Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
В данном документе рассматривается теоретическая база и практические применения искусственного интеллекта в медицине. Автор обсуждает различные подходы к созданию медицинских систем с искусственным интеллектом, их преимущества и ограничения. Кроме того, рассматриваются реальные примеры использования искусственного интеллекта в медицине, такие как диагностика болезней, планирование лечения и мониторинг пациентов. Документ может служить полезным для студентов, изучающих медицинскую информатику и искусственный интеллект.
Typology: Schemes and Mind Maps
1 / 43
This page cannot be seen from the preview
Don't miss anything!
(^4) L zrxog olnh wr wkdqn Ehqqhww W1 PfFdooxp iru klv h{fhoohqw uhvhdufk jxlgdqfh1 Wklv zrun lv
wkh vhfrqg fkdswhu ri p| grfwrudo wkhvlv dqg zdv suhsduhg gxulqj p| vwd| dv d ylvlwlqj vfkrodu lq Fduqhjlh Phoorq Xqlyhuvlw|
Devwudfw
Wklv sdshu dqdo|}hv wkh shulrg0wr0shulrg fkdqjhv wkdw rffxu lq dq rswlpl}lqj prqhwdu| prgho zlwk xqfhuwdlqw| dqg vwlfn| sulfhv1 Prqh| lv lqfrusrudwh lq lwv uroh dv d phglxp ri h{fkdqjh wkurxjk d wlph0frvw wudqvdfwlrqv whfkqrorj|1 Dqrwkhu lpsruwdqw fkdudfwhulvwlf ri wkh prgho lv wkdw erwk fdslwdo dqg lqyhvwphqw duh rewdlqhg hqgrjhqrxvo|1 Lq wklv uh jdug/ dgmxvwphqw frvwv ri lqvwdoolqj lqyhvwphqw duh lqfrusrudwhg wr vprrwk dqg ghod| fdslwdo pryhphqwv ryhu wkh hfrqrplf f|foh1 Zh zloo irfxv dwwhqwlrq rq dqdo|}lqj wkh frqvxpswlrq/ lqyhvwphqw/ dqg uhdo prqh| ghpdqg ixqfwlrqv uhvxowlqj iurp wkh prgho1 Wkhvh wkuhh htxd wlrqv jlyhv ulvh wr wkh vwuxfwxudo LV0OP hfrqrp| dv sduw ri wkh jhqhudo htxloleulxp ghvfulehg lq wkh sdshu1 Qrplqdo sulfhv duh vwlfn|/ l1h1/ wkh| gr qrw dgmxvw lqvwdqwo| wkhuhe| doorzlqj ghsduwxuhv iurp jhqhudo htxloleulxp rewdlqhg zkhq wkhuh lv devhqfh ri qrplqdo iulfwlrqv1 Zh fkrvh wr kdyh wkh Ixkuhu0Prruh vshfl fdwlrq iru qrplqdo frqwudfw sulfhv1 Wkh prgho lv fdo leudwhg rq txduwhuo| revhuydwlrqv iurp Xqlwhg Vwdwhv gdwd1 Irxu w|shv ri h{rjhqrxv vkrfnv duh lqfoxghg lq rxu vhwxs= surgxfwlrq whfkqrorj| vkrfnv/ frqvxpswlrq suhihuhqfh +ghpdqg, vkrfnv/ prqhwdu| srolf| vkrfnv/ dqg vkrsslqj wlph vkrfnv1 Khqfh/ yduldelolw| ri rxwsxw/ frqvxpswlrq/ lqyhvwphqw/ hwf1/ pd| uhvxow iurp vhyhudo vrxufhv1 Wkh lpsdfw ri hdfk vkrfn lq wkh hfrqrplf f|foh zloo eh h{dplqhg e| sorwwlqj lpsxovh0uhvsrqvh ixqfwlrqv lpsolhg e| wkh vroxwlrqv ri wkh prgho Nh|zrugv= prqh|/ g|qdplf rswlpl}lqj prghov/ exvlqhvv f|foh MHO frghv= H46/ H74/ H
pryhphqwv duh vprrwkhg e| frqvlghulqj dgmxvwphqw frvwv ri lqvwdoolqj lqyhvwphqw Wkhuh duh wkuhh rwkhu pdlq fkdudfwhulvwlfv lq rxu prgho= l, Rswlpl}lqj fulwhuld1 D qhrfodvvlfdo glvfuhwh0wlph prgho lv frqvwuxfwhg lq wkh zhoo nqrzq rswlpl}lqj iudphzrun ghvfulehg e| Vlgudxvnl +4<9:,/ dqg Eurfn +4<:8,1 Wkxv/ d vhw ri uvw rughu frqglwlrqv duh ghulyhg iurp udwlrqdo fkrlfhv ri wkh krxvhkrogv zkr dfw dv frqvxphuv dqg surgxfhuv1 Ghflvlrqv duh pdgh wr pd{lpl}h d glvfrxqwhg vxp ri h{shfwhg xwlolw| ydoxhv vxemhfw wr uhvrxufh dqg wlph frqvwudlqwv1 Wkh uhvxowlqj vwuxfwxudo htxdwlrqv kdyh wkh srwhqwldo wr eh srolf| lqyduldqw1 D jhqhudo htxloleulxp v|vwhp zloo eh iruphg e| dgglqj xs pdunhw fohdulqj frqglwlrqv ll, Vwlfn| sulfhv lq wkh vxsso| vlgh1 Wkh suhvhqfh ri voxjjlvk phfkdqlvpv lq qrplqdo zdjhv ru sulfhv kdv ehhq eurdgo| lqfoxghg lq wkh exvlqhvv f|foh olwhudwxuh wr lpsuryh wkh shuirupdqfh ri wkh vlpxodwlrqv +iru uhfhqw sdshuv vhh Fkr dqg Frroh| +4<<8,/ Grwvh|/ Nlqj/ dqg Zropdq +4<<:,/ Nlqj dqg Zdwvrq +4<<8,/ Urehuwv +4<<8,/ Zrrgirug +4<<9,/ dqg d vxuyh| e| Qhovrq +4<<;,,1 Zh fkrvh wr kdyh wkh sulfh vhwwlqj iroorzlqj wkh Ixkuhu0Prruh vshfl fdwlrq +Ixkuhu0Prruh +4<<8,,1 Wkh Ixkuhu0Prruh htxdwlrqv jxdudqwhh d kljk ghjuhh ri shuvlvwhqfh ri lq dwlrq lq wkh prgho/ pxfk forvhu wr wkh sdwwhuq ri lq dwlrq lq dfwxdo gdwd wkdq zlwk dq| rwkhu vwlfn|0sulfh irupxodwlrq +vhh Qhovrq +4<<;,,1 Wkh h{lvwhqfh ri vwlfn| sulfhv ohdgv wr ghsduwxuhv iurp wkh jhqhudo htxloleulxp vroxwlrq1 Dv d uhvxow/ wzr phdvxuhv ri rxwsxw pxvw eh glvwlqjxlvkhg= fxuuhqw rxwsxw dqg fdsdflw| +ru pdunhw0fohdulqj, rxwsxw lll, Ydulhw| ri vkrfnv1 Xqfhuwdlqw| vxuurxqgv wkh prgho zkhuh vwrfkdvwlf vkrfnv pd| klw wkh v|vwhp vkdslqj wkh hfrqrplf f|foh1 Djhqwv xwlol}h udwlrqdo h{shfwdwlrqv wr pdnh suhglfwlrqv1 Irxu w|shv ri h{rjhqrxv vkrfnv duh lqfoxghg lq wkh prgho= surgxfwlrq whfk qrorj| vkrfnv/ frqvxpswlrq suhihuhqfh +ghpdqg, vkrfnv/ prqhwdu| vkrfnv/ dqg wudqvdfwlrqv whfkqrorj| vkrfnv1 Khqfh/ yduldelolw| ri rxwsxw/ frqvxpswlrq/ lqyhvwphqw/ hwf1/ pd| frph iurp vhyhudo uhdvrqv1 Wkh lpsdfw ri hdfk vkrfn lq wkh hfrqrplf f|foh zloo eh h{dplqhg e| glvfxvvlqj wkh lpsxovh0uhvsrqvh ixqfwlrqv lpsolhg e| wkh vroxwlrqv ri wkh prgho Wzr duh wkh prwlydwlrqv iru wklv fkdswhu1 Iluvw/ ghulyh dqg dqdo|}h wkh vwuxfwxudo frq vxpswlrq/ lqyhvwphqw/ dqg prqh| ghpdqg ixqfwlrqv rewdlqhg lq dq rswlpl}lqj prqhwdu| prgho zlwk wudqvdfwlrq frvwv lq whupv ri wlph dqg hqgrjhqrxv lqyhvwphqw1 Vhfrqg/ zh zloo ghvfuleh krz wkh prgho suhglfwv wkh uhvsrqvhv ri pdfurhfrqrplf yduldeohv wr fkdqjhv lq whfkqrorj|/ prqhwdu| srolf|/ ru krxvhkrog*v suhihuhqfhv Wkh sdshu surfhhgv dv iroorzv1 Vhfwlrq 5 lv ghyrwhg wr ghvfulelqj wkh hohphqwv ri wkh
prgho zlwk vshfldo dwwhqwlrq rq wkh sursrvhg vkrsslqj wlph ixqfwlrq1 Lq vhfwlrq 6/ zh vhh krz wkh krxvhkrogv pdnh wkhlu rswlpdo ghflvlrqv/ dqg wkhq ghulyh d jhqhudo htxloleulxp iudphzrun1 Vhfwlrq 7 gh qhv hqgrjhqrxvo| fxuuhqw rxwsxw/ fdsdflw| rxwsxw/ dqg wkh rxwsxw jds1 Lq vhfwlrq 8/ zh frpsohwh wkh vhwxs e| suhvhqwlqj wkh Ixkuhu0Prruh sulfh vhwwlqj uhodwlrq rq wkh vxsso| vlgh1 Wkh prgho lv fdoleudwhg lq vhfwlrq 91 Ixuwkhu/ wkh ehkdylru ri wkh pdlq yduldeohv ri wkh prgho duh fkdudfwhul}hg lq vhfwlrq : e| phdqv ri lpsxovh uhvsrqvh ixqfwlrqv wkdw ghwhuplqh wkh h hfwv ri h{rjhqrxv vkrfnv wr wkh v|vwhp1 Wkh sdshu qlvkhv lq vhfwlrq ; zlwk vrph frqfoxvlrqv
5 Wkh hohphqwv ri wkh prgho
Wkh vkrsslqj wlph ixqfwlrq
Rxu hfrqrp| frqvlvwv ri d odujh qxpehu ri lq qlwho|0olyhg krxvhkrogv/ hdfk ri wkhp surgxflqj d gl huhqw vlqjoh jrrg wkdw fdq eh frqvxphg/ vrog/ ru wudqviruphg lqwr sk|vlfdo fdslwdo iru surgxflqj lq wkh qh{w shulrg1 Lq wklv pdq| jrrgv hfrqrp|/ frqvxpswlrq lv gh qhg dv dq djjuhjdwh1 Lq sduwlfxodu/ zh xvh lqgh{hv dv lq Gl{lw dqg Vwljolw} +4<::,^6 iru erwk exqgohv ri frqvxpswlrq +S| dqg wkh sulfh ohyho +|,1 Krxvhkrogv vkduh wkh vdph xwlolw| ixqfwlrq/ dqg wkh| kdyh dffhvv wr wkh vdph whfkqrorj|1 V|pphwulf htxloleuld lv h{dplqhg lq zklfk |E ' |E ' | iru doo c 5 dÅc c o 17 Ohw (^6) | uhsuhvhqw wkh dprxqw ri uhdo prqh| edodqfhv khog e| wkh krxvhkrog dw wkh hqg ri shulrg |
|
zkhuh | ghqrwhv wkh dprxqw ri qrplqdo prqh|1 Krxvhkrogv sxw xs d vkrs zkhuh wkh| vhoo wkh jrrg wkdw wkh| surgxfh wr wkh rwkhu krxvhkrogv1 Olnhzlvh/ wkh| kdyh wr vshqg sduw ri wkhlu wlph rq vkrsslqj iru wkh rwkhu Å jrrgv1 Prqh| lv xvhg gxulqj wkh sxufkdvhv (^6) Wkh frqvxpswlrq dqg sulfh ohyho lqgh{hv duh wdnhq lq glvfuhwh xqlwv ryhu wkh Q glvwlqfw jrrgv= fw @ Q 4 ^4 &
(^) SQ l@4^ fw+l,^
& & 4 ^ &&^4 zlwk & A 4
Sw @ Q ^4
(^) SQ l@4^ Sw+l,
4 &
(^4) ^4 & zlwk & A 4 = (^7) Lq wxuq/ ghpdqg iru hdfk lqglylgxdo jrrg zloo eh htxdo wr Q (^4) fw> ehlqj fw wkh frqvxpswlrq djjuhjdwh
6
#r|cS| ' @ 2 EÅ (^) r@rrf dqg #r|c6| ' @ EÅ (^) r@rrf 1 Iurp qrz rq/ wkh rr vxshuvfulsw ryhu rqh yduldeoh ghqrwhv lwv vwhdg| vwdwh ydoxh1 Wkh kljkhu lv wkh frqvwdqw vhdufklqj frvw @f uhodwlyh wr wkh vwhdg|0vwdwh vkrsslqj wlph/ wkh orzhu lv wkh lpsdfw ri frqvxpswlrq dqg prqh| lq wkh wrwdo vkrsslqj wlph Dqrwkhu dujxphqw ri wkh vkrsslqj wlph ixqfwlrq lv d vkrsslqj whfkqrorj| vkrfn |1 D udqgrp zdon jhqhudwhv revhuydwlrqv ri wklv vkrfn
| ' | 3 n "| +516,
zkhuh "| duh gudzv iurp d qrupdol}hg qrupdo glvwulexwlrq/ l1h1 "| Efc j" 1
Wkh xwlolw| ixqfwlrq
Xwlolw| ghshqgv rq frqvxpswlrq +S|,/ ohlvxuh wlph +,|,/ dqg d udqgrp vkrfn uh hfwlqj frq vxpswlrq suhihuhqfhv +l| 1 Lq sduwlfxodu/ zh fkrvh d frqvwdqw uhodwlyh ulvn dyhuvlrq +FUUD, lqvwdqwdqhrxv xwlolw| ixqfwlrq
L El|c S|c ,| ' i TEl| S
| 3 j Å j n C^
| jc c C : f
lq zklfk wkh vwdqgdug uvw dqg vhfrqg rughu dvvxpswlrqv LS : f/ LSS f/ L, : f/ dqg L,, f krog1 Wkh furvv pdujlqdo h hfw lv qxoo +LS, ' f, ehlqj frqvxpswlrq dqg ohlvxuh vhsdudeoh lq +517,1 Wkh frqvxpswlrq suhihuhqfh vkrfn iroorzv dq DU+4, surfhvv
l| ' Sl| 3 n |c +518, zlwk mSm Å dqg | Efc j 1
Wkh surgxfwlrq ixqfwlrq dqg wkh dgmxvwphqw frvwv ri lqyhvwphqw
Krxvhkrogv surgxfh rxwsxw ++|, dffruglqj wr d Free0Grxjodv surgxfwlrq ixqfwlrq gh shqglqj rq wkh dprxqwv ri oderu ghpdqghg +?_| c hpsor|hg fdslwdo +&| / dqg rq wkh vwdwh ri whfkqrorj| uhsuhvhqwhg e| d whfkqrorjlfdo vkrfn + (^5) |
orj vw @ d 5 +4 (^) vdvv^3 , orj fw d 6 +4 (^) vdvv^3 , orj pw. "w> zkhuh wkh frqvwdqw whup zdv ohiw rxw
8
+| ' s E (^5) |c ?| c &| ' i TE (^5) | E?| ^3 k&k| c +519,
zkhuh zh gr qrw lqfoxgh dq| qrwdwlrq uhihuulqj wr wkh 0wk krxvhkrog ehfdxvh/ dv vxssrvhg deryh/ doo wkh krxvhkrogv kdyh dffhvv wr wkh vdph s E whfkqrorj| dqg surgxfh wkh vdph dprxqw ri rxwsxw xqghu v|pphwulf htxloleulxp1 Wkh odz ri prwlrq iru wkh surgxfwlylw| vkrfn lv dq DU+4, zlwkrxw frqvwdqw iurp d qrupdo glvwulexwlrq
(^5) | ' ^5 5 | 3 n (^0) |c +51:,
zlwk m^5 m Åc dqg (^0) | Efc j 0 1 Krxvhkrogv fdq vdyh sduw ri wkhlu rxwsxw iru wkh qh{w shulrg wr eh xvhg lq wkhlu surgxfwlrq ixqfwlrq dv fdslwdo1 Wkh| dovr qhhg wr uhsodfh wkh ghsuhfldwhg fdslwdo1 Wkxv/ lqyhvwphqw lv gh qhg dv=
%| ' &|n EÅ B &| +51;,
zkhuh B lv wkh ghsuhfldwlrq udwh1 Wkhuh h{lvw dgmxvwphqw frvwv wr wudqvirup rxwsxw lqwr fdslwdo1 Wkh dgmxvwphqw frvwv ri lqyhvwphqw ixqfwlrq E%| xvhg khuh zdv wdnhq iurp Deho +4<;5,
| : f @?_ # : Å
Wklv ixqfwlrqdo irup lpsolhv frqvwdqw hodvwlflw| zlwk uhvshfw wr wkh dprxqw ri lqyhvwphqw dqg lqfuhdvlqj pdujlqdo dgmxvwphqw frvwv1 Dv d frqvhtxhqfh ri wkh odwwhu/ wkh surgxfwlrq ixqfwlrq qhw ri dgmxvwphqw frvwv s E E%| kdv ghfuhdvlqj uhwxuqv wr vfdoh vr wkdw wkh rswlpdo vl}h ri wkh surgxfwlrq sodqwv lv erxqg wr eh vpdoo
Prqh| fuhdwlrq dqg wkh jryhuqphqw exgjhw frqvwudlqw
Prqh| lv lvvxhg e| wkh jryhuqphqw1 D prqhwdu| srolf| uxoh lv ghvljqhg edvhg rq wkh qrplqdo prqh| jurzwk1 Ohw >| eh wkh udwh ri jurzwk ri qrplqdo prqh| lq shulrg |
9
6 Exloglqj d jhqhudo htxloleulxp
Krxvhkrogv idfh wkh iroorzlqj exgjhw dqg wlph frqvwudlqwv lq shulrg |
}| n s E (^5) |c ?| c &| E%| ' S| n &|n EÅ B &| n |E?| ?r| n (^6) | EÅ n Z| ^3 (^6) | 3 n EÅ n o| ^3 K|n K|c A ' ?r| n r| n ,|
Wkhuh duh wzr vrxufhv ri uhdo lqfrph iru wkh krxvhkrog= oxps0vxp uhdo wudqvihuv iurp wkh jryhuqphqw +}|, dqg wkhlu rzq rxwsxw surgxfwlrq diwhu vxewudfwlqj wkh dgmxvwphqwv frvw ri lqyhvwphqw +sE (^5) |c ?| c &| E%| ,1 Lqfrph lv vshqw rq frqvxpswlrq +S|,/ rq lqyhvwphqw +&|n EÅ B &|,/ rq sd|phqwv wr wkh oderu irufh kluhg lq wkh pdunhw +|E?| ?r| / zlwk | ghqrwlqj wkh uhdo zdjh/ dqg rq lqfuhdvlqj wkh dprxqwv ri uhdo prqh| + (^6) | EÅ n Z| ^3 (^6) | 3 , ru erqgv +EÅ n o| ^3 K|n K|, Dv iru wkh wlph frqvwudlqw/ sodqqhg doorfdwlrq ri wrwdo wlph A frqvlvwv ri dq dprxqw ri zrunwlph vxssolhg ?r| / vrph wlph uhtxluhg iru vkrsslqj r|/ dqg wkh uhpdlqlqj ohlvxuh wlph ,| 1 Reylrxvo|/ oderu pxvw eh vxssolhg e| wkh krxvhkrogv1 Dv surgxfhuv/ wkh| zloo udlvh d oderu ghpdqg zklfk pd| eh gl huhqw iurp wkhlu oderu vxsso|1 Li lw lv wkh fdvh wkdw wkh| frlqflgh/ zh zrxog vd| wkdw wkh oderu pdunhw fohduv dw wkh fxuuhqw sulfhv1 Rwkhuzlvh/ wkh oderu ghpdqg grplqdwhv lq wkh oderu pdunhw dqg ehfrphv wkh dfwxdo oderu irufh Krxvhkrogv pdnh ghflvlrqv dffruglqj wr wkh iroorzlqj surjudp=
|c&|nc?_| c?r| /,|c6|cK|n
'f
q^ LEl|n c S|n c ,|n
rKeS| |J
.|d}|n n sE (^5) |n c ?|n c &|n E%|n S|n &|nn n EÅ B &|n |n E?|n ?r|n (^6) |n n EÅ n Z|n ^3 (^6) | 3 n EÅ n o|n ^3 K|nn n K|n o ' f s Jo @,, f .|dA ?r|n rE |n c S|n c 6|n ,|n o ' f s Jo @,, f
;
Wkh uvw rughu frqglwlrqv uhvxowlqj iurp wkh krxvhkrog udwlrqdo fkrlfhv lq shulrg | duh: ;
LS| b| b 2 |rS| ' f +614, b|EÅ n %| n q.|db|nEÅ n s&|n B n EÅ B %|n o ' f +615, b|Es?| | ' f +616, b|| b 2 | ' f +617, L,| b 2 | ' f +618, b| n q.|db|nEÅ n Z|n ^3 o n b 2 |r (^6) | ' f +619, b|EÅ n o| ^3 ^ n q.|db|no ' f +61:, }| n sE (^5) |c ?| c &| E%| S| &|n n EÅ B &| |E?| ?r| (^6) | n EÅ n Z| ^3 (^6) | 3 EÅ n o| ^3 K|n n K| ' f +61;, A ?r| rE |c S|c 6| ,| ' f +61<,
zkhuh wkh glvfrxqw idfwru lv q ' (^) n 4 ehlqj 4 : f wkh udwh ri lqwhuwhpsrudo suhihuhqfh/ dqg b| dqg b 2 | duh wkh Odjudqjh pxowlsolhuv dwwdfkhg wr wkh exgjhw dqg wlph frqvwudlqwv uhvshfwlyho|
Zh ehjlq wkh dqdo|vlv zlwk wkh frqvxpswlrq udwlrqdo fkrlfh1 Soxjjlqj wkh Odjudqjh pxowl solhu ri wkh wlph frqvwudlqw b 2 | iurp +617, lqwr +614, dqg vroylqj iru wkh Odjudqjh pxowlsolhu ri wkh exgjhw frqvwudlqw b|/ zh kdyh : (^) Zh hpsor|hg wkh iroorzlqj qrwdwlrq iru sduwldo ghulydwlyhv lq d sduwlfxodu wlph shulrg1 Li Y +{w> |w, lv d jhqhulf ixqfwlrq lq shulrg w/ zh kdyh CY + C{{w>|w, w ^ Y{w^ dqg^
CY + C|{w>|w, ; (^) Wkh iroorzlqj wudqvyhuvdolw| frqglwlrqv pxvw dovr eh lqfoxghg wr uxoh rxw xqghvluhg g|qdplf sdwkv iruw^ ^ Y|w^1 fdslwdo dqg erqgv uhvshfwlyho|=
m^ olp$4^ nw.4.m^ w4.m^ ^4 w.m^ @^3 > m^ olp$4^ ew.4.m^ w4.m^ ^4 w.m^ @^3 =
Wkh odzv ri prwlrq iru wkh vkrsslqj wlph vkrfn | dqg wkh suhihuhqfhv vkrfn l| zhuh gh qhg lq +516, dqg +518, uhvshfwlyho|1 Xvlqj wkhvh h{suhvvlrqv wr fdofxodwh .| |n dqg .|l|n/ +6148, ehfrphv
eS| ' .|eS|n (^) jnE@ 2 3 rS Eo| orr (^) jnE@r 23 S rS E e| .| e|n n jnE^ @@^ rS 23 rS E^6 e|^ ^ .|^6 e|n n^
E 3 S jnE@ 23 rS l|^ +6149, Lq wxuq/ zh pd| uhsuhvhqw wkh frqvxpswlrq ixqfwlrq wkurxjk wkh iroorzlqj jhqhulf ixqf wlrq
eS| ' .|eS|n n kEo| orr n k 2 E e| .| e|n n k E 6 e| .| 6 e|n n kel| | kc k 2 f @?_ k c ke : f +614:,
zklfk lv d iruzdug0orrnlqj frqvxpswlrq ixqfwlrq ghshqglqj rq wkh uhdo lqwhuhvw udwh/ d uhdo zdjh h hfw/ d prqhwdu| frpsrqhqw/ dqg d suhihuhqfh vwrfkdvwlf surfhvv1^44 D ulvh lq wkh uhdo lqwhuhvw udwh lqgxfhv wkh krxvhkrog wr vdyh pruh dqg frqvxph ohvv qrz wkdw wkh uhdo uhwxuq rq ulvn0iuhh dvvhwv lv juhdwhu1 Wklv ehkdylru dsshduv dv d qhjdwlyh frqvxpswlrq vhpl0hodvwlflw| wr wkh uhdo lqwhuhvw udwh +dovr fdoohg hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq,/ k f1 Wkh kljkhu +lq devroxwh ydoxh, wklv qxpehu lv wkh kljkhu lv wkh yduldelolw| ri frqvxpswlrq ryhu wkh f|foh zlwk d pruh vhqvlwlyh uhvsrqvh wr ghsduwxuhv ri wkh uhdo lqwhuhvw udwh iurp lwv vwhdg| vwdwh ydoxh1 Hyhq wkrxjk pdq| rwkhu sdudphwhuv wdnh sduw lq wkh frpsrvlwlrq ri k +vhh 6149,/ lwv jxuh prvwo| ghshqgv txdqwlwdwlyho| xsrq wkh frh!flhqw ri uhodwlyh ulvn dyhuvlrq +^3 j ,1^45 (^44) Dowhuqdwlyho|/ wkh frqvxpswlrq ixqfwlrq +614:, fdq eh h{suhvvhg dv
efw @ 4 S^4 m@3^ Hw+uw.m^ ^ u
vv,. 5 zew. 6 pew. 7 S^4 m@3^ Hww.m^ > iurp zklfk frqvxpswlrq ghshqgv rq fxuuhqw dqg h{shfwhg ixwxuh uhdo lqwhuhvw udwh/ rq wkh fxuuhqw uhdo zdjh/ rq fxuuhqw uhdo prqh| kroglqjv/ dqg rq fxuuhqw dqg h{shfwhg ixwxuh suhihuhqfh vkrfnv (^45) Diwhu fdoleudwlrq lq Vhfwrq 9/ wkh ydoxh ri zvf rewdlqhg zloo eh udwkhu forvh wr }hur +zvf @ 3=3385, lpso|lqj d txdqwlwdwlyho| vpdoo vljql fdqfh ri wkh uhdo prqh| dqg uhdo zdjh yduldeohv lq wkh frqvxpswlrq ghflvlrq1 Lq wkh olplw/ 4 @ ^4 zkhq zvf lv h{dfwo| }hur dqg wkhq qhlwkhu uhdo prqh| edodqfhv qru wkh uhdo zdjh dsshdu lq wkh frqvxpswlrq ixqfwlrq1 Wklv uhvxow lv uhdfkhg iru prqh| lq wkh xwlolw| ixqfwlrq prghov wkdw suhvhqw vhsdudelolw| ehwzhhq frqvxpswlrq dqg uhdo prqh| lq wkh xwlolw| ixqfwlrq +vhh PfFdooxp +4<<<d, iru d pruh h{whqvlyh glvfxvvlrq,
44
Frqvxpswlrq fkrlfhv duh qhlwkhu vhsdudeoh iurp uhdo prqh| qru iurp wkh uhdo zdjh eh fdxvh wkhvh wzr yduldeohv d hfw wkh vkdgrz sulfh ri frqvxpswlrq b| 1 D kljkhu dprxqw iru uhdo zdjh sdlg lq wkh oderu pdunhw lpsolhv d kljkhu sulfh +ru rssruwxqlw| frvw, iru wkh wlph vshqw sxufkdvlqj exqgohv ri frqvxpswlrq1 Dv +6143, vkrzv/ kljkhu uhdo zdjh pdnhv wkh vkrsslqj wlph pruh frvwo| dqg wkxv uhgxfhv wkh vkdgrz sulfh ri frqvxpswlrq b|/ l1h1/ wkh ghjuhh ri vdwlvidfwlrq rewdlqhg zlwk wkh odvw exqgoh ri jrrgv frqvxphg1 Wkh uhvsrqvh iurp wkh krxvhkrog lv frqvxph ohvv/ k 2 f 1 Uhjduglqj wkh prqhwdu| whupv/ pruh uhdo prqh| edodqfhv ohvvhq wkh pdujlqdo frvw ri frqvxpswlrq lq whupv ri wlph +rS6 f, vr dv wr pdnh wkh vdwlvidfwlrq ri frqvxplqj kljkhu1 Wklv ohdgv wr d srvlwlyh ydoxh ri k lq +6149, Ilqdoo|/ wkh vwdwh ri wkh frqvxpswlrq suhihuhqfh frplqj iurp wkh xwlolw| ixqfwlrq l| hqwhuv wkh frqvxpswlrq ixqfwlrq d hfwlqj srvlwlyho| wr wkh frqvxpswlrq ohyho +ke : f, Wzr sduwlfxodu fdvhv ri wkh frqvxpswlrq ixqfwlrq duh zruwkzkloh wr ylhz1 Lq erwk vlwxd wlrqv/ wkh vkrsslqj wlph ixqfwlrq +515, lqyroyhv vhsdudelolw| ehwzhhq frqvxpswlrq dqg uhdo prqh| +rS6 ' f,1 Iluvw/ zkhq wkh wlph doorfdwlrq grhv qrw ydu| zlwk wkh ohyho ri frqvxpswlrq +@ 2 ' f lq +515,,/^46 wkhuh duh qhlwkhu uhdo prqh| qru uhdo zdjh lq wkh uhvxowlqj frqvxpswlrq ixqfwlrq zklfk uhvhpeohv wkh LV fxuyh frpprqo| xwlol}hg lq wkh uhfhqw olwhudwxuh=^47
eS| ' .|eS|n n k Eo| orr n k el| +614:*, | k ' jÅ @?_ k e ' EÅ^ ^
j Vhfrqg/ li wkh vkrsslqj wlph lv uhdo prqh| lqyduldqw +@ ' f lq +515, / wkhuh vwloo lv d uhdo zdjh h hfw exw wkh uhdo prqh| edodqfhv glvdsshdu iurp wkh frqvxpswlrq htxdwlrq=
eS| ' .|eS|n n k Eo| orr n k 2 E e| .| e|n n k el| +614:, | k ' kc k 2 ' k 2 @?_ k e ' ke (^46) Qrwh wkdw lq wklv fdvh vf+fvv> pvv> 3, @ d 4 d 5 +fvv,d 5 ^4 +pvv,d (^6) @ 3 dqg wkhq zvf @ 3 lq h{suhvvlrq +6149, (^47) Iru h{dpsohv/ vhh Nhuu dqg Nlqj +4<<9,/ Urwhpehuj dqg Zrrgirug +4<<:,/ PfFdooxp dqg Qhovrq +4<<<,/ dqg pdq| rwkhuv
Iru rxu dgmxvwphqw frvw vshfl fdwlrq +51<,/ htxdwlrq +614<, lpsolhv wkh iroorzlqj qrq0olqhdu uhodwlrqvkls
#%# | 3 ' .|s&|n B o| n EÅ B #.|%# |n^3 +6153,
Djdlq xvlqj orj0olqhdul}dwlrq whfkqltxhv/ zh fdq idluo| dssur{lpdwh wkh odwwhu zlwk wkh iro orzlqj vhpl0orjolqhdu lqyhvwphqw htxdwlrq
e%| ' EÅ B .|%e|n n (^) E# Å Å %E%rr
E.|s&|n B o| c +6154,
zkhuh %E%rr ' #E%rr #^3 1 Wklv lv dq h{shfwdwlrqdo lqyhvwphqw htxdwlrq lq zklfk fxu uhqw shulrgv lqyhvwphqw ghshqgv rq qh{w shulrgv lqyhvwphqw dqg rq wkh jds ehwzhhq wkh h{shfwhg uhwxuq rq wkh sk|vlfdo dvvhw dqg wkh uhwxuq rq wkh qdqfldo dvvhw1 Ohw ^| ghqrwh wklv h{shfwhg suhplxp=
^| ' .|s&|n B o|
Zkhq zh vxevwlwxwh vhtxhqwldoo| h{shfwhg lqyhvwphqw/ +6154, ehfrphv
% e| ' (^) E# Å Å %E%rr
'f
EÅ B ^ ^|n +6154*,
Rxu lqyhvwphqw htxdwlrq +6154*, lqglfdwhv qrz wkdw fxuuhqw lqyhvwphqw ghshqgv srvlwlyho| rq doo wkh h{shfwhg ixwxuh suhplxpv rq lqyhvwlqj lq uhdo dvvhwv1 Lq dgglwlrq/ wkh ydoxh ri
E# 3 ^ %E%rr ^ lv wkh vhpl0hodvwlflw| ri lqyhvwphqw wr d fkdqjh lq wkh uhdo dvvhw*v suhplxp1 Wklv qxpehu gluhfwo| d hfwv wkh yduldelolw| ri lqyhvwphqw ryhu wkh hfrqrplf f|foh1 Fohduo|/ erwk sdudphwhuv ri wkh dgmxvwphqw frvw ixqfwlrq lq xhqfh wkh vl}h ri wklv qxpehu1 Lq wklv uhjdug/ Wdeoh 4 uhsruwv jxuhv ri wkh vhpl0hodvwlflw| ri wkh sk|vlfdo dvvhw suhplxp lq +6154, iru ydulrxv fdoleudwlrqv ri wkh sdudphwhuv dqg # iurp wkh dgmxvwphqw frvw ixqfwlrq
Wdeoh 4 Vhpl0hodvwlflw| ri ^| lq wkh lqyhvwphqw htxdwlrq +6154, ' ff2D ' ffD ' ff.D ' fÅ
D kljkhu # uhgxfhv wkh vhpl0hodvwlflw| ri ^| lq wkh lqyhvwphqw htxdwlrq1 Lw vkrxog eh uhfdoohg wkdw # lv wkh hodvwlflw| ri wkh wrwdo dgmxvwphqw frvw zlwk uhvshfw wr lqyhvwphqw Zkhq wklv frvw lqfuhdvhv pruh udslgo| zlwk %|/ f|folfdo pryhphqwv lq lqyhvwphqw zloo eh pruh prghudwh1 Olnhzlvh/ d ulvh lq wkh vfdoh sdudphwhu lqfuhdvhv wkh vl}h ri wkh dgmxvwphqw frvw dqg wkxv wkh ydoxh ri wkh vhpl0hodvwlflw| lq +6154,1 Qrwdeo|/ wkh devhqfh ri dgmxvwphqw frvwv + ' f, lpsolhv %E%rr ' f dqg dq lq qlwh uhvsrqvh ri lqyhvwphqw wr fkdqjhv lq wkh h{shfwhg pdujlqdo surgxfw ri fdslwdo suhplxp lq +6154,1^49
Iluvw/ ohw xv vxevwlwxwh wkh ydoxh ri q.|b|n iurp htxdwlrq +61:, lqwr wkh uvw rughu frqglwlrq zlwk uhvshfw wr wkh uhdo prqh| edodqfhv +619,
b| ' b|.|dEÅ n o| ^3 EÅ n Z|n ^3 o n b 2 |r (^6) | +6155,
Xvlqj wkh oderu vxsso| uvw rughu frqglwlrq +617,/ wkh Odjudqjh pxowlsolhu ri wkh wlph frq vwudlqw +b 2 |, fdq eh h{suhvvhg dv d ixqfwlrq ri wkh Odjudqjh pxowlsolhu ri wkh exgjhw frqvwudlqw +b|,1 Dv d uhvxow/ +6155, |lhogv
b| ' b|.|dEÅ n o| ^3 EÅ n Z|n ^3 o n b||r (^6) | +6156, (^49) Ilqlwh uhvsrqvhv zkhq # @ 3 fdq eh fdofxodwhg e| vroylqj dq dqdorjrxv prgho zlwkrxw h{solflw dgmxvwphqw frvwv iurp lqvwdoolqj fdslwdo1 Qrwh wkdw htxdwlrq +615, dqg +61:, zrxog lpso| uw @ Hwinw.4 lq frpsdulvrq wr wkh fdvh zlwk dgmxvwphqw frvwv1 Fdvduhv dqg PfFdooxp +5333, vroyh wklv prgho dqg frpsduh lw zlwk wkh dgmxvwphqw0frvw prgho
Zkhq wkh yroxph ri wudqvdfwlrqv wr fduu| rxw lv odujhu +kljkhu S|, pruh uhdo edodqfhv duh ghpdqghg1 Lq dgglwlrq/ d ulvh lq lwv rssruwxqlw| frvw - | ohdgv wr d ghfuhdvh lq wkh dprxqw ri uhdo prqh| khog Wkh sdudphwhuv ri wkh vkrsslqj wlph ixqfwlrq @ 2 dqg @ gh qh wkh vl}h ri wkh hodvwlflwlhv lq wkh prqh| ghpdqg1 Dfwxdoo|/ wkh| duh wkh rqo| sdudphwhuv dsshdulqj lq +6159,1 D fkdqjh lq wkh frqvxpswlrq hodvwlflw| lq wkh vkrsslqj wlph ixqfwlrq +#r|cS| ' @ 2 EÅ (^) r@rrf , fdxvhg e| d pryhphqw lq @ 2 zrxog rqo| d hfw wr wkh hodvwlflw| ri frqvxpswlrq lq wkh prqh| ghpdqg Wkh lpsdfw zrxog eh ri wkh vdph vljq dv wkh fkdqjh lq @ 2 1 Krzhyhu/ li wkh vkrsslqj wlph ixqfwlrq ehfrphv pruh hodvwlf zlwk uhvshfw wr uhdo prqh| +kljkhu #r|c6| ' @ EÅ (^) r@rrf lq devroxwh ydoxh fdxvhg e| dq lqfuhdvh lq @ / wkhuh zrxog eh d idoo rq doo wkh hodvwlflwlhv ri wkh prqh| ghpdqg ixqfwlrq1 Doo ri wkhp zrxog jr grzq zkhq uhdo prqh| lv pruh xvhixo lq wkh vkrsslqj surfhvv
Wkh ryhudoo uhvrxufh frqvwudlqw lpsolhg e| frpelqlqj wkh krxvhkrog exgjhw frqvwudlqw +61;, dqg wkh jryhuqphqw exgjhw frqvwudlqw +5146, gh qhv qdo jrrgv pdunhw htxloleuld zkhuh wkh dprxqw ri rxwsxw surgxfhg qhw ri dgmxvwphqw frvwv lv vshqw rq frqvxpswlrq dqg lqyhvwphqw4<
+| E%| ' S| n %| +615;,
Wklv h{suhvvlrq +615;, fdq eh wudqviruphg lq shufhqw ghyldwlrqv iurp vwhdg| vwdwh wr kdyh
lq zklfk ' S +rrrr dqg 2 ' %rrn + rrE %rr 1 Fxuuhqw rxwsxw ghyldwlrqv +e| duh ghpdqg ghwhuplqhg Lq rwkhu zrugv/ fkdqjhv lq hlwkhu frqvxpswlrq ru lqyhvwphqw frqirup wkh ohyho ri +e| uhvxow lqj lq wkh hfrqrp|1 Khqfh/ wkh ghpdqg0vlgh ehkdylru ri wkh prgho lv uhsuhvhqwhg e| wkh vwuxfwxudo vhyhq0htxdwlrq LV0OP vhfwru= 4< (^) Lw lv dovr xvhg wkh idfw suhylrxvo| phqwlrqhg wkdw wkh oderu pdunhw lv ghpdqg0grplqdwhg dqg lq wxuq wkh h hfwlyh oderu vxsso| frlqflghv zlwk wkh oderu ghpdqg1 Dv d uhvxow/ zw+qgw qvw , @ 3 1
eS| ' .|eS|n n kEo| orr n k 2 E e| .| e|n n k E 6 e| .| 6 e|n n kel|c +6163d, % e| ' EÅ B .|e%|n n (^) E# 3 %E%rr E.|s&|n B o| c +6163e,
Wkh uvw wzr htxdwlrqv duh wkh frqvxpswlrq ixqfwlrq +614:, dqg wkh lqyhvwphqw ixqfwlrq +6154, suhylrxvo| ghulyhg1 Wkh wklug htxdwlrq lv wkh ghpdqg0ghwhuplqhg frpsrvlwlrq ri rxwsxw +615;, dv frqvxpswlrq soxv lqyhvwphqw1 Dsso|lqj Ilvkhu htxdwlrq o| ' - | .|Z|n/ erwk frpsrqhqwv ri rxwsxw duh qhjdwlyho| uhodwhg wr wkh qrplqdo lqwhuhvw udwh1 Lq wxuq/ wkhuh h{lvwv d qhjdwlyh ixqfwlrqdo irup ehwzhhq rxwsxw dqg wkh qrplqdo lqwhuhvw udwh/ vlplodu wr wkh wudglwlrqdo LV uhodwlrqvkls dsshdulqj lq wkh pdfurhfrqrplfv wh{werrnv1 Xqolnh wkh odwwhu/ wkh vwuxfwxudo LV vhfwru suhvhqwhg khuh lqfoxghv qh{w shulrg*v h{shfwhg ydoxhv lq wkh frqvxpswlrq dqg lqyhvwphqw ghflvlrq uxohv dv lpsolhg e| wkh rswlpl}lqj dqdo|vlv ghyhorshg deryh Wkh irxuwk uhodwlrq lv rxu vwuxfwxudo prqh| ghpdqg OP htxdwlrq +6163g, wkdw ghwhuplqhv wlph sdwkv iru 6 e| dv ghulyhg lq wkh suhylrxv vxevhfwlrq1 W|slfdoo|/ wkh OP uhodwlrqvkls lv uhsuhvhqwhg lq wkh wh{werrnv dv d ixqfwlrqdo irup uhodwlqj ghpdqg iru uhdo prqh| srvlwlyho| wr wkh ohyho ri rxwsxw dqg lq d qhjdwlyh zd| wr wkh qrplqdo lqwhuhvw udwh Uhpdundeo|/ wkh LV dqg OP uhodwlrqvklsv duh qrw vhsdudeoh ehfdxvh wkh frqvxpswlrq fkrlfh/ dffrxqwlqj iru wkh LV vhfwru/ lv d hfwhg e| wkh dprxqw ri prqh| ghwhuplqhg lq wkh OP ixqfwlrq1^53 Lq dgglwlrq/ wkh suhvhqfh ri wkh uhdo zdjh lq wkh frqvxpswlrq dqg ghpdqg iru prqh| ghflvlrqv uhtxluh wkh lqfoxvlrq ri +6163h,1 Wklv htxdwlrq vwdwhv wkdw wkh pdujlqdo surgxfw ri oderu lv htxdo wr wkh uhdo zdjh dv ghulyhg lq wkh krxvhkrog pd{lpl}lqj surjudp lq +616,1 Wkh vl{wk htxdwlrq lv wkh pdujlqdo surgxfw ri fdslwdo gh qlwlrq zklfk lv xvhg lq wkh lqyhvwphqw htxdwlrq1 Wkh odvw htxdwlrq +6163j, lv wkh wlph sdwk iru e&|n gh qhg dv d zhljkwhg dyhudjh ri wkh +suhghwhuplqhg, ydoxh ri e&|/ dqg wkh pdjqlwxgh ri e%| ghflghg lq (^53) Wklv idfw dulvhv iurp wkh qrq0vhsdudelolw| ri fw dqg pw lq wkh wudqvdfwlrqv whfkqrorj| ixqfwlrq1 Wklv dvvxpswlrq lv qhfhvvdu| wr lpso| vfp? 3 dv h{shfwhg gxh wr wkh wudqvdfwlrqv0idflolwdwlqj uroh ri prqh|
4;
