$ 3UDFWLFDO $SSURDFK 7R 6LJQDOV 6\VWHPV DQG
&RQWUROV
$UPDQGR $ 5RGULJXH]
$VVRFLDWH 3URIHVVRU
$UL]RQD 6WDWH 8QLYHUVLW\
'HSDUWPHQW RI (OHFWULFDO (QJLQHHULQJ
6\VWHP 6FLHQFH DQG (QJLQHHULQJ 5HVHDUFK &HQWHU
7HPSH $=
KWWSZZZHDVDVXHGX
z
DDU
DDU#DVXHGX
6HSWHPEHU
&RQWHQWV
,QWURGXFWLRQ 6\VWHPV $QG 7KH 8VH 2I 0DWKHPDWLFDO 0RGHOV
:LGHVSUHDG $SSOLFDELOLW\ $QG &RPPRQ )HDWXUHV 2I 0DWKHPDWLFDO 0RGHOV
$8QLI\LQJ)UDPHZRUN7KH&RQFHSW2I$'\QDPLFDO6\VWHP
/LQHDU6\VWHPV$*RRG7RRO)RU$SSUR[LPDWLRQ
6\VWHP7\SHV
0DWKHPDWLFDO 3UHOLPLQDULHV
&RPSOH[1XPEHUV$QG&RPSOH[$ULWKPHWLF
5HVXOWV)URP7ULJRQRPHWU\
5HVXOWV)URP&DOFXOXV
0DWKHPDWLFDO 0RGHOV )RU &RQWLQXRXV 7LPH 6LJQDOV
$ 0RGHO IRU 7DOO 7KLQ 3XOVHOLNH 6LJQDOV 'LUDF 'HOWD 'LVWULEXWLRQ
$0RGHOIRU6WHSOLNH6LJQDOV8QLW6WHS)XQFWLRQ
([SRQHQWLDO6LJQDOV
&RPSOH[ ([SRQHQWLDO 6LJQDOV &UHDWLQJ 6LQXVRLGV DQG ([SRQHQWLDO 6LQXVRLGV
$Q ,QWURGXFWLRQ 7R /DSODFH 7UDQVIRUPV
7KH8QLODWHUDO/DSODFH7UDQVIRUP$Q,QWURGXFWLRQ
(OHPHQWDU\8QLODWHUDO/DSODFH7UDQVIRUPV
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
1DWXUDO 0RGHV ([SRQHQWLDO 6WDELOLW\ (LJHQYDOXHV (LJHQYHFWRUV 0DWUL[ ([SRQHQ
WLDO$QG0RGDO(TXDWLRQV
0RGDO$QDO\VLV
7UDQVPLVVLRQ=HURV$QG(QHUJ\$EVRUSWLRQ
3URSDJDWLRQ 2I 6LJQDO 8QFHUWDLQW\ 7KURXJK $Q /7, 6\VWHP &RYDULDQFH $QDO
\VLV
$Q ,QWURGXFWLRQ 7R )HHGEDFN &RQWURO 6\VWHPV 7KH %LJ 3LFWXUH
0RWLYDWLRQ
6WDQGDUG1HJDWLYH)HHGEDFN6\VWHP
7KH%LJ3LFWXUH$6PDOO7UDFNLQJ(UURU
/7,&ORVHG/RRS&DOFXODWLRQV
6WDELOL]DWLRQ
+LJK)UHTXHQF\'\QDPLFV
,QWHUQDO0RGHO3ULQFLSOH
)HHGIRUZDUG&RPSHQVDWLRQ
6WDELOLW\5REXVWQHVV0DUJLQV
&RQWUROODELOLW\ 7UDQVIHULQJ 7KH 6WDWH 2I $ 6\VWHP
)XOO 6WDWH )HHGEDFN $ 0HWKRG )RU $OWHULQJ 7KH 1DWXUDO 0RGHV 2I $ 6\VWHP
2EVHUYDELOLW\ 5HFRQVWUXFWLQJ 7KH 6WDWH 2I $ 6\VWHP
6WDWH 2EVHUYHUV 6WDWH 5HFRQVWUXFWLRQ 2U (VWLPDWLRQ 9LD )HHGEDFN
)XQGDPHQWDO 6\VWHP ,QYDULDQFH 3URSHUWLHV
0RGHO %DVHG &RPSHQVDWRUV $QG 7KH 6HSDUDWLRQ 3ULQFLSOH
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
5HSUHVHQWLQJ 8QFHUWDLQW\
$ 1HZ 3DUDGLJP )RU &RQWURO 6\VWHP 'HVLJQ
%DFNZDUG(XOHU,QWHJUDWLRQ)RUZDUG'LmHUHQFH0HWKRG
7UDSH]RLGDO,QWHJUDWLRQ7XVWLQ%LOLQHDU0HWKRG
5HIHUHQFHV
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
&KDSWHU
,QWURGXFWLRQ 6\VWHPV $QG 7KH 8VH
2I 0DWKHPDWLFDO 0RGHOV
:LGHVSUHDG $SSOLFDELOLW\ $QG &RPPRQ )HDWXUHV 2I 0DWKH
PDWLFDO 0RGHOV
:LWK WKH DGYHQW RI SRZHUIXO FRPSXWHUV WKH XVH RI PDWKHPDWLFDO PRGHOV E\ PDWKHPDWLFLDQV VFL
HQWLVWV HQJLQHHUV DQG UHVHDUFKHUV KDV EHFRPH FRPPRQSODFH 0DWKHPDWLFDO PRGHOV DUH XVHG IRU
DQDO\VLV GHVLJQ DQG WR SUHGLFW WKH EHKDYLRU RI FRPSOH[
q SK\VLFDO V\VWHPV HJ DLUSODQHV URERWV VXEPDULQHV HFRQRPLF V\VWHPV PXVFXORVNHOHWDO V\V
WHPV KHDW H[FKDQJHUV VDWHOOLWHV HOHFWULFDO QHWZRUNV ZHDWKHU V\VWHPV PLVVLOH WDUJHW LQWHU
FHSWV HWF
q SK\VLFDO VRFLDO HFRQRPLF SROLWLFDO DQG YLUWXDO SURFHVVHV HJ VHPLFRQGXFWRU PDQXIDFWXULQJ
SURFHVVHV FKHPLFDO SURFHVVHV WKHUPDO GLmXVLRQ EUHDWKLQJ FRPEXVWLRQ VLJQDO SURFHVVLQJ
SRSXODWLRQ JURZWK VHFXULWLHV WUDGLQJ VXSSO\ DQG GHPDQG GHFLVLRQ PDNLQJ EDWWOH PDQDJH
PHQW HWF DQG
q SK\VLFDO VRFLDO HFRQRPLF DQG SROLWLFDO SKHQRPHQD HJ HOHFWULFDO PHFKDQLFDO WKHUPDO RS
WLFDO LQoDWLRQ LQWHUHVW UDWHV VWUXFWXUDO YLEUDWLRQ DQG GHIRUPDWLRQ R]RQH OD\HU GHSOHWLRQ
HWF
&RPPRQ )HDWXUHV :KLOH WKH DERYH OLVW LV LQWHQGHG WR LOOXVWUDWH WKH ZLGHVSUHDG DSSOLFDELOLW\ RI
PDWKHPDWLFDO PRGHOV LW LV PRUH LPSRUWDQW WR QRWH WKUHH FRPPRQ IHDWXUHV ZKLFK HDFK HOHPHQW RI
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
X
D
X
D
7 X
D
7 X
IRU DOO VFDODUV D
D
DQG DOO ?LQSXW HOHPHQWV X
X
ZLWKLQ WKH GRPDLQ RI 7 c
/LQHDU V\VWHPV DUH XVHG LQ DOO nHOGV RI VFLHQFH DQG HQJLQHHULQJ HJ URERWLFV VHPLFRQGXFWRU
PDQXIDFWXULQJ PLVVLOH QDYLJDWLRQ JXLGDQFH DQG FRQWURO HFRQRPLFV HWF :KLOH WKH XVH RI VXFK
PRGHOV LV ZLGHVSUHDG LW LV LPSRUWDQW WR QRWH WKDW WKH\ DUH W\SLFDOO\ XVHG DV DSSUR[LPDWLRQV WR
PRUH FRPSOH[ QRQOLQHDU V\VWHPV RU PRGHOV $V VXFK XVHUV RI V\VWHP PRGHOV DQG HVSHFLDOO\ OLQHDU
PRGHOV PXVW NHHS LQ PLQG WKH IROORZLQJ FRPSHOOLQJ WUXWK ?PDWKHPDWLFDO PRGHOV KDYH OLPLWDWLRQV
&RPSOH[ 1XPEHUV $QG &RPSOH[ $ULWKPHWLF
&RPSOH[ QXPEHUV DUH SDUWLFXODUO\ LPSRUWDQW LQ WKH VWXG\ RI OLQHDU G\QDPLFDO V\VWHPV 7KHLU
PDLQ XWLOLW\ OLHV LQ WKH IDFW WKDW WKH\ SHUPLW HQJLQHHUV WR UHSUHVHQW VLQXVRLGDO DQG H[SRQHQWLDO
VLQXVRLGDO IXQFWLRQV E\ FRPSOH[ H[SRQHQWLDOV *LYHQ WKLV FRPSOH[ H[SRQHQWLDOV JUHDWO\ IDFLOLWDWH
WKH VWXG\ RI OLQHDU V\VWHPV 0RUHRYHU WKHLU XVH OHDGV QDWXUDOO\ WR QHZ SRZHUIXO V\VWHP FRQFHSWV
HJ IUHTXHQF\ UHVSRQVH RI D V\VWHP HWF $V VXFK FRPSOH[ QXPEHUV DUH LQGLVSHQVDEOH LQ WKH
VWXG\ RI G\QDPLFDO V\VWHPV
0RWLYDWLRQ 7KH 1HHG )RU &RPSOH[ 1XPEHUV %HFDXVH HTXDWLRQV RI WKH IRUP
]
b
ZHUH IRXQG WR EH SUHVHQW HYHU\ZKHUH LQ VFLHQFH DQG HQJLQHHULQJ LW EHFDPH HVVHQWLDO WR LQYHQW FRP
SOH[ QXPEHUV 1HFHVVLW\ LV WKH PRWKHU RI LQYHQWLRQ
'HnQLWLRQ 'HnQLWLRQ &RPSOH[ 1XPEHUV
$ FRPSOH[ QXPEHU ] LV DQ RUGHUHG SDLU RI UHDO QXPEHUV [ DQG \ 7KH FRPSOH[ QXPEHU ] LV GHQRWHG
]
GHI
[ \ 7KH UHDO QXPEHU [ LV FDOOHG WKH UHDO SDUW RI WKH FRPSOH[ QXPEHU ] [ \7KHUHDO
QXPEHU \ LV FDOOHG WKH LPDJLQDU\ SDUW RI WKH FRPSOH[ QXPEHU ] [ \,I[ ZHVD\WKDWWKH
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
FRPSOH[ QXPEHU ] \ LV SXUHO\ LPDJLQDU\ ,I \ ZH VD\ WKDW WKH FRPSOH[ QXPEHU ] [
LV UHDO 7KH VHW RI DOO FRPSOH[ QXPEHUV LV GHQRWHG &
(TXDOLW\ RI &RPSOH[ 1XPEHUV 7ZR FRPSOH[ QXPEHUV ]
[
\
[
\
LV GHQRWHG ]
]
DQG GHnQHG DV IROORZV
]
]
GHI
[
[
\
\
7KDW LV WKH VXP LV REWDLQHG E\ DGGLQJ WKH UHDO SDUWV DQG E\ DGGLQJ WKH LPDJLQDU\ SDUWV WR JHW WKH
QHZ UHDO DQG LPDJLQDU\ SDUWV UHVSHFWLYHO\ 7KLV GHnQLWLRQ LPSOLHV WKDW FRPSOH[ QXPEHUV DGG OLNH
YHFWRUV $Q LQWXLWLYH JHRPHWULF LQWHUSUHWDWLRQ LV WKXV SRVVLEOH
0XOWLSOLFDWLRQ RI &RPSOH[ 1XPEHUV 7KH SURGXFW RI WZR FRPSOH[ QXPEHUV ]
[
\
[
\
:KLOH WKLV GHnQLWLRQ LV HDV\ WR XVH LW LV UDWKHU VWUDQJH DQG DUJXDEO\ XQPRWLYDWHG $Q LQWXLWLYH
JHRPHWULF LQWHUSUHWDWLRQ IRU FRPSOH[ PXOWLSOLFDWLRQ ZLOO EH JLYHQ EHORZ
&RPPXWDWLYH $VVRFLDWLYH DQG 'LVWULEXWLYH 3URSHUWLHV ,W FDQ EH VKRZQ WKDW WKH DGGLWLRQ DQG
PXOWLSOLFDWLRQ RI FRPSOH[ QXPEHUV DUH FRPPXWDWLYH DVVRFLDWLYHDQGGLVWULEXWLYH RSHUDWLRQV 7KDW
LVIRUDQ\FRPSOH[QXPEHUV]
]
DQG]
WKH IROORZLQJ UHODWLRQVKLSV KROG
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
LQ )LJXUH 7KH LGHD RI UHSUHVHQWLQJ FRPSOH[ QXPEHUV E\ SRLQWV YHFWRUV LQ WKH SODQH ZDV
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
IRUPXODWHG E\ *DXVV LQ KLV GLVVHUWDWLRQ DQG LQGHSHQGHQWO\ E\ $UJDQG LQ > SJ @
)LJXUH LV VRPHWLPHV UHIHUUHG WR DV DQ $UJDQG GLDJUDP
&RPSOH[ ]3ODQH
'UDZQ IRU [ \ !
5HDO $[LV
,PDJLQDU\ $[LV
E
E
E
E
EU
] [ M\
q
M
[
b\
)LJXUH 9LVXDOL]LQJ D &RPSOH[ 1XPEHU ] ,Q 7KH &RPSOH[ ]3ODQH
:LWK WKH DERYH QRWDWLRQ HTXDWLRQV DQG EHFRPH
]
]
GHI
>[
,Q RUGHU WR SURSHUO\ GHnQH FRPSOH[ GLYLVLRQ LW LV XVHIXO WR LQWURGXFH QHZ WHUPV
&RQMXJDWH RI D &RPSOH[ 1XPEHU 7KH FRQMXJDWH RI D FRPSOH[ QXPEHU ] [ M\ LV GHQRWHG
] DQG
GHnQHG DV IROORZV
]
GHI
[ b M\
7KH FRQMXJDWH
] RI WKH FRPSOH[ QXPEHU ] [M\ PD\ YLVXDOL]HG DV VKRZQ LI )LJXUH 0RUHRYHU
LW FDQ EH VKRZQ WKDW
] ]
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
&RPSOH[ ]3ODQH
'UDZQ IRU [ \ !
5HDO $[LV
,PDJLQDU\ $[LV
E
E
E
E
EU
M]M
E
EU
E
EL
WR QRWH WKDW
M]M
S
] ]
'LYLVLRQ *LYHQ WKH DERYH WKH GLYLVLRQ RI WZR FRPSOH[ QXPEHUV ]
[
M\
DQG ]
[
M\
]
LV GHnQHG LQ WHUPV RI WKH UDWLR
]
]
]
]
]
[
[
\
\
[
\
w
M
\
[
b [
\
[
\
!
!
WDQ
[!\!
b WDQ
ww
[\!
b WDQ
ww
ww
[\
b WDQ
ww
[!\
(QJLQHHUV RIWHQ XVH WKH ZRUG SKDVH LQVWHDG RI DQJOH )URP )LJXUH LW IROORZV WKDW WKH UHDO DQG
LPDJLQDU\ SDUWV RI D FRPSOH[ QXPEHU ] PD\ EH UHDGLO\ FRPSXWHG IURP WKH PDJQLWXGH DQG DQJOH
RI ] DV IROORZV
5HI]J M]M FRV
%
]
] [ b M\ M]MH
bM
]
q
M]M
M
[ M]M FRV
]
]
b\ M]M VLQ
]
b
]
)LJXUH 9LVXDOL]LQJ 7KH 3RODU )RUP 2I $ &RPSOH[ 1XPEHU
3RODU ([SRQHQWLDO 5HSUHVHQWDWLRQ )URP WKH DERYH GLVFXVVLRQ LW IROORZV WKDW
] M]MH
w
%
ZKHUH
H
w
%
M]
MM]
M H
w
%
%
]
]
M]
M
M]
M
H
w
%
,P
E
E
E
E
E
EU
]
(
x]
E
E
E
E
E
E
] x] 5H]
PRWLYDWH WKH GHnQLWLRQ IRU WKH FRPSOH[ H[SRQHQWLDO H
RI D FRPSOH[ QXPEHU ] [ M\
H
GHI
H
H
w
)URP HTXDWLRQ LW WKHQ IROORZV WKDW
H
H
H
w
H
FRV \ MH
VLQ \
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
)URP WKLV LW IROORZV WKDW WKH PDJQLWXGH RI H
LV JLYHQ E\
MH
M]M
P
Q
H
w
P
Q
%
{x
N Qb
([DPSOH &XEH 5RRW RI
6LQFH WKH QXPEHU b PD\ EH H[SUHVVHG DV H
w
p
H
w
p
DQG H
w
p
LW IROORZV WKDW WKH WKUHH
URRWV ZKLFK VDWVI\ ]
b DUH JLYHQ E\
] H
w
p
}
R
H
wÄ
R
%
Y
J
5HIM;MH
w
%
Y
H
}
R
wÄ
R
J
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
6LPLODUO\
[W M;MH
}
R
VLQ
}
R
wÄ
R
J
:H VD\ WKDW WKH DERYH H[SRQHQWLDO VLQXVRLGV DUH JHQHUDWHG IURP WKH FRPSOH[ H[SRQHQWLDO
] M]MH
w
%
M;MH
}
R
H
wÄ
R
%
Y
M;MH
w
%
Y
H
}
R
wÄ
R
W b
p
FRV
W
p
7KLV VLJQDO LV WKH VXP RI WZR VLQXVRLGV RI WKH VDPH IUHTXHQF\
7KH JRDO LQ WKLV H[DPSOH LV WR
LOOXVWUDWH KRZ FRPSOH[ SKDVRUV PD\ EH XVHG WR DGG WKH VLQXVRLGV WR JHW D VLQJOH FRVLQH VLJQDO RI
IUHTXHQF\
:H SURFHHG DOJHEUDLFDOO\ XVLQJ WKH WULJRQRPHWULF UHODWLRQVKLS
VLQ[ FRV[b
p
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
DQG (XOHUV LGHQWLWLHV DV IROORZV
IW FRV
W b
p
FRV
W
p
H
w
p
H
w
p
L
H
wÄ
R
J
5HI;H
wÄ
R
J
5HIM;MH
wÄ
R
%
Y
J
M;M FRV
W
%
;
ZKHUH WKH FUWLFDO FDOFXODWLRQ OLHV LQ WKH DGGLWLRQ RI FRPSOH[ SKDVRUV DVVRFLDWHG ZLWK HDFK VLQXVRLG
;
x
M;
x
M5HIH
wÄ
R
%
Y
N
J
;
x
5HIM;
x
MH
wÄ
R
%
Y
N
J
J
6LJQDOV 6\VWHPV DQG &RQWUROV $$ 5RGULJXH]
F
o
5HI
;
x
M;
x
MH
w
%
Y
N
H
wÄ
R
J
5HI
;
x
;
x
%
;
ZKHUH
; M;MH
w
%
Y
;
x
;
x
;
x
M;
x
MH
w
%
Y
N
5HVXOWV )URP 7ULJRQRPHWU\
7KH IROORZLQJ UHODWLRQVKLSV DUH XVHIXO LQ WKH VWXG\ RI G\QDPLFDO V\VWHPV
VLQ[ h \ VLQ[ FRV \ h FRV [ VLQ \
FRV[ h \ FRV[ FRV \ i VLQ [ VLQ \
WDQ[ h \
[
>FRV[@
WDQ
[
b FRV [
FRV[
VLQ
[ FRV
[
WDQ
[ VHF
[
FRW
[ FVF
[
5HVXOWV )URP &DOFXOXV
7KH IROORZLQJ UHODWLRQVKLSV DUH XVHIXO LQ WKH VWXG\ RI G\QDPLFDO V\VWHPV
I
G
G[
FRV
[ b
S
b [
G
G[
WDQ
[
[
G
G[
H
h
DH
G
G[
OQ [
d
I
[J[ b I[J
[
e
=
XGY XY b
=
YGX
7KHRUHP 0HDQ 9DOXH 7KHRUHP )RU 'LmHUHQWLDWLRQ
,I Ic LV GLmHUHQWLDEOH RQ WKH RSHQ LQWHUYDO [
[
DQG FRQWLQXRXV RQ WKH FORVHG LQWHUYDO >[
[
@
WKHQ WKHUH H[LVWV ] [
[
VXFK WKDW
I
]
[
b [
=
I~ G~
7KH QXPEHU I] LV VRPHWLPHV UHIHUUHG WR DV WKH PHDQ YDOXH RU DYHUDJH YDOXH RI I RQ WKH LQWHUYDO
>[
[
@
7KHRUHP )XQGDPHQWDO 7KHRUHP 2I &DOFXOXV
6XSSRVH I LV FRQWLQXRXV RQ WKH FORVHG LQWHUYDO >D E@
L ,I
*[
GHI
=
h
I~ G~
IRU DOO [ >D E@WKHQ* LV WKH DQWLGHULYDWLYH RI I RQ >D E@ LH
*
[