohv lq wkh prgho wr jxdudqwhh wkdw rqh ru pruh ri wkh hvwlpdwruv
duh frqvlvwhqw1
Wkh uvw vwhs lv wr dssur{lpdwh wkh odjjhg ghshqghqw yduldeoh e| d olqhdu
surmhfwlrq xsrq 5 c ghqrwhg . i+c|  m5 j Xvlqj uhfxuvlyh vxevwlwxwlrq/ rqh
fdq hdvlo| ghulyh wkdw
W
3
. i+c|  m5 j ' q
W
|3 2
[
3
'f
3
4
|32
[
k . W i%c|3 3 m5 j n C k D . W iw m5 j
'f
nk|  . i+J m5 j n
3
W
+59,
|32
[
k . W ic|3 3 m5 j
'f
Lq d uvw fdvh/ fdvh D/ zh dvvxph wkdw wkh vwduwlqj ydoxh +J lv h{rjhqrxv dqg
lqghshqghqw ri w 4< Lq d vhfrqg fdvh/ fdvh E/ zh dvvxph wkdw wkh surfhvv lv
lq htxloleulxp ru 0 htxlydohqwo| 0 wkdw lw vwduwhg lq dq lqqlwh sdvw1 Iru fdvh
E/ wkh h{suhvvlrq iru . i+c|  m5 j vlpsolhv wr
W
3
. i+c|  m5 j ' q
W
3
"
[
'f
k . i%c|
W
 m5 j
+5:,
3 3
[ 
n
. iw m5 j n
k . ic|
k
'f
"
W
W
 m5 j
3 3
4;
Ehfdxvh wklv dqg wkh iroorzlqj vhfwlrq glvfxvv gdwd jhqhudwlqj phfkdqlvpv/ udwkhu wkdq
wkh vdpsoh dydlodeoh iru hvwlpdwlrq/ lw lv dssursuldwh dqg suhihudeoh wr xvh wkh vwdqgdug
sdqho gdwd qrwdwlrq1
4<
Wkh vwduwlqj gdwh qhhg qrw frlqflgh zlwk wkh ehjlqqlqj ri wkh vdpsoh shulrg1
45
Qrwh wkdw lq wkh uvw fdvh wkh lqvwuxphqwv 5 pd| eh suhglfwlqj wkh odjjhg
ghshqghqw yduldeoh sduwo| wkurxjk wkh lqlwldo ydoxh +J Ohw xv frqvlghu wkh uhvshfwlyh hvwlpdwruv1 Iluvw/ wkh ROV hvwlpdwru uh0
txluhv wkdw wkh lqvwuxphqwv duh xqfruuhodwhg zlwk wkh htxdwlrq*v huuru whupv1
Wkdw lv
. iw m5 j ' f
. i| m5 j ' f
W
W
+5;,
+5<,
Ixuwkhu/ lw lv uhtxluhg wkdw wkh suhglfwlrq huuru lv xqfruuhodwhg zlwk %| c zklfk
uhtxluhv wkdw %c|  . i%c|  m5 j lv xqfruuhodwhg zlwk %| + ' fc c 2c Iru d wlph0ydu|lqj %| 0yduldeoh/ wklv lpsrvhv wkh vwurqj uhvwulfwlrq ri wkh
devhqfh ri dxwrfruuhodwlrq uhodwlyh wr wkh frkruw0vshflf phdq1
Wkh vwdqgdug LY hvwlpdwru/ xvlqj 5 dv lqvwuxphqwv iru %| dv zhoo/ dovr
lpsrvhv frqglwlrqv +5;,0+5<,1 Lq dgglwlrq/ wkh udqn frqglwlrq uhtxluhv wkdw
. i%| m5 j lv qrw froolqhdu zlwk . i+c|  m5 j Iru fdvh D lw lv vx!flhqw wkdw
. i%| m5 j 9' f +dqg q 9' f ru wkdw . i+J m5 j 9' f +dqg k 9' f Iru fdvh
E zh qhhg wkdw . i%| m5 j ydulhv zlwk | Iru fdvh E lw lv wkxv uhtxluhg wkdw
wkh wlph0lqyduldqw yduldeohv lq 5 kdyh d wlph0ydu|lqj uhodwlrqvkls zlwk wkh
h{rjhqrxv yduldeohv lq wkh prgho1 Iru fdvh D wklv lv qrw uhtxluhg dv wkh
{hg vwduwlqj gdwh ri wkh surfhvv lqgxfhv yduldwlrq ryhu wlph hyhq li wkh
uhodwlrqvkls ehwzhhq 5 dqg %| lv wlph0lqyduldqw153
Iru wkh dxjphqwhg LY hvwlpdwru/ zklfk h{whqgv wkh vwdqgdug LY hvwlpdwru
e| lqfoxglqj 5 lq wkh prgho/ frqglwlrq +5;, lv qr orqjhu uhtxluhg1 Lpsolflwo|/
wklv doorzv iru frkruw0vshflf phdqv lq wkh surfhvvhv iru +| dqg %| Erwk
fdvh D dqg fdvh E qrz uhtxluh wkdw . i%| m5 j ydulhv zlwk | +dqg q 9' f Iru
fdvh D/ wklv pd| djdlq eh uhsodfhg e| . i+J m5 j 9' f1 Lw vkrxog eh vwuhvvhg
wkdw/ jlyhq wkh lqfoxvlrq ri 5 lq wkh prgho/ wkh qhhg iru wlph0yduldwlrq lq
. i%| m5 j dqg . i+c|  m5 j lv pxfk vwurqjhu1
Ilqdoo|/ frqvlghu d vlpsoh yduldqw ri wkh hvwlpdwru sursrvhg e| Froodgr
+4<<;,/ zklfk uhvwulfwv dwwhqwlrq wr glvfuhwh0ydoxhg 5 Zkloh wklv hvwlpdwru
grhv qrw uhtxluh frqglwlrqv +5;,0+5<,/ lw grhv uhtxluh wkdw
W
3 3
3 3
W
W
3
W
W
W
W
W
W
W
3
. i| m5 j . ic|  m5 j
W
W
3
lv xqfruuhodwhg zlwk . ic| m5 j iru ' 2c c Qrwh wkdw wkh fdvh zlwk
d qrq0}hur wlph0lqyduldqw . i| m5 j lv douhdg| fdswxuhg e| wkh dxjphqwhg
W
3
W
53
Li lv vpdoo ru w lv odujh/ wkh lpsdfw ri wkh lqlwldo ydoxhv lv vpdoo dqg wlph0yduldwlrq
lq H i{lw mzl j lv uhfrpphqghg +dv lq fdvh E,1
46
LY hvwlpdwru wkurxjk wkh lqfoxvlrq ri 5 Wkh rqo| uhohydqw h{whqvlrq wkxv
rffxuv zlwk d wlph0ydu|lqj . i| m5 j Krzhyhu/ wkh rqo| vwdwlrqdu| surfhvv
wkdw lv frqvlvwhqw zlwk wklv frqglwlrq lv fkdudfwhul}hg e| htxl0fruuhodwlrq dqg
lpsolhv wkdw . i| m5 j lv }hur +jlyhq wkh suhvhqfh ri wkh frkruw0vshflf hhfw
.iw m5 j Dv d uhvxow/ wkh uhod{hg frqglwlrqv vhhp wr eh qr zhdnhu wkdq wkrvh
iru wkh dxjphqwhg LY hvwlpdwru dqg wkh dgglwlrqdo vwdjh ri lqvwuxphqwdwlrq
lq Froodgr +4<<;, dsshduv xqqhfhvvdu|1
W
W
8
D Prqwh Fduor vwxg|
Zh qrz suhvhqw wkh uhvxowv ri d Prqwh Fduor h{shulphqw wkdw lqyhvwljdwh
krz odujh wkh frkruw0vshflf yduldwlrq lq lqglylgxdo gdwd qhhg wr eh iru wkh
lqvwuxphqwdo yduldeohv hvwlpdwruv wr zrun vdwlvidfwrulo| zlwk uhdvrqdeoh vdp0
soh vl}hv1 Frkruw0vshflf hhfwv pd| eh suhvhqw lq d qxpehu ri sodfhv= lq
wkh vwduwlqj ydoxh ri wkh surfhvv iru +c lq wkh surfhvv wkdw jhqhudwhv wkh %0
yduldeohv/ lq wkh lqglylgxdo0vshflf khwhurjhqhlw| w dqg lq wkh lglrv|qfudwlf
huuru whupv | Zh vkdoo dvvxph wkdw | kdv qr frkruw hhfwv1 Wklv dvvxps0
wlrq lv vwurqjhu wkdq wkdw pdgh e| Froodgr/ exw lv vxevwdqwldoo| zhdnhu wkdq
zkdw lv dvvxphg lq Pr!ww +4<<6,/ Jlupd +5333, dqg lq wkh vlpxodwlrq vwxg|
ri PfNhq}lh +5334,1 Qrwh wkdw wkh wkhruhwlfdo vhfwlrq ri PfNhq}lh suhvhqwv
d jhqhudo prgho wkdw lqfrusrudwhv wkh rqh zh vlpxodwh khuh1 Krzhyhu/ kh
surylghv qr vlpxodwlrq uhvxowv iru vxfk d prgho1
Erwk Froodgr +4<<;, dqg Jlupd +5333, suhvhqw d Prqwh Fduor vwxg| wr
looxvwudwh krz zhoo wkhlu uhvshfwlyh hvwlpdwruv shuirup zlwk uhdolvwlf vdpsoh
vl}hv1 Erwk vwxglhv qg wkdw iru wkh hvwlpdwruv wr zrun zhoo lw lv qhfhvvdu|
wkdw wkh dprxqw ri yduldwlrq lq wkh gdwd wkdw lv gxh wr frkruw hhfwv lv vxe0
vwdqwldo1 Iru h{dpsoh/ lq vrph lqvwdqfhv lw qhhgv wr eh xs wr .DI ri wkh
wrwdo yduldwlrq1 Wkhuh duh vrph dgglwlrqdo ihdwxuhv ri wkh ghvljq lq Froodgr
+4<<;, wkdw duh zruwk qrwlqj1 Iluvw/ wkhuh duh qr h{rjhqrxv yduldeohv lq wkh
vlpxodwlrq1 Vhfrqg/ vkh vwduwv e| dvvxplqj dq dxwruhjuhvvlyh prgho dw wkh
frkruw ohyho dqg wkhq jhqhudwhv lqglylgxdo revhuydwlrqv dv wkh frkruw revhu0
ydwlrqv soxv d udqgrp qrlvh whup1 Zh ihho wkdw wkhuh duh vrph vkruwfrplqjv
dvvrfldwhg zlwk vxfk dq dssurdfk1 Iluvw/ zh suhihu wr wklqn ri wkh gdwd jhqhu0
dwlqj surfhvv dv vrphwklqj wkdw rshudwhv dw wkh lqglylgxdo ohyho/ zkhuh frkruw
hhfwv duh dhfwlqj vrph ri wkh lqglylgxdo yduldeohv/ udwkhu wkdq dw wkh fr0
kruw ohyho/ lqglylgxdo revhuydwlrqv ehlqj udqgrp ghyldwlrqv iurp wkh frkruw
phdqv1 Fkrrvlqj frkruwv lv vrphwklqj grqh e| wkh uhvhdufkhu zkhq kh2 vkh
47
vwduwv dqdo|}lqj wkh gdwd/ qrw vrphwklqj ghwhuplqhg e| qdwxuh1 Lq sduwlf0
xodu/ zkhq fkrrvlqj frkruwv rqh kdv wr zruu| derxw vlplodu lvvxhv dv zkhq
fkrrvlqj lqvwuxphqwv1 Vhfrqg/ wkh frkruw0ohyho DU+4, prgho lv lqfrqvlvwhqw
zlwk wkh lqglylgxdo gdwd ehlqj jhqhudwhg iurp dq DU+4, prgho +hyhq wkrxjk
Froodgr vxjjhvwv rwkhuzlvh,1 Lq idfw/ khu gdwd jhqhudwlqj phfkdqlvp lpsolhv
wkdw lqglylgxdo revhuydwlrqv duh jhqhudwhg e| dq DU+4, prgho soxv zklwh
qrlvh/ zklfk dv lv zhoo0nqrzq lq wkh wlph vhulhv olwhudwxuh/ vhh Kdplowrq
+4<<7/ Vhfw1 71:, grhv qrw fruuhvsrqg wr dq DU+4, surfhvv1
Wkh gdwd lq rxu Prqwh Fduor h{shulphqw duh jhqhudwhg dv iroorzv1 Iluvw/
zh jhqhudwh d yhfwru ri frkruw gxpplhv 5 Wkh h{rjhqrxv yduldeoh lv jhqhu0
dwhg dv
%| ' w%c|  n V| 5 n 1 | c
+63,
3
zkhuh 1 | U(Efc j21 dqg V| duh yhfwruv ri frkruw hhfwv zlwk hohphqwv
U(Efc j 2 54 Zh vkdoo dvvxph wkdw DfI ri wkh yduldwlrq lq wkhvh frkruw
hhfwv lv wlph0lqyduldqw1 Wkh wrwdo huuru yduldqfhv dqg w duh fkrvhq lq vxfk d
zd| wkdw wkh -2 ri wkh htxdwlrq lv .DIc +qrw wdnlqj lqwr dffrxqw wkh frkruw
hhfw,/ zkloh wkh wrwdo xqfrqglwlrqdo yduldqfh ri %| lv vhw wr rqh/ zlwkrxw
orvv ri jhqhudolw|1 Dq lpsruwdqw hohphqw lv wkh uhodwlyh lpsruwdqfh ri wkh
frkruw hhfwv lq +63,/ zklfk zh vkdoo ydu| ehwzhhq 58 dqg DfI1 Qrwh wkdw
lq wkh devhqfh ri frkruw hhfwv lq %| c qrqh ri wkh hvwlpdwruv lv h{shfwhg wr
kdyh jrrg surshuwlhv1
Vhfrqg/ zh jhqhudwh d vwduwlqj ydoxh iru wkh + surfhvv dv
+f ' V2 5 n J c
dqg dq| vxevhtxhqw revhuydwlrqv duh jhqhudwhg dv
+| ' k+c|
3

n q%| n V 5 n | c
zkhuh | U(Efc j2 c f U(Efc j2f 1 Erwk V2 dqg V duh yhfwruv ri
frkruw hhfwv zlwk hohphqwv U(Efc j22 dqg U(Efc j2 c uhvshfwlyho|1 Qrwh
wkdw Jlupd +5333, dqg PfNhq}lh +5334, dvvxph V ' f lq wkhlu vlpxodwlrq
vwxglhv1 Zlwkrxw orvv ri jhqhudolw|/ sdudphwhu ydoxhv duh fkrvhq vxfk wkdw
wkh wrwdo huuru yduldqfh +lqfoxglqj wkh frkruw hhfwv, lv qrupdol}hg wr rqh1
Iru wkh lqlwldo shulrg/ lw lv qrupdol}hg wr *E k2 Wkh uhodwlyh lpsruwdqfh
ri wkh frkruw hhfwv ydulhv ehwzhhq 3(/ 58( dqg 83(1 Wkh uvw 43 shulrgv
ri gdwd duh glvfdughg1
54
Wkh vwduwlqj ydoxh ri wkh {lw surfhvv lv {hg dw }hur1
48
Zlwklq wklv Prqwh Fduor vhw0xs/ wkhuh duh wkuhh vrxufhv ri frkruw hhfwv=
lq wkh surfhvv iru wkh h{rjhqrxv yduldeoh/ lq wkh vwduwlqj ydoxh +f dqg lq wkh
wlph0lqyduldqw xqrevhuyhg khwhurjhqhlw| w Lq rxu h{shulphqwv zh vkdoo ydu|
wkh sursruwlrq ri frkruw0vshflf yduldwlrq lq wkhvh wkuhh vrxufhv iurp fI/
2DI wr DfI1 Wkh ydoxhv ri k dqg q duh {hg dw 3181 Zh frqvlghu vlwxdwlrqv
zlwk A ' D dqg A ' f/ zkhuh ' 2fff ru ' Dfff lqglylgxdov155 Wkh
qxpehu ri frkruwv lv ydulhg iurp 53 wr 4331
Iru wkh fdvh zlwk yh wlph shulrgv dydlodeoh iru hvwlpdwlrq/ dyhudjh hv0
wlpdwhv ryhu 4333 uhsolfdwlrqv duh suhvhqwhg lq Wdeoh 4 iru ' 2fffc dqg
lq Wdeoh 5 iru ' Dfff Vwdqgdug huuruv +lq sduhqwkhvhv, duh frpsxwhg dv
vdpsoh vwdqgdug ghyldwlrqv ri wkh Prqwh Fduor hvwlpdwhv1 Wkh fruuhvsrqglqj
uhvxowv iru A ' f duh suhvhqwhg lq Wdeohv 6 dqg 7/ uhvshfwlyho|1 Wkh wdeohv
suhvhqw uhvxowv iru wkh LY hvwlpdwru/ ghwhuplqhg dv ROV dw wkh frkruw djjuh0
jdwhv/ dqg wkh dxjphqwhg LY hvwlpdwru +DLY,/ edvhg rq wkh zlwklq hvwlpdwru
rq frkruw dyhudjhv1
Dq h{dplqdwlrq ri wkh vlpxodwlrq uhvxowv ohdgv wr d qxpehu ri lqwhuhvwlqj
frqfoxvlrqv1 Iluvw/ wkh pdjqlwxgh ri wkh frkruw hhfwv lq wkh vwduwlqj ydoxh ri
+| kdv rqo| d vpdoo lpsdfw rq wkh hvwlpdwruv1 Wklv lv h{shfwhg dv wkh vwduwlqj
ydoxh gdwhv edfn 43 shulrgv iurp wkh ehjlqqlqj ri wkh hvwlpdwlrq vdpsoh1 Iru
wkh fdvh zlwk A ' D hvwlpdwlrq shulrgv/ wkh pdjqlwxgh ri frkruw hhfw lq wkh
vwduwlqj ydoxh ri +| kdv d eljjhu lpsdfw/ erwk lq wkh uhgxfwlrq ri wkh eldv
dqg xsrq wkh vwdqgdug huuruv ri wkh hvwlpdwruv1
Vhfrqg/ zkhq pdq| frkruwv duh xvhg/ zlwk d olplwhg qxpehu ri revhu0
ydwlrqv hdfk/ doo hvwlpdwruv kdyh d vxevwdqwldo eldv1 Wklv eldv lv pruh sur0
qrxqfhg iru wkh frh!flhqw rq wkh h{rjhqrxv yduldeoh1 Wklv eldv uhgxfhv
vrphzkdw zkhq wkh frkruw hhfwv duh pruh lpsruwdqw/ exw lv vwloo qrq qhjol0
jleoh1 Wklv frqfoxvlrq lv frqvlvwhqw zlwk fxuuhqw sudfwlfh ri xvlqj frkruwv ri
dw ohdvw 433 lqglylgxdov dqg lv dovr lq olqh zlwk wkh frqfoxvlrq ri Yhuehhn dqg
Qlmpdq +4<<5, wkdw 4330533 lqglylgxdov shu frkruw duh uhfrpphqghg lq ru0
ghu wr ljqruh eldvhv gxh wr phdvxuhphqw huuru lq wkh frkruw dyhudjhv +zklfk/
lq rxu whuplqrorj|/ fruuhvsrqgv wr hvwlpdwlrq huuru lq wkh uhgxfhg irup sd0
udphwhuv,1 Vrphzkdw vxusulvlqjo|/ kdylqj pruh wlph shulrgv dydlodeoh iru
hvwlpdwlrq lq vrph fdvhv lqfuhdvhv wkh eldv wkdw rffxuv iurp wkh vpdoo frkruw
vl}hv1 Wkh dxjphqwhg LY hvwlpdwru w|slfdoo| vxhuv vrphzkdw pruh iurp
wklv eldv wkdq grhv wkh LY hvwlpdwru1 Iru fdvhv zkhuh wkh LY hvwlpdwru lv
lqfrqvlvwhqw/ frkruw vl}hv duh kdugo| uhodwhg wr wkh dprxqw ri eldv wkdw lv
55
Qrwh wkdw W @ 8 uhtxluhv 9 furvv0vhfwlrqv +w @ 3> 4> ===> W , zkloh W @ 43 uhtxluhv 441
49
frkruw hhfwv +(,
% +f +
58 3
3
k
31777
q
31867
58 58 3
k
31797
q
31856
58 83 3
k
317:7
q
3184:
83 58 3
k
317;3
q
31846
83 83 3
k
317;7
q
31843
58 58 58 k
31;9<
q
315;4
58 83 58 k
31;63
q
31637
83 58 58 k
31;38
q
3164:
58 58 83 k
31<94
q
3155;
58 83 83 k
31<5<
q
31579
83 58 83 k
31<3:
q
3158;
83 83 83 k
31;;5
q
315:5
A ' D> ' 2fff(
' 2f
LY
DLY
+31365, 3176< +31366,
+31354, 318<5 +313;5,
+31357, 31785 +3135<,
+3134:, 318;: +313;3,
+3134<, 31793 +31358,
+31347, 318;5 +313:;,
+3134;, 317:7 +31354,
+31345, 3186< +3138;,
+31348, 317:: +3134<,
+31344, 3186: +3138:,
+313:6, 317:7 +31353,
+313;7, 318:9 +313:4,
+313:9, 317:: +3134;,
+313;;, 318:8 +3139<,
+313:7, 317;6 +31349,
+313:7, 31867 +31384,
+31396, 317;7 +31348,
+313<3, 318:5 +31394,
+3139:, 317;9 +31346,
+313<9, 318:4 +3138<,
+313:4, 317;< +31345,
+313;4, 31863 +31377,
+313:6, 317<4 +31344,
+313;7, 31863 +31376,
k ' fD( q ' fD
' ff
LY
DLY
31649 +31363, 315<9 +31366,
31947 +31353, 31:47 +31393,
3169: +31357, 31663 +3135<,
318;7 +3134:, 31:3: +31393,
316<< +31353, 3168: +3135:,
31899 +31348, 31:35 +3138<,
3174: +3134<, 316<: +31355,
31884 +31346, 31964 +31386,
31766 +31349, 3173< +31353,
31875 +31344, 31959 +31385,
31;53 +31367, 316<; +31354,
3164< +31369, 319<7 +31386,
31:<5 +31367, 31745 +31353,
31668 +3136:, 319<5 +31386,
31::7 +31367, 31765 +3134:,
3166; +31365, 3194; +3137:,
31<73 +3135;, 31769 +31349,
3157; +3136;, 319;: +31379,
31<48 +3135<, 31777 +31347,
31596 +31373, 319;9 +31378,
31;<: +31364, 31787 +31346,
31599 +31367, 3193< +31373,
31;:9 +31365, 31793 +31345,
315:; +31368, 3193: +3136<,
Wdeoh 4= Dyhudjh hvwlpdwhv dqg vwdqgdug huuruv/ A ' D/ ' 2fff
4:
frkruw hhfwv +(,
% +f +
58 3
3
k
317:9
q
31848
58 58 3
k
317;8
q
31843
58 83 3
k
317;<
q
3183:
83 58 3
k
317<4
q
31838
83 83 3
k
317<6
q
31837
58 58 58 k
31;;4
q
315:6
58 83 58 k
31;6<
q
315<:
83 58 58 k
31;46
q
31645
58 58 83 k
31<9;
q
31556
58 83 83 k
31<67
q
31575
83 58 83 k
31<45
q
31587
83 83 83 k
31;;9
q
3159<
A ' D> ' Dfff(
' 2f
LY
DLY
+31353, 317:7 +31354,
+31346, 31877 +31386,
+31348, 317:< +3134;,
+31343, 31873 +31385,
+31345, 317:< +3134;,
+3133<, 3186; +31384,
+31344, 317;< +31346,
+3133;, 3184: +3136;,
+31343, 317<4 +31345,
+3133:, 31849 +3136:,
+313:6, 317;< +31345,
+313;7, 31868 +31378,
+313:8, 317<3 +31344,
+313;<, 31867 +31378,
+313:7, 317<5 +31343,
+313:7, 31847 +31366,
+31395, 317<6 +3133<,
+313;<, 31866 +3136<,
+3139:, 317<7 +3133;,
+313<9, 31865 +3136;,
+313:4, 317<8 +3133;,
+313;4, 31846 +3135;,
+313:5, 317<9 +3133:,
+313;7, 31846 +3135:,
k ' fD( q ' fD
' ff
LY
DLY
31737 +31353, 316<: +31353,
31893 +31346, 31974 +31379,
31769 +31348, 3174: +3134;,
31874 +31344, 31966 +31378,
31786 +31345, 31765 +31349,
31863 +3133<, 3195; +31378,
31796 +31344, 31787 +31346,
31856 +3133;, 3189: +31368,
317:4 +31343, 31793 +31345,
3184< +3133:, 31897 +31367,
31;:3 +31365, 31786 +31346,
315;5 +31369, 31954 +31373,
31;67 +31366, 31793 +31344,
31637 +3136:, 3194; +3136<,
31;3< +31366, 317:3 +31343,
31649 +31365, 3188< +31364,
31<:3 +31359, 317:4 +3133<,
31556 +3136:, 31947 +31367,
31<73 +3135;, 317:8 +3133;,
31574 +3136<, 31946 +31366,
31<4; +31364, 31784 +3133;,
31583 +31367, 31887 +31359,
31;<8 +31364, 317;6 +3133:,
31597 +31368, 31886 +31358,
Wdeoh 5= Dyhudjh hvwlpdwhv dqg vwdqgdug huuruv/ A ' D/ ' Dfff
4;
frkruw hhfwv +(,
% +f +
58 3
3
k
31764
q
31888
58 58 3
k
3177:
q
31876
58 83 3
k
31789
q
31869
83 58 3
k
317:5
q
31856
83 83 3
k
317:9
q
31853
58 58 58 k
31;<7
q
314<8
58 83 58 k
31;:8
q
31543
83 58 58 k
31;73
q
31567
58 58 83 k
31<8:
q
3147:
58 83 83 k
31<77
q
3148:
83 58 83 k
31<4<
q
314:7
83 83 83 k
31<3;
q
314;6
A ' f> ' 2fff( k ' fD( q ' fD
' 2f
' ff
LY
DLY
LY
DLY
+31359, 3175; +3135:, 315;< +31356, 315:; +31359,
+31354, 318;9 +31375, 3199< +3134<, 31:5; +31366,
+31354, 31774 +31357, 31655 +31354, 31639 +31357,
+3134;, 318:< +31374, 31977 +3134;, 31:4: +31366,
+3134<, 31783 +31354, 3167: +3134<, 3165< +31355,
+31349, 318:7 +3136<, 31958 +31349, 31:3< +31366,
+31348, 3179< +31349, 316<5 +31349, 316;4 +3134;,
+31345, 31869 +3135;, 318;: +31346, 31964 +3135:,
+31346, 317:5 +31347, 31737 +31348, 316<5 +3134:,
+31344, 31867 +3135;, 318:; +31345, 31959 +31359,
+31386, 31799 +3134:, 31;74 +31358, 316:7 +3134;,
+31399, 31898 +31368, 31578 +31363, 319<7 +3135<,
+31389, 317:3 +31348, 31;5< +31359, 316;: +3134:,
+313:3, 31896 +31368, 31588 +31364, 319:3 +3135<,
+31393, 317;3 +31345, 31;3; +3135:, 3174; +31347,
+31398, 31863 +31358, 31595 +3135;, 31945 +31356,
+31373, 317:< +31345, 31<66 +3134;, 31749 +31347,
+31394, 3188; +31363, 314:8 +31359, 319:< +31358,
+31376, 317;4 +31344, 31<57 +3134<, 31757 +31346,
+31398, 3188: +3135<, 314;6 +3135;, 319:: +31358,
+31383, 317;: +3133<, 31<39 +31354, 31777 +31344,
+31394, 31858 +31354, 314;9 +31359, 318<< +31353,
+31384, 317;; +3133<, 31;<; +31355, 3177; +31343,
+31397, 31857 +31353, 314<5 +3135:, 318<: +3134<,
Wdeoh 6= Dyhudjh hvwlpdwhv dqg vwdqgdug huuruv/ A ' f/ ' 2fff
4<
frkruw hhfwv +(,
% +f +
58 3
3
k
317:3
q
31857
58 58 3
k
317:;
q
3184;
58 83 3
k
317;5
q
31848
83 58 3
k
317;<
q
3183<
83 83 3
k
317<3
q
3183;
58 58 58 k
31<39
q
314;7
58 83 58 k
31;;9
q
314<<
83 58 58 k
31;7<
q
3155:
58 58 83 k
31<96
q
31473
58 83 83 k
31<83
q
31484
83 58 83 k
31<56
q
314:3
83 83 83 k
31<45
q
314:<
A ' f> ' Dfff( k ' fD( q ' fD
' 2f
' ff
LY
DLY
LY
DLY
+31348, 3179< +31348, 316;9 +31349, 316;4 +3134:,
+31345, 3186< +31358, 318<5 +31346, 31973 +31357,
+31345, 317:8 +31346, 3173< +31347, 31733 +31348,
+31343, 3186< +31358, 318:7 +31345, 31963 +31357,
+31343, 317:< +31345, 31757 +31345, 31747 +31347,
+3133<, 31866 +31357, 31895 +31344, 31956 +31356,
+3133<, 317;: +3133<, 31783 +31343, 31779 +31343,
+3133:, 31848 +3134:, 31873 +3133;, 31898 +3134:,
+3133;, 317;< +3133;, 31789 +3133<, 31784 +31343,
+3133:, 31847 +3134:, 31868 +3133:, 31894 +3134:,
+31385, 317;9 +3133<, 31;<4 +31355, 31773 +31344,
+31397, 3185< +31354, 314<; +3135:, 31944 +31354,
+31388, 317;: +3133;, 31;:8 +31356, 3177: +31343,
+3139<, 3185; +31354, 31544 +3135<, 3193; +31353,
+3138<, 317<5 +3133:, 31;77 +31359, 31797 +3133;,
+31397, 31845 +31348, 31565 +3135:, 31887 +31348,
+3136<, 317<4 +3133:, 31<93 +31349, 31795 +3133;,
+31393, 31859 +3134;, 31478 +31357, 31933 +3134;,
+31375, 317<5 +31339, 31<7< +3134:, 31798 +3133:,
+31398, 31858 +3134;, 31487 +31359, 318<; +3134:,
+3137<, 317<8 +31338, 31<59 +31353, 317:9 +31339,
+31394, 31843 +31345, 3149: +31358, 3187: +31345,
+31384, 317<8 +31338, 31<49 +31354, 317:; +31339,
+31397, 31843 +31345, 314:8 +31359, 31878 +31345,
Wdeoh 7= Dyhudjh hvwlpdwhv dqg vwdqgdug huuruv/ A ' f/ ' Dfff
53
irxqg1
D wklug frqfoxvlrq lv uhodwhg wr wkh vl}h ri wkh vwdqgdug huuruv1 Lq jhqhudo/
wkh vwdqgdug huuruv gr qrw surylgh dq| lqglfdwlrq ri wkh ghjuhh ri eldv wkdw
fdq eh h{shfwhg1 Zkhq wkh qxpehu ri frkruwv lv lqfuhdvhg/ vwdqgdug huuruv
w|slfdoo| ghfuhdvh vrphzkdw/ zkloh wkh eldv lqfuhdvhv vxevwdqwldoo|1 Qrwh
wkdw wkh vdpsolqj yduldwlrq ri doo hvwlpdwruv lv udwkhu orz1 Rq wkh rqh kdqg/
wklv lv d srvlwlyh uhvxow dv lw lqglfdwhv wkdw wkh hvwlpdwruv duh txlwh dffxudwh/
hyhq li wkh frkruw hhfwv duh uhodwlyho| xqlpsruwdqw1 Rq wkh rwkhu kdqg/ lw
lv ri frqfhuq ehfdxvh lq fdvhv zkhuh wkh hvwlpdwruv duh vhyhuho| eldvhg/ kljk
vwdqgdug huuruv duh qrw zduqlqj djdlqvw wkhlu xvh1 Dssduhqwo|/ vdpsolqj
yduldwlrq lq wkh frkruw dyhudjhv surgxfhv hvwlpdwhv wkdw kdyh d orz vwdqgdug
huuru/ hyhq li wklv yduldwlrq lv kdugo| uhodwhg wr jhqxlqh frkruw yduldwlrq1
Irxuwk/ zkhq frkruw hhfwv duh lqwurgxfhg lq wkh pdlq htxdwlrq/ wkh LY
hvwlpdwru lv vhyhuho| eldvhg1 Zlwk wkh wuxh dxwruhjuhvvlyh frh!flhqw htxdo
wr 318/ LY surgxfhv hvwlpdwhv lq wkh udqjh ri 31;3031<8/ ghshqglqj xsrq wkh
lpsruwdqfh ri wkh frkruw hhfwv1 Vlploduo|/ wkh frh!flhqw hvwlpdwhv iru wkh
h{rjhqrxv uhjuhvvru duh lq wkh udqjh ri 314803163/ djdlq zlwk d wuxh ydoxh ri
3181 Lw lv lqwhuhvwlqj wr qrwh wkdw wkh suhflvlrq ri wkh LY hvwlpdwruv lv uhgxfhg
zkhq frkruw hhfwv duh lqwurgxfhg lq wkh pdlq htxdwlrq/ exw lqvx!flhqwo| wr
jhqhudwh d srwhqwldo zduqlqj iru lwv eldv1 Dovr qrwh wkdw wkh dxjphqwhg LY
hvwlpdwru lv kdugo| dhfwhg e| wkh suhvhqfh ri frkruw hhfwv lq wkh surfhvv
iru +| 1 Udwkhu/ lwv eldvhv dsshduv wr eh voljkwo| vpdoohu wkdq lq wkhlu devhqfh1
Iliwk/ lq wkh fdvhv zkhuh wkh hvwlpdwruv duh frqvlvwhqw/ wkhuh jhqhudoo|
dsshduv wr eh d qhjdwlyh vpdoo vdpsoh eldv lq wkh hvwlpdwlrq ri wkh dxwruh0
juhvvlyh frh!flhqw1 Vlplodu uhvxowv duh uhsruwhg iru dowhuqdwlyh hvwlpdwruv
dqg gdwd jhqhudwlqj surfhvvhv e| wkhlu uhvshfwlyh dxwkruv +Froodgr/ 4<<;/
Jlupd/ 5333,1 Zh dovr qg d srvlwlyh eldv lq wkh frh!flhqw iru wkh h{rjh0
qrxv uhjuhvvru +zklfk lv qrw suhvhqw lq Froodgr*v vlpxodwlrqv,1 Wkhvh eldvhv
ehfrph vpdoohu zkhq vdpsoh vl}hv lqfuhdvh +sduwlfxoduo| zkhq wkh qxpehu ri
revhuydwlrqv shu frkruw lqfuhdvh,/ dqg zkhq wkh frkruw hhfwv ehfrph pruh
lpsruwdqw1 Wklv lv sduwlfxoduo| wuxh ri wkh frkruw hhfw lq wkh surfhvv iru
wkh h{rjhqrxv yduldeoh1 Iru uhdvrqdeoh frkruw vl}hv/ wkh eldvhv vhhp wr eh
dffhswdeoh1
Vl{wk/ iru wkh jhqxlqh sdqho gdwd fdvh/ wkh uhvxowv ri Qlfnhoo +4<;4, lpso|
wkdw iru A ' f wkh zlwklq hvwlpdwru kdv d suredelolw| olplw +iru $ 4 ri
31661 Fohduo|/ wkh frkruw0edvhg hvwlpdwruv shuirup ehwwhu lq whupv ri vpdoo
A eldv1 Wklv frqupv rxu hduolhu uhpdun wkdw hvwlpdwlqj d g|qdplf prgho
iurp frkruw ohyho gdwd fdq eh frpsxwdwlrqdoo| vlpsohu wkdq iurp jhqxlqh
54
sdqho gdwd1
9
Frqfoxglqj uhpdunv
Vhyhudo dxwkruv kdyh dujxhg wkdw wkh hvwlpdwlrq ri g|qdplf prghov dw wkh
lqglylgxdo ohyho lv srvvleoh rq wkh edvlv ri uhshdwhg furvv0vhfwlrqv dqg suhvhqw
dowhuqdwlyh hvwlpdwruv wkdw duh frqvlvwhqw xqghu dssursuldwh frqglwlrqv1 Wkh
sursrvhg hvwlpdwruv ydu| zlgho| lq wkh ghjuhh ri frpsxwdwlrqdo frpsoh{0
lw| dqg lq wkh zd| wkh| duh prwlydwhg dqg suhvhqwhg1 Iru h{dpsoh/ Pr!ww
+4<<6, suhvhqwv d vlpsoh hvwlpdwru edvhg rq ROV zkhuh wkh odjjhg ghshqghqw
yduldeoh lv uhsodfhg e| d suhglfwhg ydoxh1 Lq frqwudvw/ Froodgr +4<<;, uhtxluhv
djjuhjdwlrq ri doo revhuydwlrqv lqwr frkruwv/ diwhu zklfk d JPP hvwlpdwru/
zlwk d phdvxuhphqw huuru fruuhfwlrq/ lv hpsor|hg xvlqj wkh frkruw djjuh0
jdwhv1 Jlupd +5333, suhvhqwv dq LY hvwlpdwru wkdw lqyroyhv wkh xvh ri rwkhu
revhuydwlrqv iurp wkh frkruw wr dssur{lpdwh wkh odjjhg ghshqghqw yduldeoh
dqg wr dfw dv lqvwuxphqwv1 Wklv sdshu uhylhzhg wkh lghqwlfdwlrq frqglwlrqv
xqghuo|lqj wkhvh dowhuqdwlyh hvwlpdwruv dqg suhvhqwhg frpsxwdwlrqdoo| dw0
wudfwlyh dowhuqdwlyhv1 Rxu pdlq irfxv zdv xsrq wkh lghqwli|lqj frqglwlrqv1
Frqvlvwhqf| uhihuv wr wkh fdvh zkhuh wkh wlph glphqvlrq lv {hg dqg wkh
qxpehu ri lqglylgxdov jurzv/ zkloh wkh qxpehu ri lqvwuxphqwv lv frqvwdqw1
Qrwh wkdw PfNhq}lh +5334, surylghv d ghwdlohg glvfxvvlrq ri wkh dv|pswrwlf
surshuwlhv ri wkh hvwlpdwruv xvlqj dowhuqdwlyh pxowlglphqvlrqdo olplwv1
Wkh wkuhh prvw lpsruwdqw frqfoxvlrqv iurp rxu dqdo|vlv duh wkh iroorzlqj1
Iluvw/ rqh fdq rqo| lghqwli| lqglylgxdo g|qdplfv zlwkrxw kdylqj lqglylgxdo
wlph0vhulhv ru sdqho gdwd/ e| dvvxplqj wkh h{lvwhqfh ri d vhw ri lqvwuxphqwv
wkdw lv +4, revhuyhg iru hdfk shuvrq lq wkh hqwluh vdpsoh/ dqg +5, dssursuldwh
iru hdfk yduldeoh lq wkh prgho ri lqwhuhvw1 Vxfk lqvwuxphqwv fdq eh xvhg
wr djjuhjdwh wkh gdwd lq d qxpehu ri pxwxdoo| h{foxvlyh jurxsv +frkruwv,1
Zkloh vxfk dq dvvxpswlrq pd| eh xquhdolvwlf lq fhuwdlq dssolfdwlrqv/ lw lv/
jhqhudoo|/ qrw whvwdeoh gxh wr lwv lghqwli|lqj qdwxuh1 Wkh Prqwh Fduor vwxg|
looxvwudwhv wkh idfw wkdw lpsrvlqj lqfruuhfw lghqwli|lqj dvvxpswlrqv pd| sur0
gxfh vhyhuho| eldvhg hvwlpdwruv wkdw h{klelw olwwoh vdpsolqj yduldwlrq1 Ryhu0
doo/ wkh vl}h ri wkh vwdqgdug huuruv lv qr lqglfdwlrq iru wkh dprxqw ri eldv
wkdw lv suhvhqw1
Vhfrqg/ li rqh dvvxphv wkh devhqfh ri frkruw hhfwv lq wkh xqrevhuydeohv
ri wkh pdlq htxdwlrq/ d vlpsoh LY hvwlpdwru edvhg xsrq ROV xvlqj wkh
frkruw djjuhjdwhv/ lv frqvlvwhqw +surylghg doo yduldeohv h{klelw jhqxlqh frkruw0
55
vshflf yduldwlrq,1 Wkh hvwlpdwru sursrvhg e| Pr!ww +4<<6, lv lqfrqvlvwhqw
lq wkh suhvhqfh ri wlph0ydu|lqj h{rjhqrxv uhjuhvvruv/ zkloh wkh hvwlpdwru
suhvhqwhg e| Jlupd +5333, iru wklv prgho lv xqqhfhvvdulo| frpsolfdwhg1 Jlyhq
prghudwh vdpsoh vl}hv/ wkh vdpsolqj yduldwlrq ri wkh LY hvwlpdwru lv pdlqo|
gulyhq e| wkh lpsruwdqfh ri wkh frkruw hhfwv lq wkh h{rjhqrxv yduldeohv1
Wkh Prqwh Fduor vwxg| uhyhdov wkdw vwdqgdug huuruv duh idluo| orz/ hyhq
zkhq rqo| 2DI ri wkh lqqrydwlrqv lq wkh surfhvv iru wkh h{rjhqrxv yduldeoh
fdq eh dwwulexwhg wr frkruw0vshflf hhfwv1 Zkhq wkh qxpehu ri lqvwuxphqwv
+frkruwv, lv odujh uhodwlyh wr wkh qxpehu ri lqglylgxdov/ d vpdoo vdpsoh eldv lv
suhvhqw lq wkh LY hvwlpdwru1 Zkhq frkruw vl}hv duh 433 ru pruh/ wkhvh eldvhv
vhhp wr eh dffhswdeoh/ surylghg vx!flhqw frkruw0vshflf yduldwlrq lv suhvhqw
lq wkh h{rjhqrxv yduldeohv1
Wklug/ li rqh doorzv iru wlph0lqyduldqw frkruw hhfwv lq wkh xqrevhuydeohv
ri wkh pdlq htxdwlrq/ dq dxjphqwhg LY hvwlpdwru/ edvhg xsrq wkh zlwklq
hvwlpdwru xvlqj frkruw0djjuhjdwhv/ lv frqvlvwhqw1 Wklv uhtxluhv wkdw doo h{rjh0
qrxv yduldeohv h{klelw jhqxlqh wlph0ydu|lqj frkruw0vshflf yduldwlrq1 Djdlq/
vx!flhqwo| odujh frkruwv duh uhtxluhg wr uhgxfh wkh vpdoo vdpsoh eldv lq wklv
hvwlpdwru1 Qrwdeo|/ wkh eldv wkdw lv suhvhqw lq wkh zlwklq hvwlpdwru iru wkh
g|qdplf prgho xvlqj jhqxlqh sdqho gdwd +vhh Qlfnhoo/ 4<;4,/ lv pxfk odujhu
wkdq zkdw lv irxqg iru vlplodu hvwlpdwruv hpsor|hg xsrq frkruw djjuhjdwhv1
Uhihuhqfhv
^4` Dohvvlh/ U1/ P1S1 Ghyhuhx{ dqg J1 Zhehu +4<<:,/ Lqwhuwhpsrudo Frq0
vxpswlrq/ Gxudeohv dqg Oltxlglw| Frqvwudlqwv= D Frkruw Dqdo|vlv/
Hxurshdq Hfrqrplf Uhylhz/ 74/ 6:08<1
^5` Dqghuvrq/ W1Z1 dqg F1 Kvldr +4<;4,/ Hvwlpdwlrq ri G|qdplf Prghov
zlwk Huuru Frpsrqhqwv/ Mrxuqdo ri wkh Dphulfdq Vwdwlvwlfdo Dvvrfld0
wlrq/ :9/ 8<;09391
^6` Dqjulvw/ M1G1 +4<<4,/ Jurxshg0gdwd hvwlpdwlrq dqg whvwlqj lq vlpsoh
oderu0vxsso| prghov/ Mrxuqdo ri Hfrqrphwulfv/ 7:/ 57605991
^7` Duhoodqr/ P1 dqg V1 Erqg +4<<4,/ Vrph Whvw ri Vshflfdwlrq iru Sdqho
Gdwd= Prqwh Fduor Hylghqfh dqg dq Dssolfdwlrq wr Hpsor|phqw Htxd0
wlrqv/ Uhylhz ri Hfrqrplf Vwxglhv/ 8;/ 5::05<:1
56
^8` Eoxqghoo/ U1/ P1 Eurzqlqj/ dqg F1 Phjklu +4<<7,/ Frqvxphu Ghpdqg
dqg wkh Olih0F|foh Doorfdwlrq ri Krxvhkrog H{shqglwxuh/ Uhylhz ri
Hfrqrplf Vwxglhv/ 94/ 8:0;31
^9` Froodgr/ P1G1 +4<<;,/ Hvwlpdwlqj G|qdplf Prghov iurp Wlph Vhulhv
ri Lqghshqghqw Furvv0Vhfwlrqv/ Mrxuqdo ri Hfrqrphwulfv/ ;5/ 6:0951
^:` Gdujd|/ M1P1 dqg S1F1 Y|wkrxondv +4<<<,/ Hvwlpdwlrq ri d G|qdplf
Fdu Rzqhuvkls Prgho= D Svhxgr Sdqho Dssurdfk/ Mrxuqdo ri Wudqv0
sruw Hfrqrplfv dqg Srolf|/ 66/ 5;606351
^;` Ghdwrq/ D1 +4<;8,/ Sdqho Gdwd iurp Wlphv Vhulhv ri Furvv0Vhfwlrqv/
Mrxuqdo ri Hfrqrphwulfv/ 63/ 43<04591
^<` Jlupd/ V1 +5333,/ D Txdvl0Glhuhqflqj Dssurdfk wr G|qdplf Prg0
hoolqj iurp d Wlph Vhulhv ri Lqghshqghqw Furvv0Vhfwlrqv/ Mrxuqdo ri
Hfrqrphwulfv/ <;/ 69806;61
^43` Kdplowrq/ M1G1 +4<<7,/ Wlph Vhulhv Dqdo|vlv/ Sulqfhwrq Xqlyhuvlw|
Suhvv/ Sulqfhwrq1
^44` PfNhq}lh/ G1M1 +5334,/ Dv|pswrwlf Wkhru| iru Khwhurjhqhrxv G|0
qdplf Svhxgr0Sdqhov/ zrunlqj sdshu/ Vwdqirug Xqlyhuvlw|1
^45` Pr!ww/ U1 +4<<6,/ Lghqwlfdwlrq dqg Hvwlpdwlrq ri G|qdplf Prghov
zlwk d Wlph Vhulhv ri Uhshdwhg Furvv0Vhfwlrqv/ Mrxuqdo ri Hfrqrphw0
ulfv/ 8</ <<04571
^46` Qlfnhoo/ V1 +4<;4,/ Eldvhv lq G|qdplf Prghov zlwk Il{hg Hhfwv/
Hfrqrphwulfd/ 7</ 474:047591
^47` Yhuehhn/ P1 +4<<9,/ Svhxgr Sdqho Gdwd/ lq= O1 Päw|äv dqg S1
Vhyhvwuh/ hgv1/ Wkh Hfrqrphwulfv ri Sdqho Gdwd= Kdqgerrn ri wkh Wkh0
ru| zlwk Dssolfdwlrqv/ 5qg uhylvhg hglwlrq/ Noxzhu Dfdghplf Sxeolvkhuv/
5;305<51
^48` Yhuehhn/ P1 dqg Wk1 Qlmpdq +4<<5,/ Fdq Frkruw Gdwd Eh Wuhdwhg dv
Jhqxlqh Sdqho GdwdB/ Hpslulfdo Hfrqrplfv/ 4:/ <0561
^49` Yhuehhn/ P1 dqg Wk1 Qlmpdq +4<<6,/Plqlpxp PVH Hvwlpdwlrq ri d
Uhjuhvvlrq Prgho zlwk Il{hg Hhfwv iurp d Vhulhv ri Furvv Vhfwlrqv/
Mrxuqdo ri Hfrqrphwulfv/ 8</ 45804691
57