LUND UNIVERSITY PO Box 117 221 00 Lund +46 46-222 00 00
The Search for Chaos and Nonlinearities in Swedish Stock Index Returns
Byström, Hans; Amilon, Henrik
1998
Link to publication
Citation for published version (APA):
Byström, H., & Amilon, H. (1998). The Search for Chaos and Nonlinearities in Swedish Stock Index Returns.
(Working Papers, Department of Economics, Lund University; No. 6). Department of Economics, Lund University. http://swopec.hhs.se/lunewp/abs/lunewp1998_006.htm
Total number of authors:
2
General rights
Unless other specific re-use rights are stated the following general rights apply:
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal
Read more about Creative commons licenses: https://creativecommons.org/licenses/
Take down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Wkh Vhdufk iru Fkdrv dqg Qrqolqhdulwlhv lq Vzhglvk Vwrfn Lqgh{ Uhwxuqv
Khquln Dplorq dqg Kdqv QH E|vwuùpW Ghsduwphqw ri Hfrqrplfv
Oxqg Xqlyhuvlw|
S1R1 Er{ :3;5 V0553 3: Oxqg
Vzhghq
skrqh= .790790555 :< 3<
H0pdlo= Khquln1DplorqCqhn1ox1vh H0pdlo= Kdqv1E|vwurpCqhn1ox1vh
Devwudfw
Qxphurxv hpslulfdo vwxglhv kdyh vkrzq hylghqfh ri qrqolqhdulwlhv lq qdqfldo wlph vhulhv/
zklfk fdq eh ri erwk d ghwhuplqlvwlf dqg d vwrfkdvwlf qdwxuh1 Fkdrv lv dq h{dpsoh ri wkh iruphu/ dqg khwhurvfhgdvwlflw| lq wkh frqglwlrqdo yduldqfh dq h{dpsoh ri wkh odwwhu1 Zh dsso| d whvw/ wkh EGV whvw/ wr Vzhglvk Vwrfn Lqgh{ uhwxuqv dqg ghwhfw odujh ghyldwlrqv iurp wkh LLG0k|srwkhvlv1 Wkhuh lv qr hylghqfh ri fkdrv/ dqg prvw ri wkh qrqolqhdulwlhv duh gxh wr frqglwlrqdoo| khwhurvfhgdvwlf huuru whupv1 Zh orrn dw prqwko|/ gdlo|/ dqg 480plqxwh uhwxuq vhulhv/ dqg qg qr vhqvlwlylw| lq wkh uhvxowv wr fkrlfh ri vdpsolqj iuhtxhqf|1 Gl huhqw JDUFK prghov riwhq vhhp wr h{sodlq wkh qrqolqhdulwlhv ghwhfwhg e| wkh EGV whvw/ zklfk lv sduwlfxoduo| wkh fdvh iru JDUFK prghov zlwk w0glvwulexwhg huuruv wwhg wr prqwko| dqg gdlo| uhwxuqv1
WWkh dxwkruv duh judwhixo wr Emùuq Kdqvvrq dqg Oduv Qrugìq iru surylglqj wkh gdwd/ dqg wr Gdylg Hgjhuwrq iru xvhixo frpphqwv rq vwdwlvwlfdo lvvxhv1
4
4 Lqwurgxfwlrq
Frpprq dvvxpswlrqv lq qdqfldo hfrqrplfv duh wkdw qdqfldo yduldeohv olnh vwrfn uhwxuqv dqg h{fkdqjh udwhv fdq eh ghvfulehg e| vwrfkdvwlf surfhvvhv/ dqg wkdw hfrqrplf v|vwhpv duh olqhdu1 Iurp d wkhruhwlfdo srlqw ri ylhz/ qrwklqj lqglfdwhv wkdw wklv pxvw eh wkh fdvh/ dqg qxphurxv hpslulfdo vwxglhv kdyh vkrzq wkdw gl huhqw qrqolqhdu ghshqghqfhv pljkw h{lvw lq qdqfldo wlph vhulhv1 Wkh kljk iuhtxhqf| ri fudvkhv dqg errpv wkdw fdq eh revhuyhg lq vwrfn pdunhwv lv dq h{dpsoh ri ehkdylru qrq0frqvlvwhqw zlwk d olqhdu prgho4zlwk qrupdoo| glvwulexwhg glvwxuedqfhv1 Qrqolqhdu prghov h{sodlqlqj wklv w|sh ri ehkdylru fdq eh erwk vwrfkdvwlf dqg ghwhuplqlvwlf1 Dq h{dpsoh ri d vwrfkdvwlf qrqolqhdu prgho lv Hqjoh*v +4<;5, dxwruhjuhvvlyh frqglwlrqdoo| khw0 hurvfhgdvwlf +DUFK, prgho zlwk h{whqvlrqv1 Wkhvh prghov kdyh ehhq vxffhvvixo lq h{sodlqlqj yrodwlolw| foxvwhulqj lq vwrfn uhwxuqv dqg lq prgholqj lqwhuhvw udwhv1 D fkdrwlf surfhvv/ rq wkh rwkhu kdqg/ lv shuihfwo| ghwhuplqlvwlf exw lwv ehkdylru lv lqglvwlqjxlvkdeoh iurp sxuh udqgrpqhvv1 Fkdrwlf v|vwhpv duh dozd|v qrqolqhdu dqg kdyh ehhq lpsruwdqw lq h{sodlqlqj vhhplqjo| udqgrp ehkdylru lq wkh vflhqfhv1 D yhu| lpsruwdqw idfw uhjduglqj fkdrwlf prghov/ zklfk lv gl huhqw iurp wkh xvxdo vlwxdwlrq lq hfrqrplfv/ lv wkdw doo pryhphqwv duh jhqhudwhg zlwklq wkh prgho1 Qr h{whuqdo vkrfnv qhhg wr eh lqwurgxfhg1
Wkh odfn ri h{sodqdwlyh srzhu lq olqhdu prghov kdv ohg qxphurxv uhvhdufkhuv wr wkh vwxg| ri fkdrv lq qdqfldo hfrqrplfv/ vhh h1j1 Vfkhlqnpdq dqg OhEdurq +4<;<,/ Shwhuv +4<<4,/ Oduudlq +4<<4,/ dqg Kvlhk +4<<4,1 Kvlhk +4<;<, lqvwhdg irfxvhv rq qrqolqhdu vwrfkdvwlf prghov/ lq sduwlfxodu JDUFK prghov1 Yduvrq dqg Grudq +4<<8, wu| wr glvwlqjxlvk fkdrv iurp udqgrp qrqolqhdulwlhv zlwk wkh Judvvehujhu0Surfdffld fruuhodwlrq glphqvlrq1 Wkh glvdgydqwdjhv ri wklv phwkrg duh wkdw yhu| orqj gdwd vhulhv duh qhhghg/ dqg wkdw lw grhv qrw frqvwlwxwh d vwdwlvwlf whvw1 Wkhuhiruh/ Eurfn/ Ghfkhuw/ dqg Vfkhlqnpdq +4<;:, sursrvhg d uhodwhg vwdwlvwlf whvw/ wkh EGV whvw/ edvhg rq wkh fruuhodwlrq lqwhjudo/ d phdvxuh ri vsdwldo fruuhodwlrq/ dqg lw fdq eh xvhg wr ghwhfw ghyldwlrqv iurp wkh LLG0k|srwkhvhv1 Vlqfh wkh EGV whvw grhv qrw glvwlqjxlvk ehwzhhq gl huhqw fdxvhv iru uhmhfwlrqv ri LLG/ Kvlhk +4<<4, uvw owhuv gdwd zlwk gl huhqw olqhdu dqg qrqolqhdu owhuv lq rughu wr ghwhfw zkdw nlqg ri prgho fdq h{sodlq wkh qrqolqhdulwlhv1
Lq wklv fkdswhu/ zh orrn dw Vzhglvk vwrfn lqgh{ uhwxuqv/ dqg wu| wr ghwhfw wkh suhvhqfh ri qrqolqhdulwlhv lq wklv pdunhw1 Fdq qrqolqhdu h hfwv/ iru lqvwdqfh/ khosh{sodlqlqj wkh odujh mxpsv lq vwrfn sulfhv wkdw rffxu zlwk idluo| kljk iuhtxhqf|B Rxu gdwd fryhuv prqwko| uhwxuqv iurp 4<4< wr 4<<9/ gdlo| gdwd iurp 4<:: wr 4<<9/ dqg lqwudgdlo| gdwd +48 plqxwhv, iurp Mdqxdu| 4<<5
4E| d olqhdu prgho/ zh phdq d prgho zlwk d olqhdu phdq dqg dgglwlyh LLG huuru whupv1
5
wr Dxjxvw 4<<61
Vlqfh wkh pryhphqwv fdqqrw eh vxffhvvixoo| ghvfulehg e| olqhdu prghov/ zh wu| wr ghwhuplqh krz wr prgho wkh qrqolqhdulwlhv wkdw pxvw eh lqwurgxfhg1 W|slfdoo|/ qrqolqhdulwlhv fdq hqwhu lq wkh surfhvv jryhuqlqj uhwxuqv +qrqolqhdulwlhv lq wkh frqglwlrqdo phdq, ru lq wkh surfhvv ri wkh wlph0ydu|lqj frqglwlrqdo yduldqfh +qrqolqhdulwlhv lq wkh frqglwlrqdo yduldqfh,1 Lq rughu wr kdyh d fkdrwlf uhwxuqv surfhvv/ wkhuh pxvw h{lvw qrqolqhdulwlhv lq wkh frqglwlrqdo phdq1 Wr whvw wklv k|srwkhvlv/ zh lqyhvwljdwh wkh suhglfwdelolw| ri olqhdu yhuvxv qrqolqhdu prghov wwhg wr gdwd e| xvlqj qhxudo qhwzrunv1 Wkh zhdn hylghqfh ri qrqolqhdulwlhv lq wkh frqglwlrqdo phdq pdnhv xv frqwlqxh rxu vhdufk iru fdxvhv ri wkh qrq0LLG vwrfn uhwxuqv1 Zh wxuq wr wkh EGV whvw/ dqg wu| wr vhsdudwh wkh gl huhqw fdxvhv ri d uhmhfwlrq ri wkh LLG0k|srwkhvlv lq rxu pdunhw> fkdrv/
khwhurvfhgdvwlf frqglwlrqdo yduldqfh/ qrqvwdwlrqdulw|/ ru vlpsoh olqhdu dxwrfruuhodwlrq1 Zh uhmhfw wkh k|srwkhvlv ri frqglwlrqdo phdq fkdqjhv ru qrqvwdwlrqdulwlhv fdxvlqj wkh EGV whvw wr uhmhfw wkh LLG k|srwkhvlv1 Lqvwhdg/ zh qg vwurqj hylghqfh ri frqglwlrqdo yduldqfh ghshqghqfhv lq wkh vwrfn lqgh{ uhwxuqv1 Zh gr qrw rqo| qg vljql fdqw DUFK0h hfwv exw dovr wkh vwurqjhu uhvxow wkdw DUFK0h hfwv dorqh frqwulexwh wr pruh ru ohvv doo wkh qrq0LLG ehkdylru lq rxu gdwd1
Lq vhfwlrq 5/ zh jlyh dq lqwurgxfwlrq wr fkdrv/ qrqolqhdulwlhv/ dqg wkh EGV whvw/ lq vhfwlrq 6 rxu gdwd lv ghvfulehg/ lq vhfwlrq 7 zh suhvhqw rxu uhvxowv/ dqg vhfwlrq 8 frqfoxghv wkh fkdswhu1
5 Fkdrv dqg Krz wr Ghwhfw lw
Zkdw lv fkdrvB D fohdu fxw gh qlwlrq kdv qrw |hw ehhq jhqhudoo| dffhswhg/ exw wkuhh ixqgdphqwdo surshuwlhv pxvw eh lqfoxghg +vhh Vwurjdw} +4<<9,,=
Fkdrwlf prwlrq pxvw kdyh dq dshulrglf orqj0whup ehkdylru1 Wklv phdqv wkdw wkhuh duh wudmhfwrulhv5 zklfk gr qrw vhwwoh grzq wr {hg srlqwv/ shulrglf ruelwv/ ru txdvlshulrglf ruelwv dv w $ 41
Fkdrwlf prwlrq h{klelwv vhqvlwlyh ghshqghqfh rq lqlwldo frqglwlrqv/ zklfk phdqv wkdw qhdue| wudmhfwrulhv glyhujh h{srqhqwldoo| idvw1
Fkdrwlf prwlrq lv sxuho| ghwhuplqlvwlf1 Wkh luuhjxodu ehkdylru vwhpv iurp wkh qrqolqhdulw|
ri wkh v|vwhp dqg qrw iurp udqgrp h hfwv1
D g|qdplf v|vwhp hyroylqj lq d fkdrwlf zd| lv dovr vdlg wr kdyh d vwudqjh dwwudfwru1 Lw lv srvvleoh wr txdqwli| wkh vhqvlwlylw| rq wkh lqlwldo frqglwlrqv deryh e| gh qlqj wkh Oldsxqry
5Wkh skdvh vsdfh lv wkh vsdfh vsdqqhg e| wkh vwdwh yduldeohv1 D wudmhfwru| lv wkh sdwk ri hyroxwlrq ri d srlqw lq wkh skdvh vsdfh1
6
h{srqhqwv iru fkdrwlf v|vwhpv lq frqwlqxrxv wlph + rzv,/ dqg glvfuhwh wlph +pdsv,1 Iru d rz lq q glphqvlrqv/ dq lq qlwhvlpdo vskhuh zloo eh glvwruwhg lqwr dq lq qlwhvlpdo hoolsvrlg gxulqj lwv hyroxwlrq lq skdvh vsdfh1 Li &+w,/ n @ 4> ===> q ghqrwhv wkh ohqjwk ri wkh nwk sulqflsdo d{hv ri wkh hoolsvrlg/ wkhq &+w, 2 &+3, hb&|/ zkhuh wkh &*v duh wkh Oldsxqry h{srqhqwv1 Wkh q gl huhqw Oldsxqry h{srqhqwv wkhuhiruh ghvfuleh wkh ghirupdwlrq ri wkh v|vwhp1 Iru d vwudqjh dwwudfwru/ dw ohdvw rqh ri wkh &*v pxvw eh srvlwlyh ehfdxvh ri wkh vhsdudwlrq ri qhljkerulqj wudmhfwrulhv1 Wkh vxp ri wkh &*v ghvfulehv wkh frqwudfwlrq ri yroxph/ zklfk pxvw eh qhjdwlyh vlqfh/ e| gh qlwlrq/
dq dwwudfwru +vwudqjh ru rwkhuzlvh, vkrxog dwwudfw doo wudmhfwrulhv vwduwlqj lq d vx!flhqwo| vpdoo rshq vhw frqwdlqlqj wkh dwwudfwru1 Fkdrwlf ehkdylru ghpdqgv d srvlwlyh Oldsxqry h{srqhqw/
zklfk lpsolhv wkh h{lvwhqfh ri d wlph krul}rq eh|rqg zklfk suhglfwlrq euhdnv grzq/ vlqfh dq|
glvfuhsdqf| lq wkh hvwlpdwh ri dq lqlwldo vwdwh zloo jurz h{srqhqwldoo|1
Iurp zkhuh grhv fkdrv frphB Lq wkh gh qlwlrq ri fkdrv/ wzr surshuwlhv vhhp kdug wr frpelqh1 Krz fdq wudmhfwrulhv rq d vwudqjh dwwudfwru glyhujh hqgohvvo| dqg |hw uhpdlq erxqghgB Wkh dqvzhu lv wkdw vwudqjh dwwudfwruv uhvxow iurp d vwuhwfklqj dqg iroglqj surfhvv1 Wr eh pruh frqfuhwh/ frqvlghu djdlq wkh vskhuh lq wkh skdvh vsdfh1 D vwudqjh dwwudfwru lv jhqhudwhg zkhq wkh v|vwhp frqwudfwv wkh vskhuh lq vrph gluhfwlrqv dqg vwuhwfkhv lw lq rwkhuv1 Wr uhpdlq erxqghg/
wkh glvwruwhg vskhuh pxvw eh iroghg edfn rqwr lwvhoi1 Diwhu d odujh qxpehu ri lwhudwlrqv/ wkh vskhuh lv vsuhdg wkurxjkrxw d erxqghg uhjlrq lq wkh skdvh sodqh1 Wkh phfkdqlvpv lqyroyhg uh hfw wkh yroxph frqwudfwlrq/ wkh vhqvlwlylw| rq lqlwldo frqglwlrqv dqg wkh erxqghgqhvv ri wkh dwwudfwru1
514 Wkh Fruuhodwlrq Lqwhjudo
Krz gr zh whvw iru wkh suhvhqfh ri fkdrvB Rqh zd| zrxog eh wr fdofxodwh wkh odujhvw Oldsxqry h{srqhqw dqg vhh li lw lv srvlwlyh/ exw wklv lv qrw hdvlo| grqh lq uhdo zruog vlwxdwlrqv1 Pruhryhu/
wkh rwkhu Oldsxqry h{srqhqwv duh hyhq kdughu wr hvwlpdwh vr wklv surfhgxuh fdqqrw glvwlqjxlvk ehwzhhq gl huhqw w|shv ri vwudqjh dwwudfwruv1 Judvvehujhu dqg Surfdffld +4<;6, wkhuhiruh sur0 srvhg d gl huhqw surfhgxuh edvhg rq vsdwldo fruuhodwlrq1
Frqvlghu d vhw i{|jA ri s rlqwv rq dq dwwudfwru1 Gh qh wkh fruuhodwlrq lqwhjudo dv F+o, @ olpA <"FA+o,>
FA+o, @ 4 W2
[
rc| +o m{| {rm,>
zkhuh +d, @ 4 li d A 3> dqg 3 rwkhuzlvh1 Wkh txdqwlw| F+o, phdvxuhv wkh iudfwlrq ri wkh wrwdo qxpehu ri sdluv +{|> {r, zkrvh glvwdqfh m{| {rm lv ohvv wkdq o1 Judvvehujhu dqg Surfdffld irxqg wkdw F+o,/ dqg lwv vdpsoh hvwlpdwh FA+o,/ duh sursruwlrqdo wr o_ zkhuh g lv fdoohg wkh fruuhodwlrq glphqvlrq1 Lq sudfwlfh/ g lv hvwlpdwhg dv wkh vors h lq d s orw ri oq FA+o, yhuvxv oq+o,1
7
Zkhq ghdolqj zlwk qlwh gdwd vhwv/ wkh srzhu odz/ FA+o, 2 o_/ rqo| krogv ryhu dq lqwhuphgldwh udqjh ri o/ vlqfh dw odujh o> doo srlqwv zloo eh zlwklq d glvwdqfh ri o/ zkloh rq wkh rwkhu kdqg/ dw h{wuhpho| vpdoo o> qr sdlu ri srlqwv zloo eh zlwklq o1 W|slfdoo|/ vwudqjh dwwudfwruv kdyh d iudfwdo vwuxfwxuh dqg wkhuhiruh/ d qrq0lqwhjhu fruuhodwlrq glphqvlrq1
515 Dwwudfwru Uhfrqvwuxfwlrq
D wlph vhulhv/ i{|j/ ri doo vwdwh yduldeohv { 5 U? lv riwhq qrw dffhvvleoh1 Lq pdq| fdvhv/ lw lv qrw hyhq srvvleoh wr vshfli| hlwkhu wkh uhohydqw frpsrqhqwv ri { ru lwv glphqvlrq/ zklfk pljkw eh kljk1 Qhyhuwkhohvv/ vxfk v|vwhpv pljkw kdyh orz0glphqvlrqdo dwwudfwruv1 Iruwxqdwho|/
Sdfndug hw do1 +4<;3, dqg Wdnhqv +4<;4, vkrz wkdw wkh g|qdplfv lq wkh ixoo skdvh vsdfh fdq eh uhfrqvwuxfwhg iurp phdvxuhphqwv ri d vlqjoh0yduldeoh wlph vhulhv ii{|j> {|5 U> { {j= Wkh pdlq lghd lv wr frqvwuxfw p0glphqvlrqdo yhfwruv
|@ +{|> {|n > ===> {|nE63 ,
iru vrph ghod| A 31 Li i{|j srvvhvvhv dq dwwudfwru wkdw fdq eh hpehgghg lq dq q0glphqvlrqdo {0vsdfh/ wkhq wkh wrsrorjlfdo vwuxfwxuh ri wkh dwwudfwru uhpdlqv xqfkdqjhg zkhq hpehgghg lq
0vsdfh/ surylghg wkdw wkh hpehgglqj glphqvlrq p lv odujh hqrxjk1 D qhfhvvdu| frqglwlrq lv p q dqg d vx!flhqw frqglwlrq lv p 5q . 41 Lq rwkhu zrugv/ lw lv srvvleoh wr pdnh vrph nlqg ri yduldeoh vxevwlwxwlrq ehwzhhq wkh xqrevhuydeoh yduldeohv lq { dqg wkh odjjhg revhuydeohv1
Vlqfh i{|j dqg i|j duh revhuydwlrqdoo| htxlydohqw/ wkh| dovr kdyh wkh vdph fruuhodwlrq glphqvlrq/ surylghg wkdw p lv odujh hqrxjk1 Wkxv/ F6A+o, fdq eh fdofxodwhg/ qrz zlwk d vxevfulsw p/ xvlqj | lqvwhdg ri {|1 Li i|j lv rewdlqhg iurp d fkdrwlf wlph vhulhv/ wkhq wkh frpsxwhg fruuhodwlrq glphqvlrq zloo ohyho r dw lwv wuxh ydoxh zkhq wkh hpehgglqj glphqvlrq lv odujh hqrxjk/ vr wkdw wkhuh lv hqrxjk urrp iru wkh dwwudfwru wr xqirog1 Wkh srlqw wr eh pdgh lv wkdw li i|j lv rewdlqhg iurp d sxuho| udqgrp +LLG, vhtxhqfh/ wkh fruuhodwlrq glphqvlrq nhhsv lqfuhdvlqj zlwk p/ vlqfh qrlvh dozd|v oov xs wkh vsdfh lq zklfk lw lv hpehgghg1 Lq sulqflsoh/
hvwlpdwhv ri g fdq wkxv eh xvhg wr glvwlqjxlvk fkdrv iurp qrlvh1
Wkh ghod| vkrxog qrw eh fkrvhq wrr vpdoo vlqfh wkhq {| {|n ===/ dqg wkh dwwudfwru pljkw kdyh s ureohpv lq glvhqwdqjolqj1 Li/ rq wkh rwkhu kdqg/ lv fkrvhq wrr odujh/ sureohpv dulvh ehfdxvh wkh vsdwldo fruuhodwlrq lv orz iru glvwdqw ydoxhv lq wkh wlph vhulhv1
Xqiruwxqdwho|/ wkh Judvvehujhu0Surfdffld phwkrg kdv vrph gudzedfnv1 Iluvw/ lw pxvw eh hpskdvl}hg wkdw wkh phwkrg euhdnv grzq zkhq wkh hpehgglqj glphqvlrq lv wrr odujh gxh wr wkh vsduvlw| ri gdwd/ zklfk fdxvhv vwdwlvwlfdo vdpsolqj sureohpv1 Vhfrqg/ Judvvehujhu dqg Surfdffld edvhg wkhlu uhvxowv rq wlph vhulhv ri 43333063333 srlqwv/ dowkrxjk wkh| dujxhg wkdw rqo| d ihz wkrxvdqg srlqwv zrxog eh qhfhvvdu| wr rewdlq uhdvrqdeoh hvwlpdwhv1 Udpvh| dqg \xdq +4<;<,
8
kdyh vkrzq wkdw wkhuh lv d whqghqf| wr xqghuhvwlpdwh wkh vorsh lq gdwd vhwv zlwk dv pdq| dv 5333 srlqwv/ wkxv lqglfdwlqj fkdrv zkhq qrqh lv suhvhqw1 Wklug/ wkh judsklfdo surfhgxuh lv qrw sxw rq up vwdwlvwlfdo jurxqg dqg lw pljkw eh gl!fxow wr lqwhusuhw zkhq qrlvh lv dgghg wr wkh v|vwhp1
516 Lq xhqfh ri Qrlvh
Lq uhdo olih/ v|vwhpv fdq vhogrp eh lvrodwhg iurp wkh xqsuhglfwdeoh hqylurqphqw1 Wkh revhuyhg g|qdplfv zloo wkhq frqvlvw ri wzr sduwv> wkh lqwulqvlf ghwhuplqlvwlf g|qdplfv ri wkh v|vwhp dqg wkh lq xhqfh ri udqgrp qrlvh1 Rqh frpprq zd| ri prgholqj wkh udqgrpqhvv lv dv g|qdplfdo qrlvh=
{|@ i+{|3> ===, . %|> +4, zkhuh %| lv LLG/ {| lv wkh revhuydeoh/ dqg i lv d ghwhuplqlvwlf/ srvvleo| qrqolqhdu/ ixqfwlrq1 Li i lv fkdrwlf/ pxfk ri zkdw kdv ehhq vdlg deryh derxw fkdrv lv vwloo ydolg zkhq qrlvh lv dgghg wr wkh v|vwhp/ surylghg wkh qrlvh ohyho lv qrw wrr kljk1 Rwkhuzlvh/ wkh wudmhfwrulhv frxog mxps rxw ri wkh edvlq ri dwwudfwlrq/ ru mxpsehwzhhq gl huhqw sduwv ri wkh dwwudfwru lq vxfk d zd|
wkdw lw ehfrphv lpsrvvleoh wr vhsdudwh wkh ehkdylru iurp wuxh qrlvh> wkh dwwudfwru kdv orvw lwv vwuxfwxuh1 Dq dffhswdeoh qrlvh ohyho lv kdug wr ghwhuplqh vlqfh gl huhqw fkdrwlf v|vwhpv h{klelw yhu| gl huhqw vhqvlwlylwlhv wr qrlvh/ exw jhqhudoo|/ rqo| vpdoo dprxqwv ri qrlvh fdq eh lqwurgxfhg1 Wkh Judvvehujhu0Surfdffld phwkrg pxvw eh prgl hg zkhq ghdolqj zlwk qrlvh +vhh Ehq0 Pl}udfkl hw do1 +4<;7,,1 Iru ohqjwk vfdohv/ o/ ehorz wkrvh zkhuh wkh udqgrp frpsrqhqw eoxuv wkh vwuxfwxuh/ wkh vorsh ri oq F6A+o, yhuvxv oq+o, lv sursruwlrqdo wr wkh hpehgglqj glphqvlrq/
zkloh iru ohqjwk vfdohv deryh/ wkh vorsh lv htxdo wr wkh fruuhodwlrq glphqvlrq ri wkh ghwhuplqlvwlf v|vwhp +F6A+o, 2 o_,1
Htxdwlrq +4, fdq dovr eh xvhg wr ghvfuleh gl huhqw vwrfkdvwlf surfhvvhv1 Li/ iru lqvwdqfh/ i lv d qrqolqhdu +ru olqhdu, pds zlwk d vwdeoh dwwudfwlqj { srlqw/ wkhq wkh rxwsxw fruuhvsrqgv wr sdvvlqj udqgrp qrlvh wkurxjk d qrqolqhdu +olqhdu, uhfxuvlyh owhu/ wkhuhe| rewdlqlqj vwdwlrqdu|
dshulrglf ehkdylru1 Lq sulqflsoh/ wkh Judvvehujhu0Surfdffld phwkrg frxog eh xvhg wr lghqwli|
wkh surshuwlhv ri i e| fdofxodwlqj wkh vorsh dw gl huhqw o dqg p/ dqg frps dulqj lw wr wkh vors h ri dq LLG surfhvv ri wkh vdph ohqjwk dqg zlwk wkh vdph prphqwv/ l1h1 wkh wlph vhulhv udqgrpo|
shupxwhg ru vfudpeohg1
Iljxuh 4 vkrzv oq+F+o,, yhuvxv oq+o, iru wkh Kìqrq +4<:9, pds
{&n @ 4 . |& d{2&> +5,
|&n @ e{&> +6,
9
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 -12
-10 -8 -6 -4 -2 0
log(l)
log(C(l))
Iljxuh 4= Sorw ri oq+F+o,, yv1 oq+o, iru wkh Kìqrq pdszlwk d @ 4=7 dqg e @ 3=6 +wklfn olqhv, dqg lwv vfudpeohg frxqwhusduw +grwwhg,/ iru p @ 5> 6/ dqg 71 Kljkhu o|lqj olqhv fruuhvsrqg wr orzhu p1 Qrlvh xqlirupo| glvwulexwhg rq +3=36> 3=36, lv g|qdplfdoo| dgghg wr wkh { 0frpsrqhqw1 Wkh fkdqjhv lq vorshv iru o ? 3=39 +oq+o, ? 5=;, duh fohduo| ylvleoh1 Wkh wlph vhulhv lv qrupdol}hg wr xqlw vdpsoh yduldqfh/ dqg frqvlvwv ri 5333 revhuydwlrqv1
iru p @ 5> 6> dqg 7 zlwk xqlirupo| glvwulexwhg qrlvh g|qdplfdoo| dgghg1 Lq wklv fdvh wkh dwwudfwru fdq eh frpsohwho| uhfrqvwuxfwhg zkhq p @ 616 Iurp Iljxuh 4/ lw lv fohdu wkdw wkh vorshv duh gl huhqw iru ohqjwk vfdohv ohvv wkdq wkh pdjqlwxgh ri wkh lpsrvhg qrlvh1 Vlqfh wkh qrlvh lv xqlirupo| glvwulexwhg lq +3=36> 3=36, > wkh vorshv iru o ? 3=39> oq+o, ? 5=;, duh htxdo wr wkh fruuhvsrqglqj hpehgglqj glphqvlrqv1 Dv h{shfwhg/ wkh fruuhodwlrq glphqvlrqv ri wkh vfudpeohg wlph vhulhv vkrz qr vdwxudwlrq dv p jurzv1
Wklv judsklfdo surfhgxuh vkrxog/ lq sulqflsoh/ dovr eh ydolg iru uhvlgxdov iurp DUFK0w|sh
6Wkh vors h iru 6 ' 2 lv rqo| voljkwo| ohvv wkdq wkh vorshv lq kljkhu hpehgglqj glphqvlrqv/ wkhuhe| vxjjhvwlqj wkdw wkh dwwudfwru kdv d qrq0wulyldo vwuxfwxuh douhdg| zkhq 6 ' 2
:
prghov7/ zkhuh %|@ j+{|3> ===,x|> x| LLG/ exw lw zrxog eh kdug wr whoo li dq| ghyldwlrq iurp wkh vfudpeohg frxqwhusduw lv fdxvhg e| wkh frqglwlrqdo phdq surfhvv i ru wkh frqglwlrqdo yduldqfh surfhvv j ru erwk +vhh Vfkhlqnpdq dqg OhEdurq +4<;<,,1 Eurfn/ Ghfkhuw/ dqg Vfkhlqnpdq +4<;:, prgl hg wkh Judvvehujhu0Surfdffld phwkrg lq rughu wr flufxpyhqw vrph ri lwv gudzedfnv e|
ghyhorslqj wkh EGV whvw1 D uhylvhg yhuvlrq ri wklv whvw lv irxqg lq Eurfn hw do1 +4<<9,1
517 Wkh EGV Whvw
Wkh whvw lv edvhg rq wkh revhuydwlrq wkdw iru dq LLG vdpsoh F6A+o, @ ^FA+o,`6=
Wkh lghqwlw| vkrxog eh xqghuvwrrg lq d vwdwlvwlfdo vhqvh1 Eurfn/ Ghfkhuw/ dqg Vfkhlqnpdq +EGV, ghulyhg d qrupdol}dwlrq idfwru8 Y6A+o, lq rughu wr pdnh d fruuhfw vwdwlvwlfdo txdqwl fdwlrq ri wkh ghsduwxuh iurp LLG1 Pruh vshfl fdoo|/ wkh| vkrzhg wkdw wkh EGV vwdwlvwlf
Z6A+o, @s W
kF6A+o, ^FA+o,`6l Y6A+o,
frqyhujhv lq wkh glvwulexwlrq wr Q+3> 4, dv W $ 4> iru o A 3 dqg p A 4/ xqghu wkh qxoo k|srwkhvlv ri LLG1
Wkh EGV whvw kdv suryhg wr eh txlwh srzhuixo lq qglqj ghsduwxuhv iurp LLG lq d qxpehu ri Prqwh Fduor vlpxodwlrqv> vhh/ iru h{dpsoh/ Eurfn/ Kvlhk/ dqg OhEdurq +4<<4, dqg Kvlhk +4<<4,1 Lw kdv dovr ehhq yhul hg wkdw wkh qlwh vdpsoh glvwulexwlrq zhoo dssur{lpdwhv wkh dv|pswrwlf glvwulexwlrq iru vdpsoh vl}hv deryh 4333 revhuydwlrqv1 Dv iru wkh Judvvehujhu0Surfdffld phwkrg/
wkh uhvxowv ghshqg rq wkh pdjqlwxgh ri wkh dgglwlyh qrlvh frpsrqhqw1 Lw lv wkhuhiruh uhfrp0 phqghg +vhh Eurfn/ Kvlhk/ dqg OhEdurq +4<<4,, wr dsso| wkh whvw iru o @ 3=8 4=8 wlphv wkh vdpsoh vwdqgdug ghyldwlrq ri wkh wlph vhulhv1
Wkh EGV whvw vx huv iurp wkh idfw wkdw d uhmhfwlrq ri wkh qxoo grhv qrw surylgh dq| klqwv derxw wkh fdxvh ri wkh uhmhfwlrq1 Lw kdv ehhq vkrzq lq Eurfn dqg Srwwhu +4<<6, dqg gh Olpd +4<<9,/ krzhyhu/ wkdw wkh qxoo glvwulexwlrq ri wkh whvw lv qrw d hfwhg e| dsso|lqj wkh EGV whvw wr wkh hvwlpdwhg uhvlgxdov iurp d jhqhudo fodvv ri sdudphwulf prghov zlwk dgglwlyh LLG huuruv=
{|@ i+{|3> ===> , . x|>
7E| DUFK0w|sh prghov/ zh phdq prghov wkdw h{klelw dxwruhjuhvvlyh frqglwlrqdo khwhurvfhgdvwlflw|1
8Wkh qrupdol}dwlrq idfwru lv d frpsolfdwhg ixqfwlrq ri fruuhodwlrq lqwhjudov lq gl huhqw glphqvlrqv1 Lw lv qrw yhu| looxplqdwlqj dqg wkhuhiruh qrw suhvhqwhg/ exw fdq eh irxqg lq Eurfn hw do1 +4<<9,1
;
zkhuh lv d sdudphwhu yhfwru dqg x| lv LLG/ surylghg wkdw d s
W 0frqvlvwhqf| hvwlpdwlrq ri wkh sdudphwhuv lv srvvleoh1 Wkh odvw uhtxluhphqw phdqv wkdw s
W +a , $ Q+3> , iru vrph fryduldqfh pdwul{ / dqg lv ixo oohg lq pd{lpxp olnholkrrg hvwlpdwlrqv1 Pruhryhu/ gh Olpd +4<<9, vkrzv wkdw wkh qxlvdqfh sdudphwhu iuhh surshuwlhv ri wkh EGV whvw uhpdlq ydolg iru uhvlgxdov iurp vrph pxowlsolfdwlyh prghov
{|@ i+{|3> ===> , . j+{|3> ===> ,x|> +7, zkhuh x| lv LLG/ zklfk fryhuv pdq|/ li qrw doo/ ri wkh nqrzq DUFK0w|sh prghov/ li wkh whvw lv dssolhg wr wkh wudqviruphg uhvlgxdov ay| @ oq+ax2|, zkhuh ax| duh wkh hvwlpdwhg uhvlgxdov lq +7,1 Wkh nh| lghd lv wkdw wkh wudqvirupdwlrq jlyhv ulvh wr d prgho zlwk dgglwlyh LLG huuruv=
{|@ oq+{| i+{|3> ===> ,,2@ oqj2+{|3> ===> ,. oq+x2|, @ i+{|3> ===> , . y|>
zkhuh y| lv LLG ehfdxvh x| lv LLG1 Lw pxvw eh nhsw lq plqg wkdw wkh dv|pswrwlf surshuwlhv ri ay| dqg y| duh htxdo rqo| li lv nqrzq/ l1h1 zkhq d JDUFK prgho lv wwhg wr wkh gdwd zlwkrxw wkh frqglwlrqdo phdq surfhvv1 Krzhyhu/ wkh eldv lqwurgxfhg lv h{shfwhg wr eh vpdoo zkhq dqg duh mrlqwo| hvwlpdwhg 111ehfdxvh ri wkh odfn ri suhglfwdelolw| ri wkh uvw prphqw frpsduhg wr wkh vhfrqg fhqwudo prphqw111 +vhh Eurfn dqg Srwwhu +4<<6,,/ dv lv riwhq wkh fdvh lq qdqfldo wlph vhulhv91
Ixuwkhupruh/ wkh EGV whvw grhv qrw uhtxluh dq| prphqwv iru wkh wlph vhulhv vwxglhg wr h{lvw/ frqwudu| wr rwkhu qrqolqhdulw| whvwv/ exw zkhq wkh whvw lv dssolhg wr prgho uhvlgxdovs
W 0 frqvlvwhqf| ri wkh sdudphwhu hvwlpdwhv vrphwlphv ghpdqg wkh h{lvwhqfh ri kljkhu prphqwv lq wkh glvwxuedqfh surfhvv/ w|slfdoo| d qlwh yduldqfh1 Wkh dvvxpswlrq ri wkh h{lvwhqfh ri kljkhu rughu prphqwv pljkw rwkhuzlvh eh vhyhuh lq wkh qrqolqhdulw| whvwlqj ri qdqfldo wlph vhulhv/
zkrvh glvwulexwlrqv duh riwhq khdylo| wdlohg1 Wkh glvdgydqwdjh lv wkdw wkh EGV whvw pd| uhtxluh orqjhu vdpsohv wr eh dv h hfwlyh dv vrph rwkhu whvwv1
Dowrjhwkhu/ wkh EGV whvw pd| qrw rqo| eh xvhixo lq ghwhfwlqj fkdrv/ exw dovr lq gldjqrvwlf whvwlqj ri hvwlpdwhg prgho uhvlgxdov1
6 Wkh Gdwd
Wkh gdwd zh xvh lv orj0uhwxuqv iurp wzr gl huhqw Vzhglvk vwrfn lqglfhv/ wkh RP[0Lqgh{ dqg wkh D çuvyçuoghqv Jhqhudolqgh{1 Wkh iruphu zdv fuhdwhg dv dq xqghuo|lqj vhfxulw| iru wudglqj lq vwdqgdugl}hg vwrfn lqgh{ rswlrqv dqg iruzdug frqwudfwv/ dqg frqvlvwv ri d ydoxh0zhljkwhg
9Zlwk wkh srvvlelolw| ri errwvwudsslqj vpdoo vdpsoh glvwulexwlrqv/ wkh dv|pswrwlf surshuwlhv ri wkh whvw duh qrzdgd|v ri ohvv lpsruwdqfh1
<
Wdeoh 4= Vdpsoh vwdwlvwlfv iru uhwxuq gdwd1
Prqwko| Gdlo| 48 plq1 4 48 plq1 5 48 plq1 6 48 plq1 doo Qr1 ri revhuydwlrqv <69 5<7< 58:< 58:4 5:9: :<4:
Phdq Euf2 317< 31356 0313348 3133:5 3133;; 31337<
Yduldqfh Eufe 54194 3163 3136 3146 31369 3139
Vnhzqhvv 03185 0314; 4138 7186 315: 7169
Nxuwrvlv ;137 <163 651:5 <9176 <18; 463144
frpelqdwlrq ri wkh 63 prvw wudghg vwrfnv dw wkh Vwrfnkrop Vwrfn H{fkdqjh1 Wkh odwwhu lv wkh prvw xvhg vwrfn lqgh{ lq Vzhghq dqg lw lv d ydoxh0zhljkwhg lqgh{ ri wkh pdmrulw| ri doo vwrfnv txrwhg rq wkh Vwrfnkrop Vwrfn H{fkdqjh1
Wkh wlph lqwhuydo ehwzhhq revhuydwlrqv lv vrphwlphv ri lps ruwdqfh lq qdqfldo wlph vhulhv1 Wkhuhiruh/ zh orrn dw vhulhv zlwk gl huhqw vdpsolqj iuhtxhqf|> prqwko|/ gdlo|/ dqg 48 plqxwhv1: Lq wklv zd|/ zh fdq ghwhfw fkdrwlf dqg qrqolqhdu ehkdylru rq gl huhqw wlph vfdohv1 Lq dgglwlrq/
wkh sureohp ri wrr vkruw ru wrr orqj ghod| wlphv lq wkh uhfrqvwuxfwlrq ri wkh fkdrwlf dwwudfwru phqwlrqhg lq vhfwlrq 515 fdq eh plwljdwhg1
Wkh prqwko| gdwd +D çuvyçuoghqv Jhqhudolqgh{, lv h{whqghg ryhu wkh shulrg 4<4<04<<9/
zkloh wkh gdlo| gdwd +RP[0Lqgh{, lv iru wkh shulrg 4<;704<<91 Wkh lqwudgdlo| vhulhv +RP[0 Lqgh{,/ zlwk gdwd froohfwhg hyhu| 48 plqxwhv/ fryhuv wkh shulrg Mdqxdu| 4<<50Dxjxvw 4<<6 dqg lv glylghg lqwr wkuhh vhulhv zlwk dssur{lpdwho| htxdo ohqjwk wr fdwfk dq| lqvwdelolw| lq wkh hvwlpdwhg sdudphwhuv1; Dw wkh vdph wlph/ wkh ohqjwk ri wkh vhulhv ehfrphv frpsdudeoh wr wkh ohqjwk ri wkh gdlo| vhulhv1 Wkh uvw revhuydwlrq hyhu| gd| kdv ehhq uhpryhg gxh wr wkh dxfwlrq0 olnh vwduw xssurfhgxuh dw wkh h{fkdqjh1 Doo gdwd lv zlwkrxw glylghqgv/ exw zh kdyh dovr orrnhg dw prqwko| +4<4<04<<9, dqg gdlo| +4<::04<<4, uhwxuqv zlwk glylghqgv lqfoxghg +D çuvyçuoghqv Jhqhudolqgh{,1< Wkh uhvxowv iru wkhvh vhulhv duh qrw uhsruwhg exw duh vlplodu wr wkrvh zlwkrxw glylghqgv dqg dq| gl huhqfhv zloo eh frpphqwhg xsrq1 Wdeoh 4 glvsod|v vrph vdpsoh vwdwlvwlfv ri wkh uhwxuq gdwd1 Wkh kljk ghjuhh ri nxuwrvlv dqg vnhzqhvv/ lq sduwlfxodu iru wkh 48 plqxwh gdwd lv revhuyhg1
:Wkh 480plqxwh wlph vsdq ehwzhhq gdwd srlqwv vkrxog eh odujh hqrxjk wkdw srvvleoh plfurpdunhw vwuxfwxuh ghshqghqfhv gr qrw vklqh wkurxjk1 Vxfk ghshqghqfhv fdq/ iru lqvwdqfh/ dulvh iurp wkh vhtxhqwldo h{hfxwlrq ri rughuv lq wkh wudghuv* olplw rughu errn/ ru e| wklq wudglqj lq vrph ri wkh xqghuo|lqj vwrfnv1
;Wkh vhfrqg vxeshulrg fryhuv wkh yrodwloh prqwkv zkhq Vzhghq dedqgrqhg wkh {hg h{fkdqjh udwh uhjlph1
<Iru prqwko| uhwxuqv/ wkh vhulhv zlwk glylghqgv kdv <69 revhuydwlrqv dqg iru wkh gdlo| uhwxuqv/ 67;< revhuyd0 wlrqv1
43
Wdeoh 5= EGV vwdwlvwlfv iru udz gdwd1 o lv wkh ohqjwk vfdoh/ lv wkh vdpsoh vwdqgdug ghyldwlrq/
dqg p lv wkh hpehgglqj glphqvlrq1 Wkh whvw vwdwlvwlf lv dv|pswrwlfdoo| glvwulexwhg dv Q+3> 4,1
6
j, 5 6 7 8 9 : ; < 43
prqwko| 318 ;166 431;6 4618: 49194 53146 5917< 66189 7;154 9:154 4 ;176 431:7 4616< 481:5 4:1;4 53187 56197 5:167 65153 418 :1:3 <1;6 45137 461:< 48153 491:6 4;157 4<1:9 5417<
gdlo| 318 45169 47199 4:188 54138 571;5 63138 691;5 791<7 8<154 4 45186 47173 49168 4;186 5317< 551;5 5815< 5;14: 641;:
418 46158 48145 491:6 4;146 4<154 53167 5417: 55189 561;9 48 plq1 4 318 44176 46146 451;5 4618; 461;4 461;4 4619: 4619: 46133 4 451:9 46186 451;; 45194 45158 441:6 44146 43175 <184 418 441;3 441:< 431;< 4314< <17; ;1;5 ;154 :17; 91:5 48 plq1 5 318 491:4 4<144 531;: 5517: 57136 5816; 59167 5:156 5:16:
4 47167 4819: 481:; 481:7 48165 471:; 47134 46158 45169 418 <184 43134 <19: <14< ;186 :1;8 :149 917; 81;4 48 plq1 6 318 ;1;< 44149 441<6 4616< 48148 491:< 4;146 4<15< 531::
4 ;169 4318< 44156 45173 4617; 4715: 471:< 471<8 48147 418 :17< ;1<; <168 43144 431:5 44133 44138 431;8 43187 48 plq1 doo 318 571<; 5<1;: 65196 69164 7317< 771:; 7<145 8618; 8;135 4 5713< 91;< 5:1:8 5;19; 5<15; 5<18: 5<198 5<185 5<154 418 53145 5418: 5416< 5415< 531<5 53173 4<1:< 4<146 4;176
7 Hpslulfdo Uhvxowv
Lq wkh iroorzlqj vhfwlrq/ zh dsso| wkh EGV whvw wr wkh udz gdwd wr whvw li wkh gl huhqw wlph vhulhv duh LLG1 Li wkh qxoo k|srwkhvlv ri LLG lv uhmhfwhg +dqg lw lv,/ zh wu| wr glvfryhu wkh fdxvh ri wkh uhmhfwlrq e| dsso|lqj wkh EGV whvw wr wkh hvwlpdwhg uhvlgxdov iurp gl huhqw prghov1 Pruh suhflvho|/ zh h{dplqh zkhwkhu wkh uhmhfwlrq ruljlqv iurp qrqolqhdu +srvvleo| fkdrwlf, ru olqhdu ghshqghqfhv lq wkh phdq surfhvvhv/ qrqvwdwlrqdulwlhv/ ru li wkh uhmhfwlrq fdq eh h{sodlqhg e|
qrqolqhdu vwrfkdvwlf prghov h{klelwlqj frqglwlrqdo khwhurvfhgdvwlflw|1
714 Whvw ri wkh Udz Gdwd
Wdeoh 5 vkrzv wkh EGV vwdwlvwlfv zkhq wkh hpehgglqj glphqvlrq lv ehwzhhq 5 dqg 43/ dqg o lv fkrvhq dv 318/ 4 dqg 418 wlphv wkh vdpsoh vwdqgdug ghyldwlrq ri wkh gl huhqw wlph vhulhv1 Dv fdq
44
eh revhuyhg/ wkh whvw vwurqjo| uhmhfwv wkh k|srwkhvlv ri LLG vwrfn uhwxuqv dw dq| frqyhqwlrqdo vljql fdqfh ohyho dqg iru doo p dqg o> hyhq wkrxjk olwwoh fdq eh vdlg derxw wkh fdxvh ri wkh uhmhfwlrq1 Li wkh wlph vhulhv fdph iurp d qrlv| fkdrwlf v|vwhp zlwk d kljkhu glphqvlrqdo dwwudfwru/
wkhq wkh whvw vwdwlvwlf zrxog w|slfdoo| eh vpdoo iru orz hpehgglqj glphqvlrqv dqg odujh lq glphqvlrqv zlwk hqrxjk urrp iru wkh dwwudfwru1 Reylrxvo|/ wklv lv qrw wkh fdvh vlqfh wkh uhmhfwlrqv duh vljql fdqw douhdg| dw p @ 51
Frqvlghulqj wkh uhvxowv/ zh surfhhg wr vwxg| krz ghshqghqfhv lq wkh frqglwlrqdo phdq pljkw fdxvh wkh uhmhfwlrq ri LLG1
715 Whvw ri Ghshqghqfhv lq wkh Frqglwlrqdo Phdq Vxssrvh vwrfn uhwxuqv duh jhqhudwhg dv=
{|@ i+{|3> ===, . %| +8,
zkhuh %| lv LLG dqg i lv wkh frqglwlrqdo phdq surfhvv1 Wkh h{suhvvlrq lq +8, fdq eh xvhg wr ghvfuleh qrlv| fkdrwlf prghov dv zhoo dv gl huhqw vwrfkdvwlf surfhvvhv1 Qrz/ zh zloo lqyhvwljdwh zkhwkhu i lv qrqolqhdu +dqg srvvleo| fkdrwlf,/ dqg zkhwkhu dq| ghshqghqfhv lq wkh frqglwlrqdo phdq fdq h{sodlq wkh uhmhfwlrq ri LLG1
Li i lv qrqolqhdu/ lw fdq eh prghohg e| qrqsdudphwulf uhjuhvvlrqv/ h1j1 duwl fldo qhxudo qhwzrunv1 Wklv srzhuixo whfkqltxh lv h{kdxvwlyho| ghvfulehg hovhzkhuh +vhh Khuw} hw do1 +4<<4, dqg Nxdq dqg Zklwh +4<<7,,/ dqg wkxv/ zh zloo khuh phuho| vd| wkdw d pxowl od|hu shufhswurq +POS,/ zlwk rqh klgghq od|hu ri vljprlg wudqvihu ixqfwlrqv/ fdq frqvlvwhqwo| prgho d wlph vhulhv jhqhudwhg e| dq duelwudu| ixqfwlrq/ surylghg wkdw wkh ixqfwlrq lv erxqghg dqg xqlirupo|
frqwlqxrxv1 Rq wkh rwkhu kdqg/ li wkh POS rqo| frqwdlqv olqhdu wudqvihu ixqfwlrqv/ lw fdq rqo|
slfn xs olqhdu ghshqghqfhv/ l1h1 wkh qrqsdudphwulf prgholqj uhgxfhv wr d olqhdu uhjuhvvlrq1 Zh wkhuhiruh surfhhg lq wkh iroorzlqj zd|=
Zh vsolw hdfk wlph vhulhv lqwr wkuhh sduwv ri xqhtxdo vl}h1 Wkh uvw 83( duh xvhg iru wkh uhjuhvvlrq/ ru wudlqlqj/ zkloh wkh frqvhfxwlyh 58( duh xvhg dv ydolgdwlrq vhwv1 Wkh odvw 58(
+w @ 4> ===> W|er|, duh xvhg dv whvw vhwv iru rxw0ri0vdpsoh hydoxdwlrq1 Iru hdfk uhwxuqv vhulhv/ zh wudlq 43 POS*v zlwk 6/ 8 dqg : klgghq qhxurqv/ hdfk zlwk vljprlg +wdqk, wudqvihu ixqfwlrqv/ lq d vlqjoh klgghq od|hu dqg d olqhdu rxwsxw qhxurq1 Dv lqsxwv zh xvh {|3> ===> {|3b fruuhvsrqglqj wr dq hpehgglqj glphqvlrq p @ 431 Iru hdfk wlph vhulhv/ zh dovr wudlq olqhdu shufhswurqv wr fdswxuh wkh olqhdu ghshqghqfhv1 Lq doo fdvhv/ wkh vxp vtxduhg huuruv ri wkh ydolgdwlrq vhwv duh plqlpl}hg/ dqg iru hdfk wlph vhulhv/ zh slfn wkh qhwzrun zlwk wkh ehvw ydolgdwlrq shuirupdqfh1 Wkh gl huhqw POSv duh wkhq hydoxdwhg xvlqj wkh rxw0ri0vdpsoh suhglfwlrq huuru qrupdol}hg zlwk
45
Wdeoh 6= Rxw0ri0vdpsoh hydoxdwlrq iru qrqolqhdu dqg olqhdu qhxudo qhwzrunv1 POS6 uhihuv wr d POS zlwk 6 klgghq qhxurqv hwf1 POSO uhihuv wr d olqhdu shufhswurq1
.R
POS6 POS8 POS: POSO
prqwko| 413845 41374; 414968 41368:
gdlo| 31<<46 413465 413379 41343;
48 plq1 4 31<<36 31<;:4 41345: 31<;85 48 plq1 5 31<<75 413785 413348 31<<76 48 plq1 6 31<<59 31<;9< 41335< 31<;53 48 plq1 doo 31<<34 31<<58 31<<3< 31<;<8
wkh rxw0ri0vdpsoh yduldqfh ri {|=
HR @
A|er|
SA|er|
|'
{| ai{|3> ===> {|3b2
a 2|er| =
Li i lv qrqolqhdu/ HR vkrxog eh vpdoohu iru qrqolqhdu wkdq iru olqhdu qhwzrunv1 Ixuwkhupruh/ li i lv fkdrwlf/ wkhq wkh shuirupdqfh ri qrqolqhdu qhwzrunv vkrxog lpsuryh dv wkh iruhfdvw krul}rq vkruwhqv/ wkdw lv/ zkhq zh jr iurp d vdpsoh lqwhuydo ri rqh prqwk grzq wr 48 plqxwhv/ dqg lw lv d frqvhtxhqfh ri wkh srvlwlyh Oldsxqry h{srqhqw+v, lq d fkdrwlf v|vwhp1
Wkh uhvxowv duh uhsruwhg lq Wdeoh 61 Wkuhh wklqjv fdq eh qrwhg1 Iluvw/ wkh rxw0ri0vdpsoh suhglfwlrq shuirupdqfh lv yhu| zhdn/ zlwk HR voljkwo| ehorz rqh +dqg rffdvlrqdoo| deryh,1 Vhf0 rqg/ wkhuh lv qr vxssruw lq wkh gdwd iru qrqolqhdu uhjuhvvlrq ehlqj vxshulru wr olqhdu uhjuhvvlrq1 Vrphwlphv/ lw lv vxshulru exw wkhq wkh lpsuryhphqw lv qhjoljleoh1 Wklug/ wkh shuirupdqfh ri wkh qrqolqhdu qhwzrunv grhv qrw lpsuryh dv wkh vdpsoh lqwhuydo lv uhgxfhg1
Zh dovr dqdo|}h wkh ghshqghqfh ri wkh rxwsxw rq wkh lqsxw yduldeohv e| lqvshfwlqj ghulydwlyhv diwhu frps ohwhg uhjuhvvlrq1 Wklv lv grqh e| fdofxodwlqj
V&+w, @ C ai+{|3> ===> {|3b,
C{|3& > n @ 4> ===> <
iru w @ 4> ===> W|er| dqg wkhq frps xwlqj V&@ 4
W|er|
[
| m V&+w, m = +9,
Wkh phdvxuh lq +9, vkrzv zklfk yduldeohv duh wkh prvw lpsruwdqw1 Iru d olqhdu qhwzrun/ wklv lv mxvw htxdo wr +wkh devroxwh ydoxhv ri, wkh frh!flhqwv iurp d olqhdu uhjuhvvlrq1
Wkh uhvxowv duh irxqg lq Wdeoh 71 Zh vhh wkdw erwk w|shv ri qhwzrunv lghqwli| dssur{lpdwho|
wkh vdph yduldeohv dv ehlqj lpsruwdqw/ riwhq wkh uvw1 Wkh pdjqlwxghv ri V& gl hu vrphzkdw
46
Wdeoh 7= Vhqvlwlylw| ghshqghqfhv lq wkh qrqsdudphwulf uhjuhvvlrqv1 POS uhihuv wr d qrqolqhdu pxowl od|hu shufhswurq1 POSO uhihuv wr d olqhdu shufhswurq1
7 72 7 7e 7D 7S 7. 7H 7b
prqwko| POS 314486 31453; 313693 313<5< 31373< 3143<4 313596 313848 31455;
POSO 314967 314464 313456 313563 313334 3149;4 31345< 31437< 31454:
gdlo| POS 31454: 3134:4 313795 31357< 31343: 3138:7 3137:8 313658 313575 POSO 314;76 313457 31389< 313439 313543 31398< 313;<; 313:;3 3135;4 48 plq1 4 POS 31386: 31383: 313844 3136;4 3134<4 313379 313539 3134;6 313438 POSO 313;45 313698 313;49 3134;< 3135<5 31339< 313386 313444 31348:
48 plq1 5 POS 313<6< 3137<< 313556 313356 313998 3138:5 313373 31347: 31354<
POSO 313:47 3137;9 3135<; 313347 313438 3134<8 3136;4 313584 313557 48 plq1 6 POS 313674 313595 313435 313388 313467 313564 31339; 313489 313389 POSO 3138;8 3133<8 313486 3135:3 31336< 313336 31343< 313395 313549 48 plq1 doo POS 314663 3136;8 3133<4 313658 3133:3 31355: 313465 313395 313636 POSO 3146:4 3136<6 313386 31367; 313387 313495 313358 3134;4 313647
ehwzhhq olqhdu dqg qrqolqhdu qhwzrunv/ exw wkh srlqw lv wkdw wkhuh vhhp wr eh qr vxevwdqwldo ghshqghqfhv lq wkh gdwd qrw fdswxuhg e| wkh olqhdu qhwzrunv1
Dowkrxjk wklv lv qrw d vwdwlvwlfdo whvw ri qrqolqhdulwlhv lq wkh frqglwlrqdo phdq surfhvv/ zh gudz wkh frqfoxvlrq wkdw wkhuh duh qrqh1 Hlwkhu wkh ghshqghqfhv lq i duh wuxo| olqhdu/ ru wkh| duh vr eoxuuhg zlwk qrlvh wkdw wkh frqglwlrqdo phdq fdq ehvw eh dssur{lpdwhg e| d olqhdu surfhvv1 Vlqfh fkdrv fdq rqo| dulvh iurp d qrqolqhdu pdsslqj/ wklv vxjjhvwv wkdw wkhuh lv qr fkdrwlf ehkdylru lq wkh phdq surfhvv1
Fdq wkh olqhdu ghshqghqfhv lq i fdxvh wkh EGV whvw wr uhmhfw LLGB Zh wu| wr dqvzhu wklv txhvwlrq e| dsso|lqj wkh EGV whvw wr wkh uhvlgxdov iurp wkh olqhdu qhwzrunv deryh1 Dv dsshduv iurp Wdeoh 8/ wkh whvw vwurqjo| uhmhfwv wkh uhvlgxdov dv ehlqj LLG dw dq| uhdvrqdeoh vljql fdqfh ohyho1 Wkh whvw vwdwlvwlfv ri wkh olqhduo| owhuhg gdwd gr qrw gl hu d juhdw ghdo iurp wkrvh ri wkh udz gdwd/ lqglfdwlqj wkdw wkh uhmhfwlrq lv qrw rqo| gxh wr olqhdu ghshqghqfhv431
Lq wkh iroorzlqj/ zh wu| wr vruw rxw zkdw nlqg ri qrqolqhdulwlhv wkh EGV whvw lq idfw ghwhfwv lq rxu gdwd1
43Zh dovr dssolhg wkh EGV whvw wr wkh uhvlgxdov iurp wkh qrqolqhdu qhwzrunv1 Dv h{shfwhg/ wkh whvw vwdwlvwlfv glg qrw gl hu d juhdw ghdo iurp wkrvh lq Wdeoh 8/ exw wr rxu nqrzohgjh/ lw lv qrw fohdu zkhwkhu wkh qxlvdqfh iuhh surshuwlhv ri wkh whvw lv ydolg iru uhvlgxdov iurp qrqsdudphwulf uhjuhvvlrqv1
47
Wdeoh 8= EGV vwdwlvwlfv iru olqhduo| owhuhg gdwd1 o lv wkh ohqjwk vfdoh/ lv wkh vdpsoh vwdqgdug ghyldwlrq/ dqg p lv wkh hpehgglqj glphqvlrq1 Wkh whvw vwdwlvwlf lv dv|pswrwlfdoo| glvwulexwhg dv Q+3> 4,1
6
j, 5 6 7 8 9 : ; < 43
prqwko| 318 :15; 4313< 4518< 48137 4:1:3 55144 5914; 64168 69199 4 :137 <1:: 45188 471;9 491;: 4<179 55169 581;8 6316<
418 9183 ;1<: 44163 4614< 47198 49155 4:1:6 4<159 531<:
gdlo| 318 441;6 47174 4:137 53187 57178 6317; 6:1:8 7:1<6 8;1;<
4 4514; 4715: 49149 4;185 5319< 56159 59136 5<154 6615;
418 451<7 48138 4919; 4;158 4<177 53198 541;: 5613< 57186 48 plq1 4 318 4318; 45154 441<; 45193 451<3 451:: 4517; 45143 44147 4 4415: 441<< 44179 4414; 431;9 43167 <1:7 ;1<< ;13;
418 43179 43167 <197 ;1<: ;165 :19< :145 9174 819<
48 plq1 5 318 47167 491;: 4;185 4<1;6 54145 55158 551:; 56166 56167 4 4715: 48178 48186 48184 48149 4719: 461<8 4614< 4515:
418 <19< <1<< <194 <13; ;175 :1:; :145 9177 81:7 48 plq1 6 318 :1<< 43154 431;3 45135 4617< 48137 49198 4:1<5 4<1:;
4 :186 <1;6 43185 4419< 451:5 46177 461<6 47138 4714;
418 919: ;16: ;1;: <1:9 43175 431:6 431:: 43189 43156 48 plq1 doo 318 55175 5:158 6313< 66176 6:145 731;; 77179 7;14< 85146 4 55185 5815< 59158 5:146 5:1:3 5:1<8 5:1<9 5:1:; 5:175 418 4<189 531;5 53195 5317< 53145 4<195 4<134 4;167 4:18<
48
716 Whvw ri Qrqvwdwlrqdulwlhv
Ilqdqfldo wlph vhulhv duh qrw qhfhvvdulo| vwdwlrqdu|1 Wkhuh pljkw eh d qxpehu ri vwuxfwxudo fkdqjhv fuhdwlqj qrqvwdwlrqdu| vhulhv= srolf| fkdqjhv/ fkdqjhv lq qdqfldo vwuxfwxuhv dv zhoo dv lpsruwdqw whfkqrorjlfdo lqqrydwlrqv1 Wkhuhiruh/ lw lv srvvleoh wkdw d uhmhfwlrq ri LLG e| wkh EGV whvw lv fdxvhg e| qrqvwdwlrqdulwlhv dqg d forvhu vwxg| ri zkhwkhu wklv lv wkh fdvh lv prwlydwhg1 Wkh w|sh ri qrqvwdwlrqdulwlhv zh glvfxvv lq wklv vhfwlrq dulvh iurp vzlwfklqj ehwzhhq gl huhqw olqhdu prghov> wkdw lv/ zh doorz wkh sdudphwhuv ri d olqhdu prgho wr fkdqjh iurp uhjlph wr uhjlph1
Wr lqyhvwljdwh wkh lq xhqfh ri qrqvwdwlrqdulw|/ zh fdq hlwkhu dsso| wkh EGV whvw wr wkh uhvlgxdov iurp d olqhdu vzlwfklqj prgho/ ru zh fdq wu| wr mxgjh lw pruh txdolwdwlyho| +iroorzlqj Kvlhk +4<<4,,1 Vlqfh zh kdyh wlph vhulhv zlwk gl huhqw h{whqvlrqv lq wlph/ zh gr wkh odwwhu1 Rxu orqjhvw uhwxuq vhulhv lv :: |hduv dqg rxu vkruwhvw lv derxw : prqwkv1 Lq ehwzhhq/ zh kdyh d wlph vhulhv ri derxw 46 |hduv1 Zkhq zh surfhhg wr lqfuhdvlqjo| vkruwhu wlph lqwhuydov/ zh h{shfw wkh h hfw ri vwuxfwxudo fkdqjhv wr glvdsshdu +zh vlpso| dvvxph wkdw ohvv dqg ohvv fkdqjhv rffxu, dqg frqvhtxhqwo|/ dq| uhmhfwlrq ri wkh LLG k|srwkhvlv gxh wr qrqvwdwlrqdulwlhv dorqh vkrxog glvdsshdu1
Wkhuh lv qr vxssruw iru wklv lghd1 Wkh olqhduo| owhuhg vwdwlvwlfv lq Wdeoh 8 duh dv kljk iru wkh 48 plqxwh gdwd dv iru wkh prqwko| dqg gdlo| gdwd1 Dv orqj dv wkh olqhdu uhjlph vkliwv rffxu zlwk d iuhtxhqf| orz hqrxjk wr ohdyh rxu 48 plqxwh gdwd xqd hfwhg/ wkh| fdqqrw eh wkh rqo|
fdxvh ri wkh EGV whvw uhmhfwlqj wkh LLG k|srwkhvlv1
717 Whvw ri Ghshqghqfhv lq wkh Frqglwlrqdo Yduldqfh
Dv phqwlrqhg/ wkhuh duh pdq| zd|v ri jhqhudwlqj qrq0LLG gdwd> zh kdyh wuhdwhg olqhdu dqg qrqolqhdu dxwruhjuhvvlrq/ fkdrv/ dqg qrqvwdwlrqdulw|1 Qrz/ zh wxuq wr prghov h{klelwlqj qrq0 olqhdulwlhv lq wkh frqglwlrqdo yduldqfh1 H{dpsohv ri prghov zlwk wklv ehkdylru duh dxwruhjuhvvlyh frqglwlrqdoo| khwhurvfhgdvwlf prghov olnh DUFK/ JDUFK/ dqg wkhlu h{whqvlrqv1
Pdq| wlph vhulhv h{klelw shulrgv ri xqxvxdoo| odujh yrodwlolw| iroorzhg e| shulrgv ri wudqtxlo0 lw|1 Xqghu vxfk flufxpvwdqfhv/ wkh dvvxpswlrq ri d frqvwdqw yduldqfh lv reylrxvo| lqdssursuldwh1 Lq rughu wr ghwhfw wkh wlph0ydu|lqj yduldqfh/ wkh frqglwlrqdo yduldqfh fdq eh prghohg dv d olqhdu ixqfwlrq ri sdvw vtxduhg huuruv1 Wklv lv wkh DUFK prgho1 Ixuwkhu h{whqvlrqv ri wkh prgho/ do0 orzlqj wkh frqglwlrqdo yduldqfh wr dovr eh d ixqfwlrq ri lwv rzq odjv/ jlyhv xv Eroohuvohy*v +4<;9, jhqhudol}hg DUFK prgho/ JDUFK+s> t,/ zkhuh s dqg t duh wkh qxpehu ri odjjhg frqglwlrqdo yduldqfh dqg vtxduhg huuru frpsrqhqwv/ uhvshfwlyho|1 Lq wkh fdvh ri dq DU+5, surfhvv zlwk wkh
49
frqglwlrqdo yduldqfh prghohg dv JDUFK+4/4, zh kdyh=
{| @ f. {|3. 2{|32. %|
2| @ !f. !%2|3. !2 2|3>
zkhuh 2| lv wkh frqglwlrqdo yduldqfh ri %|/ %| @ |x|/ dqg x| Q+3> 4,1 Prvw hpslulfdo vwxglhv vxjjhvw wkdw s dqg t odujhu wkdq rqh duh uduho| qhhghg1 Rqh sureohp zlwk JDUFK prgholqj lv wkdw wkh sdudphwhuv duh uhvwulfwhg> !f/ !/ dqg !2 pxvw doo eh odujhu wkdq }hur dqg wkh vxp ri !dqg !2 pxvw eh ohvv wkdq rqh1 Wkh vxp !. !2 phdvxuhv wkh shuvlvwhqfh ri frqglwlrqdo yduldqfh wr vkrfnv/ zklfk dssurdfkhv lq qlw| dv wkh vxp dssurdfkhv rqh iurp ehorz1 Wkh fdvh zkhuh wkh vxp ri ! dqg !2 htxdov rqh lv uhihuuhg wr dv Lqwhjudwhg lq JDUFK ru LJDUFK1
Zh surfhhg rqh pruh vwhs/ wklv wlph xvlqj dq dv|pphwulf vshfl fdwlrq iru wkh frqglwlrqdo yduldqfh surfhvv/ e| h{whqglqj wkh vwdqgdug DUFK0prgho wr Qhovrq*v +4<<4, HJDUFK0prgho=
oq 2| @ !f. !%|3
|3 . !2#%|3
|3
H%%|3
|3
&$
. oq+ 2|3,
zkhuh dv deryh 2| lv wkh frqglwlrqdo yduldqfh ri %|dqg %|m %|3 Q+3> |,1 Xqolnh vlpsoh DUFK dqg JDUFK/ HJDUFK fdq fdswxuh wkh dv|pphwulf uhvsrqvh ri wkh yduldqfh wr wkh gluhfwlrq ri %|/ wkdw lv/ d kljkhu yduldqfh zkhq %|lv qhjdwlyh/ dqg d orzhu yduldqfh zkhq %|lv srvlwlyh1 Lq wklv prgho/ ! ? 3 uhsuhvhqwv wkh dv|pphwulf h hfw zkloh !2 A 3 surgxfhv wkh DUFK h hfw1 Wkh 0sdudphwhu ghwhuplqhv wkh shuvlvwhqfh lq yduldqfh1 Xqolnh wkh JDUFK prgho/ wkhuh duh qr qrq0qhjdwlylw| uhvwulfwlrqv rq wkh sdudphwhuv lq wkh HJDUFK prgho1
D fuxfldo dvvxpswlrq lq wkh vshfl fdwlrqv deryh lv wkh qrupdolw| dvvxpswlrq rq wkh vwdq0 gdugl}hg huuru whup x|> wkh frqglwlrqdo khwhurvfhgdvwlflw| ri %| dorqh lv h{shfwhg wr h{sodlq wkh revhuyhg nxuwrvlv lq wkh uhwxuq glvwulexwlrq1 Vlqfh hpslulfdo hylghqfh vwurqjo| uhmhfwv wkh lghd wkdw qdqfldo uhwxuqv duh qrupdoo| glvwulexwhg/ zh frpsduh hvwlpdwhv zlwk qrupdoo| glvwulexwhg huuruv zlwk w0glvwulexwhg rqhv1 Lq wklv zd|/ d odujhu sduw ri wkh h{fhvv nxuwrvlv lq wkh vwrfn uhwxuqv pljkw eh fdswxuhg144
Edloolh dqg GhJhqqdur +4<<3, dv zhoo dv Srrq dqg Wd|oru +4<<5, fohduo| ghprqvwudwh wkdw wkh w0glvwulexwlrq jlyhv d ehwwhu w wr wkh huuru whup ri qdqfldo uhwxuq vhulhv1 Iru wkh w0glvwulexwlrq rqh dgglwlrqdo sdudphwhu pxvw eh hvwlpdwhg/ wkh ghjuhh ri iuhhgrp/ 1 Wkh w0glvwulexwlrq lv
44Iru wkh HJDUFK prgho/ rqh pxvw eh dzduh ri wkh idfw wkdw wkh frqglwlrqdo yduldqfh h{suhvvlrq fkdqjhv zlwk wkh glvwulexwlrq> .kj0w4w4l
ghshqgv rq wkh fkrvhq glvwulexwlrq1 Iru wkh qrupdo glvwulexwlrq .kj0w4w4l 's2
Z/ dqg iru wkh |0glvwulexwlrq/ rxu fdofxodwlrqv jlyh .kj0w4w4l
' KE 45
KE5
Iv
IZ1 Iru odujh ydoxhv ri v +ryhu 63,/ wklv h{suhvvlrq lv yhu| forvh wrs2
Z1
4:
Wdeoh 9= JDUFK zlwk qrupdoo| glvwulexwhg huuruv1 Wkh dvwhulvn lqglfdwhv wkdw wkh qxoo k|srwk0 hvlv ri dq LJDUFK fdqqrw eh uhmhfwhg1 Vpdoo qxpehuv lqglfdwh vwdqgdug huuruv1
kfu fD k k2 fu fD 2 (u fD
prqwko|W ff
fff2 fS
ff D 0313;;
ff e DH.
H f f.
ff2e fH
ff2D
gdlo| 2f
H.f f2f
ffb fe
ff22 f
fffH fH
ff2
48 plq1 4 04169
22 f2
ff2 ff.
fff2e ff
ff f2.
ff. Hb
fb
48 plq1 5 eDD
Sef f2S
ffb fb
fffSH f2f
ff2 f.
ff .D
feD2
48 plq1 6 ee
2bb fe
ff22 f2
fffS fD
ff. f e
ff2D .f
fS
48 plq1 doo bH
SDf f2
ff2 f
fff2 f2e
ff f.
fff. S D
ffbb
v|pphwulf/ exw kdv idwwhu wdlov dqg d kljkhu nxuwrvlv wkdq wkh qrupdo glvwulexwlrq1 Zkhq jrhv wr lq qlw|/ wkh w0glvwulexwlrq dssurdfkhv d qrupdo glvwulexwlrq1
Lq wklv vhfwlrq/ zh hvwlpdwh wkh sdudphwhuv lq gl huhqw JDUFK prghov1 HJDUFK prghov kdyh dovr ehhq hvwlpdwhg exw wkh uhvxowv duh qrw suhvhqwhg vlqfh wkhvh duh yhu| vlplodu wr wkrvh ri wkh JDUFK prghov1 Iru hdfk wlph vhulhv/ d qxpehu ri gl huhqw JDUFK dqg HJDUFK prghov kdyh ehhq hvwlpdwhg/ dqg wkh prghov zlwk wkh ehvw w kdyh ehhq fkrvhq/ rq edvlv ri sduvlprq|/
olnholkrrg ydoxh/ dqg ehkdylru ri wkh vwdqgdugl}hg uhvlgxdov1
Iru wkh w0glvwulexwlrq/ rqh sureohp wkdw kdv hduolhu ehhq irxqg iru wkh 480plqxwh gdwd lv wkh orz ghjuhhv ri iuhhgrp +ehorz 7,/ zklfk phdqv wkdw irxuwk0rughu prphqwv gr qrw h{lvw +Kdqvvrq dqg Kùugdko +4<<7,,1 Rqh srvvleoh h{sodqdwlrq iru wkhvh uhvxowv lv wkdw wkh ghqvlw| iru wkh 48 plqxwh uhwxuqv kdv yhu| idw wdlov/ gulylqj wkh ghjuhhv ri iuhhgrp ri wkh w0glvwulexwlrq wr orz ohyhov1 Lq idfw/ wkhvh sureohpv duh vroyhg e| dgglqj d gdlo| ryhuqljkw gxpp| wr wkh 480plqxwh vhulhv lq rughu wr dffrxqw iru wkh lqfuhdvhg yduldqfh lq ryhuqljkw uhwxuqv145 Xvlqj d surfhvv olnh wkh iroorzlqj iru wkh frqglwlrqdo yduldqfh lq wkh JDUFK prgho/
2| @ !(G . !f. !%2|3. !2 2|3
zkhuh G lv wkh gxpp| yduldeoh zlwk sdudphwhu !(/ zh fdswxuh d frqvlghudeoh sduw ri wkh nxuwrvlv dqg jhw zhoo0ehkdyhg uhvlgxdov146
Wkh pd{lpxp olnholkrrg hvwlpdwhv +wkh EKKK dojrulwkp, ri JDUFK prghov zlwk qrupdoo|
glvwulexwhg/ dv zhoo dv w0glvwulexwhg/ huuruv duh suhvhqwhg lq Wdeohv 9 dqg :1 Iru doo rxu prghov dqg uhwxuq vhulhv/ zh jhw vljql fdqw sdudphwhu hvwlpdwhv/ h{fhsw iru firu wkh 480plqxwh uhwxuqv1
45Lw zrxog dovr eh vwudljkwiruzdug wr lqfoxgh d zhhnhqg gxpp| exw lwv lpsdfw vkrxog eh vpdoohu wkdq wkh ryhuqljkw gxpp|> wkh qxpehu ri qrqwudghg gd|v hdfk zhhnhqg lv pxfk vpdoohu wkdq wkh qxpehu ri qrqwudghg 480plqxwh shulrgv ryhuqljkw1
46Lq wkh HJDUFK0fdvh/ d gxpp| lv lqwurgxfhg lq d vlplodu pdqqhu> *? j2| ' *?E n (( n fn111
4;
Wdeoh := JDUFK zlwk w0glvwulexwhg huuruv1 Wkh dvwhulvn lqglfdwhv wkdw wkh qxoo k|srwkhvlv ri dq LJDUFK fdqqrw eh uhmhfwhg1 Vpdoo qxpehuv lqglfdwh vwdqgdug huuruv1
kfu fD k k2 fu fD 2 (u fD v
prqwko|W ff
fff2 fS
ff D 0313:;
ff D SSf
2DS fS
ff fH2
ff b
2DHf
gdlo| eff
.ebf f.
ffb 03136:
ffb ffb
ff2f f
ffD fHS
ffH .2H
fS f
48 plq1 4 0415;
bf f.
ffb ffS
fffD ff
ff2e f e
ff De
fbf ebf
feDD
48 plq1 5 22e
2eS f2e
ffb fS
ff f2
ff e f2S
ff2. ...
fb2D eSS
fe H
48 plq1 6 Df
2SDS f
ffb f
ff fS
ff f D
ffe2 2f.
fDH eS2
fe H
48 plq1 doo D
e S f.
ff ff
fffS f2D
ff2 f
ffH f.
f2 D ef2
fbS
Iru wkh prqwko| gdwd/ zh fdqqrw uhmhfw wkh qxoo ri dq lq qlwh xqfrqglwlrqdo yduldqfh +zh fdqqrw uhmhfw wkh qxoo ri dq LJDUFK lq wkhvh JDUFK prghov, hyhq wkrxjk lqwxlwlrq vxjjhvwv d phdq0 uhyhuwlqj yduldqfh1 Wklv frq lfwlqj hylghqfh pljkw eh uhfrqflohg e| doorzlqj iru iudfwlrqdo rughuv ri lqwhjudwlrq +Eroohuvohy dqg Plnnhovhq +4<<9,,1 Lqfoxglqj DU+5, whupv lq wkh frqglwlrqdo phdq jlyhv ehwwhu w wr gdwd dqg ohvv +olqhduo|, fruuhodwhg vwdqgdugl}hg uhvlgxdov iru vrph gdwd vhulhv1 Lq jhqhudo/ wkh frh!flhqwv ri wkh DU+5, whupv duh yhu| vpdoo/ krzhyhu1 Wkh hvwlpdwhv ri wkh ghjuhhv ri iuhhgrp sdudphwhu iru wkh w0glvwulexwlrq duh fohduo| qlwh/ lqglfdwlqj qrq0qrupdoo|
glvwulexwhg huuruv1 Wkh ghjuhhv ri iuhhgrp sdudphwhuv gl hu ehwzhhq wlph vhulhv exw duh vlplodu iru wkh JDUFK dqg HJDUFK prghov/ uhvshfwlyho|1 Ryhudoo/ wkh sdudphwhu hvwlpdwhv duh idluo|
qrq0vhqvlwlyh wr wkh vshfl fdwlrq ri wkh huuru glvwulexwlrq1
Iru rxu gdlo| RP[0Lqgh{ gdwd/ Kdqvvrq dqg Kùugdko +4<<:, kdyh hvwlpdwhg HJDUFK prghov zkhuh wkh huuruv duh ghvfulehg e| wkh qrupdo glvwulexwlrq dv zhoo dv wkh jhqhudol}hg huuru glvwulexwlrq +JHG,1 D frpsdulvrq ri rxu hvwlpdwhv zlwk wkh uhvxowv iurp Kdqvvrq dqg Kùugdko +4<<:, vkrzv doprvw lghqwlfdo hvwlpdwhv iru !/ !2 dqg zkhq wkh huuruv duh dvvxphg wr eh qrupdo1 Wkh vdph krogv iru rxu sdudphwhuv lq wkh w0glvwulexwhg prgho/ zkhq frpsduhg wr wkh hvwlpdwhg JHG prghov e| Kdqvvrq dqg Kùugdko147
Lq sudfwlfh/ doo prghov uhwdlq vrph vnhzqhvv lq wkh uhvlgxdov/ d qglqj frpprq lq hpslulfdo vwxglhv ri frqglwlrqdoo| khwhurvfhgdvwlf prghov1 D frpsdulvrq ri wkh ydoxhv ri nxuwrvlv ehwzhhq wkh prghov vkrzv wkdw/ dv h{shfwhg/ wkh w0glvwulexwhg prgho kdv uhvlgxdov zlwk kljkhu nxuwrvlv wkdq wkh qrupdoo| glvwulexwhg rqh1 Zkloh wkh qrupdoo| glvwulexwhg vwdqgdugl}hg uhvlgxdov vkrz h{fhvv nxuwrvlv iru doo gdwd vhulhv/ wkh w0glvwulexwhg vwdqgdugl}hg uhvlgxdov vkrz d orzhu wkdq wkhruhwlfdoo| suhglfwhg nxuwrvlv ri derxw wkh vdph vl}h iru doo exw wkh gdlo| vhulhv1
47Lw lv lpsruwdqw wr qrwh wkdw/ dv Qhovrq +4<<4, vkrzv/ HJDUFK prghov zlwk d |0glvwulexwhg | jlyh ulvh wr vwulfwo| vwdwlrqdu| dowkrxjk qrw fryduldqfh vwdwlrqdu| j2| dqg 0|> l1h1 j2| dqg 0| kdyh qr qlwh xqfrqglwlrqdo prphqwv1
4<
Wdeoh ;= EGV vwdwlvwlfv iru wudqviruphg JDUFK uhvlgxdov/ qrupdo glvwulexwlrq1 o lv wkh ohqjwk vfdoh/ lv wkh vdpsoh vwdqgdug ghyldwlrq/ dqg p lv wkh hpehgglqj glphqvlrq1 Wkh whvw vwdwlvwlf lv dv|pswrwlfdoo| glvwulexwhg dv Q+3> 4,1
6
j, 5 6 7 8 9 : ; < 43
prqwko| 318 04146 04143 031<; 031:< 03176 3137 3168 0314: 05157 4 0319< 03179 0315; 03148 03146 03154 03153 0315: 03146 418 0317< 03155 0313; 0313: 03139 03157 0315< 03179 03179
gdlo| 318 31;5 414; 416: 41:7 5134 41;7 4174 3193 3146
4 3183 3198 31;3 415< 416; 415< 413< 31<: 31<5
418 319: 316: 3174 31;: 31;9 31:4 318: 3183 3178
48 plq1 4 318 314< 03159 0317; 03178 03158 03144 3135 4137 416;
4 03185 03177 0315< 0313; 3154 3188 31<4 415; 4179 418 031;< 03169 3138 316: 31;3 4153 418< 41<9 5149 48 plq1 5 318 4163 417: 416; 418< 419< 4194 31;3 03185 0519<
4 4184 416: 4166 4185 4193 41;3 41<8 5147 5133
418 41;6 41;9 41;< 41<: 5139 5155 5164 5176 516<
48 plq1 6 318 03177 031<6 04137 0319; 031:4 03176 031;: 0417< 051<7 4 0315: 031:4 031<; 031:5 0318: 03188 0318< 03193 0318<
418 0316< 031<3 031<: 0319: 0317< 03188 0318< 03195 03194 48 plq1 doo 318 3173 3136 3134 3178 31:3 31<3 31<8 31;6 31;7
4 315: 3139 3143 3196 4134 4173 4199 41<4 5137
418 316; 3163 3189 4136 4176 41;3 5144 516; 5184
71714 EGV Vwdwlvwlfv rq Wudqviruphg Vwdqgdugl}hg Uhvlgxdov
D frpsdulvrq ri EGV vwdwlvwlfv iru wkh olqhduo| owhuhg qdqfldo gdwd dqg iru wkh JDUFK uhvlgxdov/ wkh h{whqw wr zklfk wkh uhmhfwlrq ri LLG uhvlgxdov duh gxh wr DUFK0h hfwv lv fdswxuhg1 Dv glvfxvvhg lq vhfwlrq 517/ zh flufxpyhqw wkh sureohp lq JDUFK owhulqj uhodwhg wr wkh lvvxh ri qxlvdqfh sdudphwhuv e| orrnlqj dw wkh wudqviruphg vwdqgdugl}hg uhvlgxdov1 Lq Wdeoh ; dqg Wdeoh </ zh suhvhqw wkh EGV vwdwlvwlfv iru wkh wudqviruphg vwdqgdugl}hg uhvlgxdov iurp wkh wwhg JDUFK prghov lq Wdeoh 9 dqg Wdeoh :/ uhvshfwlyho|1
Wkh prvw reylrxv uhvxow lv wkh vxevwdqwldo ghfuhdvh lq uhmhfwlqj iuhtxhqf| diwhu JDUFK o0 whulqj481 Zh fdqqrw uhmhfw wkh qxoo k|srwkhvlv iru prqwko| dqg gdlo| gdwd zlwk qrupdo huuruv1
48Wkh EGV whvw lv dv|pswrwlfdoo| glvwulexwhg dv Efc dqg frqvlghulqj rxu orqj gdwd vhulhv/ zh dvvxph wkh
53
