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wkh sduwlwlrq/ h1j1 vpdoo/ phglxp/ odujh1 Deuxsw wudqvlwlrqv iurp rqh ydoxh wr
dqrwkhu rqh fdq eh dyrlghg xvlqj ix}}| vhjphqwv lqvwhdg ri fulvs rqhv1
61415 Wlph
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
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