Reliability Estimation by Advanced Monte Carlo

0 downloads 0 Views 675KB Size Report
achieve an acceptable accuracy in the estimated failure probability, with costly ... stead of random points in order to probe the failure domain of interest. .... quence of intermediate events, so that F = n F/, k = 1, 2, ..., m. ..... Nil) Ä£?(x) a jak .... Rotate the coordinate system (i.e., by means of a proper rotation matrix R ) so that the ...
          

E. Zio, N. Pedroni, “Reliability Estimation by Advanced Monte Carlo Simulation”, In: J. Faulin, A.A. Juan, S. Martorell, J.E. Ramirez-Marquez (Eds.), Simulation Methods for Reliability and Availability of Complex Systems (Springer series in Reliability Engineering), pp. 3-39, Springer-Verlag, London, United Kingdom, 2010, ISBN 978-1-84882-212-2.

&KDSWHU 5HOLDELOLW\HVWLPDWLRQE\DGYDQFHG0RQWH&DUOR VLPXODWLRQ (=LRDQG13HGURQL (QHUJ\'HSDUWPHQW3ROLWHFQLFRGL0LODQR9LD3RQ]LR0LODQ,WDO\ 3KRQHID[ (PDLODGGUHVVHQULFR]LR#SROLPLLW

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ĥLPSRU WDQWGLUHFWLRQĦLVGHWHUPLQHGZKLFKSRLQWVWRZDUGVWKHIDLOXUHGRPDLQRILQWHUHVW WKH KLJKGLPHQVLRQDO UHOLDELOLW\ SUREOHP LV WKHQ UHGXFHG WR D QXPEHU RI FRQGL WLRQDO RQHGLPHQVLRQDO SUREOHPV ZKLFK DUH VROYHG DORQJ WKH ĥLPSRUWDQW GLUHF WLRQĦ 7KHWZRPHWKRGVDUHDSSOLHGRQWZRVWUXFWXUDOUHOLDELOLW\PRGHOVRIOLWHUDWXUH LHWKHFUDFNHGSODWHPRGHODQGWKH3DULV(UGRJDQPRGHOIRUWKHUPDOIDWLJXHFUDFN JURZWK7KHHIILFLHQF\RIWKHSURSRVHGWHFKQLTXHVLVHYDOXDWHGLQFRPSDULVRQWR RWKHUVWRFKDVWLFVLPXODWLRQPHWKRGVRIOLWHUDWXUHLHVWDQGDUG0&6,PSRUWDQFH 6DPSOLQJ ,6 'LPHQVLRQDOLW\5HGXFWLRQ '5 DQG2UWKRJRQDO$[LV 2$ 

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

3 )

ZKHUH [

3 [  )

^[  [   [   [ ` ƒ 



M

Q

Q

³,

)

[ T [ G[ 

 

 LV WKH YHFWRU RI WKH XQFHUWDLQ LQSXW SDUDPH

WHUVYDULDEOHV RI WKH PRGHO ZLWK PXOWLGLPHQVLRQDO SUREDELOLW\ GHQVLW\ IXQFWLRQ 3')  T ƒ Q o > f  )  ƒ Q LVWKHIDLOXUHUHJLRQDQG , ) ƒ Q o ^` LVDQLQ GLFDWRUIXQFWLRQVXFKWKDW , ) [  LI [  ) DQG , ) [  RWKHUZLVH7KHIDLO XUHGRPDLQ)LVFRPPRQO\GHILQHGE\DVRFDOOHG3HUIRUPDQFH)XQFWLRQ 3) RU /LPLW6WDWH)XQFWLRQ /6)  J [ [ ZKLFKLVORZHUWKDQRUHTXDOWR]HURLI [ 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ĥLPSRUWDQWGLUHFWLRQĦLVRSWLPDOO\GHWHUPLQHGWRSRLQWWRZDUGVWKHIDLOXUHGR PDLQ RI LQWHUHVW DQG D QXPEHU RI FRQGLWLRQDO RQHGLPHQVLRQDO SUREOHPV DUH VROYHG DORQJ VXFK GLUHFWLRQ LQ SODFH RI WKH KLJKGLPHQVLRQDO SUREOHP 3UDGO ZDUWHUHWDO 7KHDSSURDFKKDVEHHQVKRZQWRSHUIRUPDOZD\VEHWWHUWKDQ VWDQGDUG0&6IXUWKHUPRUHLIWKHERXQGDULHVRIWKHIDLOXUHGRPDLQRILQWHUHVWDUH QRWWRRURXJK LHDOPRVWOLQHDU DQGWKHĥLPSRUWDQWGLUHFWLRQĦ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Ġ YDULDEOHVIL





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





6LPXODWLRQPHWKRGVLPSOHPHQWHGLQWKLVVWXG\

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

7KHEDVLFSULQFLSOHV )RUDJLYHQWDUJHWIDLOXUHHYHQW)RILQWHUHVWOHW ) Š ) Š  Š )P TXHQFHRILQWHUPHGLDWHHYHQWVVRWKDW )N

) EHDVH

ˆ )L N īP%\VHTXHQWLDOO\ N L 

FRQGLWLRQLQJRQWKHHYHQW)LWKHIDLOXUHSUREDELOLW\3 ) FDQEHZULWWHQDV P 



3 )

3 )P

3 )

– 3 )

L 

_ )L 

 

L 

1RWLFH WKDW HYHQ LI 3 )  LV VPDOO WKH FRQGLWLRQDO SUREDELOLWLHV LQYROYHG LQ  FDQEHPDGHVXIILFLHQWO\ODUJHE\DSSURSULDWHO\FKRRVLQJPDQGWKHLQWHU PHGLDWHHYHQWV^)LL īPĠ` 7KHRULJLQDOLGHDRI66LVWRHVWLPDWHWKHIDLOXUHSUREDELOLW\3 ) E\HVWLPDWLQJ 3 ) DQG ^3 )L _ )L  L    P  ` &RQVLGHULQJIRUH[DPSOH3 )  | DQG FKRRVLQJ P   LQWHUPHGLDWH HYHQWV VXFK WKDW 3 )  DQG ^3 )L  _ )L  L    ` |  WKHFRQGLWLRQDOSUREDELOLWLHVFDQEHHYDOXDWHGHIIL FLHQWO\E\VLPXODWLRQRIWKHUHODWLYHO\IUHTXHQWLQWHUPHGLDWHHYHQWV $XDQG%HFN   6WDQGDUG0&6FDQEHXVHGWRHVWLPDWH3 ) 2QWKHFRQWUDU\FRPSXWLQJWKH FRQGLWLRQDOSUREDELOLWLHVLQ  E\0&6HQWDLOVWKHQRQWULYLDOWDVNRIVDPSOLQJ IURPWKHFRQGLWLRQDOGLVWULEXWLRQVRI [ JLYHQWKDWLWOLHVLQ)LL PĠ LHIURP T [ _ )L T [ , ) [  3 ) ,QWKLVUHJDUG0DUNRY&KDLQ0RQWH&DUOR L

0&0&  VLPXODWLRQ SURYLGHV D SRZHUIXO PHWKRG IRU JHQHUDWLQJ VDPSOHV FRQGL





WLRQDORQWKHLQWHUPHGLDWHUHJLRQV)LL PĠ $XDQG%HFN$X DQG %HFN E  )RU FRPSOHWHQHVV RI WKH SDSHU WKH UHODWHG DOJRULWKP LV SUH VHQWHGLQ$SSHQGL[

7KHDOJRULWKP ,QWKHDFWXDO66LPSOHPHQWDWLRQZLWKQRORVVRIJHQHUDOLW\LWLVDVVXPHGWKDWWKH IDLOXUHHYHQWRILQWHUHVWFDQEHGHILQHGLQWHUPVRIWKHYDOXHRIDFULWLFDOV\VWHPUH VSRQVHYDULDEOHJ [ [ d @

>

@

3 [ M d K[ [  M



^ > M

 

@`

( [ )[ _[ K[ [  M M



M

ZKHUH )[ _ [ ˜  LV WKH &XPXODWLYH 'LVWULEXWLRQ )XQFWLRQ RI [M FRQGLWLRQDO RQ M

M

[  [   [  [   [   'UDZ 1  VDPSOHV ^[  N    1 ` ZKHUH [ [  M LH [  M 7





M 

M 

Q

N M

7

[  [   [

N M

N 

N 

N M 

 [ NM   [QN 

IURP WKH Q Ġ  GLPHQVLRQDO PDUJLQDO SUREDELOLW\ GHQVLW\ IXQFWLRQ T P [  M  LH T P [  M TP [  [    [ M  [ M    [Q

³ T [  [   [   [ G[ 

[M



M

Q

M



 8VLQJWKHODVWH[SUHVVLRQLQ  FRPSXWHDQXQELDVHGDQGFRQVLVWHQWHVWL PDWH 3Ö ) IRUWKHIDLOXUHSUREDELOLW\3 ) DVIROORZV

3Ö )



 17

¦ ) >K [ @  17

[ M _ [ M

[

N M

 

N 

,WLVZRUWKQRWLQJWKDWLQ  WKHIDLOXUHSUREDELOLW\HVWLPDWHLVFRPSXWHGDV WKH H[SHFWHG YDOXH RI WKH FXPXODWLYH GLVWULEXWLRQ IXQFWLRQ )[ _ [ ˜  RI [M FRQGL M

M

WLRQDORQWKHUHPDLQLQJ QĠ LQSXWYDULDEOHV6LQFHWKLVTXDQWLW\WDNHVYDOXHVEH WZHHQDQGWKH]HURYDOXHVFRQWDLQHGLQWKHVWDQGDUG0&6HVWLPDWRU LH,) [  LI[  ) DUHUHPRYHGWKLVDOORZVWRL UHDFKDQ\OHYHORIIDLOXUHSUREDELOLW\ HYHQYHU\VPDOO DQGLL UHGXFHWKHYDULDQFHRIWKHIDLOXUHSUREDELOLW\HVWLPDWRU +RZHYHUVXFKPHWKRGFDQQRWDOZD\VEHDSSOLHGILUVWWKHSHUIRUPDQFHIXQFWLRQ J[ x PXVWEHNQRZQDQDO\WLFDOO\VHFRQGLWPXVWKDYHWKHSURSHUW\WKDWRQHRIWKH XQFHUWDLQLQSXWYDULDEOHVFDQEHVHSDUDWHGIURPWKHRWKHUVWRDOORZUHZULWLQJWKH IDLOXUHFRQGLWLRQ J [ [ d  LQWKHIRUPRI   *LOOHDQG 





)LQDOO\ QRWLFH WKDW '5 FDQ EH FRQVLGHUHG D YHU\ VSHFLDO FDVH RI /6 6HFWLRQ  ZKHUHWKHSHUIRUPDQFHIXQFWLRQJ[ x LVDQDO\WLFDOO\NQRZQDQGWKHLPSRU WDQW GLUHFWLRQ Į FRLQFLGHV ZLWK WKH ĥGLUHFWLRQĦ RI WKH YDULDEOH [M LH Į    [ M     

7KH2UWKRJRQDO$[LVPHWKRG 7KH 2UWKRJRQDO $[LV 2$  PHWKRG FRPELQHV WKH )LVW 2UGHU 5HOLDELOLW\ 0HWKRG )250  DSSUR[LPDWLRQ 'HU .LXUHJKLDQ   DQG 0RQWH &DUOR 6LPXODWLRQ 0&6 LQDVRUWRILPSRUWDQFHVDPSOLQJDURXQGWKHĥGHVLJQSRLQWĦRIWKHSUREOHP VHH6HFWLRQ  7KH2$DOJRULWKPSURFHHGVDVIROORZV *LOOHDQG 

^[  [   [   [ ` ƒ

 7UDQVIRUP [



M



Q

Q

LHWKHYHFWRURIXQFHUWDLQSDUDPHWHUV

GHILQHG LQ WKH RULJLQDO SK\VLFDO VSDFH [  ƒ Q  LQWR WKH YHFWRU ș  ƒ Q  ZKHUH HDFKHOHPHQWRIWKHYHFWRUșMM īQLVDVVRFLDWHGZLWKDFHQWUDOXQLW *DXVVLDQ VWDQGDUG GLVWULEXWLRQ 6FKXHOOHU HW DO   VHH 6HFWLRQ   7KXVWKHMRLQWSUREDELOLW\GHQVLW\IXQFWLRQRI ș FDQVLPSO\EHZULWWHQDV

– I T 

M Q ș

 ZKHUH I T M





S H



 T M 



Q

M 

M

 

M Q

 )LQGWKHĥGHVLJQSRLQWĦ ș RIWKHSUREOHP VHH6HFWLRQ   5RWDWHWKHFRRUGLQDWHV\VWHP LHE\PHDQVRIDSURSHUURWDWLRQPDWUL[ 5 VR WKDWWKHQHZFRRUGLQDWH T Q LVLQWKHGLUHFWLRQRIWKHD[LVGHILQHGE\WKHGHVLJQ SRLQW ș   'HILQHDQHZIDLOXUHIXQFWLRQ J D[LV ș DV

J D[LV ș







a a  :ULWLQJ ș DV ș T Q ZKHUH ș

T T 

J 5ș 



 

  T Q H[SUHVVWKHIDLOXUHSUREDELOLW\

3 ) DVIROORZV





> @ @ ³ > ^ > @`

a 3 ) 3 J D[LV ș  T Q d  a a a a 3 J D[LV ș  T Q d  _ ș M Q ș Gș a (șa 3 J D[LV ș T Q d 



 

^

`

a  *HQHUDWH 17 LLG Q Ġ  GLPHQVLRQDO VDPSOHV ș N  N    1 7  ZKHUH a ș N T N T N   T QN   &RPSXWHDQHVWLPDWH 3Ö ) IRUWKHIDLOXUHSUREDELOLW\3 ) DVIROORZV 3Ö )



>

@

 17

a ¦ 3>J ș T d @  17

N

D[LV

 

Q

N 

a 7KHWHUPV 3 J D[LV ș N T Q d  N ī17DUHHYDOXDWHGZLWKDQLWHUDWLYH a DOJRULWKPZKLFKVHDUFKHVIRUWKHURRWVRIWKHHTXDWLRQ J D[LV ș N  T Q   *LOOH DQG  ,W LV ZRUWK QRWLQJ WKDW WKH LGHD XQGHUO\LQJ WKH 2$ PHWKRG LV HVVHQWLDOO\ WKH VDPHDVWKDWRI/6 6HFWLRQ +RZHYHULQ2$WKHĥLPSRUWDQWGLUHFWLRQĦLV IRUFHGWRFRLQFLGHZLWKWKDWRIWKHGHVLJQSRLQWRIWKHSUREOHPPRUHRYHU2$HP SOR\VDURWDWLRQRIWKHFRRUGLQDWHV\VWHPZKLFKFDQEHGLIILFXOWWRGHILQHLQYHU\ KLJKGLPHQVLRQDOSUREOHPV









$SSOLFDWLRQWKHFUDFNHGSODWHPRGHO 7KHFUDFNHGSODWHPRGHOLVDFODVVLFDOH[DPSOHLQ)UDFWXUH0HFKDQLFVDQGLWVUHOD WLYHVLPSOLFLW\DOORZVDGHWDLOHGDQGFRPSOHWHVWXG\RIGLIIHUHQWVLPXODWLRQWHFK QLTXHV$WKRURXJKGHVFULSWLRQRIWKLVPRGHOFDQEHIRXQGLQ $UGLOORQDQG9HQ WXULQL 

7KHPHFKDQLFDOPRGHO $PHWDOSODWHRILQILQLWHOHQJWKZLWKDGHIHFWRILQLWLDOOHQJWKHTXDOWRD>P@LVFRQ VLGHUHG 7KH SODWH LV VXSSRVHG WR EH VXEMHFW WR D XQLIRUP QRUPDO ORDGLQJ LH VWUHVV  VŎ >03D@ 7KH LQWHQVLW\ IDFWRU . >03D P @ GHWHUPLQHG E\ WKH XQLIRUP ORDGLQJLQWKHQHLJKERUKRRGRIWKHGHIHFWLVGHILQHGDVIROORZV 

.

)Vf SD 

 

ZKHUH)LVWKHVKDSHIDFWRURIWKHGHIHFW7KHSODWHLVVXSSRVHGWREUHDN LH IDLO ZKHQWKHLQWHQVLW\IDFWRU.LQ  EHFRPHVJUHDWHUWKDQRUHTXDOWRDFULWL FDOYDOXH.FLH 

.

)Vf SD t . F 

 

7KHYDULDEOHVRIWKHPHFKDQLFDOPRGHODUHVXPPDUL]HGLQ7DEOH 7DEOH1DPHVGHVFULSWLRQVDQGXQLWVRIPHDVXUHRIWKHYDULDEOHVRIWKHFUDFNHGSODWHPRGHO 9DULDEOHVRIWKHFUDFNHGSODWHPRGHO 1DPH

'HVFULSWLRQ

8QLWRIPHDVXUH

.F

&ULWLFDOVWUHVVLQWHQVLW\IDFWRU

03DōP

D

,QLWLDOOHQJWKRIWKHGHIHFW

P

)

6KDSHIDFWRURIWKHGHIHFW



V Ŏ

8QLIRUPQRUPDOORDGLQJ VWUHVV WRZKLFKWKHSODWHLVVXEMHFW

03D

7KHVWUXFWXUDOUHOLDELOLW\PRGHO )URP WKH SRLQW RIYLHZ RI D VWUXFWXUDO UHOLDELOLW\ DQDO\VLV WKH FUDFNHG SODWH PH FKDQLFDOPRGHORI6HFWLRQLVDQDO\]HGZLWKLQDSUREDELOLVWLFIUDPHZRUNLQ ZKLFKWKHYDULDEOHV.FD)DQGVŎDUHXQFHUWDLQ IRUVLPSOLFLW\RILOOXVWUDWLRQZLWK





UHVSHFW WR WKH QRWDWLRQ RI WKH SUHYLRXV 6HFWLRQV WKH IRXU YDULDEOHV DUH KHUHDIWHU QDPHG[[[DQG[UHVSHFWLYHO\  5HIHUULQJWR  WKHSHUIRUPDQFHIXQFWLRQJ[ [ RIWKHV\VWHPLV

J [ [



J [ [  [   [  [ 

[  [ [ S[ 

 

7KHIDLOXUHUHJLRQ)LVWKHQH[SUHVVHGDV 

^[  J [ d ` ^ [  [  [  [  [

)



[









`

d [ [ S[  

 

)LQDOO\WKHSUREDELOLW\RIV\VWHPIDLOXUH3 ) LVZULWWHQDVIROORZV





3 ) 3 [  ) 3>J [ [ d @ 3 [ d [ [ S[ 



 

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





&DVH

&DVH

&DVH

[ .  [ D  [ )  [ VŎ 

1  

1  

1  





1  

3 ) 

ǜ







&DVH /*   

1 ǜ  

1 ǜ  

1 ǜ  

/* ǜ 

1  

1  

1  

/*  

1  

1  

/*  

ǜ

ǜ

ǜ



5HVXOWV ,QWKLV6HFWLRQWKHUHVXOWVRIWKHDSSOLFDWLRQRI66DQG/6IRUWKHUHOLDELOLW\DQDO\ VLV RI WKH FUDFNHG SODWH PRGHO RI 6HFWLRQ  DUH LOOXVWUDWHG ZLWK UHIHUHQFH WR &DVHVWXGLHVDQGGHVFULEHGLQWKHSUHYLRXV6HFWLRQ  )RUIDLUFRPSDULVRQDOOPHWKRGVKDYHEHHQUXQZLWKDWRWDORI17 VDP SOHVLQDOOIRXUFDVHV7KHHIILFLHQF\RIWKHVLPXODWLRQPHWKRGVXQGHUDQDO\VLVLV HYDOXDWHG LQ WHUPV RI IRXU TXDQWLWLHV WKH IDLOXUH SUREDELOLW\ HVWLPDWH 3Ö )  WKH VDPSOHVWDQGDUGGHYLDWLRQ VÖ RIWKHIDLOXUHSUREDELOLW\HVWLPDWH 3Ö ) WKHFRHIIL FLHQWRIYDULDWLRQ FRY  G RI 3Ö )  GHILQHGDVWKHUDWLRRIWKHVDPSOHVWDQGDUG GHYLDWLRQ VÖ WRWKHHVWLPDWH 3Ö ) DQGWKH)LJXUH2I0HULW )20 RIWKHPHWKRG

GHILQHGDV  VÖ  W FRPS ZKHUHWFRPSLVWKHFRPSXWDWLRQDOWLPHUHTXLUHGE\WKHVLPX

ODWLRQ PHWKRG  7KH FORVHU LV WKH HVWLPDWH 3Ö )  WR WKH H[DFW LH DQDO\WLFDOO\ FRPSXWHG  IDLOXUH SUREDELOLW\ 3 )  WKH PRUH DFFXUDWH LV WKH VLPXODWLRQ PHWKRG 7KHVDPSOHVWDQGDUGGHYLDWLRQ VÖ DQGWKHFRY G RI 3Ö ) DUHXVHGWRTXDQWLI\ WKHYDULDELOLW\RIWKHIDLOXUHSUREDELOLW\HVWLPDWRULQSDUWLFXODUWKHORZHUDUHWKH YDOXHVRI VÖ DQG G WKHORZHULVWKHYDULDELOLW\RIWKHFRUUHVSRQGLQJIDLOXUHSURE DELOLW\ HVWLPDWRU DQG WKXV WKH KLJKHU LV WKH HIILFLHQF\ RI WKH VLPXODWLRQ PHWKRG DGRSWHG )LQDOO\ WKH )20 LV LQWURGXFHG WR WDNH LQWR DFFRXQW WKH FRPSXWDWLRQDO WLPHUHTXLUHGE\WKHPHWKRG7KHYDOXHRIWKH)20LQFUHDVHVDVWKHVDPSOHYDUL DQFH VÖ  RIWKHIDLOXUHSUREDELOLW\HVWLPDWH 3Ö ) DQGWKHFRPSXWDWLRQDOWLPHWFRPS UHTXLUHGE\WKHPHWKRGGHFUHDVHWKXVLQWKLVFDVHWKHKLJKHULVWKHYDOXHRIWKH LQGH[WKHKLJKHULVWKHHIILFLHQF\RIWKHPHWKRG *LOOHDQG   7KHGLIIHUHQWVLPXODWLRQPHWKRGVDUHDOVRFRPSDUHGZLWKUHVSHFWWRWZRGLUHFW SHUIRUPDQFH LQGLFDWRUV UHODWLYH WR VWDQGDUG 0&6 )LUVW WKH UDWLR RI WKH VDPSOH VWDQGDUGGHYLDWLRQ VÖ 0& REWDLQHGE\6WDQGDUG0&6WRWKDWREWDLQHGE\WKHVLPXOD WLRQ PHWKRG XQGHU DQDO\VLV VÖ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Ö )  VÖ  G  )20 VÖ 0& VÖ PHWK  DQG )20 PHWK)200&REWDLQHGE\6WDQGDUG0&666DQG/6LQ&DVHVDQG 6HFWLRQ





  WKH DFWXDO QXPEHU 1V\V RI V\VWHP UHVSRQVH DQDO\VHV LH PRGHO HYDOXD WLRQV LVDOVRUHSRUWHG1RWLFHWKDWIRUERWK66DQG/6WKH DFWXDOQXPEHU1V\VRI V\VWHP DQDO\VHV GRHV QRW FRLQFLGH ZLWK WKH WRWDO QXPEHU 17 RI UDQGRP VDPSOHV GUDZQ LH17  ,QSDUWLFXODULQWKH66PHWKRGWKHSUHVHQFHRIUHSHDWHG FRQGLWLRQDO VDPSOHV LQ HDFK0DUNRY FKDLQ XVHG WR JUDGXDOO\ SRSXODWH WKH LQWHU PHGLDWHHYHQWUHJLRQV DOORZVDUHGXFWLRQLQWKHQXPEHURIPRGHOHYDOXDWLRQVUH TXLUHGDFWXDOO\RQHHYDOXDWLRQLVHQRXJKIRUDOOLGHQWLFDOVDPSOHV VHH$SSHQGL[  ,QWKH/6PHWKRGLQVWHDGWKHDFWXDOQXPEHU1V\VRIV\VWHPDQDO\VHVLVJLYHQ E\1V\V 1Vx17LQSDUWLFXODU1V DQDO\VHVDUHSHUIRUPHGWRJHQHUDWHWKH 0DUNRYFKDLQXVHGWRFRPSXWHWKHLPSRUWDQWXQLWYHFWRUĮDVWKHQRUPDOL]HGĥFHQ WHURIPDVVĦRIWKHIDLOXUHGRPDLQ) 6HFWLRQ WKHx17DQDO\VHVDUHFDU ULHG RXW WR FRPSXWH WKH 17 FRQGLWLRQDO RQHGLPHQVLRQDO SUREDELOLW\ HVWLPDWHV 3Ö N )  N    1 7 E\OLQHDULQWHUSRODWLRQ HTXDWLRQ ģ LQ$SSHQGL[ 

^

`

7DEOH5HVXOWVRIWKHDSSOLFDWLRQRIVWDQGDUG0&666DQG/6WRWKHUHOLDELOLW\DQDO\VLVRI &DVHV 5HIHUHQFH DQGRIWKHFUDFNHGSODWHPRGHORI6HFWLRQWKHYDOXHVRIWKH SHUIRUPDQFH LQGLFDWRUV XVHG WR FRPSDUH WKH HIIHFWLYHQHVV RI WKH PHWKRGV LH VÖ 0& VÖ PHWK  DQG )20PHWK)200& DUHKLJKOLJKWHGLQEROG &DVH 5HIHUHQFH  

3Ö ) 

6WDQGDUG0&6

x

ıÖ 

x

FRYį

1V\V

)20

x

ıÖ 0&  ıÖ PHWK 

)20PHWK)200&









66

x 

x 

x









/6

x

x

x



x





3Ö ) 

ıÖ 

FRYį

1V\V

)20







&DVH  6WDQGDUG0&6



x 

x 





x 

/6

x 



x 

66

 

x 



 

x  



ıÖ 0&  ıÖ PHWK 

)20PHWK)200&











x





x

x  x 

x 



x 

FRYį

1V\V

)20

ıÖ 0&  ıÖ PHWK 



x



&DVH 

3Ö ) 

6WDQGDUG0&6

x

ıÖ 





x 

x 

x 



x



x

x

x

x



x



x

FRYį

1V\V

)20

ıÖ 0&  ıÖ PHWK 

)20PHWK)200&



x



66 /6

x 



)20PHWK)200&





&DVH 

3Ö ) 

6WDQGDUG0&6

x

ıÖ 

x

x





66

x 

x 

x 



x





/6

x

x

x



x



x











 ,WFDQEHVHHQWKDW66SHUIRUPVFRQVLVWHQWO\EHWWHUWKDQVWDQGDUG0&6DQGLWV SHUIRUPDQFH VLJQLILFDQWO\ JURZV DV WKH IDLOXUH SUREDELOLW\ WR EH HVWLPDWHG GH FUHDVHVIRULQVWDQFHLQ&DVH 5HIHUHQFH ZKHUH3 ) aWKH)20RI66 QDPHO\ )2066 LV RQO\ IRXU WLPHV ODUJHU WKDQ WKDW RI 6WDQGDUG 0&6 QDPHO\ )200&ZKHUHDVLQ&DVHZKHUH3 ) aWKHUDWLR)2066)200&LVDERXW  2Q WKH RWKHU KDQG /6 RXWSHUIRUPV 66 ZLWK UHVSHFW WR ERWK VÖ 0& VÖ PHWK  DQG )20PHWK)200& LQ DOO WKH &DVHV FRQVLGHUHG )RU LQVWDQFH LQ &DVH  ZKHUH WKH IDLOXUHSUREDELOLW\3 ) WREHHVWLPDWHGLVYHU\VPDOOLH3 )  xWKHUDWLR VÖ 0& VÖ /6  LV  WLPHV ODUJHU WKDQ WKH UDWLR VÖ 0& VÖ 66  ZKHUHDV WKH UDWLR )20/6)200&LVWLPHVODUJHUWKDQWKHUDWLR)2066)200&1RWLFHWKDWIRU WKH/6PHWKRGHYHQWKRXJKWKHGHWHUPLQDWLRQRIWKHVDPSOLQJLPSRUWDQWGLUHFWLRQ Į 6HFWLRQ DQGWKHFDOFXODWLRQVRIWKHFRQGLWLRQDORQHGLPHQVLRQDOIDLO XUHSUREDELOLW\HVWLPDWHV 3Ö N )  N    1 7  HTXDWLRQ ģ LQ$SSHQGL[  UHTXLUH PXFK PRUH WKDQ 17 V\VWHP DQDO\VHV E\ WKH PRGHO WKLV LV VLJQLILFDQWO\ RYHUZHLJKHG E\ WKH DFFHOHUDWHG FRQYHUJHQFH UDWH WKDW FDQEH DWWDLQHG E\ WKH /6 PHWKRGZLWKUHVSHFWWR66

^

`

&RPSDULVRQZLWKRWKHUVWRFKDVWLFVLPXODWLRQPHWKRGV 7KHUHVXOWVREWDLQHGE\66DQG/6DUHFRPSDUHGWRWKRVHREWDLQHGE\WKH,PSRU WDQFH 6DPSOLQJ ,6  'LPHQVLRQDOLW\ 5HGXFWLRQ '5  2UWKRJRQDO $[LV 2$  PHWKRGVDQGE\DFRPELQDWLRQRI,6DQG'5 6HFWLRQ  *LOOHDQG  )RU'5WKHYDULDEOH[LVH[SOLFLWHG  7KHYDOXHVRIWKHSHUIRUPDQFHLQGLFDWRUV VÖ 0&  VÖ PHWK DQG)20PHWK)200&RE WDLQHGE\WKHIRXUPHWKRGVLQ&DVHVDQGDUHVXPPDUL]HGLQ7DEOH





7DEOH  9DOXHV RI WKH SHUIRUPDQFH LQGLFDWRUV VÖ 0& VÖ PHWK  DQG )20PHWK)200& REWDLQHG E\ ,6'5 ZLWKYDULDEOH[VSHFLILHG 2$DQG,6'5ZKHQDSSOLHGIRUWKHUHOLDELOLW\DQDO\VLVRI &DVHV  5HIHUHQFH    DQG  RI WKH FUDFNHG SODWH PRGHO RI 6HFWLRQ  *LOOH  DQG   &DVH 5HIHUHQFH  

ıÖ 0&  ıÖ PHWK 

)20PHWK)200&

,6





'5 9DULDEOH[ 





2$



x

,6'5



x 

&DVH 

ıÖ 0&  ıÖ PHWK 

)20PHWK)200&

,6





'5 9DULDEOH[ 



x

2$



x

,6'5



x 

&DVH 

ıÖ 0&  ıÖ PHWK 

)20PHWK)200&

,6



x

'5 9DULDEOH[ 





2$



x

,6'5



x 

&DVH 

ıÖ 0&  ıÖ PHWK 

)20PHWK)200&

,6





'5 9DULDEOH[ 





2$



x

,6'5



x

 &RPSDULQJ7DEOHDQG7DEOHLWFDQEHVHHQWKDW/6SHUIRUPVVLJQLIL FDQWO\ EHWWHU WKDQ ,6 DQG '5 LQ DOO WKH FDVH VWXGLHV FRQVLGHUHG LQ SDUWLFXODU LQ &DVHV  DQG  WKH YDOXHV RI WKH SHUIRUPDQFH LQGLFDWRUV VÖ 0& VÖ /6   DQG  DQG)20/6)200& xDQGx DUHPRUHWKDQRQHRUGHURIPDJ QLWXGH ODUJHU WKDQ WKRVH UHSRUWHG LQ *LOOH  DQG   IRU ,6   DQG xIRU&DVHVDQGUHVSHFWLYHO\ DQG'5 xDQG IRU&DVHVDQGUHVSHFWLYHO\ 0RUHRYHULWLVZRUWKQRWLQJWKDWLQWKHUHIHUHQFH VWXGLHVE\ *LOOHDQG DVLJQLILFDQWQXPEHURIVLPXODWLRQVKDVEHHQUXQ WRSURSHUO\WXQHWKHSDUDPHWHUVRIWKH,6'VIRUWKH,6PHWKRG LQSDUWLFXODU





DQGVLPXODWLRQVKDYHEHHQSHUIRUPHGIRU&DVHVDQGUHVSHFWLYHO\ ZLWK DVLJQLILFDQWLQFUHDVHLQWKHDVVRFLDWHGFRPSXWDWLRQDOHIIRUW /6LVIRXQGWRSHUIRUPVOLJKWO\ZRUVHWKDQ2$LQDOOWKHFDVHVWXGLHVFRQVLG HUHGDFWXDOO\WKHYDOXHVRIERWK VÖ 0& VÖ /6 DQG)20/6)200&DUHVOLJKWO\ORZHU WKDQWKRVHUHSRUWHGLQ *LOOHDQG IRU2$+RZHYHULWVKRXOGEHFRQ VLGHUHGWKDWLQWKHVHVWXGLHVWKH2$PHWKRGKDVEHHQDSSOLHGWRDVLPSOLILHGYHU VLRQRIWKHSUREOHPGHVFULEHGLQ6HFWLRQVDQGDFWXDOO\RQO\WKUHH XQFHUWDLQYDULDEOHV LH[[DQG[ KDYHEHHQFRQVLGHUHGE\NHHSLQJYDULDEOH[ LH) IL[HGWRLWVPHDQYDOXH LH WKLVFHUWDLQO\UHGXFHVWKHYDULDELOLW\RI WKHPRGHORXWSXWDQGFRQWULEXWHVWRWKHUHGXFWLRQRIWKHYDULDELOLW\RIWKHDVVRFL DWHGIDLOXUHSUREDELOLW\HVWLPDWRU )XUWKHU/6SHUIRUPVFRQVLVWHQWO\EHWWHUWKDQWKHFRPELQDWLRQRI,6DQG'5LQ WKHWDVNRIHVWLPDWLQJIDLOXUHSUREDELOLWLHVDURXQG¸ IRULQVWDQFHLQ&DVH  VÖ 0& VÖ ,6  '5    DQG VÖ 0& VÖ /6    ZKHUHDV LQ &DVH  VÖ 0& VÖ ,6  '5    DQG VÖ 0& VÖ /6   ,QDGGLWLRQ/6SHUIRUPVFRPSDUDEO\WRWKHFRPELQDWLRQRI ,6DQG'5LQWKHHVWLPDWLRQRIIDLOXUHSUREDELOLWLHVDURXQGDFWXDOO\LQ&DVH VÖ 0& VÖ ,6  '5  DQG VÖ 0& VÖ /6  ZKHUHDVLQ&DVH VÖ 0& VÖ ,6  '5   DQG VÖ 0& VÖ /6    +RZHYHU LW KDV WR EH QRWLFHG DJDLQ WKDW LQ WKH UHIHUHQFH VWXGLHVE\ *LOOHDQG DVLJQLILFDQWQXPEHURIVLPXODWLRQVKDVEHHQUXQ WRSURSHUO\WXQHWKHSDUDPHWHUVRIWKH,6'VIRUWKH,6PHWKRG LQSDUWLFXODU DQGVLPXODWLRQVKDYHEHHQSHUIRUPHGLQ&DVHVDQGUHVSHFWLYHO\  )LQDOO\LWLVZRUWKQRWLQJWKDWLQWKHVHFDVHV66SHUIRUPVZRUVHWKDQWKHRWKHU PHWKRGVSURSRVHG





$SSOLFDWLRQWKHUPDOIDWLJXHFUDFNJURZWKPRGHO 7KHWKHUPDOIDWLJXHFUDFNJURZWKPRGHOFRQVLGHUHGLQWKLVVWXG\LVEDVHGRQWKH GHWHUPLQLVWLF3DULV(UGRJDQPRGHOZKLFKGHVFULEHVWKHSURSDJDWLRQRIDPDQXIDF WXULQJGHIHFWGXHWRWKHUPDOIDWLJXH 3DULV 

7KHPHFKDQLFDOPRGHO 7KHHYROXWLRQRIWKHVL]HDRIDGHIHFWVDWLVILHVWKHIROORZLQJHTXDWLRQ

GD G1 F



& ˜ I 5 ˜ '.  P

 

ZKHUH1FLVWKHQXPEHURIIDWLJXHF\FOHV&DQGPDUHSDUDPHWHUVGHSHQGLQJRQ WKHSURSHUWLHVRIWKHPDWHULDOI 5 LVDFRUUHFWLRQIDFWRUZKLFKLVDIXQFWLRQRIWKH PDWHULDOUHVLVWDQFH5DQGŊ.LVWKHYDULDWLRQRIWKHLQWHQVLW\IDFWRUGHILQHGDV

'V ˜ < D ˜ SD 

'.



 

,Q  ŊVLVWKHYDULDWLRQRIWKHXQLIRUPORDGLQJ VWUHVV DSSOLHGWRWKHV\V WHPDQG< D LVWKHVKDSHIDFWRURIWKHGHIHFW/HW6L ŊVLEHWKHYDULDWLRQRIWKH XQLIRUPQRUPDOVWUHVVDWF\FOHL ī1F7KHLQWHJUDWLRQRIHTXDWLRQ   JLYHV D1 F

1F

GD

³ < D



SD

D



P

& ˜ ¦ I 5 ˜ 6 L  P

 

L 

ZKHUH D DQG D 1  DUH WKH LQLWLDO DQG ILQDO VL]H RI WKH GHIHFW UHVSHFWLYHO\ ,Q F

 WKHIROORZLQJDSSUR[LPDWLRQFDQEHDGRSWHG 1F

¦ I 5 ˜ 6 | 7  7 ˜ 1 ˜ I 5 ˜ 6



P



L

F

P



 

L 

ZKHUH7DQG7DUHWKHLQLWLDODQGILQDOWLPHVRIWKHWKHUPDOIDWLJXHWUHDWPHQW RI1FF\FOHV  7KHV\VWHPLVFRQVLGHUHGIDLOHGZKHQWKHVL]HD1FRIWKHGHIHFWDWWKHHQGRIWKH 1FF\FOHVH[FHHGVDFULWLFDOGLPHQVLRQDFLH



DF  D 1 d   F



 



ZKLFKLQWKHLQWHJUDOIRUP  UHDGV

\ DF \ D 1 d  



 

F

ZKHUH

\ D



GD

D

³ < D ˜ D

SD



P



 

8VLQJ  DVDIHW\PDUJLQ 0 7 FDQWKHQEHGHILQHGDVIROORZV

0 7



DF

GD

³ < D ˜ D

SD



P

 & ˜ 7  7 ˜ 1 F ˜ I 5 ˜ 6

P



 

7KH IDLOXUH FULWHULRQ FDQ WKHQ EH H[SUHVVHG LQ WHUPV RI WKH VDIHW\ PDUJLQ  

0 7 d  



 

7KHYDULDEOHVRIWKHPRGHODUHVXPPDUL]HGLQ7DEOH 7DEOH  1DPHV GHVFULSWLRQV DQG XQLWV RI PHDVXUH RI WKH YDULDEOHV RI WKH WKHUPDO IDWLJXH FUDFNJURZWKPRGHO 9DULDEOHVRIWKHWKHUPDOIDWLJXHFUDFNJURZWKPRGHO 1DPH

'HVFULSWLRQ

8QLWRIPHDVXUH

D 

,QLWLDOVL]HRIWKHGHIHFW

>P@

D F

&ULWLFDOVL]HRIWKHGHIHFW

>P@

7

,QLWLDOWLPH

>\HDUV@

7

)LQDOWLPH

>\HDUV@

&

3DUDPHWHURIWKHPDWHULDO



P

3DUDPHWHURIWKHPDWHULDO



I 5 

&RUUHFWLRQIDFWRU



1F

1XPEHURIF\FOHVSHU\HDU



6

6WUHVVSHUF\FOH

>03D@

7KHVWUXFWXUDOUHOLDELOLW\PRGHO )RU WKH SXUSRVH RI D VWUXFWXUDO UHOLDELOLW\ DQDO\VLV WKH WKHUPDO IDWLJXH FUDFN JURZWKPRGHOLVIUDPHGZLWKLQDSUREDELOLVWLFUHSUHVHQWDWLRQRIWKHXQFHUWDLQWLHV





DIIHFWLQJWKHQLQHYDULDEOHVDDF77&PI 5 1FDQG6 KHUHDIWHUQDPHG[ [[[[[[[DQG[UHVSHFWLYHO\  )URP  WKHSUREDELOLW\RIV\VWHPIDLOXUH3 ) LVZULWWHQDV

ª 3 ) 3>0 7 d @ 3 « ¬«

DF

GD

³ < D ˜

SD

D

º P  & ˜ 7  7 ˜ 1 F ˜ I 5 ˜ 6 d »    ¼»



P

RU

3 )

3>0 7 d @

ª 3« «¬

GD

[

³ < D ˜ [

SD



[

º [  [ ˜ [  [ ˜ [ ˜ [ ˜ [ d »   »¼ 

,W LV ZRUWK QRWLQJ WKH KLJKO\ QRQOLQHDU QDWXUH RI H[SUHVVLRQV   DQG  ZKLFKLQFUHDVHVWKHFRPSOH[LW\RIWKHSUREOHP

&DVHVWXGLHV 7ZRGLIIHUHQWFDVHVWXGLHVQDPHO\&DVHDQG&DVHDUHEXLOWZLWKUHIHUHQFHWR WKHVWUXFWXUDOUHOLDELOLW\PRGHORIWKHSUHYLRXV6HFWLRQ7KHFKDUDFWHULVWLFV RI WKH 3')V RI WKH XQFHUWDLQ YDULDEOHV RI 7DEOH  DUH VXPPDUL]HG LQ 7DEOH  WKH YDOXHV RI WKH H[DFW LH DQDO\WLFDOO\ FRPSXWHG  IDLOXUH SUREDELOLWLHV 3 ) IRUERWK&DVHVDQGDUHDOVRUHSRUWHGLQWKHODVWURZRI7DEOH 7DEOH3UREDELOLW\GLVWULEXWLRQVDQGSDUDPHWHUV LHPHDQVDQGVWDQGDUGGHYLDWLRQV RIWKH XQFHUWDLQYDULDEOHV[[ī[RIWKHWKHUPDOIDWLJXHFUDFNJURZWKPRGHORI6HFWLRQIRU WKHWZRFDVHVWXGLHVFRQVLGHUHG LH&DVHVDQG WKHODVWURZUHSRUWVWKHYDOXHVRIWKHFRUUH VSRQGLQJ H[DFW LH DQDO\WLFDOO\ FRPSXWHG  IDLOXUH SUREDELOLWLHV 3 )  *LOOH  DQG   ([S H[SRQHQWLDOGLVWULEXWLRQ/* /RJQRUPDOGLVWULEXWLRQ1 1RUPDOGLVWULEXWLRQ  [ D 



&DVH

&DVH 

([S ǜ 

([S ǜ  



[ DF 

1 ǜ ǜ 

1 ǜǜ 

[ 7 





[ 7 





[ & 

/* ǜǜ 

/* ǜǜ 

[ P 





[ I 5 





[ 1F 

1  

1  

[ 6 

/*  

/*  

3 ) 

x

ǜ



5HVXOWV ,QWKLV6HFWLRQWKHUHVXOWVRIWKHDSSOLFDWLRQRI66DQG/6IRUWKHUHOLDELOLW\DQDO\ VLVRIWKHWKHUPDOIDWLJXHFUDFNJURZWKPRGHORI6HFWLRQVDQGDUHLO OXVWUDWHGZLWKUHIHUHQFHWR&DVHVDQG 7DEOHRI6HFWLRQ   $JDLQIRUIDLUFRPSDULVRQDOOVLPXODWLRQPHWKRGVKDYHEHHQUXQZLWKWKHVDPH WRWDOQXPEHURIVDPSOHV 17  LQERWK&DVHVDQG7KHHIILFLHQF\RIWKH PHWKRGVKDVEHHQHYDOXDWHGLQWHUPVRIWKHVDPHLQGLFHVDQGSHUIRUPDQFHLQGLFD WRUVGHILQHGLQ6HFWLRQ 7DEOH  UHSRUWV WKH YDOXHV RI 3Ö )  VÖ  G  )20 VÖ 0& VÖ PHWK  DQG )20 PHWK)200&REWDLQHGE\6WDQGDUG0&666DQG/6LQWKH&DVHVDQGRI6HFWLRQ  WKH DFWXDO QXPEHU 1V\V RI V\VWHP UHVSRQVH DQDO\VHV LH PRGHO HYDOXD WLRQV LVDOVRUHSRUWHG 7DEOH5HVXOWVRIWKHDSSOLFDWLRQRIVWDQGDUG0&666DQG/6WRWKHUHOLDELOLW\DQDO\VLVRI &DVHVDQGRIWKHWKHUPDOIDWLJXHFUDFNJURZWKPRGHORI6HFWLRQWKHYDOXHVRIWKHSHU IRUPDQFHLQGLFDWRUVXVHGWRFRPSDUHWKHHIIHFWLYHQHVVRIWKHPHWKRGV LH VÖ 0& VÖ PHWK DQG)20 )200& DUHKLJKOLJKWHGLQEROG

PHWK

&DVH 

3Ö ) 

ıÖ 

FRYį

1V\V

)20

ıÖ 0&  ıÖ PHWK 

6WDQGDUG0&6

x

x

x



x





66

x

x

x



x





x

x

x



x



x

/6

)20PHWK)200&



&DVH 

3Ö ) 

6WDQGDUG0&6

x

FRYį

ıÖ 

x

)20

ıÖ 0&  ıÖ PHWK 



x

)20PHWK)200&





66

x 

x 

x 



x





/6

x

x

x



x



x







1V\V



 $OVRLQWKLVDSSOLFDWLRQWKH/6PHWKRGRORJ\LVIRXQGWRRXWSHUIRUP66LQERWK &DVHVDQGIRUH[DPSOHLQ&DVHZKHUHWKHIDLOXUHSUREDELOLW\3 ) WREHHV WLPDWHGLVDURXQGWKHUDWLR)20/6)200&LVDERXWWLPHVODUJHUWKDQWKH UDWLR)2066)200&

&RPSDULVRQZLWKRWKHUVWRFKDVWLFVLPXODWLRQPHWKRGV $V GRQH IRU WKH SUHYLRXV DSSOLFDWLRQ RI 6HFWLRQ  WKH UHVXOWV REWDLQHG E\ 66 DQG/6KDYHEHHQFRPSDUHGWRWKRVHREWDLQHGE\RWKHUOLWHUDWXUHPHWKRGVLQSDU WLFXODUWKH,PSRUWDQFH6DPSOLQJ ,6 DQGDFRPELQDWLRQRI,PSRUWDQFH6DPSOLQJ





DQG 'LPHQVLRQDOLW\ 5HGXFWLRQ 6HFWLRQ   ZKLFK KDYH WXUQHG RXW WR JLYH WKH EHVWUHVXOWVLQWKHFDVHVWXGLHVFRQVLGHUHG *LOOHDQG 1RWLFHWKDWWKH 2UWKRJRQDO $[LV 2$  PHWKRG KDV QRW EHHQ LPSOHPHQWHG IRU WKLV DSSOLFDWLRQ LQ WKHUHIHUHQFHVWXG\ *LOOHDQG WKLVLVGXHWRWKHKLJKGLPHQVLRQDOLW\RI WKHSUREOHPZKLFKPDNHVWKHGHILQLWLRQRIDSURSHUURWDWLRQPDWUL[YHU\GLIILFXOW VWHSLQ6HFWLRQ   7KHYDOXHVRIWKHSHUIRUPDQFHLQGLFDWRUV VÖ 0&  VÖ PHWK DQG)20PHWK)200&RE WDLQHGE\,6DQG,6DQG'5IRU&DVHVDQGRIWKHWKHUPDOIDWLJXHFUDFNJURZWK PRGHORI6HFWLRQVDQGDUHVXPPDUL]HGLQ7DEOH 7DEOH9DOXHVRIWKHSHUIRUPDQFHLQGLFDWRUV VÖ 0&  VÖ PHWK DQG)20PHWK)200&REWDLQHGE\,6 DQG ,6  '5 ZKHQ DSSOLHG IRU WKH UHOLDELOLW\ DQDO\VLV RI &DVHV  DQG  RI WKH WKHUPDO IDWLJXH FUDFNJURZWKPRGHORI6HFWLRQ *LOOHDQG  &DVH 

ıÖ 0&  ıÖ PHWK 

)20PHWK)200&

,6





,6'5



 

&DVH 

ıÖ 0&  ıÖ PHWK 

)20PHWK)200&

,6



x

,6'5



x

 ,QWKLVDSSOLFDWLRQ/6LVIRXQGWRRXWSHUIRUPERWK,6DQGWKHFRPELQDWLRQRI,6 DQG'5IRUH[DPSOHLQ&DVHWKHUDWLR)20/6)200&LVDQGWLPHVODUJHU WKDQ)20,6)200&DQG)20,6'5)200&UHVSHFWLYHO\7KLVFRQILUPVWKHFDSD ELOLW\RIWKH/6PHWKRGWRHIILFLHQWO\SUREHFRPSOH[KLJKGLPHQVLRQDOGRPDLQVRI LQWHJUDWLRQ





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ĥJRRGĦ ,6' HJ E\ LGHQWLI\LQJĥGHVLJQSRLQWVĦ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





VLPXODWLRQDUHFRUUHODWHGE\FRQVWUXFWLRQ6LQFHLWLVGHPRQVWUDWHGWKDWDKLJKFRU UHODWLRQDPRQJFRQGLWLRQDOVDPSOHVLQFUHDVHVWKHYDULDQFHRIWKH66HVWLPDWHVD JRRG FKRLFHWXQLQJ RI WKH 66 SDUDPHWHUV LH WKH FRQGLWLRQDO SUREDELOLW\ S DQG WKH SURSRVDO 3')V IRU 0&0& VLPXODWLRQ  LV UHTXLUHG WR DYRLG LW $X DQG %HFN E  )LQDOO\ DQRWKHU GUDZEDFN RI WKH 66 PHWKRG LV WKH QHHG WR H[SUHVV WKH IDLOXUHHYHQW)LQWHUPVRIDUHDOYDOXHGSDUDPHWHUFURVVLQJDJLYHQWKUHVKROG LH ) ^ f  VDPSOH 17 YHFWRUV ^[ N  N    1 7 ` ZLWK

^[  [   [   [ `E\VWDQGDUG0&6  7UDQVIRUP WKH 1  VDPSOH YHFWRUV ^[  N    1 ` GHILQHG LQ WKH RULJLQDO [N

N 

N 

N M

N Q

N

7

7

LHSK\VLFDO VSDFHRISRVVLEO\GHSHQGHQWQRQQRUPDOUDQGRPYDULDEOHV VWHS  DERYH  LQWR 17 VDPSOHV ^ș N  N    1 7 `  GHILQHG LQ WKH VWDQGDUGQRUPDO VSDFHZKHUHHDFKFRPSRQHQWRIWKHYHFWRU ș N

^T

N 

T N   T MN  T QN ` N 

17LVDVVRFLDWHGZLWKDQLQGHSHQGHQWFHQWUDOXQLW*DXVVLDQVWDQGDUGGLVWUL EXWLRQ 6HFWLRQ   (VWLPDWH 17 FRQGLWLRQDO ĥRQHGLPHQVLRQDOĦ IDLOXUH SUREDELOLWLHV 3Ö N )  N    1 7  FRUUHVSRQGLQJ WR HDFK RQH RI WKH VWDQGDUG QRUPDO

^

VDPSOHV ^ș  N N

`

   1 7 `  REWDLQHG LQ VWHS  DERYH ,Q SDUWLFXODU IRU HDFK

UDQGRPVDPSOH ș N N ī17SHUIRUPWKHIROORZLQJVWHSV )LJXUHģ  6FKXHOOHUHWDO3UDGOZDUWHUHWDO3UDGOZDUWHUHWDO  a D 'HILQHWKHVDPSOHYHFWRU ș N N 17DVWKHVXPRIDGH WHUPLQLVWLFPXOWLSOHRI ĮDQGDYHFWRU ș N A N 17SHU SHQGLFXODUWRWKHGLUHFWLRQĮLH

a șN

 

ģ 

ZKHUHFNLVDUHDOQXPEHULQ>ŎŎ@DQG

ș N A



F N Į  ș N A N 17

ș N  Į ș N Į N 17

ģ 

,Q ģ  ș N N 17GHQRWHVDUDQGRPUHDOL]DWLRQRI WKHLQSXWYDULDEOHVLQWKHVWDQGDUGQRUPDOVSDFHRIGLPHQVLRQQ







DQG Į ș N LVWKHVFDODUSURGXFWEHWZHHQĮDQG ș N N  17 )LQDOO\ LW LV ZRUWK QRWLQJ WKDW VLQFH WKH VWDQGDUG *DXVVLDQ VSDFHLVLVRWURSLFERWKWKHVFDODUFNDQGWKHYHFWRU ș N A DUHDOVR VWDQGDUGQRUPDOO\GLVWULEXWHG 3UDGOZDUWHUHWDO  E &RPSXWHWKHYDOXH F N DVWKHLQWHUVHFWLRQEHWZHHQWKHOLPLWVWDWH a J T F N Į  ș N A  DQGWKHOLQH O N F N  Į SDVV IXQFWLRQ J T ș N



LQJWKURXJK ș N DQGSDUDOOHOWRĮ )LJXUHģ 7KHYDOXHRI F N  FDQ EH DSSUR[LPDWHG E\ HYDOXDWLQJ WKH SHUIRUPDQFH IXQFWLRQ J T ˜ DWWZRRUWKUHHGLIIHUHQWYDOXHVRIFN HJ FN  F N DQG FN  LQ )LJXUH ģ  ILWWLQJ D ILUVW RU VHFRQG RUGHU SRO\QRPLDO DQG GHWHUPLQLQJ LWV URRW )LJXUH ģ  +HQFH IRU HDFK VWDQGDUG QRUPDOUDQGRPVDPSOH ș N N ī17WZRRUWKUHHV\VWHP SHUIRUPDQFHHYDOXDWLRQVE\WKHPRGHODUHUHTXLUHG F 6ROYHWKHFRQGLWLRQDORQHGLPHQVLRQDOUHOLDELOLW\SUREOHPDVVR FLDWHGWRHDFKUDQGRPVDPSOH ș N N ī17LQZKLFKWKH RQO\ VWDQGDUG QRUPDO  UDQGRP YDULDEOH LV FN 7KH DVVRFLDWHG FRQGLWLRQDOIDLOXUHSUREDELOLW\ 3Ö N ) N ī17LVJLYHQ E\

3Ö N ) 3>1  ! F N @

  3>1  d F N @ 



  ) F



N

)  F

N

ģ 



ZKHUH ) ˜ GHQRWHVWKHVWDQGDUGQRUPDOFXPXODWLYHGLVWULEXWLRQ IXQFWLRQ  8VLQJ WKH LQGHSHQGHQW FRQGLWLRQDO ĥRQHGLPHQVLRQDOĦ IDLOXUH SUREDELOLW\ HVWL PDWHV 3Ö N )  N    1 7 LQ ģ  VWHSFDERYH FRPSXWHWKHXQELDVHG HVWLPDWRU 3Ö ) IRUWKHIDLOXUHSUREDELOLW\ 3 ) DV

^

`

3Ö )



 17

17

¦ 3Ö )  N

ģ 

N 

7KHYDULDQFHRIWKHHVWLPDWRU ģ LV





V  3Ö )



 1 7 1 7  

¦ 3Ö )  3Ö ) 17

N





ģ 

N 









:LWKWKHGHVFULEHGDSSURDFKWKHYDULDQFHRIWKHHVWLPDWRU 3Ö ) RIWKHIDLOXUH

SUREDELOLW\ 3 ) LVFRQVLGHUDEO\UHGXFHG,QJHQHUDODUHODWLYHO\ORZQXPEHU17 RI VLPXODWLRQV KDV WR EH FDUULHG RXW WR REWDLQ D VXIILFLHQWO\ DFFXUDWH HVWLPDWH $ VLQJOHHYDOXDWLRQZRXOGVXIILFHIRUWKHLGHDOFDVHLQZKLFKWKHOLPLWVWDWHIXQFWLRQ LV OLQHDU DQG D /LQH 6DPSOLQJ GLUHFWLRQ Į SHUSHQGLFXODU WR LW KDV EHHQ LGHQWLILHG .RXWVRXUHODNLVHWDO  

 )LJģ7KH/LQH6DPSOLQJSURFHGXUH 3UDGOZDUWHUHWDO