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Wkh prvw fodvvlfdo wlph yduldeoh wdnhv lwv ydoxhv lq wkh vhw ri srvlwlyh uhdo qxp0

ehuv +frqwlqxrxv wlph,1 Lq glvfuhwh wlph v|vwhpv/ wkh vhw ri srvlwlyh lqwhjhuv +ru

dq| vhw lvrprusklf wr wkdw rqh, lv xvhg zkhq vdpsohg gdwd v|vwhpv duh frqvlg0

huhg1 Wklv wlph uhsuhvhqwdwlrq lv fdoohg v|qfkurqhrxv vlqfh sudfwlfdo vdpsolqj

v|vwhpv duh gulyhq e| d forfn1 Rq wkh frqwudu|/ lq hyhqw gulyhq v|vwhpv/ wlph

lv frqvlghuhg rqo| dw hdfk hyhqw rffxuhqfh1

61416 Frqvwudlqwv

Zh frqvlghu frqwuroohg v|vwhpv zklfk reh| wkh Pdunry surshuw|1 Wklv phdqv

wkdw wkh zkroh sdvw klvwru| ri wkh v|vwhp xqwloo wlph w fdq eh vxppdul}hg e|

wkh ydoxhv ri d vhw ri yduldeohv dw wlph w +wkh v|vwhp*v vwdwh,/ dqg wkdw wkh

wudmhfwrulhv duh vxfk wkdw wkh hyroxwlrq ri wkh vwdwh dw wlph w rqo| ghshqgv rq

wkh ydoxh ri wkh vwdwh dqg lqsxwv dw wlph w1

Wkh hyroxwlrq ri wkh v|vwhp lv ghvfulehg wkurxjk d vhw ri frqvwudlqwv zklfk

dsso| wr wkh v|vwhp yduldeohv1 Dffruglqj wr wkh gl�huhqw ghvfulswlrqv ri wkh

yduldeohv dqg ri wlph/ wkh frqvwudlqwv kdyh gl�huhqw irupv1 Frqwlqxrxv wlph

wudmhfwrulhv ri frqwlqxrxv yduldeohv duh prvw frqflvho| ghvfulehg e| wkh dojheudlf

dqg gl�huhqwldo htxdwlrqv wkh| kdyh wr vdwlvi|1 Lq wkh olqhdu fdvh/ wkrvh htxdwlrqv

fdq eh wudqviruphg iurp wkh wlph grpdlq lqwr wkh v|perolf grpdlq xvlqj wkh

9

Odsodfh wudqvirup1 Zkhq phdvxuhphqw/ frqwuro dqg vxshuylvlrq lv shuiruphg

e| frpsxwhuv/ glvfuhwh wlph lv prvw dssursuldwh dqg wkh v|vwhp ehkdylrxu prgho

lv ehvw ghvfulehg e| dojheudlf dqg gl�huhqfh htxdwlrqv/ zklfk fdq dovr lq wkh

olqhdu fdvh eh wudviruphg iurp wkh wlph grpdlq lqwr wkh v|perolf grpdlq xvlqj

wkh }0wudqvirup1 Li txdolwdwlyh yduldeohv duh frqvlghuhg +dv iru h{dpsoh lq orjlf

edvhg v|vwhpv ru lq txdolwdwlyh sk|vlfv,/ glvfuhwh v|qfkurqhrxv ru dv|qfkurqhrxv

wlph prghov uhvw rq orjlf htxdwlrqv/ dxwrpdwd/ Shwul qhwv/ vhwv ri uxohv1 Qrwh

wkdw txdolwdwlyh yduldeohv fdq eh frqvlghuhg zlwk fulvs ru ix}}| prgdolwlhv/ wkxv

ohdglqj wr fulvs ru ix}}| ehkdylrxu prghov1

Lq wkh vhtxho/ zh rqo| frqvlghu frqwlqxrxv yduldeohv 2 frqwlqxrxv wlph/ frq0

wlqxrxv yduldeohv 2 glvfuhwh wlph dqg elqdu| yduldeohv 2 glvfuhwh wlph ehkdylrxu

prghov1

��� ��������� #���%���&�������� ���

Lq frqwlqxrxv yduldeoh v|vwhpv/ wkh lqsxw/ vwdwh dqg rxwsxw yduldeohv duh gh0

�qhg iru d frqwlqxxp ri ydoxhv lq U= Wkh vljqdov zklfk hqwhu vxfk v|vwhpv duh

frqwlqxrxv0wlph vljqdov/ l1h1 wkh lqghshqghqw yduldeoh lv frqwlqxrxv/ dqg wkxv

vxfk vljqdov duh gh�qhg ryhu d frqwlqxxp ri wlph ydoxhv1

61514 Qrupdo rshudwlrq prgho

Lq qrupdo rshudwlrq/ wkh v|vwhp frpsrqhqwv uhdoo| ehkdyh dv wkh ghvljqhu h{0

shfwhg zkhq kh vhohfwhg ru ghvljqhg wkhp/ l1h1 wkh frqvwudlqwv wkh| dsso| wr wkh

v|vwhp yduldeohv duh wkh qrplqdo rqhv1 D prgho ri wkhvh frqvwudlqwv lv rewdlqhg

e| xvlqj wkh sk|vlfdo odzv zklfk duh reh|hg e| wkh surfhvv frpsrqhqwv1 Lq wkh

frqwlqxrxv yduldeohv 2 frqwlqxrxv wlph rswlrq/ d txlwh jhqhudo prgho iru wkh

ghvfulswlrq ri wkh qrplqdo frqvwudlqwv frqvlvwv ri d vhw ri gl�huhqwldo htxdwlrqv

=

b{+w, @ i+{+w,> x+w,> y+w,> ��+w,, +9,

|+w, @ j+{+w,> ��+w,> %+w,, +:,

% � S3 +;,

zkhuh {+w, 5 Uq/ x+w, 5 Up/ |+w, 5 Us ghqrwh uhvshfwlyho| wkh vwdwh yhfwru/

wkh yhfwru ri nqrzq lqsxwv/ dqg wkh yhfwru ri phdvxuhg rxwsxwv1 y+w, 5 Ug

vwdqgv iru wkh yhfwru ri xqnqrzq lqsxwv ru glvwxuedqfhv dfwlqj rq wkh surfhvv/

dqg %+w, vwdqgv iru wkh phdvxuhphqw qrlvh zklfk fruuxswv wkh vhqvruv/ zlwk

suredelolw| glvwulexwlrq S3 xqghu qrupdo rshudwlrq1 ��

5 Ut lv vrph qrplqdo

sdudphwhu yhfwru1 Wkh ixqfwlrqv i dqg j duh uhvshfwlyho| Uq0ydoxhg dqg Us0

ydoxhg dqg wkh| duh dvvxphg wr eh vprrwk1 Qrwh wkdw wkhvh frqvwudlqwv duh

dojheudlf2gl�huhqwldo htxdolwlhv/ zklfk duh vdwlv�hg li dqg rqo| li wkhuh lv qr

lqwhuqdo idxow/ zklfk lv h{suhvvhg e| =

qr lqwhuqdo idxow+, +9,/ +:, dqg ; krog +<,

:

Lq qrupdo rshudwlrq/ wkh v|vwhp yduldeohv dovr kdyh wudmhfwrulhv zklfk vwd|

lq vrph grpdlq frpsdwleoh zlwk wkh v|vwhp*v remhfwlyhv1 Lq wkh frqwlqxrxv

yduldeohv 2 frqwlqxrxv wlph rswlrq/ d vlpsoh prgho iru wkh ghvfulswlrq ri wklv

grpdlq frqvlvwv ri d vhw ri dojheudlf lqhtxdolwlhv =

k+{+w,> x+w,, � 3 +43,

zkhuh k lv d Ulq0ydoxh vprrwk ixqfwlrq/ wkxv ohdglqj wr wkh vwdwhphqw =

qr h{whuqdo idxow+, +43, krogv +44,

Qrwh wkdw lqwhuqdo dqg h{whuqdo idxowv duh frpsohwho| lqghshqghqw = wkh lqhtxdo0

lw| frqvwudlqwv pd| eh vdwlv�hg e| d idxow| v|vwhp/ zkloh wkh| pd| eh ylrodwhg

e| d khdowk| rqh1

D qduurzhu fodvv ri v|vwhpv lv riwhq frqvlghuhg/ qdpho| olqhdu wlph lqyduldqw

+OWL, v|vwhpv1 Wkh olqhdulw| surshuw| dulvhv hlwkhu iurp wkh v|vwhp ehlqj uhdoo|

olqhdu ru iurp wkh dssur{lpdwlrq ri wkh qrq0olqhdu prgho lq vrph uhjlrq ri wkh

vwdwh vsdfh e| d olqhdu rqh1 Wkh lqyduldqfh surshuw| dulvhv iurp wkh dvvxpswlrq

wkdw wkh v|vwhp sdudphwhuv duh frqvwdqw gxulqj qrupdo rshudwlrq1 OWL prghov

duh ghvfulehg e| wkh iroorzlqj olqhdu vwdwh htxdwlrqv =

b{+w, @ D+��,{+w, .E+��,x+w, .H+��,y+w, +45,

|+w, @ F+��,{+w, . %+w, +46,

zkhuh wkh yduldeohv {+w,> x+w,> y+w,> |+w,/ dqg %+w, duh gh�qhg dv deryh/ dqg

D+��,> E+��,> F+��,> hwf= duh pdwulfhv ri dssursuldwh glphqvlrqv zlwk frqvwdqw

hqwulhv lq U +vlqfh ��

lv frqvwdqw,1 Wkh| zloo eh qrwhg D>E>F> hwf= iru vlpsolf0

lw| zkhq lw lv qrw qhfhvvdu| wr h{solflw wkh ghshqghqf| ryhu ��

1 Wkh dgplvvleoh

rshudwlqj grpdlq lv ghvfulehg e| wkh olqhdu lqhtxdolwlhv =

K{{+w, .Kxx+w, � k +47,

zkhuh K{> Kx>duh pdwulfhv ri dssursuldwh glphqvlrq zlwk frqvwdqw hqwulhv lq U

dqg k 5 Ulq1

61515 Idxow| rshudwlrq

Gxh wr wkh olplwhg vl}h ri wkh sdshu/ rqo| lqwhuqdo idxowv zloo eh frqvlghuhg1

Zkhq lqwhuqdo idxowv duh suhvhqw/ wkh frqvwudlqwv lqwurgxfhg e| wkh v|vwhp frp0

srqhqwv fkdqjh/ dqg vr gr wkh htxdwlrqv zklfk ghvfuleh wkh qrupdo ehkdylrxu1

Wkuhh gl�huhqw nlqgv ri prgl�fdwlrqv duh frqvlghuhg e| wkh frqwuro frppxqlw| =

pxowlsolfdwlyh idxowv/ dgglwlyh idxowv/ dqg fkdqjhv lq wkh ehkdylrxu ri vwrfkdvwlf

yduldeohv1

Pxowlsolfdwlyh idxowv Pxowlsolfdwlyh idxowv duh fkdqjhv lq wkh v|vwhp sd0

udphwhuv/ zklfk wdnh ydoxhv gl�huhqw iurp wkh qrplqdo rqhv1 Wkh pxowlsolfdwlyh

idxow prgho lv wkxv =

�+w, @ ��+w, . ��+w, +48,

;

lq zklfk ��+w, lv dq xqnqrzq yhfwru1 Wkh vwdwh dqg phdvxuhphqw htxdwlrqv

qrz duh =

b{+w, @ i+{+w,> x+w,> y+w,> ��+w, . ��+w,, +49,

|+w, @ j+{+w,> ��+w, . ��+w,> %+w,, +4:,

zkloh wkh rshudwlqj frqvwudlqwv duh vwloo ghvfulehg e| =

k+{+w,> x+w,, � 3 +4;,

Iru OWL v|vwhpv/ +45,/ +46, ehfrph =

b{+w, @ D+�� .��+w,,{+w, .E+�� .��+w,,x+w, .H+�� .��+w,,y+w, +4<,

|+w, @ F+�� .��+w,,{+w, . %+w, +53,

zkloh +47, uhpdlqv xqfkdqjhg1 Vlqfh ��+w, lv dq xqnqrzq yhfwru/ +4<, dqg +53,fdq dozd|v eh uhzulwwhq dv =

b{+w, @ +D.�D+w,,{+w, . +E .�E+w,,x+w, . +H .�H+w,,y+w, +54,

|+w, @ +F .�F+w,,{+w, . %+w, +55,

zklfk jlyhv =

b{+w, @ D{+w, .Ex+w, .Hy+w, . �D+w,{+w, . �E+w,x+w, . �H+w,y+w,+56,

|+w, @ F{+w, . �F+w,{+w, . %+w, +57,

Lq +56, dqg +57,/ lw fdq eh vhhq wkdw pxowlsolfdwlyh idxowv uhvxow lq dgglwlrqqdo

whupv lq wkh vwdwh dqg phdvxuhphqw htxdwlrqv/ hdfk rw wkhp ehlqj frqqhfwhg

zlwk vrph vshfldo nlqg ri idxow1 Lqghhg/ surfhvv frpsrqhqw idxowv jhqhudoo| ohdg

wr �D 9@ 3/ zkloh dfwxdwru idxowv ohdg wr �E 9@ 3 dqg vhqvru idxowv jlyh �F 9@ 3=�H lv jhqhudoo| qrw frqvlghuhg vlqfh/ zkhq xqnqrzq lqsxwv duh frqvlghuhg/ H

lv vxssrvhg qrw wr ghshqg rq wkh v|vwhp sdudphwhuv1

Dgglwlyh idxowv Vlqfh wkh �D>�E> hwf= duh xqnqrzq ixqfwlrqv ri wlph/ rqh

pd| ghflgh wr vlpsoli| wkh idxow prgho/ qrw frqvlghulqj wkh h{solflw irup ri

wkh dgglwlrqdo whupv zklfk lqwhuyhqh lq +56, dqg +57,/ dqg uhsodflqj wkhp e| d

xqltxh idxow yhfwru/ xqghu wkh iroorzlqj irup =

b{+w, @ D{+w, .Ex+w, .Hy+w, . *{+w, +58,

|+w, @ F{+w, . *|+w, . %+w, +59,

<

ru pruh jhqhudoo| =

b{+w, @ D{+w, .Ex+w, .Hy+w, . I{*+w, +5:,

|+w, @ F{+w, . I|*+w, . %+w, +5;,

zkhuh *+w, 5 Uilv vrph xqnqrzq idxow yhfwru/ dqg I{> I| duh pdwulfhv ri

vxlwdeoh glphqvlrqv/ zkrvh hqwulhv duh uhdo qxpehuv/ dqg zklfk wudgxfh wkh

idxow lq xhqfh/ uhvshfwlyho| rq wkh vwdwh dqg rq wkh phdvxuhphqw htxdwlrqv1

Vxfk d idxow prgho lv fdoohg dgglwlyh1 Sduwlfxodu fdvhv duh =

4,

�I{

I|

�@

�E

3

�/ zklfk ghvfulehv dfwxdwru idxowv1 Lqghhg/ lq wkdw fdvh/

wkh whup x+w,.*+w, lv idfwrul}hg lq wkh vwdwh htxdwlrq/ dqg wkh idxow *+w, dsshduvdv d �zurqj� frqwuro hqwhulqj wkh v|vwhp/ zklfk lv d phdqv ri prghoolqj dfwxdwru

idxowv +d idxow| dfwxdwru lv prghoohg dv d khdowk| rqh gulyhq e| vrph zurqj

lqsxw,1

5,

�I{

I|

�@

�3F

�> zklfk ghvfulehv vhqvru idxowv1 Lqghhg/ lq wklv fdvh wkh

whup {+w,.*+w, lv idfwrul}hg lq wkh phdvxuhphqw htxdwlrq/ zklfk lv d phdqv ri

prghoolqj vhqvru pdoixqfwlrq +wkh idxow| vhqvru phdvxuhv d zurqj vwdwh lqvwhdg

ri phdvxulqj wkh dfwxdo rqh,1

Fkdqjh lq vwdwlvwlfdo glvwulexwlrqv Zkhq wkh v|vwhp ehkdylrxu prgho

pdnhv xvh ri vwrfkdvwlf yduldeohv +zklfk lv wkh fdvh lq wkh phdvxuhphqw prghov

frqvlghuhg khuh,/ d idxow pd| dovr eh prghoohg e| d fkdqjh ri wkh vwrfkdvwlf

yduldeohv glvwulexwlrq1 Lqghhg/ wkh phdvxuhphqw htxdwlrq ri wkh khdowk| v|vwhp

ehlqj =

|+w, @ j+{+w,> ��+w,> %+w,, +5<,

% � S3 +63,

wkh fkdqjh iurp wkh glvwulexwlrq S3 wr vrph rwkhu glvwulexwlrq S4 lv lqghhg wreh frqvlghuhg dv d fkdqjh lq wkh v|vwhp ehkdylrxu1 Dv dq h{dpsoh/ frqvlghuwkh phdvxuhphqw v|vwhp zkrvh khdowk| ehkdylrxu lv ghvfulehg e|

|+w, @ F{+w, . %+w,

% � Q+3>3,

zkhuh Q+3>3, lv wkh qrupdo glvwulexwlrq zlwk phdqv }hur dqg fryduldqfh pd0wul{ 3= Dq lqfuhdvh lq wkh phdvxuhphqw qrlvh frxog eh ghvfulehg e| wkh fkdqjhri wkh qrlvh glvwulexwlrq iurpQ+3>3, wrQ+3>4,/ zkloh d eldv * 0 ru d guliw *+w,0 pljkw eh ghvfulehg e| wkh fkdqjh ri glvwulexwlrq iurp Q+3>3, wr Q+*>3, 0Q+*+w,>3,1 Qrwh wkdw wklv nlqg ri idxow pd| dovr eh ghvfulehg e| wkh dgglwlyhidxow prgho

43

|+w, @ F{+w, . I|*+w, . %+w,

% � Q+3>3,

��� ��������� #���%��� & ����������

Prvw vxshuylvlrq v|vwhpv dqg frqwuroohuv ehlqj lpsohphqwhg lq frpsxwhuv/ wkhvljqdov kdyh wr eh vdpsohg/ vlqfh frpsxwhuv duh rqo| deoh wr surfhvv gdwd vh0txhqfhv1 Doo vljqdov duh dvvxphg wr eh vdpsohg v|qfkurqrxvo| dw d �{hg vdp0solqj shulrg Wv1 Wkh wudglwlrqdo wkhru| ri vdpsohg gdwd v|vwhpv uholhv rq wkhdvvxpswlrq wkdw wkh lqsxw vljqdov duh frqvwdqw ryhu Wv1 Wklv krogv wuxh iru wkhfrqwuro yduldeohv/ dv wkh rxwsxw ri gljlwdo wr dqdorj frqyhuwhuv kdyh wklv surshuw|1Lw lv krzhyhu dqg dssur{lpdwlrq iru glvwxuedqfhv dqg idxowv1 E| lqwhjudwlqj wkhvwdwh htxdwlrq +9, ryhu rqh vdpsolqj shulrg/ wkh iroorzlqj glvfuhwh0wlph prghori wkh v|vwhp qrupdo ehkdylrxu fdq eh ghgxfhg iurp +9,/ +:,=

{+n . 4, @ i+{+n,> x+n,> y+n,> ��+n,, +64,

|+n, @ j+{+n,> ��+w,> %+n,, +65,

% � S3 +66,

zkhuh n +dfwxdoo| vwdqglqj iru nWv, ghqrwhv wkh glvfuhwh0wlph lqvwdqwv1 Vlploduo|/wkh qrupdo rshudwlqj udqjh lv ghvfulehg e| wkh lqhtxdolwlhv

k+{+n,> x+n,, � 3

dqg hyhu| ghyhorsphqw suhvhqwhg deryh zlwk frqwlqxrxv wlph w fdq eh uhshdwhgzlwk glvfuhwh wlph n/ mxvw uhsodflqj b{+w, e| {+n . 4,/ dqg {+w, e| {+n, lq wkhhtxdwlrqv1 Ilqdoo|/ rqh pd| ri frxuvh xvh w lqgl�huhqwo| iru frqwlqxrxv ru glv0fuhwh wlph1 Iru h{dpsoh/ olqhdu v|vwhpv zlwk dgglwlyh idxowv zrxog eh ghvfulehglq glvfuhwh wlph e|

{+w. 4, @ �D{+w, . �Ex+w, . �Hy+w, . �I{*+w, +67,

|+w, @ �F{+w, . �I|*+w, . %+w, +68,

zkhuh wkh pdwulfhv �D> �E> hwf= duh rewdlqhg e| lqwhjudwlqj htxdwlrq +5:, rq rqhvdpsolqj shulrg1

��� '������#� #���%��� & ������ ���

Ehkdylrxu prghov ri v|vwhpv zkrvh yduldeohv duh txdolwdwlyh fdq eh ghvfulehgxvlqj vshfl�f wrrov olnh dxwrpdwd/ Shwul qhwv/ hwf1 Lq rughu wr hpskdvl}h wkhjhqhudolw| ri wkh suhylrxv prghov/ ohw xv frqvlghu d fodvv ri vxfk v|vwhpv/ qdpho|vhtxhqwldo olqhdu v|vwhpv/ vxfk dv jdwh duud|v gulyhq e| d v|qfkurqhrxv forfn1

44

Vxfk v|vwhpv duh lqghhg txdolwdwlyh yduldeohv glvfuhwh wlph rqhv = wkh yduldeohvwdnh wkhlu ydoxh lq wkh vhw i3> 4j/ zklfk lv d Jdorlv vhw zkhq surylghg zlwkwkh wzr rshudwruv � dqg +prgxor 5 dgglwlrq dqg pxowlsolfdwlrq,/ dqg wkhv|qfkurqhrxv forfn lqwurgxfhv d frqvwdqw vdpsolqj shulrg dw zklfk wkh ydoxhvri wkh yduldeohv duh frqvlghuhg1 D olqhdu vhtxhqwldo v|vwhp ^5` lv ghvfulehg lqqrupdo rshudwlrq e| wkh vwdwh dqg phdvxuhphqw htxdwlrqv =

{+w. 4, @ D {+w,�E x+w,

|+w, @ F {+w,

zkhuh w lv wkh glvfuhwh wlph/ dqg {+w,> x+w,> |+w,> D>E>F kdyh wkh vdph phdqlqjdv ehiruh/ exw duh yhfwruv dqg pdwulfhv zkrvh hqwulhv ehorqj wr i3> 4j = Idxowvduh prghoohg e| dgglwlyh yhfwruv1 Lqghhg/ d idxow lq dq lqsxw ru surfhvvlqj jdwh/zklfk d�hfwv wkh lwk elw ri yhfwru {+w. 4, lv prghoohg wkurxjk

{+w. 4, @ ^D {+w,�E x+w,`� hl

zkhuh hl lv wkh yhfwru doo wkh frpsrqhqwv ri zklfk duh }hur h{fhswlqj wkh lwk

frpsrqhqw zklfk lv 41 Vlploduo|/ d idxow rq wkh lwk elw ri wkh rxwsxw lv prghoohge|

|+w, @ F {+w,� hl

� (�� ����� )��%���

Wklv sduw vhwv wkh wzr ghflvlrq sureohpv zklfk duh ri lqwhuhvw/ qdpho| idxowghwhfwlrq dqg idxow lvrodwlrq1 Wkh lqsxw gdwd iru wkhvh sureohpv duh wkh revhuy0deohv zklfk duh lvvxhg iurp wkh v|vwhp/ l1h1 wkh lqsxwv x+w, dqg wkh rxwsxwv|+w,= Wkh ghflghu pd| ri frxuvh xvh vhyhudo vxffhvvlyh revhuydwlrqv lq rughu wrpdnh klv ghflvlrq1 Ohw

Xv+w, @ +xw+w,> xw+w� 4,> ===xw+w� v,,w +69,

\v+w, @ +|w+w,> |w+w� 4,> ===|w+w� v,,w

eh wkh revhuydwlrqv dydlodeoh dw wlph w +xvlqj d wlph zlqgrz ri vl}h v,1 Wzrghflvlrq iudphv fdq eh glvwlqjxlvkhg/ qdpho| wkh iudph ri k|srwkhvhv fkhfnlqj+riwhq fdoohg wkh r�0olqh iudph,/ dqg wkh iudph ri fkdqjh ghwhfwlrq +riwhq fdoohgwkh rq0olqh iudph,1

4, K|srwkhvhv fkhfnlqj1 Edvhg rq wkh nqrzohgjh ri wkh gdwd +69,/ idxow

ghwhfwlrq frqvlvwv ri vhohfwlqj rqh ri wzr k|srwkhvhv =

K3 = wkh gdwd kdyh ehhq surgxfhg e| d khdowk| v|vwhp/

K4 = wkh gdwd fdqqrw kdyh ehhq surgxfhg e| d khdowk| v|vwhp

Idxow lvrodwlrq h{whqgv wkh wzr k|srwkhvhv sureohp wr d pxowl0k|srwkhvhv

rqh/ qdpho| vhohfw rqh k|srwkhvhv Kl dprqj d jlyhq vhw l 5 L @ i4> ===Qj/ hdfk

lqgh{ l ehlqj wkh odeho ri rqh vshfl�f idxow zklfk kdv wr eh gldjqrvhg1

45

Lw fdq ri frxuvh eh qrwhg wkdw erwk idxow ghwhfwlrq dqg idxow lvrodwlrq fdq eh

vwdwhg lq d pruh jhqhudo vhwwlqj/ lqfoxglqj K3 lq wkh pxowl0k|srwkhvlv sureohp/

wkh vhw ri vlwxdwlrqv wr eh glvwlqjxlvkhg ehlqj qrz L @ i3> 4> ===Qj =

5, Fkdqjh ghwhfwlrq1 Edvhg rq wkh nqrzohgjh ri wkh gdwd +69,/ idxow

ghwhfwlrq frqvlvwv qrz ri vhohfwlqj rqh ri wzr k|srwkhvhv =

K3 = wkh gdwd kdyh ehhq surgxfhg e| d khdowk| v|vwhp/

K4 = wkh gdwd kdyh ehhq surgxfhg e| d khdowk| v|vwhp xqwloo vrph wlph

wi dqg wkh| kdyh ehhq surgxfhg e| d idxow| v|vwhp diwhu wlph wi >wkh xqnqrzq

ydoxh ri wi 5 iw� v. 4> ===> w� 4j ehlqj wr hvwlpdwh1

Dv ehiruh/ idxow lvrodwlrq h{whqgv wkh wzr k|srwkhvhv sureohp wr d pxowl0

k|srwkhvhv rqh1

Wklv sdshu rqo| frqvlghuv wkh k|srwkhvhv fkhfnlqj sureohp/ vwduwlqj zlwk

wkh vlpsohvw fdvh +qr qrlvh/ qr xqfhuwdlqw|/ qr xqnqrzq lqsxwv, dqg lqwurgxflqj

judgxdoo| pruh frpsoh{ fdvhv1

��� �������� �������

71414 Qr xqfhuwdlqw| dqg qr xqnqrzq lqsxw

Frqvlghu wkh vlpsohvw irupxodwlrq ri wkh qrupdo ehkdylrxu ri d wlph lqyduldqw

glvfuhwh wlph ghwhuplqlvwlf v|vwhp +64, dqg +65,1

{+w. 4, @ i+{+w,> x+w,, +6:,

|+w, @ j+{+w,, +6;,

Qrwh wkdw wklv prgho lqfoxghv qhlwkhu xqnqrzq lqsxwv/ qru phdvxuhphqw

qrlvh/ dqg vlqfh wkh sdudphwhuv �� duh dvvxphg wr eh shuihfwo| nqrzq/ wkh

ixqfwlrqv i dqg j duh shuihfwo| nqrzq1 Wklv prgho lv hqrxjk wr vwdwh wkh idxow

ghwhfwlrq sureohp1 Lq rughu wr vwdwh wkh idxow lvrodwlrq sureohp/ wkh gl�huhqw

nlqgv ri idxowv zklfk kdyh wr eh lvrodwhg kdyh �uvw wr eh gh�qhg1 Ohw

{+w. 4, @ i+{+w,> x+w,> *+w,, +6<,

|+w, @ j+{+w,> *+w,, +73,

eh wkh prgho ri wkh idxow| v|vwhp1 Gh�qh wkh +vlqjoh ru pxowlsoh, idxowv zklfk

kdyh wr eh glvwlqjxlvkhg e| phdqv ri qrq lqwhuvhfwlqj vxevhwv vxfk wkdw =

l 5 L> *+w, 5 !l +, idxow l +74,

Iru h{dpsoh/ d idxow rq vhqvru l zrxog eh ghvfulehg e| wkh dgglwlyh prgho

|+w, @ j+{+w,, . *+w,

zkhuh *+w, @ �+w,hl> ;�+w, 5 U> ohdglqj wr !l @ i*+w,> udqn ^*+w,> hl` @ 4j =

46

Idxow ghwhfwlrq Ohw ]v+w, eh d vkruw qrwdwlrq iru

�Xv+w,\v+w,

�= Wkh idxow gh0

whfwlrq sureohp fdq eh vwdwhg dv iroorzv =

Ilqg dq lqwhjhu v dqg d vhw K3 � Up+v.4,=Us+v.4, vxfk wkdw ]v+w, 5 K3 +,

<{3 5 Uq vxfk wkdw v|vwhp +6:,/ +6;, vwduwlqj iurp lqlwldo frqglwlrq {3 surgxfhv

wkh rxwsxwv \v+w, zkhq lwv lqsxwv duh Xv+w,1Lq rwkhu zrugv/ wkh ghwhfwlrq sureohp lv wr �qg d vxevhw K3 lq wkh re0

vhuydwlrq vsdfh vxfk wkdw wkh revhuydeohv ehorqj wr wkdw vxevhw zkdwhyhu wkh

+xqnqrzq, v|vwhp lqlwldo frqglwlrq zkhq wkh v|vwhp ehkdylrxu reh|v +6:,/ +6;,

1 Zkhq wkh revhuydeohv gr qrw ehorqj wr K3 wkh lqwhusuhwdwlrq lv wkdw wkh| fdq

e| qr phdqv kdyh ehhq surgxfhg e| v|vwhp +6:, +6;,/ zklfk lqghhg ghwhfwv d

idxow| ehkdylrxu dqg mxvwl�hv wkh surgxfwlrq ri dq dodup1

Idxow lvrodwlrq Wkh idxow lvrodwlrq sureohp fdq eh vwdwhg dv =

Iru hdfk idxow fdvh l/ �qg dq lqwhjhu v dqg d vhw Kl � Up+v.4,=Us+v.4, vxfk

wkdw ]v+w, 5 Kl +, <{3 5 Uq dqg < *+w, 5 !lvxfk wkdw v|vwhp +6<, +BB,

vwduwlqj iurp lqlwldo frqglwlrq {3 surgxfhv wkh rxwsxwv \v+w, zkhq lwv lqsxwv

duh Xv+w, dqg wkh idxow lv *+w,1Lq rwkhu zrugv/ wkh lvrodwlrq sureohp lv wr �qg d vxevhw Kl lq wkh revhuydwlrq

vsdfh vxfk wkdw wkh revhuydeohv ehorqj wr wkdw vxevhw zkdwhyhu wkh +xqnqrzq,

v|vwhp lqlwldo frqglwlrq dqg zkdwhyhu wkh idxow lq wkh fodvv !l1 Zkhq wkh re0

vhuydeohv gr qrw ehorqj wr Kl wkh lqwhusuhwdwlrq lv wkdw wkh| fdq e| qr phdqv

kdyh ehhq surgxfhg e| v|vwhp +6<, +BB, xqghu dq| idxow ri fodvv !l/ zklfk lq0

ghhg lvrodwhv wkh lwk idxow| ehkdylrxu dqg mxvwl�hv wkh surgxfwlrq ri d ghglfdwhg

dodup1

71415 Qr xqfhuwdlqw|

Xqnqrzq lqsxwv y+w, duh qrz frqvlghuhg/ dqg wkh vhw Y vxfk wkdw y+w, 5 Y iru

doo w lv lqwurgxfhg1 Wklv vhw doorzv wr ghvfuleh vrph nqrzohgjh rqh pljkw kdyh

derxw xqnqrzq lqsxwv/ iru h{dpsoh Y lv d jlyhq vxevsdfh ri Ug/ Y lv d erxqghg

vhw lq Ug/ hwf1 Zkhq qr nqrzohgjh dw doo lv dydlodeoh/ rqh kdv reylrxvo| Y @ Ug1

Wkh qrupdo dqg idxow| ehkdylrxuv ri v|vwhp +6:, +6;, duh qrz ghvfulehg e|

=

{+w. 4, @ i+{+w,> x+w,> y+w,, +75,

|+w, @ j+{+w,, +76,

y+w, 5 Y +77,

dqg =

{+w. 4, @ i+{+w,> x+w,> y+w,> *+w,, +78,

|+w, @ j+{+w,> *+w,, +79,

y+w, 5 Y> *+w, 5 ^l5L!l +7:,

47

Idxow ghwhfwlrq Wkh idxow ghwhfwlrq sureohp fdq eh vwdwhg dv iroorzv =

Ilqg dq lqwhjhu v dqg d vhw K3 � Up+v.4,=Us+v.4, vxfk wkdw ]v+w, 5 K3 +,

<{3 5 Uq dqg ;w><y+w, 5 Y vxfk wkdw v|vwhp +75, +76, vwduwlqj iurp lqlwldo

frqglwlrq {3 surgxfhv wkh rxwsxwv \v+w, zkhq lwv nqrzq lqsxwv duh Xv+w, dqg

lwv xqnqrzq lqsxwv duh y+w,1Lq rwkhu zrugv/ wkh ghwhfwlrq sureohp lv wr �qg d vxevhw K3 lq wkh re0

vhuydwlrq vsdfh vxfk wkdw wkh revhuydeohv ehorqj wr wkdw vxevhw zkdwhyhu wkh

+xqnqrzq, v|vwhp lqlwldo frqglwlrq dqg zkdwhyhu wkh xqnqrzq lqsxw1 Zkhq

wkh revhuydeohv gr qrw ehorqj wr K3 wkh lqwhusuhwdwlrq lv wkdw wkh| fdq e| qr

phdqv kdyh ehhq surgxfhg e| v|vwhp +75, +76,/ zklfk lqghhg ghwhfwv d idxow|

ehkdylrxu dqg mxvwl�hv wkh surgxfwlrq ri dq dodup1

Idxow lvrodwlrq Wkh idxow lvrodwlrq sureohp fdq eh vwdwhg dv =

Iru hdfk idxow fdvh l/ �qg dq lqwhjhu v dqg d vhw Kl � Up+v.4,=Us+v.4, vxfk

wkdw ]v+w, 5 Kl +, <{3 5 Uq dqg <y+w, 5 Y> *+w, 5 !lvxfk wkdw v|vwhp

+78, +79, vwduwlqj iurp lqlwldo frqglwlrq {3 surgxfhv wkh rxwsxwv \v+w, zkhq lwv

lqsxwv duh Xv+w,/ wkh xqnqrzq lqsxwv duh y+w, dqg wkh idxow lv *+w,1Lq rwkhu zrugv/ wkh lvrodwlrq sureohp lv wr �qg d vxevhw Kl lq wkh revhuydwlrq

vsdfh vxfk wkdw wkh revhuydeohv ehorqj wr wkdw vxevhw zkdwhyhu wkh +xqnqrzq,

v|vwhp lqlwldo frqglwlrq/ zkwhyhu wkh xqnqrzq lqsxw dqg zkdwhyhu wkh idxow lq

wkh fodvv !l1 Zkhq wkh revhuydeohv gr qrw ehorqj wr Kl wkh lqwhusuhwdwlrq lv

wkdw wkh| fdq e| qr phdqv kdyh ehhq surgxfhg e| v|vwhp +78, +79, xqghu dq|

idxow ri fodvv !l/ zklfk lqghhg lvrodwhv wkh lwk idxow| ehkdylrxu dqg mxvwl�hv wkh

surgxfwlrq ri d ghglfdwhg dodup1

71416 Ghwhuplqlvwlf xqfhuwdlq v|vwhpv

Xqfhuwdlqwlhv duh dozd|v suhvhqw lq wkh prghov ri dfwxdo v|vwhpv1 Lqghhg/ wkh

qrplqdo ydoxhv ri wkh v|vwhp sdudphwhuv/ ��/ duh wkh uhvxow ri wkh v|vwhp ghvljq

vwhs +iru h{dpsoh/ d srodul}dwlrq uhvlvwru vkrxog eh 53n ,/ exw wkhlu dfwxdo ydoxhlv riwhq gl�huhqw iurp wkh qrplqdo rqh/ wkh gl�huhqfh ehlqj d idxow rqo| li lw lvodujhu wkdq vrph suhgh�qhg ydoxh +iru h{dpsoh/ uhvlvwruv ehlqj surgxfhg zlwk d53( wrohudqfh/ wkh suhylrxv srodul}dwlrq uhvlvwru zrxog eh idxow| rqo| li lwv ydoxholhv rxwvlgh wkh lqwhuydo ^49n > 53n `,1 Rwkhu h{dpsohv duh sdudphwhuv zkrvhydoxh lv lghqwl�hg/ wkxv uhvxowlqj lq vrph frq�ghqfh lqwhuydo zlwklq zklfk wkh|olh/ ru sdudphwhuv zklfk qrupdoo| olh lq d erxqghg vhw/ olnh wkh ryhudoo pdvv rivrph yhklfoh1 Lq rughu wr wdnh xqfhuwdlqwlhv lqwr dffrxqw/ wkh khdowk| v|vwhpprgho lqwurgxfhv xqnqrzq sdudphwhuv � 5 �/ zkhuh � lv d erxqghg vhw zklfkfrqwdlqv doo wkh srvvleoh ydoxhv ri � zkhq wkh v|vwhp rshudwhv qrupdoo| =

{+w. 4, @ i+{+w,> x+w,> y+w,> �, +7;,

|+w, @ j+{+w,> �, +7<,

� 5 �> y+w, 5 Y +83,

48

Wkh idxow| ehkdylrxu ri v|vwhp +7;, +7<, duh qrz ghvfulehg e| = =

{+w. 4, @ i+{+w,> x+w,> y+w,> �> *+w,, +84,

|+w, @ j+{+w,> �> *+w,, +85,

� 5 �> y+w, 5 Y>*+w, 5 ^l5L!l +86,

Idxow ghwhfwlrq Wkh idxow ghwhfwlrq sureohp fdq eh vwdwhg dv iroorzv =Ilqg dq lqwhjhu v dqg d vhw K3 � Up+v.4,=Us+v.4, vxfk wkdw ]v+w, 5 K3 +,

<{3 5 Uq><�� 5 � dqg ;w><y+w, 5 Y vxfk wkdw v|vwhp +7;, +7<, zlwk sdudphwhuv��> vwduwlqj iurp lqlwldo frqglwlrq {3 surgxfhv wkh rxwsxwv \v+w, zkhq lwv nqrzqlqsxwv duh Xv+w, dqg lwv xqnqrzq lqsxwv duh y+w,1

Lq rwkhu zrugv/ wkh ghwhfwlrq sureohp lv wr �qg d vxevhw K3 lq wkh re0vhuydwlrq vsdfh vxfk wkdw wkh revhuydeohv ehorqj wr wkdw vxevhw zkdwhyhu wkh+xqnqrzq, v|vwhp lqlwldo frqglwlrq/ zkdwhyhu wkh +xqnqrzq, sdudphwhu yhfwrulq � dqg zkdwhyhu wkh xqnqrzq lqsxw lq Y 1 Zkhq wkh revhuydeohv gr qrw eh0orqj wr K3 wkh lqwhusuhwdwlrq lv wkdw wkh| fdq e| qr phdqv kdyh ehhq surgxfhge| v|vwhp +7;, +7<,/ zklfk lqghhg ghwhfwv d idxow| ehkdylrxu dqg mxvwl�hv wkhsurgxfwlrq ri dq dodup1

Idxow lvrodwlrq Wkh idxow lvrodwlrq sureohp fdq eh vwdwhg dv =Iru hdfk idxow fdvh l/ �qg dq lqwhjhu v dqg d vhw Kl � Up+v.4,=Us+v.4, vxfk

wkdw ]v+w, 5 Kl +, <{3 5 Uq><�� 5 �/ dqg <y+w, 5 Y><*+w, 5 !l vxfk wkdwv|vwhp +84, +BB, zlwk sdudphwhuv ��> vwduwlqj iurp lqlwldo frqglwlrq {3 surgxfhvwkh rxwsxwv \v+w, zkhq lwv lqsxwv duh Xv+w,/ wkh xqnqrzq lqsxwv duh y+w, dqgwkh idxow lv *+w,1

Lq rwkhu zrugv/ wkh lvrodwlrq sureohp lv wr �qg d vxevhw Kl lq wkh revhuydwlrqvsdfh vxfk wkdw wkh revhuydeohv ehorqj wr wkdw vxevhw zkdwhyhu wkh +xqnqrzq,v|vwhp lqlwldo frqglwlrq/ zkdwhyhu wkh xqnqrzq sdudphwhu lq �/ zkdwhyhu wkhxqnqrzq lqsxw lq Y dqg zkdwhyhu wkh idxow lq wkh fodvv !l1 Zkhq wkh re0vhuydeohv gr qrw ehorqj wr Kl wkh lqwhusuhwdwlrq lv wkdw wkh| fdq e| qr phdqvkdyh ehhq surgxfhg e| v|vwhp +84, +BB, xqghu dq| idxow ri fodvv !l/ zklfk lq0ghhg lvrodwhv wkh lwk idxow| ehkdylrxu dqg mxvwl�hv wkh surgxfwlrq ri d ghglfdwhgdodup1

71417 Ghwhuplqlvwlf ghwhfwdelolw| dqg lvrodelolw|

Wkh frpprq ihdwxuh ri doo wkuhh deryh sureohp vhwwlqjv lv wkdw rqh kdv wr �qgd sduwlwlrq ri wkh revhuydwlrq vsdfh lqwr Q . 4 fodvvhv vxfk wkdw wkh Q . 4

ehkdylrxu prghov dvvrfldwhg zlwk wkh qrupdo dqg wkh Q idxow| vlwxdwlrqv zloosurgxfh revhuydwlrqv +lqsxw0rxwsxw vhtxhqfhv, hdfk ri zklfk olhv lq rqh ri wkhQ . 4 gl�huhqw fodvvhv1

Wklv lv qrw dozd|v srvvleoh/ ghshqglqj rq wkh vhwv Y>�> !l l 5 L> dqg wkxvwkh ghwhfwlrq dqg lvrodwlrq sureohpv gr qrw dozd|v kdyh d vroxwlrq1 Wklv lvpdlqo| wkh frqvhtxhqfh ri wkh lpsrvvlelolw| wr �qg vhwv K3 vxfk wkdw wkh htxly0

49

dohqfh

]v+w, 5 K3 +, khdowk| v|vwhp

lv vdwlv�hg iru doo idxowv ri lqwhuhvw1Qrq0ghwhfwdeoh idxowv irup d vxevhw M � L vxfk wkdw wkh rqo| htxlydohqfh

zklfk fdq eh suryhg lv =

]v+w, 5 K3 +, khdowk| v|vwhp RU idxow !l 5 M

Vlploduo|/ qrq0lvrodeoh idxowv duh vxfk wkdw wkh rqo| htxlydohqfh zklfk fdqeh suryhg lv =

]v+w, 5 Klm +, idxow l RU idxow m

Frqglwlrqv xqghu zklfk idxowv duh qrq0ghwhfwdeoh dqg2ru qrq0lvrodeoh fdqeh h{klelwhg jlyhq vshfl�f fodvvhv ri prghov/ dqg vshfl�f vhwv Y>�> !l l 5 L=

Wkh| zloo eh dgguhvvhg lq Sduw LLL1

��� !������ �������

71514 Phdvxuhphqw qrlvh

Vhqvruv duh frpsrqhqwv zkrvh rxwsxw vljqdo lv xqiruwxqdwho| dozd|v fruuxswhge| phdvxuhphqw qrlvh1 Lq vrph fdvhv/ phdvxuhphqw qrlvh fdq eh qhjohfwhg/

dqg ghwhuplqlvwlf prghov duh vx!flhqw dssur{lpdwlrqv ri wkh v|vwhp ehkdylrxu1Lq rwkhu fdvhv/ vwrfkdvwlf yduldeohv kdyh wr eh lqwurgxfhg lq wkh phdvxuhphqw

htxdwlrq wr dffrxqw iru wkh phdvxuhphqw qrlvh1 Wkh khdowk| dqg idxow| v|vwhp

ehkdylrxuv duh wkxv ghvfulehg e| =

{+w. 4, @ i+{+w,> x+w,> y+w,> ��, +87,

|+w, @ j+{+w,> ��> %+w,, +88,

�� 5 �> y+w, 5 Y +89,

%+w, 5 H3/ %+w, � S3 +8:,

dqg

{+w. 4, @ i+{+w,> x+w,> y+w,> ��> *+w,, +8;,

|+w, @ j+{+w,> ��> %+w,> *+w,, +8<,

�� 5 �> y+w, 5 Y>*+w, 5 ^l5L!l +93,

%+w, 5 Hl> %+w, � Sl l 5 L +94,

zkhuh %+w, lv wkh phdvxuhphqw qrlvh/ H3 +uhvs1 Hl, lv wkh vhw ri lwv ydoxhv/dqg S3 +uhvs1 Sl, lv lwv suredelolw| glvwulexwlrq lq wkh khdowk| +uhvs1 lq wkh

4:

lwk idxow|, fdvh1 Lq wklv vlwxdwlrq/ wkh revhuydwlrqv \v+w, duh uhdol}dwlrqv ri

vwrfkdvwlf yduldeohv zkrvh suredelolw| glvwulexwlrqv ghshqg rq S3 +uhvs1 Sl, lqwkh khdowk| +uhvs1 lq wkh lwk idxow|, fdvh1

Zkhq H3 dqg Hl/ l 5 L duh erxqghg vhwv/ rqh srvvleoh dssurdfk lv wr frp0

sohwho| irujhw derxw wkh glvwulexwlrqv/ dqg wr wuhdw wkh phdvxuhphqw qrlvh h{0dfwo| wkh vdph zd| dv wkh xqnqrzq lqsxwv zhuh wuhdwhg1 Lq wklv fdvh/ wkh

sureohp vhwwlqjv dqg wkh dvvrfldwhg ghwhfwdelolw| dqg lvrodelolw| frqglwlrqv duhmxvw dv h{sodlqhg deryh1 Krzhyhu/ wklv dssurdfk lv yhu| frqvhuydwlyh/ vlqfh lw

grhv qrw wdnh lqwr dffrxqw wkh nqrzohgjh zklfk lv dydlodeoh derxw wkh suredelo0

lw| glvwulexwlrqv1 Lq pdq| fdvhv +dqg hvshfldoo| zkhq wkh vhwv H3 dqg Hl/ l 5 L

duh wkh zkroh vsdfh Us> zklfk lv wkh fdvh zkhq Jdxvvldq qrlvh lv frqvlghuhg/ iru

h{dpsoh,/ qr idxow lv ghwhfwdeoh dqg lvrodeoh/ vlqfh hyhu| srlqw lq wkh revhuydwlrq

vsdfh frxog kdyh ehhq surgxfhg e| dq| ri wkh khdowk| ru idxow| prghov1Wdnlqj lqwr dffrxqw wkh vwrfkdvwlf qdwxuh ri wkh revhuydwlrqv ohdgv wr uhsodfh

wkh ghwhuplqlvwlf vwdwhphqw

]v+w, 5 K3 +, khdowk| v|vwhp +95,

e| d vwrfkdvwlf rqh1 Lqghhg/ +95, fdq eh uhdg dv = wkh suredelolw| wkdw d khdowk|v|vwhp surgxfhv revhuydwlrqv lq K3 lv htxdo wr rqh/ wkh suredelolw| wkdw revhu0

ydwlrqv zklfk olh lq K3 kdyh ehhq surgxfhg e| d khdowk| v|vwhp lv rqh1 Wklv fdqqr orqjhu eh dvfhuwdlqhg/ vlqfh revhuydwlrqv rxwvlgh K3 pljkw eh rewdlqhg dv

wkh rxwsxwv ri khdowk| exw qrlvh0fruuxswhg vhqvruv/ zklfk phdvxuh gdwd uhdoo|

o|lqj lq K31 Frqyhuvho|/ revhuydwlrqv lq K3 pljkw eh rewdlqhg hyhq zkhq wkhv|vwhp lv idxow|/ dv wkh uhvxow ri qrlvh dfwlqj rq wkh phdvxuhphqw ghylfhv1

Wkh ghflvlrq sureohp vhwwlqj wkxv pryhv wr wkh vwdwlvwlfdo ghflvlrq �hog1 Lq

wkh ghwhuplqlvwlf vwdwhphqw/ hdfk srlqw ri wkh revhuydwlrq vsdfh ehorqjv +lqwkh lghdo fdvh, wr rqo| rqh fodvv ri wkh revhuydwlrq vsdfh sduwlwlrq1 Lw lv wkxv

dvvrfldwhg zlwk d fkdudfwhulvwlf yhfwru � ri Q .4 elwv/ vxfk wkdw elw q�l lv htxdowr rqh li idxow q�l lv suhvhqw1 Wkh frqfoxvlrq iroorzv lpphgldwho| e| =

idxow l+, ]v+w, 5 Kl +, �+]v+w,, @ hl

Lq wkh vwrfkdvwlf vwdwhphqw/ hdfk srlqw ] lq wkh revhuydwlrq vsdfh lv dvvrfldwhg

zlwk d suredelolw| yhfwru +�3+],> �4+],> ==�Q +],,w/ zkhuh

�l+], @ Su+]v+w, @ ]@idxow l lv suhvhqw,

+suredelolw| ghqvlwlhv fdq dovr eh xvhg,1 Vlqfh wkh ghflvlrq kdv wr lghqwli| rqhdqg rqo| rqh idxow fdvh +uhphpehu wkdw d idxow fdvh lv d vlqjoh ru d pxowlsoh idxow,

dvvrfldwhg zlwk hdfk revhuydwlrq/ wkh ghflvlrq sureohp lv wr �qg wkh sduwlwlrq

ri wkh revhuydwlrq vsdfh lqwr Q . 4 fodvvhv zklfk lv wkh prvw frqvlvwhqw zlwkwkh Q . 4 suredelolw| glvwulexwlrqv1 Frqvlvwhqf| lv hydoxdwhg xvlqj gl�huhqw

fulwhuld vxfk dv huuru suredelolwlhv/ phdq huuru frvw/ hwf1/ dqg �qglqj wkh prvw

frqvlvwhqw sduwlwlrq lv irupxodwhg dv dq rswlpl}dwlrq sureohp1 Wkh zhoo nqrzqEd|hv dqg Qh|pdq Shduvrq ghflvlrq uxohv duh surylghg e| vxfk dq dssurdfk ^4`1

4;

71515 Jhqhudol}dwlrq

Phdvxuhphqw qrlvh lv dq xqnqrzq v|vwhp lqsxw/ derxw zklfk vrph sduwldo

nqrzohgjh lv dydlodeoh wkurxjk lwv suredelolw| glvwulexwlrq1 Lw pd| kdsshq wkdw

rwkhu xqnqrzqv fdq eh fkdudfwhul}hg lq vxfk d zd|1 Wkh xqnqrzq ydoxh ri d

uhvlvwru/ wkh xqnqrzq h�ruwv ri zdyhv rq d vkls duh wzr h{dpsohv ri xqfhuwdlq

sdudphwhuv dqg xqnqrzq lqsxwv zkrvh vwdwlvwlfdo glvwulexwlrq pljkw eh nqrzq1

Lq vxfk vlwxdwlrqv/ wkh h{wud nqrzohgjh fdq reylrxvo| eh ljqruhg +wklv ohdgv wr

wkh iruphu ghwhuplqlvwlf dssurdfkhv, ru lw fdq eh xvhg/ e| phdqv ri wkh vdph

vwrfkdvwlf dssurdfk zklfk kdv ehhq dssolhg wr wkh wuhdwphqw ri phdvxuhphqw

qrlvhv1

71516 Vwrfkdvwlf ghwhfwdelolw| dqg lvrodelolw|

Lw reylrxvo| iroorzv iurp wkh deryh glvfxvvlrq wkdw qr ghflvlrq fdq eh wdnhq zkhq

wkh gl�huhqw idxow vlwxdwlrqv duh lghqwlfdoo| glvwulexwhg1 Wkxv qrq0ghwhfwdeoh

idxowv duh idxowv vxfk wkdw

�l+], @ �3+], ;] lq wkh revhuydwlrq vsdfh

Wzr idxowv l> m 9@ 3 zklfk duh qrw lvrodeoh duh vxfk wkdw

�l+], @ �m+], ;] lq wkh revhuydwlrq vsdfh

* ����� ������ ��� ��������

IGL dojrulwkpv kdyh wr exlog d sduwlwlrq ri wkh revhuydwlrq vsdfh vxfk wkdw hdfk

fodvv pd| eh dvvrfldwhg +lq wkh vwulfw vhqvh ru lq wkh suredelolvwlf rqh,/ zlwk rqh

idxow vlwxdwlrq ri wkh v|vwhp l 5 i3> 4> ===Qj =

*�� ����� �������

Ohw xv �uvw frqvlghu wkh sureohp ri ghflglqj ehwzhhq wkh wzr k|srwkhvhv l @ 3ru l 5 i4> ===Qj/ zklfk lv wkh idxow ghwhfwlrq sureohp1 Wkh gl!fxow| lv frqqhfwhg

zlwk wkh xqnqrzqv zklfk duh suhvhqw lq wkh v|vwhp ehkdylrxu prgho =

{3 = xqnqrzq lqlwldo frqglwlrq

y+w, = xqnqrzq lqsxwv

�� = xqnqrzq qrupdo sdudphwhuv

%+w, = phdvxuhphqw qrlvh

Ohw xv uhfdoo wkdw wkh ghflvlrq sureohp vwdwhphqw qhhgv wkh revhuydwlrqv

lvvxhg iurp wkh khdowk| dqg wkh idxow| v|vwhp wr eh fkdudfwhul}hg lqghshqghqwo|

ri wkrvh xqnqrzqv1 Wzr wrrov duh dydlodeoh dqg kdyh ehhq xvhg e| wkh frqwuro

frppxqlw| iru wklv dlp1

4, Holplqdwlrq surfhgxuhv wudqvirup d vhw ri htxdwlrqv lqwr dqrwkhu htxly0

dohqw rqh/ lq zklfk wkh xqnqrzq yduldeohv gr qrw dsshdu lq vrph vxevhw ri

4<

htxdwlrqv/ wkxv surylglqj frqglwlrqv zklfk gr qrw ghshqg rq wkhp1 Holplqd0

wlrq surfhgxuhv duh wkh edvlv ri wkh vr0fdoohg Sdulw| Vsdfh dssurdfk/ dv idu dv

lqsxw 0 rxwsxw uhodwlrqv duh rewdlqhg e| holplqdwlqj wkh xqnqrzq lqlwldo vwdwh1

5, Hvwlpdwlrq surfhgxuhv uhsodfh wkh xqnqrzq yduldeohv e| wkhlu hvwlpdwlrq/

zklfk lv rewdlqhg xvlqj d vxevhw ri wkh htxdwlrqv1 Gxh wr wkh v|vwhp uhgxq0

gdqf|/ wkhuh duh uhpdlqlqj htxdwlrqv lq zklfk xqnqrzq yduldeohv gr qr orqjhu

lqwhuyhqh/ zklfk surylghv wkh h{shfwhg uhvxow1 Hvwlpdwlrq surfhgxuhv duh wkh

edvlv ri wkh vr0fdoohg Revhuyhu Edvhg Dssurdfk1

Lq wkh fdvh ri olqhdu prghov/ wkh wzr dssurdfkhv kdyh ehhq suryhg wr eh

htxlydohqw ^<`/ dqg uhfhqw zrunv duh ghyrwhg wr wkhlu frpsdulvrq lq wkh qrq0

olqhdu iudph ^6`1

81414 Xqnqrzq vwdwh

Lq wklv vhfwlrq/ rqo| v|vwhpv lq zklfk qr xqnqrzq lqsxw dqg qr xqfhuwdlqw| lv

suhvhqw duh frqvlghuhg1 Wkh ehkdylrxu prgho lv wkh iroorzlqj =

{+w. 4, @ i+{+w,> x+w,, +96,

|+w, @ j+{+w,> %+w,, +97,

Wkhuh duh wzr pdlq dssurdfkhv iru frslqj zlwk wkh xqnqrzq vwdwh/ qdpho|

wkh sdulw| vsdfh dssurdfk/ lq zklfk lw lv holplqdwhg dqg wkh revhuyhu0edvhg

dssurdfk lq zklfk lw lv hvwlpdwhg1

Sdulw| vsdfh dssurdfk Vxssrvh wkh phdvxuhphqwv duh vxffhvvlyho| phpr0

ul}hg rq d wlph zlqgrz ri vl}h v/ vr wkdw wkh iroorzlqj revhuydwlrq lv dydlodeoh

=

\v+w, @

5997

|+w,|+w� 4,===

|+w� v,

6::8

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|+w� v, @ j^{+w� v,> %+w� v,` / j3^{+w� v,> %+w� v,`

Wkhq/ e| +97, dqg +96,/ rqh kdv

|+w� v. 4, @ j^{+w� v. 4,> %+w� v. 4,`

@ j^i+{+w� v,> x+w� v,,> %+w� v. 4,`

/ j4^{+w� v,> x+w� v,> %+w� v. 4,`

dqg lwhudwlqj wklv surfhgxuh/

53

|+w� v. n, / jn^{+w� v,> Xn+w� v. n,> %+w� v. n,`

wkh iroorzlqj v|vwhp lv rewdlqhg

\v+w, @

5997

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j4^{+w� v,> X4+w� v. 4,> %+w� v. 4,`j3^{+w� v,> %+w� v,`

6::8 =

Wklv lv d v|vwhp ri s+v. 4, qrq0olqhdu htxdwlrqv/ zkhuh rqo| wkh q frpsr0

qhqwv ri {+w� v, duh xqnqrzq1 Ohw

\v+w, @ Jv^{+w� v,> Xv+w,> Hv+w,`

eh d frqflvh zulwlqj ri wklv v|vwhp1 Lw lv dvvxphg wkdw wkhuh h{lvwv d qrq0olqhdu

wudqvirupdwlrq � vxfk wkdw

�^\v+w,` @ � �Jv^{+w� v,> Xv+w,> Hv+w,` +98,

zlwk wkh iroorzlqj surshuw|

� �Jv^{+w� v,> Xv+w,> Hv+w,` @

��3v^Xv+w,>Hv+w,`

�{v ^{+w� v,> Xv+w,> Hv+w,`

�

zkhuh �3v^Xv+w,> Hv+w,` lv ri glphqvlrq u @ s+v. 4,� udqn

CJv

C{=

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olwhudwxuh1 Irupdo dojrulwkpv iru �qglqj lw duh dydlodeoh rqo| iru vshfldo fodvvhv

ri v|vwhpv +iru h{dpsoh iru sro|qrpldo v|vwhpv/ d fodvv wr zklfk olqhdu dqg

el0olqhdu v|vwhpv ehorqj,1

Lw qrz dsshduv wkdw wkh �uvw u urzv ri +98, duh htxdwlrqv lq zklfk wkh

xqnqrzq vwdwh {+w� v, grhv qrw dsshdu1 Wkh| fdq eh zulwwhq

$^]v+w,> Hv+w,` @ 3 +99,

zkhuh $ lv d u�glphqvlrqdo ixqfwlrq ri wkh revhuydeohv rq wkh wlph zlqgrz

^w> w� v`1 Lw iroorzv wkdw wkh vhw lq wkh revhuydwlrq vsdfh gh�qhg e|

K3 @ i]v+w,>$^]v+w,> 3` @ 3j +9:,

zrxog fkdudfwhul}h wkh qrupdo ehkdylrxu ri v|vwhp +96,/ +97, zkhq qr qrlvh

zrxog eh suhvhqw1

Revhuyhu edvhg dssurdfk D yhu| vwudljkwiruzdug dqg srsxodu dssurdfkwr Prgho Edvhg IGL lv edvhg rq wkh vlpxodwlrq ri wkh v|vwhp ehkdylrxu1 Dvlpxodwlrq surjudp lv ghvljqhg iurp wkh v|vwhp prgho +96,/ +97,/ dqg wkh glv0fuhsdqflhv ehwzhhq wkh vlpxodwhg dqg wkh dfwxdo phdvxuhphqwv duh wkh edvlv

54

ri idxow ghwhfwlrq dqg lvrodwlrq1 Ohw vwduv +{�> |�> hwf=, eh dvvrfldwhg zlwk wkhvlpxodwhg yduldeohv/ wkhq wkh +ghwhuplqlvwlf, vlpxodwru htxdwlrqv duh =

{�+w. 4, @ i+{�+w,> x+w,, +9;,

|�+w, @ j+{�+w,, +9<,

Wkh glvfuhsdqflhv duh fdofxodwhg e| wkh uhvlgxdo yhfwru

u+w, @ |�+w,� |+w,

zklfk lv h{shfwhg wr eh }hur lq wkh devhqfh ri qrlvh dqg idxowv1 Krzhyhu/wklv lv qrw wuxh/ vlqfh wkh dfwxdo v|vwhp dqg lwv vlpxodwlrq surjudp duh g|qdplfv|vwhpv/ zklfk duh qrw lqlwldol}hg lghqwlfdoo|1 Lqghhg/ wkh lqlwldo vwdwh ri wkhdfwxdo v|vwhp lv xqnqrzq dqg wkh fruuhfw lqlwldo frqglwlrq ri wkh vlpxodwlrqfdqqrw eh ghwhuplqhg iurp wkh vlqjoh nqrzohgjh ri |+3,1 Rqo| lq vrph sduwlfxodufdvhv zrxog wkh rxwsxwv ri wkh wzr v|vwhpv zlwk gl�huhqw lqlwldo frqglwlrqvfrqyhujh/ dv vkrzq e| wkh iroorzlqj olqhdu h{dpsoh/

{+w. 4, @ D{+w, .Ex+w, +:3,

|+w, @ F{+w, +:4,

{�+w. 4, @ D{

�+w, .Ex+w, +:5,

|�+w, @ F{

�+w, +:6,

lq zklfk wkh vxevwudfwlrq ri +:3, dqg +:5, rq wkh rqh kdqg/ ri +:4, dqg +:6,rq wkh rwkhu kdqg jlyhv wkh g|qdplf prgho ri wkh vwdwh dqg rxwsxw huuruvh{+w, @ {+w,� {

�+w,> h|+w, @ |+w,� |�+w, =

h{+w. 4, @ Dh{+w, +:7,

h|+w, @ Fh{+w, +:8,

Iurp +:7,/ +:8,/ lw lv vhhq wkdw wkh vwdwh dqg rxwsxw huuruv frqyhujh wr }hurli dqg rqo| li pdwul{ D lv vwdeoh1

Xvlqj wkh wkhru| ri ihhgedfn frqwuro/ lw lv jhqhudoo| srvvleoh wr irufh wkhfrqyhujhqfh +ru wr lpsuryh wkh frqyhujhqfh udwh, ri wkh vlpxodwru1 Vxfk dsurjudp lq zklfk wkh frqyhujhqfh lv rewdlqhg dv wkh uhvxow ri d vshfl�f ghvljqlv fdoohg dq revhuyhu1 Wkh lghd fdq eh looxvwudwhg e| dgglqj d sursruwlrqdoihhgedfn wr wkh vlpxodwru htxdwlrqv +:5,/ +:6, =

{�+w. 4, @ D{

�+w, .Ex+w, .N+|+w,� |�+w,, +:9,

|�+w, @ F{

�+w, +::,

Wkh vwdwh dqg rxwsxw huuru htxdwlrqv ehfrph qrz =

55

h{+w. 4, @ ^D�NF`h{+w, +:;,

h|+w, @ Fh{+w, +:<,

dqg pdwul{ N lv fkrvhq vxfk wkdw +:;,/ +:<, lv vwdeoh1 Lq wkh jhqhudo fdvh/ +9;,/+9<, duh uhsodfhg e| wkh qrq olqhdu revhuyhu

{�+w. 4, @ i

�+{�+w,> x+w,>N, +;3,

|�+w, @ j

�+{�+w,, +;4,

lq zklfk wkh ixqfwlrqv i�> j� dqg wkh sdudphwhuv N kdyh wr eh fkrvhq vr dv wrlqvxuh wkh frqyhujhqfh ri wkh rxwsxw huuru lq wkh devhqfh ri idxowv1

Uhpdun = exloglqj d vlpxodwru lq zklfk rqo| nqrzq yduldeohv lqwhuyhqh lvqrwklqj hovh wkdq +dqg fdq rqo| eh grqh e|, xvlqj wkh sdulw| vsdfh dssurdfk1Lqghhg/ vxfk d vlpxodwru zrxog zulwh

|+w. 4, @ z^|+w,> |+w� 4,> ===|+w� �,> x+w,> x+w� 4,> ===x+w� �,`

zklfk lv dq h{solflw irup ri htxdwlrq +99, zklwk qr qrlvh/ dqg wkh fkdqjh riqrwdwlrqv w. 4 lqvwhdg ri w/ dqg � lqvwhdg ri v� 4=

81415 Xqnqrzq lqsxwv

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{+w. 4, @ i+{+w,> x+w,> y+w,, +;5,

|+w, @ j+{+w,> y+w,> %+w,, +;6,

zkhuh y+w, 5 Ug uhsuhvhqwv wkh xqnqrzq lqsxwv zklfk dfw rq wkh v|vwhp1

Orrnlqj dw wkh sdulw| vsdfh dssurdfk/ lw lv hdvlo| vhhq wkdw wkh v|vwhp ris+v. 4, qrq0olqhdu htxdwlrqv rewdlqhg e| phprul}lqj wkh v|vwhp rxwsxwv rq dwlph zlqgrz ri vl}h v.4/ zloo qrz kdyh qrw rqo| wkh q frpsrqhqwv ri {+w�v, dvxqnqrzqv exw dovr wkh g+v.4, frpsrqhqwv ri Yv+w,1 Wkh revhuydwlrq htxdwlrqlv qrz =

\v+w, @ Jv^{+w� v,> Xv+w,> Yv+w,> Hv+w,`

dqg rqh kdv wr �qg d wudqvirupdwlrq�zklfk holplqdwhv doowrjhwkhu wkh xqnqrzqlqlwldo vwdwh dqg wkh xqnqrzq lqsxwv =

� �Jv^{+w� v,> Xv+w,> Yv+w,> Hv+w,` @

��3v^X

v+w,>Hv+w,`

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v^{+w� v,> Xv+w,> Yv+w,> Hv+w,`

�

56

ohdglqj wr fkdudfwhul}h wkh qrupdo ehkdylrxu ri wkh v|vwhp e| wkh urexvw dqdo|wlf

uhgxqgdqf| uhodwlrqv =

$^]v+w,> Hv+w,` @ 3 +;7,

Krzhyhu/ frqwudu| wr wkh vwdwh holplqdwlrq zklfk fdq dozd|v eh shuiruphg/

vxfk d wudqvirupdwlrq grhv qrw dozd|v h{lvw1 Wkh srvvlelolw| wr holplqdwh wkh

xqnqrzq lqsxwv uhvwv rq wkh h{lvwhqfh ri d vxev|vwhp ri v|vwhp +;5, +;6, zklfk lv

qrw ghshqghqw rq wkhp/ l1h1 rq wkh h{lvwhqfh ri d wudqvirupdwlrq }+w, @ W +{+w,,

vxfk wkdw wkh v|vwhp prghov ehfrphv =

�}4+w. 4,

}5+w. 4,

�@

��i4+}4+w,> x+w,,

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�+;8,

�|4+w,

|5+w,

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�+;9,

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vsdfh dssurdfk wr wkh +}4> |4, sduw ri wkh prgho +;8, +;9,1 Qrwh wkdw vlqfh pruh

yduldeohv kdyh wr eh holplqdwhg/ rqh pxvw phprul}h pruh lqsxw0rxwsxw sdluv lq

rughu wr jhw hqrxjk revhuydwlrq htxdwlrqv1 Wkh vl}h ri wkh wlph zlqgrz zklfk

lv qhfhvvdu| lv qrz vxfk wkdw =

s+v. 4, A q. g+v. 4,

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qrw eh hvwlpdwhg/ vlqfh wkhuh lv qr prgho wr ghvfuleh wkhlu ehkdylrxu1 Wkxv/ wkh

revhuyhu0edvhg dssurdfk fdq eh xvhg rqo| li prgho +;5,/ +;6, fdq eh sxw xqghu

wkh irup +;8, +;9,/ l1h1 li wkhuh h{lvwv d vxev|vwhp zklfk lv qrw ghshqghqw rq

wkh xqnqrzq lqsxwv1 Zkhq wklv krogv wuxh/ dq revhuyhu fdq eh ghvljqhg dv lq

wkh suhylrxv vhfwlrq/ xvlqj wkh +}4> |4, sduw ri wkh prgho +;8, +;9,1

Wkh fdvh lq zklfk wkh xqnqrzq lqsxwv fdqqrw eh ghfrxsohg iurp dq| vxe0

v|vwhp ri +;5,/ +;6, kdv qrz wr eh frqvlghuhg1 Lq wkdw fdvh/ qr urexvw revhuyhu

dqg qr urexvw dqdo|wlf uhgxqgdqf| uhodwlrq fdq eh ghvljqhg1 Krzhyhu/ ghflvlrq

lv vwloo srvvleoh lq vrph fdvhv zkhuh wkh xqnqrzq lqsxwv duh nqrzq wr ehorqj wr

d jlyhq erxqghg vhw =

y+w, 5 Y � Ug

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xqnqrzq lqsxw dfwv rq wkh v|vwhp/ wkh vhw ri doo srvvleoh qrupdo wudmhfwrulhv lv

fkdudfwhul}hg e| =

K3 @ i]v+w,> $^]v+w,> 3` @ 3j +;:,

Zkhq xqnqrzq lqsxwv fdqqrw eh ghfrxsohg iurp dq| vxev|vwhp/ wkh +qrq ur0

exvw, dqdo|wlf uhgxqgdqf| uhodwlrqv duh =

57

$^]v+w,> Yv+w,> 3` @ 3 +;;,

dqg wkh vhw ri doo srvvleoh qrupdo wudmhfwrulhv lv fkdudfwhul}hg e| =

K3

3 @�]v+w,> <Yv 5 Y v.4 v=w= $^]v+w,> Yv> 3` @ 3

�+;<,

Wkh gl!fxow| lv lq jhqhudo wr ghwhuplqh wkh vhw K3

31 Iru olqhdu wlph lqyduldqw

v|vwhpv/ wklv lv txlwh vwudljkwiruzdug/ vlqfh wkh dqdo|wlf uhgxqgdqf| uhodwlrqv

duh olqhdu zlwk uhvshfw wr doo wkhlu dujxphqwv/ dqg +;;, lv ri wkh irup =

S\v+w, .TX

v+w, .UY

v+w, @ 3

Wkxv K3

3 lv vlpso| rewdlqhg iurp K3 e| shuiruplqj wkh Plqnrzvnl vxpp =

K3

3 @ K3 �UY v.4

81416 Xqfhuwdlqwlhv

Wklv vhfwlrq h{whqgv wkh fodvv ri v|vwhpv frqvlghuhg lq wkh suhylrxv rqh e|

lqwurgxflqj dovr xqfhuwdlqwlhv lq wkh v|vwhp prgho1 Qrwh wkdw iru d jlyhq prgho

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