Detection theory: applications and digital signal processing

Detection Theory Applications and Digital Signal Processing

Detection Theory Applications and Digital Signal Processing Ralph D. Hippenstiel

CRC PR E S S Boca Raton London New York Washington, D.C.

0434/disclaimer Page 1 Wednesday, October 17, 2001 8:48 AM

Library of Congress Cataloging-in-Publication Data Hippenstiel, Ralph Dieter, 1941Detection theory : applications and digital signal processing / Ralph D. Hippenstiel. p. cm. Includes bibliographical references and index. ISBN 0-8493-0434-2 (alk. paper) 1. Signal processing—Digital techniques. 2. Signal detection—Mathematics. 3. Electronic surveillance. 4. Detectors. 5. Signal theory (Telecommunication)—Mathematics. I. Title. TK5102.9 .H58 2001 621.382¢2—dc21

2001043859

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2002 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-0434-2 Library of Congress Card Number 2001043859 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

!701,.0 7KLV ERRN KDV LWV RULJLQ LQ OHFWXUH QRWHV FUHDWHG DW WKH 1DYDO 3RVWJUDGXDWH 6FKRRO 136  ZKRVH VWXGHQW DQG IDFXOW\ VXSSRUW PDGH WKLV ZRUN SRVVLEOH 7KH ERRN LV ZULWWHQ IRU VHQLRU XQGHUJUDGXDWHV DQG nUVW\HDU JUDGXDWH VWX GHQWV 3UDFWLFLQJ HQJLQHHUV VKRXOG HVSHFLDOO\ nQG WKH ERRN XVHIXO LQ REWDLQLQJ WKH EDVLF NQRZOHGJH RI GHWHFWLRQ DQG HVWLPDWLRQ WKHRU\ 7KH PDLQ SDUW RI WKH WH[W GHDOV ZLWK GHWHFWLRQ DVSHFWV ZKLOH WKH HVWLPDWLRQ SDUW LV SUHVHQWHG DV QHHGHG 0DQ\ PRGHUQ VLJQDO SURFHVVLQJ LGHDV DUH HPEHGGHG LQ WKH DSSURDFK WR GH WHFW HYHQWV 0DQ\ DVSHFWV RI WKH IDVW )RXULHU WUDQVIRUP ))7 DQG WKH ZDYHOHW WUDQVIRUP :7 DUH H[DPLQHG DQG GLVFXVVHG LQ WKH FRQWH[W RI GHWHFWLRQ DQG SDUDPHWHU HVWLPDWLRQ 'LJLWDO VLJQDO SURFHVVLQJ KDV KDG D WUHPHQGRXV LQo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n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nFDWLRQ SUREOHPV %DVLF WRSLFV WKDW DUH LQWURGXFHG DUH 3HULRGRJUDP 6SHFWURJUDP &RUUHODWLRQ ,QVWDQWDQHRXV &RUUHODWLRQ )XQFWLRQ :LJQHU9LOOH 'LVWULEXWLRQ 6SHFWUDO &RUUHODWLRQ $PELJXLW\ )XQFWLRQ &\FOR 6WDWLRQDU\ 3URFHVVLQJ +LJKHU 2UGHU 0RPHQWV DQG 3RO\6SHFWUD &RKHUHQFH 3URFHVVLQJ :DYHOHW 3URFHVVLQJ DQG $GDSWLYH 7HFKQLTXHV $ W\SLFDO FODVV UHYLHZV DVSHFWV RI $SSHQGLFHV $^& DQG &KDSWHU  &KDS WHUV  ^ DQG  DQG 6HFWLRQV  DQG  DUH FRPSOHWHO\ FRYHUHG LQ

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nH WR ZKRP WKH ERRN LV GHGLFDWHG IRU WKHLU HQFRXUDJHPHQW DQG XQGHUVWDQGLQJ 7KH GHGLFDWLRQ RI WKLV ERRN LV PDGH UHDOL]LQJ WKDW ZRUGV DORQH FDQQRW PDNH XS IRU WKH ORVW WLPH LQ WKHLU OLYHV 5DOSK ' +LSSHQVWLHO 0RQWHUH\ &DOLIRUQLD

$XWKRU

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

&RQWHQWV  ,QWURGXFWLRQ  *(1(5$/ 3+,/2623+<          '(7(&7,21 $1' (67,0$7,21 3+,/2623+<                 'HWHFWLRQ                (VWLPDWLRQ               '(6&5,37,21 2) 63$&(6 ,192/9('  6800$5<                 

                  ,1  

                              7+( '(&,6,21           

 5HYLHZ RI 'HWHUPLQLVWLF DQG 5DQGRP 6\VWHP DQG 6LJQDO &RQFHSWV  620( 0$7+(0$7,&$/ $1' 67$7,67,&$/ %$&. *5281'                             6^@ 7KH UHIHUHQFH OLVW LV E\ QR PHDQV H[KDXVWLYH EXW LW LV PHDQW WR SURYLGH DGGLWLRQDO KHOSIXO PDWHULDO

$W WKLV WLPH UHIHUHQFHV >^^@ UHPDLQ LQ

SULQW 7KHVH ERRNV DUH H[FHOOHQW UHIHUHQFHV EXW DUH VRPHWLPHV D OLWWOH GLpFXOW WR UHDG 2XU WH[W DWWHPSWV WR TXHVWLRQ ZK\ WKLQJV DUH VHW XS LQ D SDUWLFXODU ZD\ DQG KRZ WKH UHVXOWV DUH GHULYHG

2XU VWXGHQWV SULPDULO\ PLOLWDU\ RpFHUV

DUH LQWHUHVWHG LQ WKH WRSLFV IURP D SUDFWLFDO SRLQW RI YLHZ EHFDXVH JRRG XQ GHUVWDQGLQJ ZLOO SRWHQWLDOO\ Dm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mHUHQW VFHQDULRV

7KLV DOORZV D JHQHUDO GHVFULSWLRQ DQG VROXWLRQV

:LWK WKLV DSSURDFK WKH H[WHQVLRQV WR RWKHU EDVLV

IXQFWLRQV VXFK DV .DUKXQHQ/Rt HYH LV D QDWXUDO RQH DOORZLQJ D VWUDLJKWIRUZDUG LQWHUSUHWDWLRQ

,QWURGXFWLRQ 



'(7(&7,21 $1' (67,0$7,21 3+,/2623+
F W @

 D FRV >   QW

IRU 

F

UW

›W›7

UDGDU VRQDU SUREOHP

 ' W b ~  t@  QW IRU ~

›W›~

7

ZKHUH D UHSUHVHQWV VLJQDO DWWHQXDWLRQ WKH FRHpFLHQW t LV DQ XQNQRZQ SKDVH VKLIW QW LV WKH DGGLWLYH QRLVH FRPSRQHQW  UHSUHVHQWV WKH XQNQRZQ 'RSSOHU VKLIW DQG ~ LV WKH XQNQRZQ WLPH GHOD\ SURSRUWLRQDO WR WKH URXQG WULS GLVWDQFH WR WKH WDUJHW  '

F

&ODVV ,,, UDQGRP VLJQDOV LQ DGGLWLYH QRLVH 

2QH ZLOO GHFLGH

EDVHG RQ WKH UHFHLYHG GDWD UW  ZKHWKHU RU QRW WKH PHVVDJH RU VLJQDO VZ W LV SUHVHQW RU QRW

([DPSOH 

 V W  QW   QW Z

UW



LI WKH WDUJHW LV SUHVHQW



LI WKH WDUJHW LV QRW SUHVHQW IRU 

›W›7

ZKHUH V W LV D UHDOL]DWLRQ RI D UDQGRP SURFHVV DQG D SUREDELOLVWLF GHVFULS WLRQ RI V W LV DYDLODEOH Z

Z



(VWLPDWLRQ

6RPH W\SLFDO H[DPSOHV DUH D )UHTXHQF\ HVWLPDWLRQ LH GHWHUPLQH WKH DFWXDO 'RSSOHU VKLIW E 'LmHUHQWLDO WLPH GHOD\ HVWLPDWLRQ LH GHWHUPLQH WKH GLVWDQFH RU GLI IHUHQWLDO GLVWDQFH WR D WDUJHW F $PSOLWXGH HVWLPDWLRQ LH WDUJHW VWUHQJWK VL]H GLVWDQFH G 3KDVH DQGRU EDQGZLGWK HVWLPDWLRQ LH WDUJHW LGHQWLnFDWLRQ H 6SHFWUDO HVWLPDWLRQ LH WDUJHW LGHQWLnFDWLRQ I $QDORJ ZDYH IRUP HVWLPDWLRQ LH RULJLQDO QRLVHIUHH UHFRYHU\ RI WKH RULJLQDO ZDYH IRUP XVLQJ RSWLPDO nOWHULQJ

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



)LJXUH 



'HFLVLRQ PRGHO

'(6&5,37,21 2) 63$&(6 ,192/9(' ,1 7+( '(&,6,21

7KH JHQHUDO VLJQDOGDWD oRZ IRU GHWHFWLRQHVWLPDWLRQ LV VKRZQ LQ )LJXUH  7KH JHQHUDO VLJQDO oRZ VKRZV KRZ WKH LQIRUPDWLRQ PRYHV IURP WKH HYHQW VSDFH YLD WKH VLJQDO DQG REVHUYDWLRQ VSDFHV WR WKH GHFLVLRQ VSDFH ZLWK WKH VSDFHV GHQRWHG E\ ( 6 5 DQG ' UHVSHFWLYHO\ 7KH HYHQWV DUH HQFRGHG DV VLJQDOV LH TXDQWLWLHV WKDW SURSDJDWH WKURXJK WKH FKDQQHO  ZKHUH RQH VLJQDO W\SLFDOO\ FRUUHVSRQGV WR RQH SDUWLFXODU HYHQW ,Q VRPH FDVHV WKH QXPEHU RI VLJQDOV - FDQ EH GLmHUHQW IURP WKH QXPEHU RI HYHQWV ,  $Q H[DPSOH RI WKLV W\SH LV JLYHQ E\ WKH IROORZLQJ VFHQDULR $ VLQJOH HYHQW ?WDUJHW SUHVHQW PD\ EH UHSUHVHQWHG E\ VHYHUDO GLm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

( (YHQW 6SDFH

,Q WKH HYHQW VSDFH ( RQH RI WKH , SRVVLEOH HYHQWV FDQ KDSSHQ :H PD\ RU PD\ QRW NQRZ

D SULRUL SUREDELOLWLHV RI WKH HYHQW H

L

LH 3U

IHL J IRU L

 c c c  , b  

7\SLFDO HYHQWV PD\ EH

D 0HVVDJHV DOSKDEHWLFDO QXPHULF VWDWLRQ NHHSLQJ 0RUVH FRGH

,QWURGXFWLRQ



E 3DUDPHWHUV IUHTXHQF\ SKDVH DQJOH REMHFW PRYLQJ RU QRW KHQFH REMHFW VSHHG HWF F 7DUJHWV WDUJHW SUHVHQWWDUJHW DEVHQW IULHQGIRH HWF 1RWH

7KH PHVVDJH PD\ EH LQ D GLVFUHWH RU FRQWLQXRXV IRUP PD\ EH D UDQ

GRP YDULDEOH UDQGRP YHFWRU RU D UHDOL]DWLRQ RI D UDQGRP SURFHVV

6 6LJQDO 6SDFH 7KH HYHQWV DUH FRQYHUWHG LQWR UHSUHVHQWDWLYH VLJQDOV 2I FRXUVH RQH PDSSLQJ PLJKW XVH WKH RULJLQDO GDWD HYHQWV DV WKH VLJQDO GLUHFWO\

([DPSOH  3KDVH 0RGXODWLRQ (YHQWV 

6LJQDOV  F

H

t

F

H

t

 

V W

$ FRV W  t $ FRV W  t 

V W

([DPSOH  (YHQW (QFRGLQJ (YHQWV 

›W›7

6LJQDOV 

$

H

%

H

&

H

 

IRU 

  

V W

VLJQDO UHSUHVHQWLQJ HYHQW $

V W

VLJQDO UHSUHVHQWLQJ HYHQW %

V W

VLJQDO UHSUHVHQWLQJ HYHQW &

   IRU  › W › 7 ([DPSOH  5DGDU6RQDU 3UREOHP )LJXUH  SURYLGHV D W\SLFDO WDUJHW GHWHFWLRQ VFHQDULR 7KH VLJQDO VW RI GXUDWLRQ 7 LV WUDQVPLWWHG $ WDUJHW LV 5 XQLWV LQ UDQJH DZD\ 7KH REVHUYDWLRQ 5 &  ZKHUH & LV WKH UW LV UHFHLYHG E\ WKH UHFHLYHU 7KH GHOD\ WLPH LV W SURSDJDWLRQ YHORFLW\ 1R QRLVH RU 'RSSOHU LV FRQVLGHUHG 5

(YHQWV  H QR WDUJHW H WDUJHW SUHVHQW

6LJQDOV 

 

~ W

V W



~ W

V W

D VW

b W5



IRU 



IRU WU

›W › W5 › W5  7

ZKHUH D LV WKH VLJQDO ORVV FRHpFLHQW ([DPSOH  3DVVLYH 'HWHFWLRQ 1DUURZ EDQG GHWHFWLRQ SDVVLYH GHWHFWLRQ DGGUHVVHV WKH TXHVWLRQ LV D SDUWL FXODU VSHFWUDO FRPSRQHQW DW D FHUWDLQ IUHTXHQF\ SUHVHQW RU QRW "  (YHQWV  H QR WDUJHW H WDUJHW SUHVHQW

6LJQDOV 

 

V W



V W

$ FRVF W  tF  IRU W

› W › W  7

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



)LJXUH 

7\SLFDO WDUJHW GHWHFWLRQ SUREOHP

([DPSOH  &RPPXQLFDWLRQ 6\VWHP

%LQDU\ FRPPXQLFDWLRQ SUREOHP LH IUHTXHQF\ VKLIW NH\LQJ )6. (YHQWV  H H

 

6LJQDOV  V W

$ VLQ W



IRU 

V W

$ VLQ W



IRU 

›W›7 ›W›7

([DPSOH  ,QWUXVLRQ $ODUP LH PRWLRQ GHWHFWRU IRU  › W (YHQWV  H QR LQWUXGHU H LQWUXGHU SUHVHQW

ZKHUH 

' W

6LJQDOV 

 

VW

$ FRV W  t

VW

$ FRV W  ' W  t

LV D WLPH GHSHQGHQW IUHTXHQF\ VKLIW

5 2EVHUYDWLRQ 6SDFH 7KH VLJQDOV DUH REVHUYHG LH REWDLQHG DIWHU KDYLQJ SDVVHG WKURXJK VRPH FKDQQHO LH WKH PHGLXP 

8VXDOO\ WKLV DFFRXQWV IRU WKH DGGLWLYH QRLVH

FRPSRQHQW QW  ZKLFK PD\ EH GXH WR WKH PHGLXP WKH HOHFWURQLFV DQG PD\ DOVR SRWHQWLDOO\ EH GXH WR D MDPPHU LH PDQPDGH QRLVH  ,Q JHQHUDO WKH UHFHLYHG GDWD LV WKH WUDQVPLWWHG VLJQDO SOXV DQ DGGLWLYH QRLVH FRPSRQHQW UW

VW  QW 

,QWURGXFWLRQ

)LJXUH 



7ZR DUUD\ SURFHVVLQJ

([DPSOH  7DUJHW 'HWHFWLRQ (YHQWV 

5HFHLYHG 'DWD 

H QR WDUJHW H WDUJHW SUHVHQW

 

›W›7 b W5  QW  IRU

U W

QW  IRU 

U W

D VW

W5

›W›7

 W5

ZKHUH D LV WKH VLJQDO ORVV FRHpFLHQW LH QR 'RSSOHU VKLIW DVVXPHG LQ WKLV SDUWLFXODU H[DPSOH DQG W LV WKH SURSDJDWLRQ GHOD\ ZKLFK DFFRXQWV IRU WKH URXQG WULS WR DQG IURP WKH WDUJHW ([DPSOH  7ZR $UUD\ 3URFHVVLQJ 7ZR DUUD\V DUH UHFHLYLQJ GDWD IURP D SDUWLFXODU JHRJUDSKLFDO DUHD 7KH WDUJHW PD\ RU PD\ QRW EH SUHVHQW DQG PD\ EH RU QRW EH REVHUYHG DW HLWKHU RU ERWK ORFDWLRQV 'HWDLOV FDQ EH VHHQ LQ )LJXUH  'HQRWH E\ U W WKH GDWD UHFHLYHG DW DUUD\ L JLYHQ +  WKH M WK K\SRWKHVLV LV WUXH 5

LM

M

(YHQWV  H QR WDUJHW H WDUJHW SUHVHQW

5HFHLYHG 'DWD 

 

U W

Q W  IRU 

U W

Q W  IRU 

U W U W

›W›7 ›W›7

D VW b W  Q W  IRU  D VW b W  Q W  IRU 

›W›7 ›W›7

ZKHUH D LV WKH ORVV FRHpFLHQW IRU WKH VLJQDO FRPSRQHQW DW WKH LWK DUUD\ DVVXPLQJ D ]HUR 'RSSOHU VKLIW DQG W UHSUHVHQWV WKH GHOD\ WR UHDFK WKH LWK DUUD\ IRU L   DQG 7 LV WKH GXUDWLRQ RI WKH VQDSVKRW XVHG LQ WKH GHFLVLRQ L

L

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



([DPSOH  %LQDU\ )6. 3UREOHP

7KLV H[DPSOH UHSUHVHQWV DQ )6. VLJQDO RYHU D IDGLQJ FKDQQHO PXOWLSOLFDWLYH IDGLQJ LQ DGGLWLYH QRLVH 7KH PXOWLSOLHU PW UHSUHVHQWV WKH PXOWLSOLFDWLYH IDGLQJ WHUP DQG QW LV WKH DGGLWLYH REVHUYDWLRQ QRLVH (YHQWV 



H

5HFHLYHG 'DWD  U W

;

PQ W FRV > W

b ~Q W @  QW

Q



H

U W

;

PQ W FRV > W

b ~Q W @  QW

Q

ZKHUH P W LV WKH WLPHYDU\LQJ DWWHQXDWLRQ IDFWRU DVVRFLDWHG ZLWK WKH QWK SURSDJDWLRQ SDWK DQG ~ W LV WKH FRUUHVSRQGLQJ SURSDJDWLRQ GHOD\ ,Q WKLV VLPSOH PRGHO LW LV DVVXPHG WKDW QR LQWHUV\PERO LQWHUIHUHQFH ,6, LV SUHVHQW Q

Q

' 'HFLVLRQ 6SDFH 7KH GLPHQVLRQ RI ' LV XVXDOO\ WKH VDPH DV WKH GLPHQVLRQ RI ( EXW QRW DOZD\V ,I GLP'

GLP(  WKHQ ZH WU\ LQ D EHVW WKDW LV LQ DQ RSWLPDO IDVKLRQ

WR HVWLPDWH WKH RULJLQDO PHVVDJH

$Q H[DPSOH IRU GLP' WR EH OHVV WKDQ

GLP( LV WKH HVWLPDWLRQ RI WDUJHW SDUDPHWHUV OHDGLQJ WR WKH GHFLVLRQ \HV RU QR EDVHG RQ PXOWLSOH SLHFHV RI LQIRUPDWLRQ VXFK DV PXOWLSOH VSHFWUDO OLQHV LQ WKH SDVVLYH 621$5 GHWHFWLRQ SUREOHP 2XU WH[W ZLOO IRFXV RQ ?KRZ WR REWDLQ D PDSSLQJ IURP 5 WR ' 7KLV UXOH PDSSLQJ PXVW EH VXFK WKDW D $Q XQDPELJXRXV GHFLVLRQ LV PDGH E (YHU\ RXWFRPH PXVW OHDG WR VRPH GHFLVLRQ

([DPSOH  0XOWLSOH 'HFLVLRQ VHH )LJXUH  GL

 FKRRVH +  WKH K\SRWKHVLV WKDW WKH HYHQW H RFFXUUHG ZKLFK PHDQV WKDW WKH REVHUYDWLRQ PXVW OLH LQ WKH VSDFH UHJLRQ 5  IRU L     L

L

L



6800$5
@ :KDOHQ $ 3UHVV 

>@ 9DQ 7UHHV +/

2UODQGR )/ $FDGHPLF

'HWHFWLRQ (VWLPDWLRQ DQG 0RGXODWLRQ 7KHRU\ 3DUW

 1HZ @ .D]DNRV ' DQG 3DSDQWRQL.D]DNRV 3 1HZ @ 3RRU 9+

'HWHFWLRQ DQG (VWLPDWLRQ

$Q ,QWURGXFWLRQ WR 6LJQDO 'HWHFWLRQ DQG (VWLPDWLRQ 1HZ

@ +HOVWURP &:

(OHPHQWV RI 6LJQDO 'HWHFWLRQ DQG (VWLPDWLRQ (QJOH

ZRRG &OLmV 1- 3UHQWLFH+DOO  >@ %DUNDW 0 +RXVH 

6LJQDO 'HWHFWLRQ DQG (VWLPDWLRQ

>@ :HEHU &/

1RUZRRG 0$ $UWHFK

(OHPHQWV RI 'HWHFWLRQ DQG 6LJQDO 'HVLJQ

0F*UDZ+LOO %RRN &R  DQG 6SULQJHU9HUODJ 

1HZ @ 0HOVD 7/ DQG &RKQ '/ @ 6ULQDWK 0' 5DMDVHNDUDQ 3. DQG 9LVZDQDWKDQ 5 3UHQWLFH+DOO 

&KDSWHU 

5HYLHZ RI 'HWHUPLQLVWLF DQG 5DQGRP 6\VWHP DQG 6LJQDO &RQFHSWV 

620( 0$7+(0$7,&$/ $1' 67$7,67,&$/ %$&.*5281'

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

VQ  QQ

 ZKHUH Q GHQRWHV GLVFUHWH WLPH DW LQWHUYDOV Q7 DQG 7 LV WKH VDPSOLQJ LQWHUYDO 7KH 7 GHSHQGHQF\ LV XVXDOO\ VXSSUHVVHG 

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



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nQLWH QXPEHU RI FKRLFHV LV DYDLODEOH EXW LQ SUDFWLFH RQO\ D IHZ VHWV DUH XVHG $ ZHOONQRZQ H[SDQVLRQ XVHV DV EDVLV VHW WKH RUWKRJRQDO GLUHFWLRQ FRP SRQHQWV RI WKH VLJQDO +HUH ZH FKRRVH DV WKH SULPDU\ EDVLV IXQFWLRQ RQH RI WKH VLJQDOV $IWHU WKH QRUPDOL]DWLRQ RI WKLV FKRVHQ EDVLV IXQFWLRQ ZH VHOHFW DV D VHFRQG EDVLV IXQFWLRQ WKH RUWKRJRQDO FRPSRQHQW RI D VHFRQG VLJQDO $I WHU QRUPDOL]LQJ WKH VHFRQG EDVLV IXQFWLRQ ZH UHSHDW WKH SURFHVV DV RIWHQ DV QHFHVVDU\ WR VSDQ WKH WRWDO VLJQDO VSDFH +HQFH VHTXHQWLDOO\ ZH GHVLJQ HYHU\ EDVLV IXQFWLRQ LQ DQ RUWKRQRUPDO IDVKLRQ UHODWLYH WR DOO SUHYLRXVO\ VHOHFWHG EDVLV IXQFWLRQV LH WKH *UDP6FKPLGW GHFRPSRVLWLRQ VHH &KDSWHU  6HFWLRQ   :H FRXOG XVH DQ\ RUWKRJRQDO GHFRPSRVLWLRQ RI WKH VLJQDO DV D EDVLV IXQF WLRQ $FWXDOO\ WKH PRVW FRPPRQO\ XVHG RQH LV WKH )RXULHU W\SH GHFRPSR VLWLRQ %XW ZH UHDOL]H WKDW D KRVW RI RWKHU GHFRPSRVLWLRQV LH +HUPLWH /HJHQGUH &KHEE\FKHm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tHYH H[ SDQVLRQ  ,Q WKLV H[SDQVLRQ ZH DVVXPH WKDW ZH KDYH FRUUHFW NQRZOHGJH DERXW WKH QRLVH FRYDULDQFH IXQFWLRQ (DFK RQH RI WKH DSSURDFKHV GLVFXVVHG VR IDU KDV LWV RZQ FKDUDFWHULVWLFV PHULWV DQG GLVDGYDQWDJHV ZKLFK ZH WU\ WR SRLQW RXW DV ZH JR DORQJ c

c

5HYLHZ RI 6\VWHPV 6LJQDOV



6LQFH ZH ZRUN ZLWK SURMHFWLRQ RQWR FHUWDLQ EDVLV IXQFWLRQV ZH QHHG WR HVWDEOLVK WKH FKDUDFWHULVWLFV RI WKH JHQHUDO YHFWRU VSDFH +LOEHUW VSDFH WKDW ZH DUH XVLQJ 7KH YHFWRU VSDFH PXVW KDYH DOO WKH SURSHUWLHV DVVLJQHG WR D OLQHDU YHFWRU VSDFH :H ZLOO LQGLFDWH YHFWRU RU PDWUL[ TXDQWLWLHV E\ UHVRUWLQJ WR EROG IDFHG OHWWHUV 'HnQLWLRQ  9HFWRU VSDFH RYHU WKH nHOG RI UHDO RU FRPSOH[ QXPEHUV :H QHHG D VHW . RI REMHFWV FDOOHG YHFWRUV WRJHWKHU ZLWK D YHFWRU DGGLWLRQ RQ . DQG D VFDODU PXOWLSOLFDWLRQ RI YHFWRUV E\ QXPEHUV ZLWK SURSHUWLHV

9HFWRU $GGLWLRQ

 9HFWRU DGGLWLRQ LV FRPPXWDWLYH DQG DVVRFLDWLYH  7KHUH LV D XQLTXH YHFWRU  ]HUR LQ . VXFK WKDW   [ [ IRU DOO [ LQ .  7KHUH LV D XQLTXH YHFWRU [ LQ . VXFK WKDW [   [  b

b

6FDODU 0XOWLSOLFDWLRQ

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nHOG YHFWRU VSDFH DOJHEUD FRPSOHWH QRUPHG VSDFH LH %DQDFK VSDFH FRPSOHWH LQQHU SURGXFW VSDFH UHDO RU FRPSOH[ YHFWRU VSDFH ZLWK LQQHU SURGXFW LH +LOEHUW VSDFH  7KDW LV DV RQH LPSRVHV PRUH DQG PRUH VWUXFWXUH RQH PRYHV IURP WKH VSHF LnFDWLRQ RI D JURXS WR WKRVH RI D ULQJ WR WKRVH RI D nHOG DQG VR RQ 6RPH DGGLWLRQDO LQIRUPDWLRQ FDQ EH IRXQG LQ $SSHQGL[ & DQG LQ UHIHUHQFHV >@ $ VXPPDU\ RI PDQ\ RI WKH SUREDELOLW\ SURSHUWLHV LV SUHVHQWHG LQ $SSHQGL[ $ 0RVW RI RXU ZRUN GHDOV ZLWK VWDWLVWLFDO PRPHQWV 7KH nUVW PRPHQW LV WKH '& FRPSRQHQW ZKLOH WKH VHFRQG PRPHQW GHVFULEHV WKH WRWDO SRZHU RI WKH UDQ GRP YDULDEOHV LQ TXHVWLRQ 7KH YDULDQFH RI WKH UDQGRP YDULDEOH ; LV JLYHQ 













'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

E\ WKH H[SUHVVLRQ YDULDQFH ( ; P;  ( ;  P;  ,W GHVFULEHV WKH VSUHDG RI WKH GHQVLW\ IXQFWLRQ RI WKH UDQGRP YDULDEOH DERXW LWV PHDQ 7KHVH FRQFHSWV DUH H[WHQGHG WR SURYLGH DXWRFRUUHODWLRQ DQG FURVVFRUUHODWLRQ IXQF WLRQV RI UDQGRP SURFHVVHV RU UDQGRP VHTXHQFHV  0DQ\ RI WKH SURSHUWLHV RI WKH FRUUHODWLRQ IXQFWLRQV DQG WKHLU )RXULHU WUDQVIRUPV LH WKH SRZHU VSHF WUDO GHQVLW\ 36' DUH JLYHQ LQ $SSHQGL[ $ &KDSWHU  6HFWLRQ  H[DPLQHV SRZHU VSHFWUDO GHQVLW\ HVWLPDWLRQ ZLOO SURYLGH VRPH PRUH LQVLJKW b

b



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

5HYLHZ RI 6\VWHPV 6LJQDOV



WKUHVKROG UHSUHVHQWV WKH SUREDELOLW\ RI GHWHFWLRQ 3'  7KH VWDWLVWLFDO OLWHUDWXUH UHIHUV WR 3' DV WKH SRZHU RI WKH WHVW :KHQ QR WDUJHW LV SUHVHQW WKDW LV WKH QRLVH RQO\ FRQGLWLRQ LV WUXH ZH VD\ WKDW WKH VRFDOOHG + K\SRWKHVLV LV WUXH  WKHQ WKH DUHD WR WKH ULJKW RI D SDUWLFXODU WKUHVKROG RI WKH GHQVLW\ IXQFWLRQ UHSUHVHQWV WKH SUREDELOLW\ RI IDOVH DODUP 3)$  DOVR FDOOHG WKH VL]H RI WKH WHVW  2I FRXUVH RQH FDQ DOVR FRPSXWH WKH DUHD WR WKH OHIW RI D JLYHQ WKUHVKROG 7  ZKLFK LV MXVW WKH RQHVFRPSOHPHQW RI WKH GHQVLW\ DUHD FRPSXWHG IRU WKH DUHD WR WKH ULJKW RI WKH WKUHVKROG 7  7KLV SUREDELOLW\ LV DOVR RI LQWHUHVW ZKHQ GHVFULELQJ WKH GHWHFWRU SHUIRUPDQFH ,Q WKH DSSURSULDWH VHFWLRQ ZH ZLOO DGGUHVV WKHVH TXDQWLWLHV :H ZLOO DOVR EULHo\ GLVFXVV WKH TXDQWLWLHV WKH SRZHU DQG VL]H RI WKH WHVW IRU VRPH RI WKH FRPPRQO\ RFFXUULQJ GHQVLWLHV 

*DXVVLDQ 'HQVLW\

*DXVVLDQ UDQGRP YDULDEOHV DUH FORVHG XQGHU VFDODU PXOWLSOLFDWLRQ DQG W\SLFDO OLQHDU WUDQVIRUPDWLRQV 7KHVH SURSHUWLHV DUH HDVLO\ YHULnHG XVLQJ WKH WUDQVIRU PDWLRQ RI UDQGRP YDULDEOHV RU E\ XVLQJ WKH FKDUDFWHULVWLF IXQFWLRQ 7KLV LP SOLHV WKDW nOWHULQJ RSHUDWLRQV FRQYROXWLRQ LQWHJUDWLRQ GLmHUHQWLDWLRQ DV ZHOO DV WKH FRPPRQ WUDQVIRUPV LH +LOEHUW )RXULHU /DSODFH DQG =WUDQVIRUP WR PHQWLRQ D IHZ UHWDLQ WKH FKDUDFWHU RI *DXVVLDQ UDQGRP YDULDEOHV WKDW LV WKH WUDQVIRUPHG UDQGRP YDULDEOHV VWD\ *DXVVLDQ ,I GDWD LV nOWHUHG DQG WKH HmHFWLYH LQWHJUDWLRQ WLPH RI WKH nOWHU LV UHODWLYHO\ ORQJ FRPSDUHG WR WKH FRUUHODWLRQ WLPH RI WKH QRLVH WKH 3') RI WKH RXWSXW FRPSDUHG WR WKH LQSXW YDULDEOHV WHQGV WR EH FORVHU WR D *DXVVLDQ 3') 7KLV LV D FRQVHTXHQFH RI WKH FHQWUDO OLPLW WKHRUHP RU RQH RI LWV PRUH JHQHUDO YHUVLRQV LQ WKDW WKH nOWHULQJ RSHUDWLRQ FRQVWLWXWHV D VXPPDWLRQ RI PDQ\ UDQGRP YDULDEOHV ZKHQ LQWHUSUHWLQJ WKH LQWHJUDWLRQ XVLQJ 5LHPDQQ VXPV 7KH GHQVLW\ IXQFWLRQ IRU D VLQJOH *DXVVLDQ QRUPDO UDQGRP YDULDEOH [ 1 P;  }  LV JLYHQ E\ )RU UHDO [ >@  H[S [ P;  I; [  } {} )RU FRPSOH[ [ >@  H[S [ P;  I; [  {} } 7KH DUHD WR WKH ULJKW RI D WKUHVKROG 7 IRU D ]HUR PHDQ UHDO YDOXHG QRUPDOL]HG *DXVVLDQ UDQGRP YDULDEOH LV t u =   47 H[S [ G[  { 7 6RPH WH[WERRNV IRU H[DPSOH VHH >@ UHIHU WR WKH LQWHJUDO H[SUHVVHG LQ  DV WKH FRPSOHPHQWDU\ HUURU IXQFWLRQ LH HUIF7  )LJXUH  VKRZV D W\SLFDO *DXVVLDQ 3') DQG WKH DUHD UHIHUUHG WR LQ   z

S

b

S

b

b

M

b

b

M



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH  *DXVVLDQ GHQVLW\ IXQFWLRQ 7KH IRXUWK RUGHU PRPHQW RI WKH ]HUR PHDQ UHDO YDOXHG *DXVVLDQ ZKLFK LV YHU\ XVHIXO LQ FRPSXWLQJ WKH YDULDELOLW\ RI WKH VHFRQG RUGHU PRPHQW RI *DXVVLDQ EDVHG YDULDEOHV FDQ HDVLO\ EH FRPSXWHG DQG LV RI WKH IRUP ([ [ [ [ ([ [ ([ [  ([ [ ([ [  ([ [ ([ [      (YHQ LI [ [ [ [ [ WKHQ ([ ([ ([  ([ ([     ([ ([ };  :H QRWH LI OLQHDU RSHUDWLRQV DUH SHUIRUPHG RQ *DXVVLDQ UDQGRP YDULDEOHV WKH UHVXOWDQW VWD\V *DXVVLDQ 7KLV UHVXOW LV HVSHFLDOO\ XVHIXO ZKHQ GHDOLQJ ZLWK GLVFUHWH WLPH V\VWHPV VXFK DV WKRVH XVHG ZKHQ ZRUNLQJ ZLWK GLJLWDO SURFHVVLQJ LPSOHPHQWDWLRQV $ W\SLFDO H[DPSOH LV WKH DYHUDJLQJ SURFHGXUH 6XSSRVH ZH WDNH D ER[FDU DYHUDJHU WKDW LV D GHYLFH RU DOJRULWKP WKDW VXPV XS 1 FRQWLJXRXV VDPSOHV RI [ DV VKRZQ LQ )LJXUH  DQG XQLIRUPO\ ZHLJKWV WKH GDWD DV GHVFULEHG E\ \

b 1; L



[L

7KLV H[SUHVVLRQ LV VRPHWLPHV QRUPDOL]HG E\ WKH QXPEHU RI WHUPV LQYROYHG DQG LV WKHQ FDOOHG WKH VDPSOH PHDQ :H NQRZ WKDW LI WKH LQSXW VDPSOHV DUH

5HYLHZ RI 6\VWHPV 6LJQDOV



)LJXUH  %R[FDU DYHUDJHU *DXVVLDQ WKH RXWSXW \ ZLOO EH *DXVVLDQ WRR +HQFH DOO WKDW QHHGV WR EH HVWDEOLVKHG LV WKH QHZ PHDQ RI \ DQG LWV YDULDQFH 6XSSRVH WKH LQSXW [L DUH LLG *DXVVLDQ UDQGRP YDULDEOHV ZLWK PHDQ P; DQG YDULDQFH }; WKHQ WKH RXWSXW \ LV *DXVVLDQ ZLWK PHDQ (\

(

b 1; L

b 1; L

ZLWK VHFRQG PRPHQW (\ 

(







M



L

P;

b b 1; 1; L

b 1;

[L

([L

1P;

b b 1; 1;

[L [M

1  1P; 



L

b 1;

b

L





M



([L [M

([L [L

1  1P;  1 };  P;  1  P;  1}; b

DQG YDULDQFH +HQFH \

z

  1 1P;  1};



5D\OHLJK 'HQVLW\

}@ /HW WKH YDULDEOH & EH D FKLVTXDUHG UDQGRP YDULDEOH ZLWK  GHJUHHV RI IUHHGRP +HQFH z

ccc

F

[  [



 H[S F 8 F  } }  ZKHUH 8 F LV WKH XQLW VWHS IXQFWLRQ 1RWH WKLV LV DOVR FDOOHG WKH H[SRQHQWLDO SUREDELOLW\ GHQVLW\ IXQFWLRQ 7KH DUHD WR WKH ULJKW RI D WKUHVKROG 7 LV u t =   H[S F GF H[S 7  } } 7 }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

b

b

b



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH  6SHFWUXP DQDO\]HU LQ )LJXUH  7KLV YHUVLRQ RI VSHFWUDO HVWLPDWLRQ LV FDOOHG WKH SHULRGRJUDP ZKLFK GXH WR LWV UREXVWQHVV HDVH RI LQWHUSUHWDWLRQ DQG HpFLHQF\ LV XVHG PRVW RI WKH WLPH ZKHQ GHDOLQJ ZLWK WKH GHWHFWLRQ RI VSHFWUDO FRPSRQHQWV RU WKH HVWLPDWLRQ RI WKH SRZHU VSHFWUDO GHQVLW\ 7R REWDLQ D SURSHUO\ VFDOHG SRZHU VSHFWUDO GHQVLW\ WKH YDULDEOH F VKRXOG EH QRUPDOL]HG E\ WKH QXPEHU RI WHUPV LQ WKH ))7 ,Q PRGHUQ SURFHVVRUV WKLV LV DFFRPSOLVKHG E\ XVLQJ WKH ))7 VLPSO\ EH FDXVH LW LV VR IDVW UHOLDEOH DQG DFFXUDWH 2QFH WKH GDWD KDV EHHQ SURSHUO\ VDPSOHG LH WKH $' FRQYHUVLRQ REH\V WKH 1\TXLVW FULWHULRQ  LW LV VLPSO\ SURFHVVHG DV JLYHQ E\ P; 1 b

) ) 7 I[Q J

Q P

[Q HbM {NQ 1

ZKHUH 1 LV WKH WUDQVIRUP OHQJWK DOVR FDOOHG WKH FRKHUHQW LQWHJUDWLRQ WLPH  N FRUUHVSRQGV WR WKH IUHTXHQF\ LH I N IVDPSOLQJ 1  Q LV WKH WLPH LQGH[ W cW Q  cW LV WKH VDPSOH VSDFLQJ LQ VHFRQGV P LV WKH VWDUW WLPH LQ VDPSOHV DQG IVDPSOLQJ  cW ,I ZH GHQRWH WKH RXWSXW RI WKH ))7 E\ ; N  WKHQ WKH VSHFWUDO HVWLPDWH 3; N LV GHQRWHG E\ 3; N



1

n n 1 b n  nnnP; {QN 1 n b M [Q H n

; N M



M

1n Q P

n

7KLV HVWLPDWH RI WKH VSHFWUDO GHQVLW\ LV FDOOHG WKH SHULRGRJUDP >@ DV HDUOLHU PHQWLRQHG ,I QR VSHFLDO GDWD ZHLJKWLQJ LV XVHG ZH FDQ HDVLO\ VKRZ WKDW WKH PHDQ RI WKH HVWLPDWH HTXDOV WR RQH VWDQGDUG GHYLDWLRQ RI WKH HVWLPDWH ZKHQ RQO\ ZKLWH *DXVVLDQ QRLVH LV SUHVHQW LH WKH + K\SRWKHVLV LV WUXH  7KLV LV WKH $FKLOOHV KHHO RI WKH SHULRGRJUDP QDPHO\ WKH YDULDQFH LV UHODWLYHO\ ODUJH DQG GRHV QRW GHFUHDVH DV WKH LQWHJUDWLRQ WLPH 1 LQFUHDVHV $ W\SLFDO H[SRQHQWLDO GHQVLW\ XVLQJ D } RI  LV VKRZQ LQ )LJXUH  ,Q JHQHUDO LI F LV WKH VXP RI 1 LLG *DXVVLDQ VTXDUHG UDQGRP YDULDEOHV F

1 ; L



[L



[L

DUH LLG



5HYLHZ RI 6\VWHPV 6LJQDOV



)LJXUH  ([SRQHQWLDO 3') ZKHUH WKH [L

7KH 3') WKHQ EHFRPHV r F s  I& F F1  b H[S   1 1 } 8 F  } b1  6RPH W\SLFDO PHPEHUV RI WKLV 3') XVLQJ D } RI  DUH VKRZQ LQ )LJXUH  :H QRWH WKDW WKLV FRUUHVSRQGV WR LQFRKHUHQW DOVR FDOOHG SRZHU DYHUDJ LQJ RI SHULRGRJUDPV 7\SLFDOO\ RQH VHHNV WR LPSURYH WKH YDULDQFH RI WKH EDVLF SHULRGRJUDP E\ DYHUDJLQJ VHTXHQWLDO VSHFWUDO RXWSXWV 7KHVH RXWSXWV DUH VHTXHQWLDO LQ WLPH DQG IRU WKH SXUSRVH RI RXU GLVFXVVLRQ DVVXPHG WR EH LQGHSHQGHQW $V\PSWRWLFDOO\ IRU ODUJH QXPEHUV RI DYHUDJHV RQH H[SHFWV DERXW D  G% GHFLEHO LPSURYHPHQW SHU GRXEOLQJ RI WKH QXPEHU RI WHUPV XVHG LQ WKH DYHUDJLQJ SURFHGXUH 7KH EDVLF nOWHU EDQGZLGWK RI WKH VSHFWUXP DQDO\]HU ))7 UHPDLQV FRQVWDQW URXJKO\ IVDPSOLQJ 1 SURYLGLQJ VRPH UR EXVWQHVV 2Q WKH RWKHU KDQG XVLQJ FRKHUHQW DYHUDJLQJ ZRXOG KDYH UHVXOWHG LQ D  G% JDLQ SHU GRXEOLQJ RI WKH QXPEHU RI WHUPV LQ WKH ))7 RSHUDWLRQ EXW WKHQ WKH nOWHU EDQGZLGWK VKULQNV IVDPSOLQJ 1 ZLWK WKH LQFUHDVLQJ 1  'HSHQGLQJ RQ ZKDW WKH VLJQDO RI LQWHUHVW GRHV LW PD\ EH ORVW E\ EHLQJ RYHU UHVROYHG WKDW LV WKH VLJQDO VSLOOV LQWR PDQ\ DGMDFHQW VSHFWUDO ELQV 2QH QHHGV WR FDUHIXOO\ HYDOXDWH WKH WUDGHRmV EHWZHHQ YDULDQFH UHGXFWLRQ EDQGZLGWK z

1  }  

b



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH  &KLVTXDUHG 3') ZLWK  GHJUHHV RI IUHHGRP SURFHVVLQJ JDLQ DQG RYHUUHVROYLQJ WR H[WUDFW VSHFWUDO LQIRUPDWLRQ 7KH DUHD WR WKH ULJKW RI D WKUHVKROG 7 LV JLYHQ E\ =  r F s  F1  b H[S  1 1 } GF  , X S 7  } b1  b

b

ZKHUH , X S LV 3HDUVRQ V LQFRPSOHWH o IXQFWLRQ >@ 

b1 

= 7 



HbZ Z1

b

 

r , 7 1 b   1

GZ

DQG S X

1



}

b



7 S   



s

  b



5HYLHZ RI 6\VWHPV 6LJQDOV 



5LFLDQ 'HQVLW\ 1RQ&HQWUDO 5D\OHLJK

7KH 5LFLDQ GHQVLW\ LV D PRGLnHG 5D\OHLJK GHQVLW\ ZKLFK LV REWDLQHG DV WKH HQYHORSH RI WZR LQGHSHQGHQW *DXVVLDQ UDQGRP YDULDEOHV WKDW KDYH RQH RU WZR VLJQDO UHODWHG FRPSRQHQWV LQ RQH RU ERWK YDULDEOHV UHVSHFWLYHO\ 7KLV FRUUHVSRQGV WR WKH GHQVLW\ RI WKH RXWSXW RI DQ HQYHORSH GHWHFWRU ZKLFK LV IHG E\ WKH VLJQDO FRPSRQHQWV HPEHGGHG LQ *DXVVLDQ EDQGSDVV QRLVH 62 5LFH DIWHU ZKLFK WKH GHQVLW\ LV QDPHG LV FUHGLWHG ZLWK WKH RULJLQDO ZRUN >@ 7KH GHWHFWRU IRUP LV WKH VDPH DV LQ )LJXUH  7KH GHQVLW\ LV H[SUHVVHG DV  HbX P } X , r XP s I X  8

IRU SRVLWLYH X ZKHUH DQG



}

S

X

}

[  \

 H[S [ P   \ {}  }  6RPH W\SLFDO GHQVLW\ IXQFWLRQ H[DPSOHV DUH JLYHQ LQ )LJXUH  7KH DUHD WR WKH ULJKW RI D JLYHQ WKUHVKROG 7 LV u t =   HbXP } X , r XP s GX 4 P  7 3 I;< [ \

'

b



7 } 

b

}



} }

ZKHUH 4  LV WKH 0DUFXP 4IXQFWLRQ ZKLFK LV WDEXODWHG LQ >@ 

1RQ&HQWUDO &KL6TXDUHG 'HQVLW\

7KH PHFKDQLVP WR FUHDWH WKLV GHQVLW\ LV WKH VDPH DV IRU WKH UHJXODU FKL VTXDUHG GHQVLW\ 7KH GLmHUHQFH FRPHV IURP WKH HPEHGGHG VLJQDO FRPSRQHQWV WKDW VKLIW WKH FHQWUDO GHQVLW\ WR D QRQFHQWUDO RQH )RU D EORFN GLDJUDP GHVFULSWLRQ FRQVXOW )LJXUH F RU G UHPRYLQJ WKH VTXDUH URRW RSHUDWLRQ  7KH GHQVLW\ LV FUHDWHG E\ WKH IROORZLQJ RSHUDWLRQ ZKHUH WKH [L DUH LLG 1  }  z

Y I9 Y

1 ; L





$  [L 

}

r Y s1 b

w



H[S

t b

u wY }  ,1

b

 

›

w

Yw    }

IRU Y  ZLWK w $ 1  $ W\SLFDO VHW RI 3')V LV VKRZQ LQ )LJXUH  7KH DUHD WR WKH ULJKW RI D JLYHQ WKUHVKROG 7 LV JLYHQ E\ u › w t =  r Y s1 b    Yw   wY 3' H[S }  ,1  b }  GY 7 } w w

b



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH  5LFLDQ 3') 40

tT u S 1$   7 } 



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

5HYLHZ RI 6\VWHPV 6LJQDOV



)LJXUH  1RQFHQWUDO FKLVTXDUHG 3') 6RPH UHVXOWV DUH DYDLODEOH XVLQJ KLJKHU RUGHU PRPHQWV RU FXPXODQWV &X PXODQWV RI RUGHU WKUHH DQG KLJKHU DUH ]HUR IRU WUXH *DXVVLDQ YDULDWHV DOORZLQJ WKH HVWLPDWLRQ DQG LGHQWLnFDWLRQ RI WKH GHJUHH RI *DXVVLDQLW\ LH KRZ PXFK RI WKH QRLVH LV *DXVVLDQ >@ 5HFHQWO\ VHYHUDO SDSHUV DSSHDULQJ LQ WKH ,((( 6LJQDO 3URFHVVLQJ 7UDQVDFWLRQV GHDO ZLWK WKHVH W\SHV RI SUREOHPV >@ :H QRWH DOVR WKDW WKH W\SHV RI GHQVLWLHV GHVFULEHG HDUOLHU LQ WKLV FKDSWHU DUH W\SLFDO IRU WKRVH REVHUYHG DW WKH RXWSXW RI ))7W\SH SURFHVVRUV ZKLFK DUH XVHG WR REWDLQ VSHFWUDO GHWHFWLRQ LH SRZHU VSHFWUXP RU VSDWLDO GHWHFWLRQ LH LQWHQVLW\ RU SRZHU RXW RI D EHDP IRUPHU DV D IXQFWLRQ RI ORRN GLUHFWLRQ  

6800$5
@ 5LFH 62 ?0DWKHPDWLFDO DQDO\VLV RI UDQGRP QRLVH %HOO 6\VW 7HFK - 9RO  SS ^  >@ 0HQGHO -0 ?7XWRULDO RQ KLJKHURUGHU VWDWLVWLFV VSHFWUD LQ VLJQDO SURFHVVLQJ DQG V\VWHP WKHRU\ 7KHRUHWLFDO UHVXOWV DQG VRPH DSSOLFD WLRQV 3URF ,((( 9RO  1R  SS ^ 0DUFK  >@ 1LNLDV &/ DQG 5DJKXYHHU 05 ?%LVSHFWUXP HVWLPDWLRQ $ GLJLWDO VLJQDO SURFHVVLQJ IUDPHZRUN 3URF ,((( 9RO  1R  SS ^ -XO\  >@ 7KH 0DWKZRUNV ,QF 1DWLFN 0$ >@ 7KHUULHQ &: 'LVFUHWH 5DQGRP 6LJQDOV DQG 6WDWLVWLFDO 6LJQDO 3UR FHVVLQJ (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ .D\ 60 0RGHUQ 6SHFWUDO (VWLPDWLRQ 7KHRU\ DQG $SSOLFDWLRQ (Q JOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ 0DUSOH 6/ 'LJLWDO 6SHFWUDO $QDO\VLV (QJOHZRRG &OLmV 1- 3UHQWLFH +DOO  >@ &KHQ &7 /LQHDU 6\VWHPV 7KHRU\ DQG 'HVLJQ 1HZ @ 0RUWHQVHQ 5( 5DQGRP 6LJQDOV DQG 6\VWHPV 1HZ @ .DLODWK 7 /LQHDU 6\VWHPV (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO 

5HYLHZ RI 6\VWHPV 6LJQDOV



>@ *ROXE *+ DQG 9DQ/RDQ &) 0DWUL[ &RPSXWDWLRQV QG (G %DO WLPRUH DQG /RQGRQ 7KH -RKQV +RSNLQV 8QLYHUVLW\ 3UHVV  >@ 6KDQPXJDQ .6 DQG %UHLSRKO $0 5DQGRP 6LJQDOV 'HWHFWLRQ (VWLPDWLRQ DQG 'DWD $QDO\VLV 1HZ @ 1LNLDV &/ DQG 3HWURSXOX $3 +LJKHU2UGHU 6SHFWUD $QDO\VLV $ 1RQOLQHDU 6LJQDO 3URFHVVLQJ )UDPHZRUN (QJOHZRRG &OLmV 1- 3UHQWLFH +DOO  >@ /DUVRQ +- DQG 6KXEHUW %2 3UREDELOLVWLF 0RGHOV LQ (QJLQHHULQJ 6FLHQFHV 9ROXPH  5DQGRP 9DULDEOHV DQG 6WRFKDVWLF 3URFHVVHV 1HZ @ 3HDUVRQ . 7DEOHV RI WKH ,QFRPSOHWH b)XQFWLRQ /RQGRQ DQG 1HZ @ 0DUFXP -, ?7DEOH RI 4)XQFWLRQV 5DQG &RUSRUDWLRQ 5HSRUW 50  -DQXDU\  >@ 3URDNLV -* 'LJLWDO &RPPXQLFDWLRQV UG (G 1HZ @ 7KH GDWD VHTXHQFHV FDQ DOVR EH UHSUHVHQWHG LQ YHFWRU IRUP $ W\SLFDO GDWD oRZ VFHQDULR LV VKRZQ LQ )LJXUH  )RU PRUH LQIRUPDWLRQ WKH UHDGHU LV HQFRXUDJHG WR FRQVXOW >@ 6XSSRVH WKDW ZH KDYH WKH VHTXHQFH I[L JL1>

I[  [  [  c c c  [1> J





WKHQ ZH FDQ DOVR UHSUHVHQW WKH VHTXHQFH XVLQJ WKH YHFWRU QRWDWLRQ [7

I[  [  c c c  [1> J



,I WKH LQWHQW LV FOHDU ZH FDQ ZULWH WKH VHTXHQFH DV I[Q

I[  [  c c c  [1> J



Q

  c c c  1 b J

RU LQ VKRUW KDQG QRWDWLRQ DV [Q RU [Q E\ GURSSLQJ WKH FXUO\ EUDFNHWV *LYHQ WZR HTXDO OHQJWK VHTXHQFHV [Q DQG XQ  ZH FDQ RSHUDWH RQ WKHP LQ WKH IROORZLQJ ZD\V 3URGXFW ]L [L XL  IRU L   c c c  1 b  SRLQW E\ SRLQW 6XP SRLQW E\ SRLQW

ZL

[L  XL 

&RQYROXWLRQ

\Q

&RUUHODWLRQ

FQ

,QQHU SURGXFW RU 6FDODU SURGXFW

F

L [7 X VL]H  d   ZKHUH [ DQG \ DUH RI

2XWHU SURGXFW

4

[X7 

;L L ;L

IRU L   c c c  1 b 

[ XQ>L

[Q

[ X L Q

[Q

e

FRQY XQ



e FRUU XQ

GLPHQVLRQ 1 d 

VL]H 1 d 1  ZKHUH [ DQG \ DUH RI GLPHQVLRQ 1 d 

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ 



',6&5(7(7,0( 75$16)250$7,216

7KH ELODWHUDO =WUDQVIRUP RI WKH VHTXHQFH [Q LV GHnQHG E\ = I[Q J

; ]

;

Q >

[Q ] >Q



ZKHUH ] UHM    >b{ {@ DQG U LV D SRVLWLYH UHDO QXPEHU :H QRWH WKDW LQ FLUFXLW DQG V\VWHP DQDO\VLV ZKHQ JRLQJ WR WKH WUDQVIRUP GRPDLQ FRQYROXWLRQ DQG FRUUHODWLRQ RSHUDWLRQV DUH UHGXFHG WR SURGXFW RSHUDWLRQV 2I FRXUVH RQH KDV WR SHUIRUP ERWK WKH IRUZDUG DQG LQYHUVH WUDQVIRUPDWLRQV 7KH LQYHUVH LV JLYHQ E\  ; ] ]Q> G] [Q = > I; ] J  {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nQHG LQ   7KH '7)7 SDLU LV GHnQHG E\ WKH IRUZDUG WUDQVIRUP

,

;

; HM

Q >

[Q H>MQ



ZKLFK LV MXVW WKH =WUDQVIRUP HYDOXDWHG RQ WKH XQLW FLUFOH LH ] U H[S>M  ZLWK U   ZKLOH WKH LQYHUVH WUDQVIRUP LV JLYHQ E\  { ; HM HMQ G [Q { >{ ,I WKH GDWD VHWV DUH nQLWH LQ GXUDWLRQ I[Q  Q  c c c  1 b J WKHQ WKH '7)7 SDLU EHFRPHV WKH GLVFUHWH )RXULHU WUDQVIRUP ')7 SDLU JLYHQ E\

=

; N

DQG [Q

;

1> Q



;

[Q H>M { 1 NQ

 1> ; N HM { 1 NQ 1 N







'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

ZKHUH { 1 N FRUUHVSRQGV WR WKH NWK IUHTXHQF\ FRPSRQHQW DOVR H[SUHVVHG DV N  7KH ')7 LV WKH GLVFUHWH WLPH DQDORJ RI WKH FRQWLQXRXV WLPH )RXULHU VHULHV H[SDQVLRQ WKDW LV ; N LV WKH SURMHFWLRQ RI [Q RQWR D FRPSOH[ VLQXVRLG ZKLFK KDV H[DFWO\ N SHULRGV RYHU 1 VHTXHQWLDO SRLQWV LQ WLPH 7R VLPSOLI\ WKH PDWKHPDWLFDO H[SUHVVLRQV ZH VRPHWLPHV UHSODFH H[S bM {N 1 ZLWK WKH V\PERO :1N  7KH ')7 DQG LWV LQYHUVH FDQ WKHQ EH ZULWWHQ DV ; N

DQG [Q

;

1> Q



[Q :1NQ

;

 1> ; N : >NQ 1 1 Q



7KH QRUPDOL]DWLRQ E\  1 FDQ EH DSSOLHG LQ HLWKHU GRPDLQ RU FDQ EH VSOLW S LH  1 DQG EH DSSOLHG LQ ERWK GRPDLQV 7KHUH LV D EUXWH IRUFH ZD\ WR HYDOXDWH WKH ')7  DQG D IDVW ZD\ 7UDGLWLRQDOO\ LI 1 LV D SRZHU RI  LH 1 y  IRU y SRVLWLYH LQWHJHU WKHQ  LV FRPSXWHG XVLQJ WKH ))7 5HODWLYH WR WKH EUXWH IRUFH ZD\ ZKLFK UHTXLUHV 1  PXOWLSOLFDWLRQV ZKHQ PDQLSXODWLQJ 1 LQSXW GDWD SRLQWV WKH ))7 LV IDVWHU LH WKH W\SLFDO FRVW LV 1  ORJ 1 PXOWLSOLFDWLRQV  ZKLOH SUHVHUYLQJ H[DFWO\ WKH VDPH DFFXUDF\ 7KHUH DUH PDQ\ RWKHU ZD\V WR LPSOHPHQW WKH ))7 RSHUDWLRQ LH WKH :LQRJUDG WUDQVIRUP QXPEHU WKHRUHWLF WUDQVIRUP HWF >@ 

),/7(5,1*

2QH RI WKH PDLQ VLJQDO FRQGLWLRQLQJ DQG VLJQDO SURFHVVLQJ RSHUDWLRQV SHU IRUPHG LV WKH nOWHULQJ RSHUDWLRQ 5RXJKO\ VSHDNLQJ JLYHQ D GDWD VHW ZLWK VRPH SDUWLFXODU FKDUDFWHULVWLFV ZH ZDQW WR LVRODWH EORFN RU SDVV FHUWDLQ W\SHV RI FKDUDFWHULVWLFV 8VXDOO\ ZH GHDO ZLWK D VSHFWUDO GLVFXVVLRQ LQ ZKLFK ZH WDON DERXW D VSHFWUDO UHJLRQ IUHTXHQF\ UHJLRQV WKDW ZH ZDQW WR VKDSH WKDW LV DQQLKLODWH RU SDVV $ VLPSOH H[DPSOH UHODWHV WR WKH UDGLR VSHFWUXP VD\ WKH )0 SRUWLRQ LQ ZKHUH ZH ZDQW WR SDVV WKDW LV WR OLVWHQ WR RQH JLYHQ )0 UDGLR VWDWLRQ 7\SLFDOO\ ZH WXQH D nOWHU EDQGSDVV WR WKH VSHF WUDO UHJLRQ VD\ IRU H[DPSOH  0+] DQG SDVV VLJQDOV WKDW H[LVW LQ WKH QHLJKERUKRRG RI WKH FDUULHU IUHTXHQF\  0+]  )RU FRPPHUFLDO )0 VWD WLRQV WKH QHLJKERUKRRG LV h .+] LH EDQGZLGWK DQG WKH FDUULHU VSDFLQJ LH SRWHQWLDOO\ WKH PLQLPXP VSHFWUDO GLVWDQFH WR DQRWKHU )0 VWDWLRQ LV  .+] 7\SLFDO nOWHU IXQFWLRQV DUH RI D ORZ SDVV KLJK SDVV EDQGSDVV RU EDQGVWRS QDWXUH $V WKH QDPH LPSOLHV ORZ IUHTXHQF\ KLJK IUHTXHQF\ DQG LQWHUPHGLDWH IUHTXHQF\ FRPSRQHQWV DUH SDVVHG RU LQWHUPHGLDWH IUHTXHQF\ FRPSRQHQWV DUH

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ



)LJXUH  1 VL]HG ),5 nOWHU UHPRYHG ,Q VRPH DSSOLFDWLRQV WKH nOWHU IUHTXHQF\ UHVSRQVH LQ WKH SDVV EDQG UHJLRQ LV IXQFWLRQDOO\ UHODWHG WR WKH VLJQDO DQG QRLVH SRZHU VSHFWUDO GHQVLWLHV LH WKH PDWFKHG nOWHU  

),1,7( ,038/6( 5(63216( ),/7(5

)LQLWH LPSXOVH UHVSRQVH ),5 nOWHUV KDYH DQ LPSXOVH UHVSRQVH WKDW ZLOO EH ]HUR DIWHU VRPH nQLWH WLPH WKDW LV KQ  IRU Q w 1  $ W\SLFDO 1 VL]HG ),5 nOWHU LV VKRZQ LQ )LJXUH  $Q LQVSHFWLRQ RI WKH GLDJUDP UHYHDOV WKDW DW WLPH Q WKH RXWSXW \Q LV JLYHQ E\ WKH FRQYROXWLRQDO VXP [Q K  [Q b  K  c c c  [Q b 1   K1 b  \ Q

;

1> L



[Q b L KL



,I ZH LQMHFW D .URQHFNHUGHOWD IXQFWLRQ VD\ [Q pQ  LQWR WKH UHOD[HG QHWZRUN LH WKH LQLWLDO FRQGLWLRQV DUH ]HUR  WKHQ WKH RXWSXW ZLOO EH IK  K  K c c c  K1 b      c c cJ IK  K  K c c c  K1 b  J 7KLV GHPRQVWUDWHV ZK\ WKH QDPH ),5 nOWHU LV DWWDFKHG WR WKH VWUXFWXUH RI )LJXUH  DQG DOVR LOOXVWUDWHV WKH FRQYROXWLRQDO SURSHUWLHV LH  LV WKH GLVFUHWH WLPH FRQYROXWLRQ RI WKH LQSXW DQG WKH LPSXOVH UHVSRQVH :H QRWH WKHUH LV QR IHHGEDFN KHQFH WKH nOWHU ZLOO DOZD\V EH VWDEOH $Q HTXLYDOHQW ZD\ RI QRWLQJ WKLV LV WR REVHUYH WKDW WKH WUDQVIHU IXQFWLRQ RI WKH nOWHU KDV QR SROHV RWKHU WKDQ DW WKH RULJLQ  EXW KDV ]HURV RQO\ RU WKDW D ERXQGHG LQSXW UHVXOWV LQ D ERXQGHG RXWSXW ERXQGHG LQSXWERXQGHG RXWSXW %,%2 VWDELOLW\  &ODVVLFDO DQDORJ nOWHU WKHRU\ KDV PHPRU\ IHHGEDFN JLYLQJ ULVH WR SROHV LQ WKH WUDQVIHU IXQFWLRQ 7KHUH DUH DOVR GLJLWDO nOWHUV RI WKLV IRUP LH ,,5



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

nOWHUV ,,5 LQnQLWH LPSXOVH UHVSRQVH nOWHUV KDYH DQ LPSXOVH UHVSRQVH WKDW JRHV RQ IRUHYHU LH KQ mQ 8 Q    m    )RU PRUH EDFNJURXQG RQ ,,5 nOWHUV WKH UHDGHU LV HQFRXUDJHG WR FRQVXOW >^@ :H FDQ WDNH WKH )RXULHU WUDQVIRUP RI  < HM

; HM + HM

j

k

DQG REWDLQ WKH RXWSXW LQ WKH WLPH GRPDLQ YLD WKH LQYHUVH WUDQVIRUP JLYHQ E\ ) > < HM

\ Q



,I WKLV LV GRQH XVLQJ ))7V ZH FDOO WKH RSHUDWLRQ RI  ) ) 7 > < N

\Q

) ) 7 > I) ) 7 I[Q J ) ) 7 IKQ JJ



WKH IDVW FRQYROXWLRQ )RU ORQJ nOWHUV LH 1 LV ODUJH  LW PD\ EH PXFK PRUH HFRQRPLFDO WR WDNH WKH UHTXLUHG IRUZDUG DQG LQYHUVH WUDQVIRUPV WR REWDLQ WKH WLPH GRPDLQ LH \Q GDWD 6RPH FRQVLGHUDWLRQV PXVW EH JLYHQ WR DYRLG WKH ZUDSDURXQG PRGXOXV 1  DOVR FDOOHG PRG1 SUREOHP ZKLFK LV FDXVHG E\ WKH FLUFXODU FRQYROXWLRQV >@

%R[FDU $YHUDJHU 7KLV EXLOGLQJ EORFN LV DOVR FDOOHG DQ LQWHJUDWH DQG GXPS nOWHU )LJXUH  VKRZV WKH DYHUDJHU LQ D EORFN GLDJUDP DQG E ),5 nOWHU IRUP ZKLOH F GHSLFWV WKH IUHTXHQF\ UHVSRQVH %\ LQVSHFWLQJ )LJXUH F  WKH LPSXOVH UHVSRQVH LV WKH VHTXHQFH 1 WLPHV I  c c c  J ,WV IUHTXHQF\ UHVSRQVH LV ([DPSOH 

] `_ ^

;  >MQ

1>

+ HM

Q

H



 b H>M1  b H>M

VLQ1   VLQ  7KH UDWLR RI WKH WZR VLQXVRLGDO IXQFWLRQV LV DOVR NQRZQ DV WKH GLJLWDO VLQF IXQFWLRQ 7KH PDJQLWXGH VTXDUHG IUHTXHQF\ UHVSRQVH EHFRPHV VLQ 1   + HM VLQ   H>M1 >

nn

nn



,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ



)LJXUH 

%R[FDU DYHUDJHU LQWHJUDWH DQG GXPS nOWHU  D EORFN GLDJUDP E ),5 nOWHU IRUP DQG F PDJQLWXGH RI WKH IUHTXHQF\ UHVSRQVH

ZKLFK LV SORWWHG LQ )LJXUH F  ,W VKRZV WKDW WKH ER[FDU DYHUDJHU LV D ORZ SDVV nOWHU VLQFH IUHTXHQF\ FRPSRQHQWV DURXQG   DUH SDVVHG ZLWK SRZHU JDLQ HTXDO WR 1   DQG IUHTXHQF\ FRPSRQHQWV DZD\ IURP   LH '& DUH DWWHQXDWHG 7KH DWWHQXDWLRQ WHQGV WR LQFUHDVH ZLWK IUHTXHQF\ LH WKH IXUWKHU DZD\ IURP   WKH ODUJHU WKH DWWHQXDWLRQ  ,Q WHUPV RI SDVVEDQG GHnQLWLRQ RQH FRXOG XVH WKH  G% SRLQWV WKH QXOOWRQXOO GLVWDQFH RU WKH FRQFHSW RI QRLVH HTXLYDOHQW EDQGZLGWK 2I DOO SRVVLEOH ORZ SDVV nOWHU LPSOHPHQWDWLRQV WKH ER[FDU DYHUDJHU WHQGV WR EH WKH ZRUVW RQH LQ WHUPV RI VLGH OREH OHDNDJH DQG UROO Rm LQ WKH WUDQVLWLRQ EDQG 

7+( )$67 )285,(5 75$16)250

7KH IDVW )RXULHU WUDQVIRUP ))7  ZKLFK LV MXVW D IDVW LPSOHPHQWDWLRQ RI WKH ')7 DV LQWURGXFHG LQ  KDV PDQ\ GHVLUDEOH DWWULEXWHV DQG PDQ\ LQWHUHVW LQJ DQG XVHIXO LQWHUSUHWDWLRQV ,I ZH YLVXDOL]H WZR EODFN ER[HV RQH FRQWDLQLQJ WKH ')7 DQG WKH RWKHU FRQWDLQLQJ WKH ))7 DQG ZH HYDOXDWH WKH RXWSXW RI HDFK ER[ DW VSHFWUDO ORFDWLRQV {N 1 UDG RQH FDQQRW GHWHUPLQH ZKLFK ER[ LH ZKLFK DOJRULWKP LV UHVSRQVLEOH IRU WKH RXWSXW XQGHU REVHUYDWLRQ 7KH UHVSRQVHV DUH LQGLVWLQJXLVKDEOH 2QH FDQ DVVRFLDWH WKH VSHHG RU FRQYHUVHO\



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

WKH VORZQHVV RI WKH )RXULHU WUDQVIRUP DOJRULWKP  ZLWK WKH QXPEHU RI PXOWLSOHV FRPSOH[ YDOXHG  :H VHH WKDW WKH ')7 LPSOHPHQWDWLRQ UHTXLUHV 1 FRPSOH[ YDOXHG 1 UHDO YDOXHG PXOWLSOLHV SHU IUHTXHQF\ ORFDWLRQ LH SHU ELQ  7R REWDLQ WKH WUDQVIRUP IRU DOO IUHTXHQFLHV N   c c c  1 b   1  FRPSOH[ YDOXHG PXOWLSOLHV DUH UHTXLUHG 7KH ))7 LPSOHPHQWDWLRQ VHH /XGHPDQ >@ RU 6WUXP DQG .LUN >@ UHTXLUHV 1  ORJ 1 FRPSOH[ YDOXHG PXO WLSOLHV 6XSSRVH ZH KDYH D GDWD VHTXHQFH RI OHQJWK 1  WKHQ WKH ')7 YHUVLRQ ZRXOG UHTXLUH  {  FRPSOH[ YDOXHG PXOWLSOLHV ZKLOH WKH ))7 UHTXLUHV RQO\  c   FRPSOH[ YDOXHG PXOWLSOLHV $Q LQFUHDVH LQ VSHHG E\ PRUH WKDQ WZR RUGHUV ZLWKRXW D ORVV LQ DFFXUDF\ LV REWDLQHG 7KH ODUJHU WKH WUDQVIRUP VL]H WKH ODUJHU WKH VSHHG XS ZLOO EH UHODWLYH WR WKH WLPH D ')7 LPSOHPHQWDWLRQ ZRXOG UHTXLUH 6SHHGXSV LQ WKH RUGHU RI  WR  LH  WR  DUH QRW XQXVXDO 7KLV SURSHUW\ KDV PDGH WKH ))7 WKH PRVW SRSXODU WRRO LQ VSHFWUDO HVWLPDWLRQ FRUUHODWLRQ DQG PDQ\ nOWHULQJ RSHUDWLRQV 2QH FRXOG KDYH XVHG GLm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mHUHQWLDO RU GLmHUHQFH HTXDWLRQ '(  6ROXWLRQV WR D VHFRQG RUGHU '( W\SLFDOO\ OHDG WR D VLQXVRLGDO VROXWLRQ 6LQXVRLGV DUH DOVR HLJHQIXQFWLRQV RI OLQHDU WLPH LQYDULDQW V\VWHPV >@ 7KDW LV RQFH D FRPSOH[ VLQXVRLG LH H[Sb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mHUHQW LQWHUSUHWDWLRQV RI WKH ')7))7 6RPH RI WKH LQWHUSUHWDWLRQV DUH GHVFULEHG LQ >@ 

),5 )LOWHU ZLWK &RPSOH[ 9DOXHG :HLJKWV ,QWHUSUHWDWLRQ

)LJXUH  GHSLFWV WKH ),5 nOWHU LQWHUSUHWDWLRQ RI WKH ')7))7 ZKHUH :1N A H>M { 1 N

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ

)LJXUH 



),5 nOWHU LQWHUSUHWDWLRQ

:H n[HG WKH WLPH LQGH[ DW Q EXW LW VKRXOG EH REYLRXV ZH FRXOG DVN IRU WKH RXWSXW DW WLPH Q  L LQ ZKLFK FDVH DOO GDWD VDPSOHV LQ WKH GHOD\ OLQH PRYH ?L SODFHV WR WKH ULJKW LH WKH \RXQJHVW GDWD VDPSOH ZRXOG EH [Q  L  ZKLOH WKH ROGHVW GDWD VDPSOH ZRXOG EH [L  8VXDOO\ GDWD LV SURFHVVHG LQ VHJPHQWV EORFNV RI VL]H 1  ZKHUH 1 FDQ EH WKH OHQJWK RI D GDWD VHJPHQW RU D VKRUWHU SLHFH RI LW 2QH FRXOG DOVR ]HUR SDG D VKRUWHU VHJPHQW VD\ ZH XVH 0 GDWD SRLQWV ZKHUH 0  1  DQG DSSHQG WKH GDWD YHFWRU WR VL]H 1 XVLQJ 1 b 0 ]HURV )RU PRUH GHWDLOV VHH &KDSWHU  6HFWLRQ  DQG &KDSWHU  6HFWLRQ  7KH DFWXDO GDWD OHQJWK PD\ EH PXFK ORQJHU WKDQ WKH ')7))7 VL]H LH UHDO WLPH SURFHVVLQJ VR RQH XVHV VHJPHQWV RI WKH GDWD LQ D VHTXHQWLDO PDQQHU ,I WKH VHJPHQWV DUH FRQWLJXRXV WKDW LV QRQRYHUODSSHG ZLWK QR GDWD SRLQW EHLQJ RPLWWHG  WKHQ WKH ),5 nOWHU RXWSXW UDWH DW HDFK VSHFWUDO ELQ N LV  1 WLPHV WKH LQSXW VDPSOH UDWH ,Q WKLV VHQVH WKH ))7 ')7 FRUUHVSRQGV WR D PXOWLUDWH nOWHU LH nOWHU DQG GHFLPDWH LQ WLPH VHH 6HFWLRQ  &KDSWHU  6HFWLRQ  DQG $SSHQGL[ ( >@ 

&RPSOH[ 'HPRGXODWRU ,QWHUSUHWDWLRQ

,I ZH ORRN DW   ZH FDQ VHH WKDW LW LV WKH SURGXFW RI WZR VHTXHQFHV IROORZHG E\ D VXPPDWLRQ GHYLFH )LJXUH  SUHVHQWV WKLV W\SH RI LQWHU SUHWDWLRQ :H QRWH WKDW H>M { 1 NQ LV SHULRGLF LQ 1 LH PRGXOXV 1  KHQFH ZH FDQ ORRN DW LW DV WKH RXWSXW RI DQ RVFLOODWRU LH H>M { 1 NQ FRV{ 1 NQ b M VLQ{ 1 NQ ZKHUH HYHU\ N FRUUHVSRQGV WR D FRPSOH[ YDO XHG VLQXVRLGDO VLJQDO WKDW KDV H[DFWO\ N SHULRGV RYHU 1 GDWD SRLQWV ,I ZH LQWHUSUHW WKH 1 GDWD SRLQWV DV RQH SHULRG RI D ODUJHU GXUDWLRQ GDWD VHJPHQW WKHQ ZH VHH WKDW WKLV LQWHUSUHWDWLRQ LV WKDW RI WKH GLVFUHWH WLPH )RXULHU VHULHV GHFRPSRVLWLRQ 7KH VHWXS LQ )LJXUH  LV WKH VDPH DV LQ DQ\ FRQYHQWLRQDO KHWHURG\QH UHFHLYHU WKH GLmHUHQFH LV LQ WKH QDWXUH RI WKH RVFLOODWRU VLJQDO D FRPSOH[ YDOXHG VLQXVRLG UDWKHU WKDQ D UHDO YDOXHG VLQXVRLG  7KH PXOWLSOLFD WLRQ FUHDWHV WKH VXP DQG GLmHUHQFH IUHTXHQF\ WHUPV ZKLOH WKH VXPPHU D ORZ SDVV nOWHU VHH ([DPSOH  UHPRYHV WKH KLJK LH VXP IUHTXHQF\ WHUPV



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

&RPSOH[ GHPRGXODWRU LQWHUSUHWDWLRQ

)LJXUH  , 4 GHPRGXODWRU 

,4 'HPRGXODWRU ,Q3KDVH 4XDGUDWXUH 'HPRGXODWRU ,QWHUSUHWDWLRQ

,I RQH SORWV WKH UHDO DQG LPDJLQDU\ FRPSRQHQW VHFWLRQV RI )LJXUH  VHSD UDWHO\ RQH REWDLQV WKH IRUP DV VKRZQ LQ )LJXUH  7KH WRS OHJ FRUUHVSRQGV WR WKH LQSKDVH WHUP , WHUP  ZKLOH WKH ERWWRP OHJ FRUUHVSRQGV WR WKH TXDGUD WXUH WHUP 4WHUP  6R ZH VHH WKH UHDO DQG LPDJLQDU\ SDUWV RI WKH ))7 ')7 FRUUHVSRQG WR WKH FRQYHQWLRQDO GLVFUHWH WLPH , 4 GHPRGXODWRU 7KH GHPRGXODWRUV DUH IROORZHG E\ GHFLPDWRUV >@ VLQFH 1 GDWD VDPSOHV UHVXOW LQ RQH RXWSXW VDPSOH SHU SURFHVVLQJ EUDQFK 

&RUUHODWRU ,QWHUSUHWDWLRQ

7KH H[SUHVVLRQ JLYHQ LQ  FDQ DOVR EH VNHWFKHG DV VKRZQ LQ )LJXUH  )LJXUH  LV WKH FODVVLF FRUUHODWRU FURVVFRUUHODWRU WKDW HYDOXDWHV WKH FURVV FRUUHODWLRQ EHWZHHQ WKH GDWD DQG D NQRZQ RU DVVXPHG UHIHUHQFH VLQXVRLGDO IXQFWLRQ 7KH FRUUHODWRU GRHV QRW FRPSXWH FRUUHODWLRQ YDOXHV RWKHU WKDQ IRU

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ

)LJXUH 



&RUUHODWRU

WKH ]HUR ODJ EXW LW FRXOG HDVLO\ EH PRGLnHG WR GR VR :H DOVR QRWH WKDW WKH FURVVFRUUHODWLRQ LV MXVW WKH LQQHU SURGXFW ZKLFK LV WKH SURMHFWLRQ RI WKH GDWD RQWR D VHW RI RUWKRJRQDO VLQXVRLGDO IXQFWLRQV 7KLV RSHUDWLRQ LV DOVR FDOOHG D VFDODU SURGXFW 

&RQYROYHU ,QWHUSUHWDWLRQ

,Q JHQHUDO WKH RXWSXW RI D OLQHDU WLPH LQYDULDQW nOWHU FDQ EH ZULWWHQ DV

;Q

\ Q

L



[L KQ b L



/RRNLQJ DW   ZH VHH LI ZH OHW ; N EH WKH RXWSXW RI WKH nOWHU DW WLPH

1 b  GHQRWHG E\

; N

; N 1 b 

;

1> Q

:H FDQ DOVR ZULWH  DV \ Q

,I ZH OHW KL



;Q L

H>M { 1 N1>>L \1 b 

;

1> L





\ 1 b 

[Q H>M { 1 NQ

KL [Q b L

DQG Q





1 b  WKHQ  EHFRPHV

H>M { 1 N1>>L [1 b  b L



ZKLFK HTXDOV  E\ XVLQJ D FKDQJH RI YDULDEOHV DQG UHYHUVLQJ WKH XSSHU DQG ORZHU OLPLWV RI WKH VXPPDWLRQ 7KLV LQWHUSUHWDWLRQ LV JLYHQ LQ )LJXUH 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH  

&RQYROYHU

0DWFKHG )LOWHU ,QWHUSUHWDWLRQ

$V ZLOO EH VKRZQ LQ &KDSWHU  WKH RSWLPXP GHWHFWRU IRU D NQRZQ VLJQDO HPEHGGHG LQ ZKLWH *DXVVLDQ QRLVH LV WKH PDWFKHG nOWHU ZKHUH WKH PDWFKHG nOWHU UHVSRQVH LV WKH WLPH UHYHUVHG LH PLUURUHG VLJQDO :H VHH WKDW WKH KQ GHnQHG LQ 6HFWLRQ  GHQRWHV D FRPSOH[ YDOXHG VLQXVRLG KHQFH WKH ))7 ')7 LV WKH RSWLPDO UHFHLYHU GHWHFWRU IRU VLQXVRLGV HPEHGGHG LQ ZKLWH *DXVVLDQ QRLVH $V VXFK WKH ))7')7 FRQVWLWXWHV D EDQN RI PDWFKHG nOWHUV ZLWK IUHTXHQFLHV ELQV ORFDWHG DW VSHFWUDO ORFDWLRQV IV N 1  ZKHUH N LV WKH ELQ LQGH[  › N › 1 b   1 LV WKH WUDQVIRUP VL]H DQG IV LV WKH VDPSOLQJ IUHTXHQF\ 

&RRUGLQDWH 7UDQVIRUPDWLRQ ,QWHUSUHWDWLRQ

:H FDQ DOVR XVH OLQHDU DOJHEUD YHFWRUV DQG PDWULFHV WR GLVFXVV WKH ))7 RSHUDWLRQ /HW [7 >[  [  c c c  [1> @ :

      

ZLWK



H>M { 1 H>M { 1 

H>M { 1  H>M { 1 

 

 

ccc ccc ccc



H>M { 1 1> H>M { 1 1>

 

H>M { 1 1> H>M { 1 1>  c c c H>M { 1 1> 1>

;7

WKHQ DQG



  

>;   ;   c c c  ; 1 b  @ ;

:[

))7 I[Q J



 :> ; ))7> I; N J  1 (TXDWLRQ  LV LQ WKH IRUP RI DQ\ VWDQGDUG PDWUL[ HTXDWLRQ \ $[ ZKHUH \ LV D OLQHDU WUDQVIRUPDWLRQ RI [ >@ [

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ 



)$67 &255(/$7,21

&RUUHODWLQJ WZR GDWD VHTXHQFHV SURYLGHV D PHDVXUH RI OLNHQHVV EHWZHHQ WKH WZR VHTXHQFHV ,I RQH QRUPDOL]HV WKH FRUUHODWLRQ RXWSXWV WKH YDOXHV WDNH RQ D YDOXH EHWZHHQ PLQXV DQG SOXV RQH 7KH QRUPDOL]DWLRQ LV D GLYLVLRQ RI WKH VTXDUH URRW E\ WKH SURGXFW RI WKH HQHUJ\ RI WKH WZR VHTXHQFHV LQYROYHG $ PLQXV RQH DQG SOXV RQH FRUUHVSRQG WR b SHUFHQW DQG  SHUFHQW UHVSHF WLYHO\ :KHQ WKH QRUPDOL]HG FURVVFRUUHODWLRQ YDOXH LV  WKH WZR VHTXHQFHV DUH LGHQWLFDO &RQYHUVHO\ D YDOXH RI b LQGLFDWHV WKDW WKH WZR VHTXHQFHV DUH LGHQWLFDO LQ D PDJQLWXGH VHQVH EXW GLmHU LQSKDVH E\ F  $ FRUUHODWLRQ RI D YDOXH RI ]HUR LQGLFDWHV WKDW WKH WZR VHTXHQFHV DUH XQFRUUHODWHG $VVXP LQJ WKDW WKH VHTXHQFHV DUH RI HTXDO OHQJWK D W\SLFDO FRUUHODWLRQ H[SUHVVLRQ GLVUHJDUGLQJ WKH QRUPDOL]DWLRQ LV JLYHQ E\ 5;< C

; Q

[Q \Q  C



7KH GDWD VHTXHQFHV DV XVHG LQ   GR QRW KDYH WR EH RI LGHQWLFDO OHQJWK 7R JHW PHDQLQJIXO UHVXOWV ZKHQ XVLQJ D QXPHULFDO HYDOXDWLRQ URXWLQH ZH ZRXOG ]HUR SDG HDFK VHTXHQFH WR EH RI OHQJWK HTXDO WR RU ODUJHU WKDQ WKH VXP RI OHQJWK [ SOXV OHQJWK \ b  6RPH VRIWZDUH LH ;&255 LQ 0$7/$% DXWRPDWLFDOO\ WDNHV FDUH RI WKLV UHTXLUHPHQW 7KLV RSHUDWLRQ DOVR FRUUHVSRQGV WR WKH SURMHFWLRQ RI RQH RI WKH YHFWRUV RQWR WKH VHFRQG YHFWRU :H FDQ DOVR REWDLQ WKLV UHVXOW E\ 5;< C

))7> I; N < A N J



ZKHUH ; N < N

))7 I[Q J ))7 I\Q J

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pFXOWLHV )RU H[DPSOH FRUUHODWLQJ WZR VHTXHQFHV RI OHQJWK 1 UHVXOWV LQ D FRUUHODWLRQ IXQFWLRQ RI OHQJWK 1 b  +HQFH ERWK VHTXHQFHV VKRXOG EH ]HUR H[WHQGHG DW OHDVW WR D OHQJWK RI 1 b  7KLV LV LQ DGGLWLRQ WR WKH ]HUR SDGGLQJ UHTXLUHG E\ WKH ))7 RSHUDWLRQV



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



3(5,2'2*5$0 32:(5 63(&75$/ '(16,7< (67,0$7(

$ YHU\ FRPPRQ ZD\ WR REWDLQ DQ HVWLPDWH RI WKH SRZHU VSHFWUDO GHQVLW\ RI WKH VHTXHQFH [Q LV YLD WKH IROORZLQJ H[SUHVVLRQ

nnn nn

nnn ; nn Q

 1>  { 1 NQ >M [Q Z Q H 3; N A 1 



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nHG WR 3;< N

‚;

 ‚1>;





 1> [Q H>M { 1 NQ 1 Q



Q

\ Q H>M { 1 NQ

A



0RUH DSSOLFDWLRQRULHQWHG PDWHULDO FDQ EH IRXQG LQ &KDSWHU  7KHUH DUH PDQ\ RWKHU ZD\V WR REWDLQ D VSHFWUDO HVWLPDWH 6RPH RI WKHVH DUH DXWR UHJUHVLYH $5  PRYLQJ DYHUDJH 0$  DQG DXWRUHJUHVVLYH PRYLQJ DYHUDJH $50$ PRGHOLQJ VXEVSDFH PHWKRGV DQG 3URQ\ V DQG &DSRQ V PHWKRG >^ @ ([WHQVLRQV WKDW GHDO ZLWK WLPH YDU\LQJ VFHQDULRV VXFK DV WKH VSHFWURJUDP DQG WKH :LJQHU9LOOH GLVWULEXWLRQ HWF DUH LQWURGXFHG LQ &KDSWHU  

:$9(/(76



,QWURGXFWLRQ

:DYHOHWV KDYH FDXJKW WKH DWWHQWLRQ RI WKH VLJQDO SURFHVVLQJ DQG FRPPXQLFD WLRQV FRPPXQLW\ RQ D ODUJH VFDOH ,QJULG 'DXEHFKLHV >@ DGYRFDWHG WKH XVH RI EDVLV IXQFWLRQV KDYLQJ D VKRUW VXSSRUW GXUDWLRQ  7KH HmHFWLYH OHQJWK RI WKH VXSSRUW LV GLmHUHQW DW GLmHUHQW IUHTXHQFLHV VFDOHV  +HU UHVHDUFK IROORZHG HDUOLHU ZRUN E\ 0RUOHW *URVVPDQQ 0H\HU 0DOODW DQG RWKHUV IRU PRUH GH WDLO WKH UHDGHU LV HQFRXUDJHG WR FRQVXOW >@  ,I WKH UHDGHU LV LQWHUHVWHG LQ D

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ



'63 UHODWHG GLVFUHWH WLPH ZDYHOHW UHIHUHQFH WKH ERRNV E\ 6WUDQJ DQG 1JX\HQ >@ DQG 9HWWHUOL DQG .RYD›FHYLu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nOWHULQJ RFWDYH EDQG SUR FHVVLQJ FRQVWDQW 4n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mHUHQWLDO HTXDWLRQV FRPSUHVV GDWD LH SLFWXUHV YLGHR PXVLF VSHHFK  RU GHQRLVH GDWD LH UHPRYH XQZDQWHG GLVWXUEDQFHV  MXVW WR PHQWLRQ D IHZ (DFK RQH RI WKHVH DUHDV KDV D ODUJH ERG\ RI UHIHUHQFH PDWHULDO :H ZLOO GHYHORS WKH EDVLF LGHDV RI ZDYHOHW EDVHG SUR FHVVLQJ nUVW LQ D EUXWH IRUFH ZD\ DQG WKHQ LQ WHUPV RI D IDVW YHUVLRQ VLPLODU WR WKH QRWLRQ RI FODVVLFDO ')7 DQG ))7 SURFHVVLQJ  $V IDU DV WKH ZDYHOHW WRSLF LV FRQFHUQHG ZH XVH ),5 nOWHUV OLQHDU DQG WLPH LQYDULDQW WKDW DUH W\SLFDOO\ RI VKRUW OHQJWK DQG DUH IROORZHG E\ D GHFLPDWLRQ LQWHJHU UHVDPSOLQJ SURFHGXUH 0RUH RIWHQ WKDQ QRW WKH PXOWLSOLHUV RI WKH ),5 nOWHU DUH UHDO YDOXHG 

5HYLVLWLQJ ),5 )LOWHUV DQG WKH ')7))7

/HW XV ORRN DW D GDWD VHTXHQFH [Q  Q w  DQG D QHWZRUN nOWHU ZLWK LPSXOVH UHVSRQVH KQ VLQ  Q IRU  › Q › 1 b  ZKHUH  {C 1 DQG C LV VRPH n[HG LQWHJHU 7KH IXQFWLRQ KQ  LI SORWWHG ZLOO JR H[DFWO\ WKURXJK C FRPSOHWH SHULRGV RYHU 1 GDWD SRLQWV 7KH IUHTXHQF\ UHVSRQVH + HM LV WKH )RXULHU WUDQVIRUP RI KQ GHQRWHG E\ ) IKQ J 7KLV LV WKH VDPH VFHQDULR DV LQ )LJXUH F  H[FHSW WKDW WKH nOWHU KDV LWV FHQWHU IUHTXHQF\ DW ORFDWLRQV SOXV DQG PLQXV C DQG RI FRXUVH WKH VSHFWUDO KHLJKW LH HQHUJ\ GHQVLW\ ZLOO EH VFDOHG /HW XV LGHDOL]H WKH UHVSRQVH E\ DVVXPLQJ LW UROOV Rm YHU\ IDVW DW WKH HGJHV RI WKH EDQGSDVV DQG WKDW WKH VLGH OREHV DUH QHJOLJLEOH LH FORVH WR ]HUR 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

%DQGSDVV LGHDO nOWHU

nn

nn

7KHQ WKH VSHFWUDO UHVSRQVH LV DSSUR[LPDWHG DV VKRZQ LQ )LJXUH  :H QRWH WKDW WKH nOWHU UHVSRQVH LV JLYHQ E\ + HM   ZKHUH WKH FHQWHU IUHTXHQF\ LV  h{C 1  ,I ZH HYDOXDWHG WKH RXWSXW RI WKLV nOWHU DW LWV FHQWHU IUHTXHQF\ ZH FDQ UHIHU WR + HM {C 1  RU M+ C M  ZLWK WKH XQGHUVWDQGLQJ WKDW WKH LQWHJHU C FRUUHVSRQGV WR WKH VSHFWUDO ORFDWLRQ {C 1 RU C  7KH ))7 VHH 6HFWLRQ  LV D FROOHFWLRQ RI ),5 EDQGSDVV nOWHUV ZLWK FRPSOH[ YDOXHG LPSXOVH UHVSRQVHV ZKRVH RXWSXWV DUH GHFLPDWHG LQ WLPH (DFK nOWHU KDV ZKHQ ZH LGHDOL]H WKH SHUIRUPDQFH D UHVSRQVH MXVW DV WKH nOWHU GLVFXVVHG LQ 6HFWLRQV  DQG  7KH UHVXOW LV D nOWHU EDQN ZLWK WKH nOWHU V FHQWHU IUHTXHQFLHV SRVLWLRQHG DW ORFDWLRQV C C   c c c  1 b   7KH ]HUR ORFDWLRQ KDV QR FRPSOHPHQWDU\ VSHFWUDO WHUP LH h ]HUR LV ]HUR DQG KDV WKH VDPH VSHFWUDO ZLGWK DV DQ\ RI WKH RQHVLGHG EDQGSDVV nOWHUV 7KH PDJQLWXGH VTXDUHG UHVSRQVH LV WKH VDPH DW DOO VSHFWUDO ORFDWLRQV H[FHSW DW ORFDWLRQV C  DQG C 1  6LQFH DOO ORFDWLRQV WLPH DQG IUHTXHQF\ DUH F\FOLF LH PRGXOXV 1  VSHFWUDO ORFDWLRQV 1 b N FRUUHVSRQG WR QHJDWLYH IUHTXHQF\ ORFDWLRQV bN  )RU UHDO YDOXHG LQSXW GDWD ZH H[SHFW +HUPLWLDQ V\PPHWU\ LQ WKH DPSOLWXGH VSHFWUXP WKDW LV ; N ; A bN ; A 1 b N  )RU FRPSOH[ YDOXHG LQSXW GDWD WKHUH LV QR V\PPHWU\ DW DOO LQ WKH DPSOLWXGH VSHFWUXP ,I ZH VHJPHQW WKH GDWD WKHQ HYHU\ 1 GDWD SRLQW VHJPHQW SURYLGHV D VLQ JOH RXWSXW SRLQW RQH IURP HYHU\ nOWHU  6OLGLQJ WKH VHJPHQW E\ RQH GDWD SRLQW DORQJ WKH GDWD VHTXHQFH DJDLQ SURGXFHV RQH GDWD SRLQW IURP HDFK nOWHU ORFDWLRQ PDNLQJ WKH RXWSXW UDWH HTXDO WKH LQSXW UDWH 2I FRXUVH LQ D JLYHQ VSHFWUDO ELQ WKH RXWSXW GDWD ZRXOG EH KHDYLO\ FRUUHODWHG ,I WKH VHJPHQWV DUH FKRVHQ WR EH QRQRYHUODSSLQJ DQG FRQWLJXRXV WKHQ WKH RXWSXW GDWD UDWH LV  1 WLPHV WKH GDWD VDPSOLQJ UDWH 'LmHUHQW RYHUODS IDFWRUV EHWZHHQ VXF FHVVLYH WUDQVIRUPV OHDG WR GLmHUHQW RXWSXW GDWD UDWHV )RU H[DPSOH RYHUODS IDFWRUV RI  SHUFHQW  RYHUODS DQG  SHUFHQW  RYHUODS OHDG WR RXWSXW UDWHV RI IV  1 DQG IV  1  UHVSHFWLYHO\ ,Q WKLV FDVH WKH ')7))7 LV D EDQN FRPELQDWLRQ RI EDQGSDVV nOWHUV IROORZHG E\ D GHFLPDWLRQ VFKHPH

nn

nn

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ



VHH )LJXUH   )LJXUH E DWWHPSWV WR VKRZ WKDW WKH ')7))7 LV D EDQGSDVV ),5 nOWHU EDQN IROORZHG E\ D GHFLPDWRU WKDW WKURZV DZD\ 1 b  RXWSXW GDWD SRLQWV DQG NHHSV WKH 1 WK RXWSXW GDWD SRLQW ,I PRUH VHJPHQWV DUH WR EH SURFHVVHG WKHQ IURP HDFK nOWHU RQH REWDLQV RQH RXWSXW GDWD SRLQW SHU VHJPHQW )URP )LJXUH D DQG 6HFWLRQ  LW LV FOHDU WKDW HDFK nOWHU KDV WKH VDPH EDQGZLGWK 

),5 )LOWHUV DQG :DYHOHW 7UDQVIRUPV

6XSSRVH WKDW ZH OHW WKH EDQGZLGWK RI HDFK nOWHU EH SURSRUWLRQDO WR LWV FHQWHU IUHTXHQF\ DQG ZH UHVDPSOH WKH RXWSXW DW D UDWH JRYHUQHG E\ WKH EDQGZLGWK 7KH nOWHUV nOWHUV LQ D EDQN DUH DUUDQJHG VXFK WKDW WKH VSHFWUDO UHJLRQ LV FRQ WLJXRXVO\ FRYHUHG KDYLQJ QR KROHV DQG QR UHGXQGDQW FRYHUDJH 7KH PDMRULW\ RI ZDYHOHW nOWHUV DUH ),5 nOWHUV KDYLQJ D UHDO YDOXHG LPSXOVH UHVSRQVH :H FDQ FRQVWUXFW ZDYHOHWV WKDW GR KDYH D FRPSOH[ YDOXHG LPSXOVH UHVSRQVH EXW ZH ZLOO RQO\ XVH LW WR GHVFULEH D SRRU PDQ V YHUVLRQ RI WKH ZDYHOHW WUDQVIRUP 7R LOOXVWUDWH WKH GLVFXVVLRQ ZH VKDOO FKRRVH D VPDOO GDWD VL]H VD\ 1  )LJXUH D DQG E VKRZV WKH GHVLUHG DUUDQJHPHQWV IRU VL]H 1  ,I RQH ZHUH WR H[HFXWH RQH ))7 RI OHQJWK   VSHFWUDO ELQ nOWHU RXWSXWV ZRXOG EH REWDLQHG HDFK RQH FRQVLVWLQJ RI RQH GDWD SRLQW 7KH QXPEHU RI GDWD SRLQWV LV SUHVHUYHG ZH VWDUWHG ZLWK  WLPH VDPSOHV DQG ZH HQGHG XS ZLWK  nOWHUHG RXWSXW SRLQWV $ VLPLODU FRQFOXVLRQ FDQ EH GUDZQ IURP )LJXUH  ,I ZH VXP XS WKH RXWSXW GDWD SRLQWV               ZH DJDLQ SUHVHUYH WKH QXPEHU RI GDWD SRLQWV 7KH VFKHPH LQ )LJXUH  KDV EHHQ DURXQG IRU D ORQJ WLPH DQG LV NQRZQ DV SURSRUWLRQDO EDQGSDVV nOWHULQJ RFWDYH nOWHULQJ DQG FRQVWDQW 4nOWHULQJ 4 LV WKH UDWLR RI FHQWHU IUHTXHQF\ WR EDQGZLGWK ,W LV DOVR FDOOHG WKH TXDOLW\ IDFWRU LQ WKH HOHFWURQLF FLUFXLW OLWHUDWXUH 7KH FLUFOH V\PERO ZLWK WKH GRZQSRLQWLQJ DUURZ DQG LQWHJHU , UHSUHVHQWV WKH GHFLPDWLRQ RSHUDWLRQ LH NHHS HYHU\ , WK VDPSOH  1RWH ZKLOH RXU H[DPSOH XVHV D GDWD VL]H RI 1  LW FDQ EH DQ\ DUELWUDU\ SRZHU RI WZR 7KH QXPEHU RI EDQGSDVV nOWHUV ZLOO DOZD\V EH ORJ 1  LH ORJ   ZKLOH WKHUH LV RQO\ RQH ORZ SDVV nOWHU :H KDYH HVWDEOLVKHG D ZDYHOHW WUDQVIRUPHU FRQVLVWLQJ RI ORJ 1 LH VL[ EDQGSDVV nOWHUV DQG RQH ORZ SDVV nOWHU /HW XV XVH WKH H[DPSOH GHVFULEHG LQ )LJXUH  WR PDNH VRPH REVHUYDWLRQV 7KH nUVW RQH LV WKDW WKH QXPEHU RI GDWD SRLQWV LH D FRRUGLQDWH WUDQVIRUPDWLRQ LV SUHVHUYHG 1HHGOHVV WR VD\ LI ZH SURSHUO\ LQYHUW WKH VHYHQ DYDLODEOH nOWHU RXWSXWV ZH ZLOO JHW EDFN WKH RULJLQDO GDWD VHTXHQFH 7KH nOWHU EDQG ZLGWK EHFRPHV SURJUHVVLYHO\ VPDOOHU VWDUWLQJ ZLWK WKH KDOIEDQG nOWHU WRS OHJ LQ )LJXUH E  ,W LV FXW LQ KDOI LQ HDFK VXFFHVVLYH VWDJH LH SURJUHVVLQJ GRZQ WKH nOWHU EDQN  6LQFH WKH EDQGSDVV nOWHUV UHGXFH WKH EDQGZLGWK RI WKHLU UHVSHFWLYH RXWSXWV ZH FDQ UHVDPSOH LH FKDQJH WKH VDPSOLQJ UDWH DV LQGLFDWHG  7KH nUVW RXWSXW LH WKH WRS RI )LJXUH E  GHQRWHG E\ 6& KDV DQ RXWSXW VDPSOLQJ UDWH RQH KDOI RI LWV LQSXW VDPSOLQJ UDWH 7KH VHFRQG RXWSXW GHQRWHG E\ 6& KDV DQ



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

D 6SHFWUDO UHJLRQV RI D EDQN RI EDQGSDVV nOWHUV DQG E EDQGSDVV nOWHUV IROORZHG E\ GHFLPDWLRQ

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ

D 6SHFWUDO UHJLRQV RI D EDQG RI SURSRUWLRQDO EDQGSDVV nOWHUV 1  DQG E VL[ EDQGSDVV nOWHUV DQG RQH ORZ SDVV nOWHU IROORZHG E\ GHFLPDWRUV

)LJXUH 





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

RXWSXW VDPSOLQJ UDWH RQH IRXUWK RI WKH LQSXW VDPSOLQJ UDWH 7KH QRWDWLRQ 6& UHIHUV WR ZKDW LV FRPPRQO\ FDOOHG VFDOHV ZLWK WKH ZLGHVW VSHFWUDO EDQGZLGWK DVVRFLDWHG ZLWK VFDOH  5DWKHU WKDQ XVLQJ D ELQ ELQ QXPEHUV DV LQ WKH ')7 ZH ODEHO WKH nOWHU DV VFDOHV XVXDOO\ VWDUWLQJ ZLWK WKH QXPEHU RQH IRU WKH KLJKHVW IUHTXHQF\ DQG LQFUHDVLQJ LQ FRXQW DV WKH IUHTXHQF\ ORFDWLRQ EHFRPHV VPDOOHU :H QRWH WKDW VRPH DXWKRUV ODEHO WKH VFDOHV LQ WKH VDPH GLUHFWLRQ DV WKH IUHTXHQF\ LH WKH n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nOWHU WR REWDLQ WKH GHVLUHG UHVXOWV 7R REWDLQ WKH ORZSDVV RXWSXW RQH GDWD SRLQW ZH VXP RYHU WKH ZKROH GDWD VHW DV JLYHQ E\

; 

/3 

Q



[Q

RQH RXWSXW SRLQW

7R REWDLQ WKH VFDOH RXWSXWV ZH VXP XS ZHLJKWHG GDWD SRLQWV ZKHUH WKH VWDUW VWRS DQG VXPPDWLRQ WLPHV GHSHQG RQ WKH VFDOH 7KH VFDOH RXWSXWV DUH REWDLQHG DV JLYHQ E\ 6&  6&  6& 

;  >M { Q ;  >M { Q ;  >M { 

[Q H

    Q





[Q H

    Q





Q



[Q H

! !! !! Q ! 

 RXWSXW SRLQW

   

 RXWSXW SRLQWV

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ

; Q ; Q ; Q ; 

6& 





6& 





6& 





6& 

Q



! !!! !!! Q ! !! !! Q ! !!! !!! Q ! 

[Q H>M {

 Q

[Q H>M {



[Q H>M {



[Q H>M {





 RXWSXW GDWD SRLQWV

,I ZH XVH D FRPSDFW QRWDWLRQ ZH FDQ ZULWH WKH nOWHU RXWSXWV DV 6& L 6& L

DQG nQDOO\ 6& L

; Q L? L ;>

[Q H>M {

 Q

L

  c c c    RXWSXW SRLQWV

[Q H>M {

 Q

L

  c c c    RXWSXW SRLQWV

;

[Q H>M {

 Q

L

  c c c    RXWSXW SRLQWV

L > 

   

Q L?

L >

Q L?

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nOWHU DV ORQJ DV LW FDQ EH FRQVWUXHG WR EH WKH EDQGSDVV nOWHUV ZLWK WKH GHVLUHG SDUDPHWHUV $FWXDOO\ PRVW ZDYHOHW WUDQVIRUPV XVH UHDO YDOXHG ZHLJKWV LQ WKHLU ),5 nOWHUV 7KLV UHGXFHV WKH FRPSXWDWLRQDO EXUGHQ UHODWLYH WR WKH nOWHUV ZH KDYH FKRVHQ WR PDNH WKH LOOXVWUDWLRQ $W KLV SRLQW ZH UHDOL]H WKDW WKH ZDYHOHW WUDQVIRUP LV MXVW D FROOHFWLRQ RI SURSRUWLRQDO EDQGZLGWK OLQHDU WLPH LQYDULDQW EDQGSDVV nOWHUV ZLWK D FOHYHUO\ FKRVHQ EDQGZLGWK DQG DQ RSWLPDO GHFLPDWLRQ RI WKH RXWSXW GDWD UDWH 7KH VSHFWUDO UHVSRQVH RI WKH EDQGSDVV nOWHUV LV W\SLFDOO\ QRW YHU\ JRRG 7KLV LV D GLUHFW FRQVHTXHQFH RI PDNLQJ WKHVH ),5 nOWHUV UHODWLYHO\ VKRUW LQ GXUDWLRQ LH IHZ nOWHU ZHLJKWV



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

WKHUHIRUH IHZ SURFHVVLQJ PXOWLSOLHV  ,Q 6HFWLRQV  DQG  ZH DVVXPHG WR KDYH D XQLW JDLQ LQ WKH SDVVEDQG UHJLRQ 7KLV LV XVXDOO\ QRW WUXH UDWKHU WKDQ WKDW WKH JDLQ LQFUHDVHV DV WKH IUHTXHQF\ EDQGV VWHS WRZDUGV WKH '& UHJLRQ $ ZKLWH LQSXW QRLVH SURFHVV ZRXOG WHQG WR KDYH WKH VDPH SRZHU HQHUJ\ LQ HDFK GHWDLO VHTXHQFH 

0DOODW V $OJRULWKP

-XVW DV LQ WKH FDVH RI WKH ')7 WKHUH LV D FOHYHU ZD\ WR VSHHG XS WKH FRPSX WDWLRQ RI WKH ZDYHOHW WUDQVIRUP $JDLQ ZH XVH )LJXUH  WR KHOS ZLWK WKH GHVFULSWLRQ :H QRWH WKDW H[WHQVLRQV WR DQ\ VL]H 1 LV VWUDLJKWIRUZDUG LH MXVW DGG PRUH EDQGSDVV nOWHUV KHQFH VFDOHV  :H VWDUW ZLWK D EDQGSDVV DQG ORZ SDVV nOWHU VHFWLRQ WKDW VSOLWV WKH VSHFWUDO UHJLRQ LQWR HTXDO VHJPHQWV )LJ XUH  VKRZV WKH GHWDLOV RI WKH DOJRULWKP ZKLFK LV DOVR NQRZQ DV 0DOODW V DOJRULWKP >@ $OO KLJK SDVV +3 nOWHUV DUH LGHQWLFDO DQG DOO ORZ SDVV /3 nOWHUV DUH LGHQWLFDO 7KH QXPEHU RI nOWHU ZHLJKWV LV DQ HYHQ QXPEHU 7KH +3 nOWHU LV MXVW WKH /3 nOWHU VSHFWUDOO\ UHORFDWHG WR WKH IROG RYHU IUH TXHQF\ 7KDW LV WKH ORZ SDVV nOWHU WUDQVIHU IXQFWLRQ LV PRGXODWHG WR WKH IROG RYHU IUHTXHQF\ DQG WKH ZHLJKWV DUH WLPH UHYHUVHG LH PLUURUHG K+3 Q b Q K/3 0 b  b Q IRU Q   c c c  0 b  JLYHQ WKDW WKH ),5 nOWHUV KDYH 0 ZHLJKWV 7\SLFDOO\ WKH nOWHUV KDYH YHU\ IHZ ZHLJKWV ZKLFK IRU D JLYHQ ZDYHOHW WUDQVIRUP GR QRW FKDQJH LQ QXPEHU QRU LQ YDOXH 7KH QXPEHU RI ZHLJKWV LV DW OHDVW WZR DQG PD\EH DV KLJK DV  WR  7KH ODUJHU WKH QXPEHU RI ZHLJKWV WKH EHWWHU WKH nOWHU V VSHFWUDO FKDUDFWHULVWLFV EXW WKH SRRUHU LWV DELOLW\ WR ORFDOL]H D SRVLWLRQ LQ WLPH 7KHUH DUH FRQVWUDLQWV RQ WKH nOWHU ZHLJKWV LI UHFRQVWUXFWLRQ LH WKH LQYHUVH ZDYHOHW WUDQVIRUP LV GHVLUHG >^@ )RU WKH SXUSRVHV RI WKLV ERRN JRRG FKDUDFWHULVWLFV LQ WKH ZDYHOHW GRPDLQ DUH GHVLUDEOH ZKLOH ZH DUH OHVV LQWHUHVWHG LQ XVLQJ WKH LQYHUVH ZDYHOHW WUDQVIRUP ,:7  ,Q PRVW FDVHV ZH XVH D UHDO YDOXHG LPSXOVH UHVSRQVH IRU WKH nOWHUV EXW ZH QRWH WKDW WKHUH DUH VHYHUDO ZDYHOHW FDQGLGDWHV DOORZLQJ FRPSOH[ YDOXHG ZHLJKWV :H ZDQW WR DOVR FKHFN RXW WKH FRPSXWDWLRQDO EXUGHQ RI WKH :7 )URP WKH SUHYLRXV GLVFXVVLRQ LW VKRXOG EH FOHDU WKDW RQH QHHGV ORJ 1 VWDJHV ZKHUH HDFK VWDJH FRQVLVWV RI D EDQGSDVV DQG D ORZ SDVV nOWHU (DFK nOWHU DW WKH VWDUW RI WKH FKDLQ WUHH SURFHVVHV 1 VDPSOHV XVLQJ 0 UHDO YDOXHG FRHpFLHQWV LQ ERWK WKH +3 DQG /3 nOWHUV 7KH FRPSXWDWLRQDO FRVW LQ WKH nUVW VWDJH LV  c 0 c 1  $V ZH SURFHHG WR WKH QH[W VWDJH ZH QRWLFH WKDW WKH nOWHU VL]HV VWD\ WKH VDPH EXW WKH QXPEHU RI VDPSOHV KDV EHHQ UHGXFHG WR DERXW 1  +HQFH WKH FRVW RI PXOWLSOLFDWLRQ RI WKH VHFRQG VWDJH LV  c 0 c 1  $GGLQJ XS DOO WKH FRVWV PXOWLSOLFDWLRQV IRU WKH ZKROH WUHH ZH JHW LQ JHQHUDO &RVW :7 01         c c c  1  01

t

u

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ

)LJXUH 

0DOODW V DOJRULWKP IDVW ZDYHOHW WUDQVIRUP 





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)RU RXU FDVH ZH KDYH &RVW :7 › 0  0 7KH JHQHUDO UHVXOW LV YDOLG IRU DQ\ VL]H RI GDWD VHW DVVXPLQJ LW LV LQSXWHG LQ VHJPHQWV RI VL]H DV D SRZHU RI WZR ,I ZH FRPSDUH WKH PXOWLSOLFDWLRQ SURFHVVLQJ FRVW RI WKH ))7 YHUVXV WKH IDVW :7 ZH VHH WKDW IRU WKH ))7 ORJ 1 1 ))7 FRPSOH[ YDOXHG PXOWLSOLFDWLRQV &RVW ))7   ORJ 1 1 ))7 UHDO YDOXHG PXOWLSOLFDWLRQV ZKHUH 1 ))7 VL]H GDWD VL]H ZKLOH IRU WKH IDVW ZDYHOHW WUDQVIRUP :7 &RVW 0 1 :7 UHDO YDOXHG PXOWLSOLFDWLRQV ZKHUH 0 LV nOWHU VL]H 1 GDWD VL]H )RU VPDOO 0 WKH VPDOOHVW 0 SRVVLEOH LV WZR FRUUHVSRQGLQJ WR WKH +DDU ZDYHOHW ZKLFK LV DOVR FDOOHG 'DXEHFKLHV RI RUGHU  DQG ODUJH 1  WKH :7 FDQ HDVLO\ RXWSHUIRUP WKH ))7 :H ZLOO JHW PRUH H[SRVXUH WR WKH :7 LQ &KDSWHU  DQG LQ $SSHQGL[ (

t



u

6800$5
@ /XGHPDQ /& )XQGDPHQWDOV RI 'LJLWDO 6LJQDO 3URFHVVLQJ 1HZ @ (OOLRWW ') DQG 5DPDPRKDQ5DR . )DVW 7UDQVIRUPV $OJRULWKPV $QDO\VLV $SSOLFDWLRQV 2UODQGR )/ $FDGHPLF 3UHVV 

,QWURGXFWLRQ WR 6LJQDO 3URFHVVLQJ



>@ %XUUXV &6 *RSLQDWK 5$ DQG *XR + ,QWURGXFWLRQ WR :DYHOHWV DQG :DYHOHW 7UDQVIRUPV $ 3ULPHU 8SSHU 6DGGOH 5LYHU 1- 3UHQWLFH +DOO  >@ 6WUDQJ * /LQHDU $OJHEUD DQG ,WV $SSOLFDWLRQV QG HG 1HZ @ +DUULV )* ?7KH GLVFUHWH )RXULHU WUDQVIRUP DSSOLHG WR WLPH GRPDLQ VLJQDO SURFHVVLQJ ,((( &RPPXQLFDWLRQV 0DJD]LQH 0D\  SS ^ >@ 1REOH % DQG 'DQLHO -: $SSOLHG /LQHDU $OJHEUD UG HG (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ +DUULV )* ?2Q WKH XVH RI ZLQGRZV IRU KDUPRQLF DQDO\VLV ZLWK WKH GLVFUHWH )RXULHU WUDQVIRUP 3URF ,((( 9RO  1R  SS ^ -DQ  >@ .D\ 60 0RGHUQ 6SHFWUDO (VWLPDWLRQ 7KHRU\ DQG $SSOLFDWLRQV (Q JOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ 0DUSOH 6/ 'LJLWDO 6SHFWUDO $QDO\VLV (QJOHZRRG &OLmV 1- 3UHQWLFH +DOO  >@ 6WRLFD 3 DQG 0RVHV 5 ,QWURGXFWLRQ WR 6SHFWUDO $QDO\VLV 8SSHU 6DG GOH 5LYHU 1- 3UHQWLFH+DOO  >@ 1DLGX 36 0RGHUQ 6SHFWUXP $QDO\VLV RI 7LPH 6HULHV %RFD 5DWRQ )/ &5& 3UHVV  >@ 7KHUULHQ &: 'LVFUHWH 5DQGRP 6LJQDOV DQG 6WDWLVWLFDO 6LJQDO 3UR FHVVLQJ (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ 'DXEHFKLHV , ?2UWKRQRUPDO EDVHV RI FRPSDFWO\ VXSSRUWHG ZDYHOHWV &RPPXQLFDWLRQV RQ 3XUH DQG $SSOLHG 0DWKHPDWLFV 9RO  SS ^  1RY  >@ 'DXEHFKLHV , ?7HQ OHFWXUHV RQ ZDYHOHWV 6,$0 3KLODGHOSKLD 3$  >@ %XUUXV &6 *RSLQDWK 5$ DQG *XR + ,QWURGXFWLRQ WR :DYHOHWV DQG :DYHOHW 7UDQVIRUPV $ 3ULPHU 8SSHU 6DGGOH 5LYHU 1- 3UHQWLFH +DOO  >@ 6WUDQJ * DQG 1JX\HQ 7 :DYHOHWV DQG )LOWHU %DQNV :HOOHVOH\ 0$ :HOOHVOH\&DPEULGJH 3UHVV  >@ 9HWWHUOL 0 DQG .RYD›FHYLuF - :DYHOHWV DQG 6XEEDQG &RGLQJ (QJOH ZRRG &OLmV 1- 3UHQWLFH+DOO 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

>@ 0DOODW 6* ?$ WKHRU\ IRU PXOWLUHVROXWLRQ VLJQDO GHFRPSRVLWLRQ 7KH ZDYHOHW UHSUHVHQWDWLRQ ,((( 7UDQV RQ 3DWWHUQ 5HFRJQLWLRQ DQG 0D FKLQH ,QWHOOLJHQFH 9RO  1R  SS ^ -XO\  >@ 0DOODW 6* ?0XOWLUHVROXWLRQ DSSUR[LPDWLRQ DQG ZDYHOHW RUWKRQRUPDO EDVHV RI /  7UDQV RI $PHU 0DWK 6RF 9RO  SS ^  >@ 9DLG\DQDWKDQ 33 0XOWLUDWH 6\VWHPV DQG )LOWHU %DQNV (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO 

&KDSWHU 

+\SRWKHVLV 7HVWLQJ 

,1752'8&7,21

+\SRWKHVLV WHVWLQJ DV SUHVHQWHG LQ WKLV FKDSWHU GHDOV ZLWK D nQLWH QXPEHU RI VDPSOHV 7KH VDPSOHV DUH DOVR FDOOHG GDWD RU REVHUYDWLRQV 7KH GDWD LV XVHG WR GHFLGH ZKLFK K\SRWKHVLV WKDW LV ZKLFK VLJQDO RU V\PERO LV WUXH LH ZKLFK RQH LV SUHVHQW DQG KHQFH ZDV WUDQVPLWWHG 7KURXJKRXW 6HFWLRQV ^ ZH DVVXPH WKDW DOO SDUDPHWHUV DUH NQRZQ 7KDW LV ZH NQRZ ZKHUH DQG ZKHQ WR ORRN IRU WKH VLJQDOV ,Q WKH n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n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



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

SUREDELOLW\ ODZ $Q\ W\SH RI SUREDELOLW\ GHQVLW\ IXQFWLRQ LV DOORZHG ,Q PRVW SUREOHPV ZH ZDQW WR GHWHFW WKH SUHVHQFH RU DEVHQFH RI D VLJQDO ZLWK D FHUWDLQ PHDVXUH RI UHOLDELOLW\ 2I FRXUVH XVXDOO\ GHWHFWLRQ DQG HVWLPDWLRQ DUH SHUIRUPHG ZKHQ DGGLWLYH QRLVH LV SUHVHQW ,I LW ZDV QRW IRU WKH QRLVH GLVWXUEDQFH WKHQ LW ZRXOG EH QR SUREOHP WR GHFLSKHU WKH PHVVDJH V\PERO \HVQR ]HURRQH HWF  2QH ZRXOG VLPSO\ QHHG WR GHFLGH ZKLFK RQH RI WKH WZR K\SRWKHVHV + RU + LV WUXH LH ZKLFK VLJQDO LV SUHVHQW E\ MXVW UHDGLQJ LW Rm WKH GDWD ,Q DOO FDVHV D GHWHFWLRQ VFKHPH LQ WHUPV RI WKH VLPSOHVW IXQFWLRQ RU IXQF WLRQDO RI WKH REVHUYDWLRQ LH WKH UHFHLYHG GDWD LV DWWHPSWHG 7KLV OHDGV LQ PDQ\ FDVHV WR ZKDW LV FDOOHG D VXpFLHQW VWDWLVWLF ZKLFK FRQWDLQV DOO WKH LQIRU PDWLRQ DV IDU DV WKH SUREOHP RI LQWHUHVW LV FRQFHUQHG EXW LV QXPHULFDOO\ DQG DQDO\WLFDOO\ HDVLHU WR GHDO ZLWK WKDQ WKH RULJLQDO YDULDEOH RU YDULDEOHV 5HGXF WLRQ WR D VLPSOH IRUP RI WKH REVHUYDWLRQ DOVR HQVXUHV WKDW WKH FRPSXWDWLRQDO Hm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mHUHQW SUREDELOLW\ GHQVLW\ IXQFWLRQ ,Q WKHVH FDVHV RQH HVWDEOLVKHV GHFLGHV ZKLFK JHQHUDWLQJ 3') HYHQW K\SRWKHVLV SURGXFHV WKH REVHUYDWLRQ GDWD  %HIRUH WKH EDVLF GHWHFWRU VWUXFWXUH LV GHULYHG WKH FRQFHSW RI WKH ?D SRVWH ULRUL SUREDELOLW\ LV H[DPLQHG 6XSSRVH WZR GLmHUHQW HYHQWV DUH SRVVLEOH VXFK DV D ELQDU\ ?RQH RU ?]HUR LH WDUJHW SUHVHQW RU QRW  7KH OHWWHU L L   LV XVHG WR LQGLFDWH ZKLFK K\SRWKHVLV HYHQW LV WUXH 7KH SUREDELOLW\ RI WKH HYHQW ?L LH K\SRWKHVLV +  JLYHQ WKH REVHUYDWLRQ \ LV GHVFULEHG E\ WKH ?D SRVWHULRUL SUREDELOLW\ 3UI+ M\J 7KH ?D SULRUL SUREDELOLW\ RI WKH HYHQW ?L LH SUREDELOLW\ RI K\SRWKHVLV + LV JLYHQ E\ 3UI+ J 7KH VLWXDWLRQ LV EHVW LOOXVWUDWHG XVLQJ WKH JHQHUDO VLJQDO o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

L

L

L

+\SRWKHVLV 7HVWLQJ



)LJXUH  %LQDU\ WUDQVPLVVLRQ FKDQQHO 8VLQJ %D\HV UXOH WKLV LV HDVLO\ REWDLQHG DV >\M+ @ 3U>+ @  3U >+ M\@ 3U3U> \@ ZKHUH \ LV JLYHQ E\ \ \  \  c c c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oRZ GLDJUDP LV JLYHQ LQ )LJXUH  3UI5J 3UI5M7 J 3UI7 J  3UI5M7 J 3UI7 J  d    d   3UI5J 3UI5M7 J 3UI7 J  3UI5M7 J 3UI7 J  d    d      3UI7 M5J 3UI5M7 J 3UI7 J 3UI5J  d    3UI7 M5J 3UI5M7 J 3UI7 J 3UI5J  d    L

L

1

7

L

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



)LJXUH  6LJQDO oRZ GLDJUDP IRU ([DPSOH 

)LJXUH  7UHOOLV UHSUHVHQWDWLRQ

1RWH 3UI7 5J 3UI7 M5J 3UI5J  d   3UI7 5J 3UI7 M5J 3UI5J  d   1RWH $QRWKHU ZD\ RI REWDLQLQJ WKH UHVXOWV LV E\ XVLQJ D WUHOOLV UHSUHVHQWDWLRQ DV VKRZQ LQ )LJXUH  )RU H[DPSOH WKH SUREDELOLW\ RI UHFHLYLQJ D RQH LV 3UI5J  d    d  DQG RI FRXUVH DV EHIRUH ZH FDQ FRPSXWH 3UI7 M5J 3UI5M7 J 3UI7 J 3UI5J  c    8VXDOO\ RQH QHHGV WR HYDOXDWH KRZ ZHOO D GHWHFWLRQ VFKHPH GHWHFWRU ZRUNV ,Q WKH ELQDU\ GHWHFWLRQ VLWXDWLRQ IRXU SUREDELOLWLHV DUH RI LQWHUHVW 7KHUH DUH WZR SURSHU DQG WZR LPSURSHU GHWHFWLRQV 7KH SURSHU GHWHFWLRQ RI WKH ?RQH K\SRWKHVLV LV GHQRWHG E\ 3  7KH WZR SRVVLEOH HUURUV DUH GHQRWHG E\ 3 DQG 3  3 FRUUHVSRQGV WR WKH GHWHFWLRQ RI WKH ?RQH HYHQW JLYHQ WKDW WKH ?]HUR HYHQW LV WUXH DQG 3 FRUUHVSRQGV WR GHWHFWLQJ WKH ?]HUR HYHQW JLYHQ WKDW WKH ?RQH HYHQW LV WUXH 7KH VXP RI 3 DQG 3 PXVW '

)$

0

)$

0

'

0

+\SRWKHVLV 7HVWLQJ



)LJXUH  3UREDELOLWLHV RI ELQDU\ GHWHFWLRQ HTXDO  ,Q WKH VWDWLVWLFDO OLWHUDWXUH 3  3  DQG 3 DUH DOVR GHQRWHG E\ WKH HUURU RI WKH nUVW NLQG HUURU RI WKH VHFRQG NLQG DQG SRZHU RI WKH WHVW UHVSHFWLYHO\ 7R LOOXVWUDWH WKHVH GHn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nQG VLPLODU H[SUHVVLRQV DQG DGGLWLRQDO H[DPSOHV LQ UHIHUHQFHV >^@ )$

0

'

)$

0

'

)$



%$3 & b & I \ b 3 & b & I \ @ G\ ^]

YDULDEOH FRVW WR EH RSWLPL]HG

`

 (YHU\ LQGLYLGXDO WHUP LQ  LV QRQQHJDWLYH ZLWK WKH nUVW WZR WHUPV FRQ VWLWXWLQJ D JLYHQ n[HG SRVLWLYH FRVW 7KH RSWLPDO VWUDWHJ\ WR PLQLPL]H WKH RYHUDOO DYHUDJH FRVW LV WR VHOHFW 5 WR PDNH WKH FRQWULEXWLRQ IURP WKH LQ WHJUDO PLQLPDO RU LI SRVVLEOH QHJDWLYH 7KLV LV DFFRPSOLVKHG E\ DVVLJQLQJ \ WR 5 KHQFH VD\ K\SRWKHVLV + LV WUXH ZKHQ WKH QHJDWLYH WHUP LQ WKH LQWH JUDO LV GRPLQDQW UHGXFLQJ WKH n[HG FRVW &RQYHUVHO\ LI WKH nUVW WHUP LQ WKH LQWHJUDQG LV GRPLQDQW \ LV DVVLJQHG WR EHORQJ WR UHJLRQ 5 VD\ K\SRWKHVLV + LV WUXH  KHQFH QR DGGLWLRQDO FRVW LV DGGHG WR WKH n[HG FRVW 7KH GHFLVLRQ FULWHULRQ EHFRPHV

+ 3 >& b &@ I \ ! 3 >& b &@ I \ +

2U LQ PRUH FRQYHQWLRQDO IRUP WKH %D\HV IRUPXODWLRQ EHFRPHV

+ I \ ! 3 >& b & @ I \  3 >& b & @ +

%D\HV GHWHFWRU 



1RWH DOO SULRU SUREDELOLWLHV DQG DOO WHUPV ZLWKLQ WKH VTXDUH EUDFNHWV DUH SRVL WLYH KHQFH WKH UHDUUDQJHPHQW DFURVV WKH LQHTXDOLWLHV LV VLPSOH 7KH TXDQWLW\ RQ WKH OHIWKDQG VLGH LV FDOOHG WKH OLNHOLKRRG UDWLR /5 DQG LV GHQRWHG E\

+\SRWKHVLV 7HVWLQJ



e \  ZKLOH WKH ULJKWKDQG TXDQWLW\ LV GHQRWHG E\ w D SRVLWLYH VFDODU 6R ZH FDQ ZULWH

+ e \ !  w +

ZKHUH



w 33 && bb && (TXDWLRQ  LV LQWHUSUHWHG DV IROORZV ,I e \ ! w GHFLGH RQ +  FRQYHUVHO\ LI e \  w GHFLGH RQ +  ,Q WKH XQOLNHO\ HYHQW WKDW WKHUH LV DQ HTXDOLW\

LH WKH OHIWKDQG VLGH RI  HTXDOV WKH ULJKWKDQG VLGH WKHQ RQH DUELWUDULO\ FDQ DVVLJQ DQ\ K\SRWKHVLV RU OHW QDWXUH GHFLGH E\ oLSSLQJ DQ XQELDVHG FRLQ DQG DVVLJQ WKH GHFLVLRQ DFFRUGLQJ WR WKH FRLQ WRVV 1RWH ERWK VLGHV RI WKH LQHTXDOLW\ DUH QRQQHJDWLYH KHQFH LI RQH WDNHV WKH ORJDULWKP RI ERWK VLGHV LH PRQRWRQLF IXQFWLRQ  WKH LQHTXDOLW\ LV SUHVHUYHG 7KH QDWXUDO ORJDULWKP RI WKH OLNHOLKRRG UDWLR /5 LV GHQRWHG E\ //5

+ OQ e \ !  OQ w +



([DPSOH  8QGHU +  D FRQVWDQW SRVLWLYH YROWDJH VD\ ?P YROWV LV WUDQV PLWWHG RYHU D ZLUH 8QGHU +  QRWKLQJ LH  YROW LV WUDQVPLWWHG 1 VDPSOHV DUH XVHG LQ PDNLQJ D GHFLVLRQ $GGLWLYH *DXVVLDQ QRLVH Q z 1  }  LLG LV LQWURGXFHG E\ WKH FKDQQHO VHH )LJXUH  IRU D W\SLFDO OLNHOLKRRG IXQFWLRQ SORW ZKHQ 1  IRU Q  c c c  1 +  \ Q  Q

Q

+  \

Q

Q

PQ 

IRU Q  c c c  1

Q

7KH MRLQW 3') RI WKH QRLVH LV JLYHQ E\ 

I1Q {}

1

H[S b 

Q } 

1 ; Q

Q

/RRNLQJ DW \  ZH UHDOL]H WKDW LW LV WKH VXP RI WKH UDQGRP YDULDEOH Q DQG WKH GHWHUPLQLVWLF TXDQWLW\ P KHQFH \ KDV D *DXVVLDQ GLVWULEXWLRQ $OO WKDW ZH QHHG WR nQG LV WKH PHDQ DQG YDULDQFH RI \ WR KDYH D FRPSOHWH VWDWLVWLFDO GHVFULSWLRQ IRU WKLV UDQGRP YDULDEOH 7KH PHDQ RI \ LV JLYHQ E\ ( \ ( P  Q P Q

Q

Q

Q

Q

Q

Q



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH  /LNHOLKRRG IXQFWLRQV RI ([DPSOH 

7KH YDULDQFH LV JLYHQ E\ }Q ( \ b (\  ( P  Q b P  (Q  } +HQFH WKH LQGLYLGXDO 3')V IRU WKH QWK VDPSOH JLYHQ + RU + DUH WUXH DUH JLYHQ E\  H[S b \ b P  I\ } {}     H[S b \ I\ }  {}    $FFRXQWLQJ IRU DOO WKH REVHUYDWLRQV WKLV EHFRPHV Q

\

Q

Q

Q

Q

Q

Q

1
@ DQG $EUDPRZLW] DQG 6WHJXQ >@ 7KH HUURU LV OHVV WKDQ     OLPLWLQJ WKH XVHIXOQHVV VRPHZKDW 7KDW LV IRU SUREOHPV ZKHUH WKH DUHD XQGHU WKH WDLO VHFWLRQ RI D *DXVVLDQ LV RQ WKH RUGHU RI WKLV HUURU RU VPDOOHU PRUH VRSKLVWLFDWHG SURFHGXUHV PXVW EH XVHG ([DPSOH  *LYHQ WKH VDPH VFHQDULR DV LQ WKH SUHYLRXV SUREOHP H[FHSW WKDW QRZ WKHUH LV DFFHVV WR QLQH LQGHSHQGHQW VDPSOHV LH 1   8QGHU +  \ z 1    Q   c c c   +  \ z 1    Q   c c c   )URP ([DPSOH  ZH KDYH '

b

Q Q

]

+ \ ! 1 P  c    +

1 ;

Q

Q

6LQFH \ LV *DXVVLDQ XQGHU HDFK K\SRWKHVLV LW LV REYLRXV WKDW ] ZLOO EH *DXV VLDQ XQGHU HDFK K\SRWKHVLV 7R FKDUDFWHUL]H I ]  IRU L   DOO WKDW LV QHHGHG LV WKH PHDQ RI ] XQGHU HDFK K\SRWKHVLV DQG WKH YDULDQFH RI ]  7KH PHDQV DUH Q

L

( ]

 ;

( ]

( \

c 

( \

 c  

Q

Q



 ;

Q

Q



ZKHUH WKH VXEVFULSW RI ( LQGLFDWHV ZKLFK VLJQDO LV DVVXPHG WR EH SUHVHQW LH ZKLFK K\SRWKHVLV LV WUXH 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

:H UHDOL]H WKDW WKH '& FRPSRQHQW KDV QR HmHFW RQ WKH YDULDQFH FRPSX WDWLRQ KHQFH ZH FRPSXWH WKH YDULDQFH IRU WKH ]HUR K\SRWKHVLV LH QR '& FRPSRQHQW  7KHUHIRUH WKH VHFRQG PRPHQW HTXDOV WKH YDULDQFH

} ( ] ]

(

  ; ;

 ; Q

6R XQGHU +HQFH WKH WHVW EHFRPHV

\ \ Q

Q



P 

P

}  c   \

+  ] z 1   +  ] z 1   + ] !  +

DQG WKH GHWHFWLRQ SHUIRUPDQFH EHFRPHV u t u t =       b  4 b  {  3 I] G] 4    u t = 3 I ] G] 4  4  c     1RWH WKDW ZH FRXOG KDYH XVHG D QRUPDOL]HG VWDWLVWLF WKDW LV WKH VDPSOH PHDQ 7KHQ WKH WHVW DQG WKUHVKROG ZRXOG EH DV IROORZV 

'





b

)$



+

 ;  6DPSOH PHDQ Z 1 ]  \ !    Q

Q

VR

+ Z !  +

+

7KH VDPSOH PHDQ ZLOO EH  DQG  XQGHU WKH ]HUR DQG RQH K\SRWKHVLV UHVSHF WLYHO\ 7KH YDULDQFH XQGHU HDFK K\SRWKHVLV ZLOO EH  6R ZH KDYH +  Z z 1    +  Z z 1   

+\SRWKHVLV 7HVWLQJ



2I FRXUVH WKH GHWHFWLRQ SHUIRUPDQFH RI WKH VWDWLVWLFV RI Z UHODWLYH WR WKH ] GHWHFWLRQ VWDWLVWLF ZLOO QRW FKDQJH DW DOO DV WKH LQWHUHVWHG UHDGHU PLJKW TXLFNO\ YHULI\ ([DPSOH  $Q H[SHULPHQW LV SHUIRUPHG DQG WKH QXPEHU RI HYHQWV DUH FRXQWHG 7KH QXPEHU RI HYHQWV REH\ D 3RLVVRQ ODZ 8QGHU +  3UIQ HYHQWVJ P Q H[S bP +  3UIQ HYHQWVJ P Q H[S bP  IRU Q     c c c 7KH %D\HV GHWHFWRU LV JLYHQ E\ WKH OLNHOLKRRG UDWLR WHVW /57 ZKLFK LV FRP SDUHG WR WKH WKUHVKROG w ZKLFK GHSHQGV RQ SULRU SUREDELOLWLHV DQG WKH FRVW DVVRFLDWHG ZLWK HDFK GHFLVLRQ )RU WKLV H[DPSOH ZH REWDLQ Q Q

Q HYHQWVM+ e Q 3U 3U Q HYHQWVM+

t

P u H P Q

b P

P

+ ! w  +

7KH ORJ OLNHOLKRRG UDWLR WHVW //57 EHFRPHV

+ u P  Q OQ P  P b P ! OQ w + t

'HSHQGLQJ RQ ZKLFK SDUDPHWHU LV WKH ODUJHU RQH WKH PXOWLSOLHU DVVRFLDWHG ZLWK Q LV SRVLWLYH RU QHJDWLYH 7KH GHWHFWLRQ VFKHPH EHFRPHV LI P ! P  WKHQ + Q ! OQ w tPPbuP + OQ P LI P ! P  WKHQ



+ Q ! OQ w tPPbuP + OQ P

$Q HYHQW PD\ RU PD\ QRW KDYH RFFXUUHG 7KH FRVWV DQG SULRU SUREDELOLWLHV DUH JLYHQ E\ & &  & &  DQG 3 3  UHVSHFWLYHO\ +HQFH WKH WKUHVKROG w  1 HTXDOV  WKDW LV RQH VDPSOH LV DYDLODEOH 7KH GHQVLW\ IXQFWLRQV DUH WZRVLGHG H[SRQHQWLDO GHQVLWLHV 7KH ([DPSOH 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

FRQGLWLRQDO 3')V DUH JLYHQ E\

  H[S b M\M  H[S b M\M  

I\ I\ 7KH OLNHOLKRRG UDWLR WHVW EHFRPHV

+

 !  H[SM\M b M\ M  7

e\



+

)XUWKHU VLPSOLnFDWLRQ LH WDNH QDWXUDO ORJDULWKPV OHDGV WR

+ !  OQ  M\M  +

]



(TXLYDOHQWO\ WKLV FDQ DOVR EH ZULWWHQ DV ,I b  \   WKHQ GHFLGH RQ +  RWKHUZLVH GHFLGH RQ +  7KH QXPHULFDO YDOXHV IRU WKH SHUIRUPDQFH FDQ EH REWDLQHG IURP WKH VLPSOH H[SRQHQWLDO LQWHJUDOV JLYHQ EHORZ  = \ 3 H[S b G\     

'

=

3 3

)$

0







H[S b \ G\

b3

'

$Q HYHQW PD\ RU PD\ QRW KDYH RFFXUUHG 7KH FRVWV DQG SULRU SUREDELOLWLHV DUH JLYHQ E\ & &  & &  DQG 3 3  UHVSHFWLYHO\ +HQFH WKH WKUHVKROG w  1 LQGHSHQGHQW VDPSOHV DUH DYDLODEOH 7KH GHQVLW\ IXQFWLRQV DUH RQHVLGHG H[SRQHQWLDO GHQVLWLHV 7KH FRQGLWLRQDO 3')V JLYHQ WKDW + RU + DUH WUXH DUH JLYHQ E\ I\ H[S b \ 8 \ \  I\  H[S b  8 \ ([DPSOH 

Q

Q

Q

Q

Q

Q

+\SRWKHVLV 7HVWLQJ



$FFRXQWLQJ IRU DOO WKH REVHUYDWLRQV WKLV EHFRPHV 1
b  { 1

t



t

3U VD\ + M+ LV WUXH  ] b P  1 @ G] 4 rs b P S1 s  u  H[S>b  

{ 1

3U VD\ + M+ LV WUXH  b 3U VD\ + M+ LV WUXH

+\SRWKHVLV 7HVWLQJ =

s

b



 ] b P  1 @ G] H[S> b u  t  { 1 r

S

s

HUI s b P 1 S  b 4Q b P 1 ([DPSOH  *LYHQ WKDW 1 VDPSOHV RI \ LH \  \  c c c \ DUH *DXVVLDQ 1

LLG UDQGRP YDULDEOHV ZLWK ]HUR PHDQ DQG D YDULDQFH ZKLFK FRQYH\V WKH PHV VDJH ZH ZDQW WR GHWHFW ZKLFK RQH RI WZR PHVVDJHV ZDV WUDQVPLWWHG 'HVLJQ WKH GHWHFWRU XVLQJ WKH %D\HV HUURU FULWHULRQ 8QGHU +  \ KDV YDULDQFH }  IRU Q   c c c  1 +  \ KDV YDULDQFH }  IRU Q   c c c  1 ZKHUH } ! }  3 3  & &  & &  DQG w  KHQFH OQ w  8VLQJ  RU HTXLYDOHQWO\ %D\HV WHVW IURP ([DPSOH  ZH KDYH Q Q

+ t u    }  }  ! \  } b } 1 OQ }} ] ]     + 6LQFH WKH REVHUYDWLRQV \ DUH LLG *DXVVLDQ UDQGRP YDULDEOHV WKH UDQGRP YDULDEOH ] LV  FKLVTXDUHG ZLWK 1 GHJUHHV RI IUHHGRP 7KH SUREDELOLW\ GHQVLW\ IXQFWLRQ IRU ] LV VHH $SSHQGL[ $ DQG >@    I ]  ]}b1  H  L  IRU L   1 ;

Q

Q

Q

1

L

1

b

b]

}

M

7KH GHWHFWLRQ VWDWLVWLFV DUH JLYHQ E\

3

=



)$

3

]

=

]

0

3

b

I] G] I] G]

b3 7R HYDOXDWH WKHVH H[SUHVVLRQV ZH QHHG WR KDYH WDEOHV RI 3HDUVRQ V LQFRPSOHWH JDPPD IXQFWLRQ >@ RU XVH D QXPHULFDO DSSUR[LPDWLRQ WHFKQLTXH WKDW UHTXLUHV HYDOXDWLRQ RQ D FRPSXWHU '

0



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH  $YHUDJH FRVW YHUVXV !  

0,10$; &5,7(5,21

,I WKH FRVW LQIRUPDWLRQ LH & L M   LV DYDLODEOH EXW QRW WKH ?D SULRUL SUREDELOLW\ 3 RU 3  WKHQ WKH EHVW %D\HV UHODWHG GHWHFWLRQ VFKHPH XQGHU WKH ZRUVW SRVVLEOH FKRLFH RI +  L   LV GHVLUDEOH 7KLV DSSURDFK DVVXPHV WKDW QDWXUH SOD\V WKH UROH RI DQ DGYHUVDU\ DQG PRYHV RU SLFNV WKH ZRUVW SRVVLEOH YDOXH IRU 3 DQG 3 LQ RUGHU WR PDNH WKH DYHUDJH FRVW WKH ODUJHVW RQH SRVVLEOH :H NQRZ WKLV SKHQRPHQD XQGHU D GLmHUHQW QDPH 0DQ\ WLPHV ZH FDOO LW 0XUSK\ V ODZ ZKLFK VD\V 0RWKHU 1DWXUH NQRZV ZKDW KXUWV WKH PRVW DQG XVHV LW UXWKOHVVO\ :H FDQ SORW WKH DYHUDJH FRVW YHUVXV WKH D SULRUL SUREDELOLW\ VD\ 3 DV VKRZQ LQ )LJXUH  :H DVVXPH ZLWKRXW ORVV RI JHQHUDOLW\ WKDW WKH FRVWV &  & DUH VPDOOHU WKDQ WKH FRVW RI PDNLQJ DQ HUURU DQG WKH FRVW RI DQ\ GHFLVLRQ LV VRPH QRQ QHJDWLYH QXPEHU ,I ZH WDNH %D\HV FRVW UHDUUDQJLQJ   ZH FDQ UHSODFH WKH LQWHJUDOV XVLQJ WKH V\PEROV RI 3 DQG 3  LM

L

&

3 &

)$

=

3 &

5

0

I\ G\  3 &

=

5

=

I \ G\  3 &

5

I\ G\

= 5

I\ G\

7KH DYHUDJH FRVW %D\HV DSSURDFK EHFRPHV & 3&  3 &  3& b & 3  3 & b & 3 :H QRWH WKDW DW WKH HQG SRLQWV ZH KDYH WKH IROORZLQJ FRQGLWLRQV



0

)$



+\SRWKHVLV 7HVWLQJ



:KHQ 3  LH FHUWDLQW\ RI HYHQW ?]HUR  WKHQ ZH PXVW KDYH 3   7KH DYHUDJH FRVW EHFRPHV & & & b & 3 DQG 3 & q :KHQ 3  LH FHUWDLQW\ RI HYHQW ?RQH  WKHQ ZH PXVW KDYH 3   DQG 3  7KH DYHUDJH FRVW EHFRPHV & &  & b 3 & 3 & ,I ZH DVVXPH WKDW & &  DV LV WKH FDVH LQ PRVW UHDO OLIH VLWXDWLRQV RI LQWHUHVW WKHQ DV ZH SORW WKH DYHUDJH FRVW & DV IXQFWLRQ RI 3 ZH REWDLQ D JUDSK RI WKH IRUP JLYHQ LQ )LJXUH  7KH PD[LPXP RFFXUV ZKHUH WKH VORSH RI WKH FXUYH LV ]HUR 7KH H[FHSWLRQ LV ZKHQ WKH DYHUDJH FRVW EHFRPHV D VWUDLJKW OLQH WKHQ WKH PD[LPXP RFFXUV DW RQH RI WKH HQG SRLQWV 7R REWDLQ WKH PD[LPXP RI WKH FXUYH ZH WDNH D SDUWLDO GHULYDWLYH RI & ZLWK UHVSHFW WR 3 DQG VHW LW WR ]HUR LH ZH nQG ZKHUH WKH VORSH LV ]HUR  6R JLYHQ WKH PD[LPXP DYHUDJH FRVW ZH PLQLPL]H LW KHQFH WKH QDPH 0LQ0D[ FULWHULRQ $IWHU ZH H[SUHVV HYHU\ WHUP LQ WHUPV RI 3  WDNLQJ D SDUWLDO GHULYDWLYH OHDGV WR q

)$

)$

)$

0

0

#& # #3 #3 > b 3 &  3 &  3 & b & 3   b 3 & b & 3 @ 0

)$

6HWWLQJ WKH SDUWLDO GHULYDWLYH WR ]HUR OHDGV WR & b &  & b & 3 b & b & 3 ,I & &  ZH JHW 0

& 3



)$





&3

 7KH GHWHFWRU IRUP XVLQJ WKH %D\HV FRVW IRUPXOD LV WKH %D\HV GHWHFWRU ZLWK D WKUHVKROG w WKDW UHTXLUHV  WR EH WUXH  IRU L    ([DPSOH  7KH FRVW IRU D FRUUHFW GHFLVLRQ LV ]HUR & 7KH QXPEHU RI VDPSOHV LV RQH 1   & & . !  8QGHU +  \   Q  ZKHUH Q z 1   +  \   Q 3  2U LQ WHUPV RI LQWHJUDOV ZH 7KH 0LQ0D[ WHVW EHFRPHV  3 REWDLQ  = H[S \ b   G\ S  = H[S \ b   G\ S   { { 0

)$

LL

)$

w

b

0



w

:LWK HUURU IXQFWLRQ UHSODFHPHQW WKLV EHFRPHV HUI w b  4w b   8VLQJ WKH HUURU IXQFWLRQ WDEOHV ZH VHH WKDW WKH VROXWLRQ ZLOO EH D w



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

RI  ,Q JHQHUDO LI & & DQG & & WKHQ ZH XVH & b & 3 & b & 3 & b & RU LQ WHUPV RI WKH GHWHFWLRQ VWDWLVWLF

0

b

)$

=

3



)$

=

w

b

 

w

I\ G\

I \ G\ && bb &&  && bb &&

1(8 \ b 8  b \ @ &OHDUO\ QHJDWLYH YDOXHG GDWD LV LPSRVVLEOH )$

'

b

b\

+ I\ ! I\ +

RU

H !  >8 \ b 8  b \ @  IRU \ w  &OHDUO\ WKH OHIWKDQG VLGH /+6 ! ULJKWKDQG VLGH 5+6 LI \ !  VLQFH I\ LV ]HUR IRU DUJXPHQWV JUHDWHU WKDQ   b\

,I ZH UHVWULFW RXUVHOYHV WR WKH LQWHUYDO >@ WKHQ

+ ! OQ  b\   +

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



RU

+ \ ! OQ   +

+HQFH WKH GHFLVLRQ UHJLRQV DUH ,I \   LI \ !  LI   \   =

3U ' M+ 3

'

OQ 



=

WKHQ + LV WUXH WKHQ + LV WUXH WKHQ + LV WUXH

H G\ 

=



b\

H G\  b\



 G\    6XSSRVH WKDW ZH ZDQW WR n[ 3U ' M+ DW D OHYHO RI  WKHQ ZH VKLIW WKH RULJLQDO WKUHVKROG OQ DQ LQFUHPHQW WR WKH OHIW VR WKDW =  G\    +HQFH 7  7KH QHZ WKUHVKROG DOVR FKDQJHV 3U ' M+ WR 3U ' M+

OQ 

7

=

3U ' M+ 

 

H G\ 

=

b\



H G\  b\



08/7,3/( +

0 b

5

5 WKH REVHUYDWLRQ VSDFH

L

L



1RZ 3UFKRRVH + M+ GHQRWHG E\ 3U' M+ FDQ EH ZULWWHQ DV L

M

3U' M+ L

L

=

=

M

ccc

M

5

L

I \ G\ M

7KH DYHUDJH FRVW FDQ WKHUHIRUH EH ZULWWHQ DV 0 b

;

&

&3 LL

L

8VLQJ 5

L



= L 5

L

I \ G\ 

5b6

0 b M M  L

b ; 0;

0 b

L

& 3 LM



L

LM

M

= M 5

L

I \ G\ M



5 DQG M

=

I \ G\  M

5

ZH JHW n[HG FRVW

&

] `_ 0 b ;

^

&3 LL

L



0 = ;

0 b

;

L

L

5



'HnQH , \ DV L

L

M L

0 b

;

, \ L

WKHQ & LV JLYHQ E\

3 & M

M

3 & M

M

LM

& n[HG FRVW 

LM

b



& I \



, \ G\



;= 

5

b

MM

0 b

L

& I \ G\ M

LM

L

L

MM

M

7KH nUVW WHUP LQ  LV D n[HG FRVW WHUP ZKLOH WKH VHFRQG WHUP FRQVWLWXWHV D YDULDEOH FRVW $V LQ WKH ELQDU\ GHFLVLRQ FDVH ZH DVVLJQ HDFK REVHUYDWLRQ \ WR WKH UHJLRQ ZKLFK ZLOO PDNH WKH YDULDEOH FRVW WKH VPDOOHVW 7R GR WKLV OHW

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



XV GHnQH RXU GHFLVLRQ UXOH WKDW LV ?FKRRVH WKDW K\SRWKHVLV WKDW FRUUHVSRQGV WR WKH PLQLPXP RI , \ DV WKH FRUUHFW RQH LH VD\ + LV WUXH  'LYLGLQJ , \ RI  E\ I \  ZH JHW WKH DYHUDJHG ZHLJKWHG E\ FRVW DQG SULRU SUREDELOLWLHV OLNHOLKRRG IXQFWLRQ - \ DV IROORZV , \ - \ I\ L

L

L

L

L

L

0 b

;

3 & M

M L

M

ZKHUH

LM

b

& e \ G\ MM



M

e \ II \\  7KH GHFLVLRQ GHWHFWLRQ FULWHULRQ EHFRPHV ?FKRRVH WKH K\SRWKHVLV IRU ZKLFK - \ LV PLQLPXP LH VD\ + LV WUXH  )RU H[DPSOH FKRRVH + LI DOO WKH IROORZLQJ LQHTXDOLWLHV DUH WUXH M

M

L

L

-  - -  -  

;

0 b

M

 M M  0 b

;

 M M 

0 b



;

MM

M

MM

M

MM

;

0 b

M

3 & b & e \ M

 M M  0 b

;

M

3 & b & e \  M

 M M 

M

3 & b & e \  M

0 b

-  -

3 & b & e \ 

M

;

0 b

M

M

MM

3 &

0 b M

M

M



M 0 b

M

3 & b & e \ M

 M M 

MM

M

b

& e \ MM

M



7KH GHFLVLRQ ZLOO EH PDGH LQ D OLNHOLKRRG VSDFH RI GLPHQVLRQ 0 LH DQ 0 GLPHQVLRQDO K\SHU VSDFH  ([DPSOH  6XSSRVH 0  ?

,I

-  - -  - WKHQ \  5



FKRRVH +

,I

-  - -  - WKHQ \  5



FKRRVH +

,I

-  - -  - WKHQ \  5



FKRRVH +

? ?

ZKHUH 7 UHSUHVHQWV WKH ORJLFDO ?DQG RSHUDWLRQ

+\SRWKHVLV 7HVWLQJ



([DPSOH  6XSSRVH RQH RI WKUHH FRQVWDQW VLJQDOV LV SUHVHQW P RU P RU P  7KH VLJQDO LV HPEHGGHG LQ *DXVVLDQ QRLVH LLG z 1  }  ZKHUH 1 GDWD VDPSOHV DUH DYDLODEOH +  \ z 1 P } +  \ z 1 P } +  \ z 1 P } ZKHUH Q    1 P  P  P  3   &  IRU L    &  ZKHQ L M  0LQLPL]H WKH SUREDELOLW\ RI HUURU EDVHG RQ XVLQJ DOO 1 VDPSOHV 8QGHU Q Q Q

L

LL

LM

+  I \ L

1
@ DV IROORZV = I [[M[[ G[  I [M[[ 

b

7R UHPRYH D ULJKWKDQG YDULDEOH ZH PXOWLSO\ E\ WKH FRQGLWLRQDO GHQVLW\ RI WKH YDULDEOH WR EH UHPRYHG ZUW WKH UHPDLQLQJ ULJKWKDQG YDULDEOHV DQG LQWHJUDWH RXW WKH XQGHVLUHG QXLVDQFH SDUDPHWHU DV IROORZV

I [[M[ DQG

I [[

= =

=

I [[M[ [ I [M[ G[



I [[ M[[ I [M[ I [ G[ G[





b

7KHVH LGHQWLWLHV FDQ EH YHULnHG YHU\ HDVLO\ E\ XVLQJ %D\HV UXOH :KHQ QXL VDQFH SDUDPHWHUV DUH LQYROYHG ZH FDOO WKH K\SRWKHVLV WHVW D FRPSRVLWH K\ SRWKHVLV WHVW ,I QR XQNQRZQ SDUDPHWHUV DUH LQYROYHG ZH FDOO WKH K\SRWKHVLV WHVW D VLPSOH K\SRWKHVLV WHVW :H QRWH ZKHQ WKH OLPLWV RI LQWHJUDWLRQ RU VXPPDWLRQ DUH XQGHUVWRRG WKHQ WKH\ DUH VRPHWLPHV OHIW Rm 

1XLVDQFH 3DUDPHWHUV ZLWK .QRZQ RU 8QNQRZQ 3UREDELOLW\ 'HQVLW\ )XQFWLRQ RU ZLWK 8QNQRZQ EXW )L[HG 9DOXHV

7KH OLNHOLKRRG UDWLR WHVW LV WKH RSWLPXP GHWHFWLRQ VWUDWHJ\ WKDW LV RQH FRP SXWHV e\ DQG FRPSDUHV LW WR WKH DSSURSULDWH GHWHFWLRQ WKUHVKROG ZKLFK LV D IXQFWLRQ RI WKH GHWHFWRU LPSOHPHQWDWLRQ LH 0$3 0/ HWF 

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

 7\SH 

:H KDYH

e\ II \\ 

= =

= ccc ccc

=J 

J

 \MJ I J GJ + \MJ I J GJ

! w  +



:KHQ WKH YHFWRU J LV UDQGRP ZLWK XQNQRZQ RU SDUWLDOO\ NQRZQ GHQVLW\ ZH DSSURDFK WKH SUREOHP DV IROORZV 'HVLJQ WKH GHWHFWRU DV IRU 7\SH  DVVXPLQJ WKH ZRUVW FDVH SUREDELOLW\ GHQVLW\ IXQFWLRQ LH WKH XQLIRUP 3') UHo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e \ PD[J I \ 7\SH 

J

+ I \MJA ! w I \MJA  +



ZKHUH JA LV WKH 0/ HVWLPDWH RI WKH SDUDPHWHU IRU K\SRWKHVLV L   7KHVH SUREOHPV DUH EHVW LOOXVWUDWHG XVLQJ H[DPSOHV 7KH H[DPSOHV XVH RQO\ D RQHGLPHQVLRQDO SDUDPHWHU 7KLV VHUYHV ZHOO WR LOOXVWUDWH WKH SULQFLSOH LQYROYHG 7KH H[WHQVLRQ WR PRUH WKDQ RQH QXLVDQFH SDUDPHWHU LV VWUDLJKWIRUZDUG LH UHSODFH WKH RQHGLPHQVLRQDO DYHUDJH ZLWK WKH FRUUHVSRQGLQJ PXOWLGLPHQVLRQDO DYHUDJH L

+\SRWKHVLV 7HVWLQJ



7\SH  NQRZQ QXLVDQFH 3') 6XSSRVH WKDW XQGHU WKH + DVVXPSWLRQ ZH KDYH D VLPSOH K\SRWKHVLV ZKLOH XQGHU WKH + DVVXPSWLRQ ZH KDYH D FRPSRVLWH K\SRWKHVLV 7KLV PHDQV WKDW XQGHU + ZH NQRZ WKH OLNHOLKRRG IXQFWLRQ IRU WKH ]HUR HYHQW H[DFWO\ ZKLOH XQGHU + WKHUH LV XQFHUWDLQW\ DQG D VWDWLVWLFDO GHVFULSWLRQ RI WKH SDUDPHWHU ,Q WKLV H[DPSOH ZH KDYH DV D QXLVDQFH SDUDPHWHU WKH PHDQ RI WKH SURFHVV P 8QGHU +  \ z 1  } +  \ z 1 P } ZKHUH P KDV D *DXVVLDQ 3') JLYHQ E\ P z 1  }  7KH OLNHOLKRRG UDWLR LV JLYHQ E\ ([DPSOH 

P

=

e\

I\ I\



b

I \MP I P GP +!  w I\ +

8VLQJ WKH DYDLODEOH VWDWLVWLFDO LQIRUPDWLRQ WKLV EHFRPHV =  \ b P  S  H[S b P GP + S H[S b }  } {} {}  ! w e\    \ S H[S b } +   {} 7KLV FDQ EH UHGXFHG WR 

b

P

P

+    t t  uu }  ! } }  } OQ w   OQ   } \  } + P

P

P

3UREOHPV RI W\SH  DUH KDQGOHG WKH VDPH ZD\ DV SUREOHPV RI W\SH  WKDW LV XVH ZKDWHYHU VWDWLVWLFDO LQIRUPDWLRQ LV DW KDQG RU DVVLJQ D ZRUVWFDVH RQH 2WKHU WKDQ WKDW WKH SUREOHP RI W\SH  LV KDQGOHG WKH VDPH ZD\ DV SUREOHPV RI W\SH  ([DPSOH  7\SH D 803 W\SH SUREOHP 8QGHU +  \ z 1   +  \ z 1 P  ZKHUH P LV DQ XQNQRZQ SRVLWLYH TXDQWLW\ 7KH /57 LV ZULWWHQ DV

+ u P e\ H[S P\ b  !  w + t



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

$IWHU VRPH VLPSOLnFDWLRQ WKLV FDQ EH H[SUHVVHG DV

+ \ ! P  P OQ w w +

1RWH ZH FRXOG RQO\ UHGXFH LW WR WKLV VLPSOH IRUP VLQFH ZH NQHZ LQ DGYDQFH WKDW P !  DQG WKH WHVW OHIWKDQG VLGH RI WKH LQHTXDOLW\ FDQ EH PDGH LQGHSHQGHQW RI WKH QXLVDQFH SDUDPHWHU )RU WKH 1H\PDQ3HDUVRQ WHVW WKH IDOVH DODUP UDWH LV n[HG DW VRPH GHVLUHG OHYHO RI 3  6LQFH RXU GHWHFWLRQ VWDWLVWLF LV VLPSO\ WKH REVHUYDWLRQ WKH H[SUHVVLRQ IRU WKH 3 EHFRPHV MXVW WKH DUHD XQGHU WKH LQWHJUDO RI I \ IURP w WR LQnQLW\ 7KDW LV =  \ S H[S b G\ 4w 3  {

:H VHH WKDW WKH GHWHUPLQDWLRQ RI WKH WKUHVKROG w LV LQGHSHQGHQW RI WKH QXL VDQFH SDUDPHWHU P DQG WKDW WKH 1H\PDQ3HDUVRQ WHVW LH WKH OHIWKDQG VLGH RI WKH LQHTXDOLW\ LV DOVR LQGHSHQGHQW RI WKH QXLVDQFH SDUDPHWHU P 7KLV VD\V WKDW WKLV WHVW LV D 803 WHVW ZLWK UHVSHFW WR WKH SDUDPHWHU P ([DPSOH  7\SH E JHQHUDOL]HG /57 SUREOHP $ VLPLODU SUREOHP LV JLYHQ LQ ([DPSOH  ZLWK WZR H[FHSWLRQV D ZH GR QRW NQRZ WKH SRODULW\ RI WKH VLJQDO DQG E ZH KDYH 1 VDPSOHV LH 1 !  DYDLODEOH 8QGHU +  \ z 1  } IRU Q   c c c  1 +  \ z 1 P } ZKHUH P LV D QRQ]HUR FRQVWDQW RI XQNQRZQ SRODULW\ VLJQ ,Q WHUPV RI OLNHOLKRRG IXQFWLRQ ZH KDYH <   S H[S b  \  VLPSOH K\SRWKHVLV I \ }  } { DQG IRU WKH FRPSRVLWH K\SRWKHVLV DVVXPLQJ P LV NQRZQ ZH KDYH <   S H[S b  \ b P  I \MP }  } { :H XVH D PD[LPXP OLNHOLKRRG 0/ HVWLPDWLRQ SURFHGXUH WR REWDLQ DQ HVWLPDWH RI P GHQRWHG E\ PA  7KH GHULYDWLRQ DQG LGHD EHKLQG WKH 0/ HVWLPDWH ZLOO EH GLVFXVVHG LQ &KDSWHU  )RU QRZ ZH XVH WKH VDPSOH PHDQ DV WKH 0/ HVWLPDWH RI WKH WUXH PHDQ 7KH H[SUHVVLRQ IRU WKH VDPSOH PHDQ LV  ;\ PA )$

)$



)$

w

Q Q

1

Q

Q

1

Q

Q

0/

1

0/

1

Q

Q



+\SRWKHVLV 7HVWLQJ KHQFH WKH /57 EHFRPHV e\

I \MPA I\



0/

1 < Q





‚

H[S b }  \ 1 < Q



Q



b

1 ;

1

\

P

P

H[S b }  \



Q



+ ! w  +

$IWHU VRPH PDQLSXODWLRQ WKLV FDQ EH VLPSOLnHG n n 1 n n  ; n nS \ n Qn n 1Q  n

+ ! o  +

S

} OQ w

:H LQWURGXFHG WKH LGHD RI QXLVDQFH SDUDPHWHUV XVLQJ VLPSOH H[DPSOHV :H ZLOO UHWXUQ WR WKH LGHD RI DYHUDJLQJ RXW XQGHVLUHG LQoXHQFHV LQ &KDSWHU  

5(&(,9(5 23(5$725 &+$5$&7(5,67,& &859(6 $1' 3(5)250$1&(



*HQHUDO %DFNJURXQG

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nQHG WHUPV 3  3  DQG 3  3HUKDSV VRPH EHWWHU '

)$

0

'

0

)$

0

'

)$

0

'

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



VXLWHG GHnQLWLRQV DUH 3U' M+ IRU 3  3U' M+ IRU 3  DQG 3U' M+ IRU 3  %XW VLQFH WKH PDMRULW\ RI SUREOHPV GHDO ZLWK GHWHFWLRQ RI WDUJHWV ZH XVH WKH UDGDUVRQDU GHnQLWLRQV PRVW RI WKH WLPH 2QO\ ZKHQ PRUH FODULW\ LV SURYLGHG E\ WKH JHQHUDO GHnQLWLRQ ZLOO ZH UHVRUW WR LW 7KHVH TXDQWLWLHV FDQ EH XVHG WR SURYLGH D JUDSKLFDO UHSUHVHQWDWLRQ RI WKH SDUWLFXODU GHWHFWLRQ VFKHPH ,W VHHPV D OLWWOH VWUDQJH WR WDON DERXW GLmHUHQW LPSOHPHQWDWLRQV RI GHWHFWRUV DW WKLV WLPH EXW WKHUH DUH PDQ\ UHDVRQV WR FRPSDUH GHWHFWRU SHUIRUPDQFH )RXU YHU\ W\SLFDO UHDVRQV DUH OLVWHG EHORZ 7KH\ DUH D 2QH FRXOG FKRRVH WR XVH RU GLVUHJDUG FHUWDLQ SLHFHV RI LQIRUPDWLRQ ZKHQ GHVLJQLQJ WKH GHWHFWRU $ W\SLFDO VFHQDULR LV QRW WR XVH SULRU SUREDELO LWLHV HYHQ ZKHQ WKH\ DUH DYDLODEOH DQG WKHQ REWDLQ D PHDVXUH RI WKH WUDGHRm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mHUHQW GHWHFWRU LPSOHPHQWDWLRQV ,Q WKHVH FDVHV WKH WUDGHRmV FDQ EH VWXGLHG XVLQJ WKH GHWHFWLRQ TXDQWLWLHV )$

0

'



52& &XUYHV

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

)$

'

)$

'

)$

+\SRWKHVLV 7HVWLQJ

)LJXUH  3UREDELOLW\ GHQVLW\ IXQFWLRQV DW WKH RXWSXW RI D GHWHFWRU

)LJXUH  7\SLFDO 52& FXUYH





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

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e\  ZKLFK XVXDOO\ LV D VLQJOH UDQGRP YDULDEOH XQGHU HDFK K\SRWKHVLV LV FRPSDUHG WR WKH WKUHVKROG w 7KH GHWHFWRU SHUIRUPDQFH FDQ WKHQ EH GHVFULEHG E\ )$

'

)$

'

)$

)$

'

'

3

3U e ! wM+

'

3

3U e ! wM+

)$

=



Ie  z Gz



Ie z Gz



M+

=

w 

M+

w

:H FDQ DOVR UHZULWH 3  WKH SUREDELOLW\ RI GHWHFWLRQ XVLQJ Ie  z IIe  zz Ie z ez Ie z e DV = 3 Ie  z Gz '

M+

M+

M+

M+

M+



'

M+

w

=



ez Ie z Gz



M+

w

7DNLQJ WKH GHULYDWLYHV RI  DQG  ZLWK UHVSHFW WR w XVLQJ /HLEQLW] V UXOH $SSHQGL[ '  OHDGV WR

#3 #w #3 #w

)$

'

Ie w b



M+

ew Ie w b M+

w Ie w

b

M+



+\SRWKHVLV 7HVWLQJ



ZKHUH ew w LV VXEVWLWXWHG 7KH UDWLR RI WKH WZR SDUWLDO GHULYDWLYHV WKDW LV WKH VORSH RI WKH 52& FXUYH EHFRPHV

#3 #3

)$ '

w



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

)$

)$

b



b

6800$5
@ +HOVWURP &: (OHPHQWV RI 6LJQDO 'HWHFWLRQ DQG (VWLPDWLRQ (QJOH ZRRG &OLmV 1- 3UHQWLFH+DOO  >@ *UDGVKWH\Q ,6 DQG 5\]KLN ,0 7DEOH RI ,QWHJUDOV 6HULHV DQG 3URG XFWV nIWK HG 6DQ 'LHJR &$ $FDGHPLF 3UHVV 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

>@ 0HOVD -/ DQG &RKQ '/ 'HFLVLRQ DQG (VWLPDWLRQ 7KHRU\ 1HZ @ 6ULQDWK 0' 5DMDVHNDUDQ 3. DQG 9LVZDQDWKDQ 5 ,QWURGXFWLRQ WR 6WDWLVWLFDO 6LJQDO 3URFHVVLQJ ZLWK $SSOLFDWLRQV (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ %DUNDW 0 6LJQDO 'HWHFWLRQ DQG (VWLPDWLRQ /RQGRQ $UWHFK +RXVH  >@ $EUDPRZLW] 0 DQG 6WHJXQ ,$ +DQGERRN RI 0DWKHPDWLFDO )XQF WLRQV 1HZ Q  VQ ! e eV     + +

|

| |

3

3

+  /HW - GHQRWH WKH QXPEHU RI SRVLWLYH REVHUYDWLRQV WKHQ WKH /57 FDQ EH

ZULWWHQ DV

eV

S-



 b S ->-    -

> b S @-

t uS bS

 + !  +

e

7R VLPSOLI\ LW WR WKH SRLQW ZKHUH LW LV LQ LWV PRVW XVHIXO IRUP ZH WDNH WKH ORJDULWKP WR WKH EDVH S  b S  7KH H[SUHVVLRQ FDQ WKHQ EH UHGXFHG WR + !  ORJS >S e b - ORJS >S > b S @  +

e



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

-

LV WKH VXP RI %HUQRXOOL GLVWULEXWHG UDQGRP YDULDEOHV ,I WKH HYHQW ZLWK SDUDPHWHU P LV WUXH WKHQ WKH ELQRPLDO GLVWULEXWLRQ -  KDV SDUDPHWHUV DQG  7KDW LV

b

3U - 

QMP

c t ut  uQ t  u->Q   t ut  uQ

b

Q



7KH SUREDELOLW\ RI VD\LQJ +  JLYHQ WKDW + LV WUXH ZKLFK LQ UDGDUVRQDU WHUPV LV FDOOHG WKH 3)$  LV JLYHQ E\

;- t ut  u-

3)$

Q

Q e 



,I WKH HYHQW ZLWK SDUDPHWHU P LV WUXH WKHQ WKH ELQRPLDO GLVWULEXWLRQ -  KDV SDUDPHWHUV - DQG S

b

3U - 

QMP

c t u Q

SQ

 b S ->Q

DQG WKH SUREDELOLW\ RI VD\LQJ +  JLYHQ WKDW + LV WUXH ZKLFK LQ UDGDUVRQDU WHUPV LV FDOOHG WKH 3'  LV JLYHQ E\ 3'

;- t u Q

Q e

Q

S

b S ->Q

7KHVH FRQFHSWV DUH EHVW LOOXVWUDWHG ZLWK DQ H[DPSOH ([DPSOH  *LYHQ WKDW HLJKW VDPSOHV \  \  c c c  \ DUH XVHG DQG LQVLVW LQJ RQ D

3)$

RI  GHULYH DOO SHUWLQHQW LQIRUPDWLRQ WKDW JRYHUQV D VLJQ

GHWHFWRU DQG DQDO\]H WKH SHUIRUPDQFH 8QGHU

 +  +

ZKHUH

QQ

\Q

QQ

\Q

P  QQ

Q

  c c c 

DUH LLG ]HUR PHDQ *DXVVLDQ UDQGRP YDULDEOHV

WKH VLJQDO LV D SRVLWLYH FRQVWDQW 6ROXWLRQ

-

 3)$  3)$

; t  ut  u

Q e 

Q



› 

QQ z 1  } 

DQG



1RQ3DUDPHWULF DQG 6HTXHQWLDO /LNHOLKRRG 5DWLR 'HWHFWRUV

; t  u   Q e  t u 

RU 1RZ

tu

tu





tu







L N L b N 

L N

KHQFH

›

Q



 



; t  u     Q  ; t  u      

DQG

Q 





›

Q

ZKLOH

tu

!

Q

7KLV PDNHV e  +HQFH WKH /57 EHFRPHV

;

-

Q 

8 \Q

+ !  +



; t  ut  u  Q  ; t  u Q

3)$

Q

3'

Q 

Q

S

 

 › 

b S >Q

7R REWDLQ D IHHO IRU KRZ WKH SUREDELOLW\ RI GHWHFWLRQ YDULHV ZLWK WKH SUREDELOLW\ S XQGHU WKH + K\SRWKHVLV LH SUREDELOLW\ RI WKH UDQGRP YDULDEOH \Q  IRU Q   c c c  -  WR EH SRVLWLYH  ZH FRPSXWH WKH GHWHFWLRQ SUREDELOLW\ IRU GLmHUHQW YDOXHV RI S 6XSSRVH S   WKHQ 3U FKRRVH + M+ LV WUXH LV JLYHQ E\

3'

          



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

6XSSRVH S  WKHQ 3U FKRRVH + M+ LV WUXH LV JLYHQ E\

       

3'

6XSSRVH S  WKHQ

3U FKRRVH + M+ LV WUXH ZRXOG UHVXOW LQ 

         

3'

:,/&2;21 '(7(&725

,Q WKH SUHYLRXV VHFWLRQ ZH GHYHORSHG D FRXQWLQJ SURFHGXUH WR nQG WKH QXPEHU RI WLPHV WKH REVHUYDWLRQ LV SRVLWLYH 7KLV DSSURDFK DOVR SRLQWV RXW KRZ WKH GHWHFWLRQ VFKHPH FDQ EH PRGLnHG 7KLV PRGLnFDWLRQ LV D UDQNLQJ RU ZHLJKW LQJ DFFRUGLQJ WR WKH GLVWDQFH IURP WKH OLQH RI GLFKRWRP\ LH ]HUR LQ WKLV FDVH XQGHU WKH + K\SRWKHVLV $JDLQ WKH GHWHFWRU LV XVHG ZLWK D WKUHVKROG JXDUDQWHHLQJ D &)$5 W\SH SHUIRUPDQFH 7KH GHWHFWRU LV PRGLnHG WR FRXQW DFFRUGLQJ WR WKH UDQN RI WKH REVHUYDWLRQ %\ WKLV ZH PHDQ

;- Q

Q 

G

IRU Q

+ ! 7  +



ZKHUH

M\N M › M\N M › M c c c › M\N- M

ZKHUH

GQ

  c c c  -

|

IRU \NQ   IRU \NQ w 

Q

7R LOOXVWUDWH WKH UDQNLQJ ZH VKRZ D W\SLFDO GDWD VHW DQG H[WUDFW WKH GHWHFWLRQ VWDWLVWLF ([DPSOH  *LYHQ WKH GDWD VHW ZKHUH  I\L JL



I b

  b  J

WKH PDJQLWXGH RUGHUHG VHW LV JLYHQ E\

I

  IGQ J

I

MbM  MbM 

 J

     J

1RQ3DUDPHWULF DQG 6HTXHQWLDO /LNHOLKRRG 5DWLR 'HWHFWRUV

] ZKHUH

7=

; Q

Q 

G





+ ! 7  = +

VWLOO KDV WR EH GHWHUPLQHG

1RWH WKLV UDQNLQJ LV GLUHFWO\ SURSRUWLRQDO WR WKH SRVLWLRQ RI WKH SRVLWLYH YDOXHV ZKLOH DFFRXQWLQJ IRU WKH PDJQLWXGH RI WKH QHJDWLYH YDOXHV 7KLV WHFK QLTXH FDQ EH PRGLnHG WR DOORZ RWKHU UDQNLQJV ZHLJKWLQJV  $QDO\VLV RI WKH UDQN GHWHFWRU LV QRW DV VLPSOH DV WKDW RI WKH VLJQ GHWHFWRU )RU PRUH LQIRUPD WLRQ UHJDUGLQJ QRQSDUDPHWULF GHWHFWLRQ VFKHPHV ZH UHIHU WKH UHDGHU WR WKH UHIHUHQFHV TXRWHG DW WKH HQG RI WKH FKDSWHU $ QHZ H[FLWLQJ DUHD RI VLJQDO SURFHVVLQJ GHDOV ZLWK KLJKHU RUGHU PRPHQWV DQG FXPXODQWV >@ ,Q VRPH DSSOLFDWLRQV RQH FDQ XVH WKH PRPHQW RU FXPX ODQW WR GHWHFW WKH DEVHQFH RU SUHVHQFH RI D VLJQDO RU FRPSRQHQW )RU H[DPSOH $NWDV DQG +LSSHQVWLHO >@ XVHG WKUHVKROGV EDVHG RQ IRXUWK RUGHU PRPHQWV WR UHMHFW QRLVH LQ WKH UHFRQVWUXFWLRQ RI D ZDYHIRUP LQ D ZDYHOHW EDQG SURFHVVLQJ VFKHPH 

6(48(17,$/ '(7(&7,21

7KH VHTXHQWLDO OLNHOLKRRG UDWLR GHWHFWLRQ FULWHULRQ ZDV LQWURGXFHG E\ $ :DOG   ZKRVH QDPH LW FDUULHV LH :DOG V VHTXHQWLDO /57  2QH FDQ LQWHUSUHW WKLV WHFKQLTXH DV D PRGLnHG 1H\PDQ3HDUVRQ WHVW LQ ZKLFK WZR WKUHVKROGV DUH HVWDEOLVKHG 7HVWLQJ LV GRQH XQWLO RQH RI WKH WZR WKUHVKROGV LV FURVVHG 7KLV PRGLnHG 1H\PDQ3HDUVRQ WHVW n[HV WKH SUREDELOLW\ RI D PLVV LQ DGGL WLRQ WR WKH SUREDELOLW\ RI D IDOVH DODUP ,Q WKH FODVVLFDO 1H\PDQ3HDUVRQ WHVW ZH FRPSDUH WKH /5 ZLWK D WKUHVKROG ZKLFK LQ WXUQ LV JRYHUQHG E\ WKH 3)$  D JLYHQ FRQVWDQW ,Q WKH PRGLnHG 1H\PDQ3HDUVRQ WHVW DOVR FDOOHG WKH VHTXHQWLDO SUREDELOLW\ UDWLR WHVW 6357  ZH XVH WKH OLNHOLKRRG UDWLR /5 DQG FRPSDUH LW DW HYHU\ XSGDWH WLPH LH DW HDFK QHZ VHTXHQWLDO REVHUYDWLRQ SRLQW ZLWK WZR WKUHVKROGV 7KHVH WKUHVKROGV DUH GHQRWHG E\ s DQG s  7KH WKUHVKROGV DUH GHWHUPLQHG E\ VSHFLI\LQJ D 3)$ n[HG DW D YDOXH RI m DQG D 30 n[HG DW D YDOXH RI n  ,I WKH /5 LV ODUJHU WKDQ s  ZH GHFLGH RQ +  ZKLOH LI WKH /5 LV VPDOOHU WKDQ s  ZH GHFLGH RQ +  ,I HLWKHU WKUHVKROG LV FURVVHG WKH WHVW LV WHUPLQDWHG DQG WKH DSSURSULDWH GHFLVLRQ LV GHFODUHG ,I WKH /5 IDOOV EHWZHHQ WKH WZR WKUHVKROGV WKHQ ZH GHIHU WKH GHFLVLRQ WKDW LV ZH WDNH DQRWKHU VDPSOH DQG UHSHDW WKH WHVW /HW \  \  c c c  \,  UHSUHVHQW WKH REVHU YDWLRQV DVVXPHG WR EH LQGHSHQGHQW DQG LGHQWLFDOO\ GLVWULEXWHG GHQRWHG E\ \, \  \  c c c  \,  7KHQ WKH /57 EHFRPHV e\,

I \, I \,

 e\, ZKHUH WKH EROGIDFH TXDQWLWLHV DUH YHFWRUV DQG WKH QRQEROGIDFH TXDQWLWLHV DUH VFDODUV 7KH YHFWRUV KDYH DQ LQGH[ WKDW LV XVHG WR LQGLFDWH WKDW WKH\ FKDQJH LQ WKH VL]H RI WKHLU GLPHQVLRQ LH LQFUHDVH DV WKH WHVW SURJUHVVHV  +HQFH ZH KDYH D UHFXUVLYH DUUDQJHPHQW IRU WKH /57 ZLWK LQLWLDO FRQGLWLRQ e\ e\ ZKHUH e\ LV JLYHQ E\ e\ e\ e\ e\ e\ )RU D VSHFLnHG n[HG 3)$ DQG 30  ZH QHHG WR GHULYH WKUHVKROGV s DQG s VR WKDW ZH PHHW WKH FRQVWUDLQWV m 3)$ 3U FKRRVH + M+ LV WUXH

= = ccc

= =

ZKHUH 3'

I \Q G\Q

 IRU Q   c c c  ,



 b 3'

30

n

5

ccc

3'

5

= = = =

3U FKRRVH + M+ LV WUXH

I \Q G\Q

 IRU Q   c c c  ,



,I ZH PXOWLSO\ DQG GLYLGH WKH LQWHJUDQG RI  E\ I \Q  ZH REWDLQ I \Q ccc 3' I \Q G\Q 5 I \Q ccc

5

e\Q I \Q G\Q IRU Q   c c c  ,

7R GHFODUH GHWHFWLRQ WKH VHTXHQWLDO DUUDQJHPHQW RI WKH /57 LV GHQRWHG E\ 6/57 e\Q e\Q> e\Q> c c c e\ w s PXVW EH WUXH +HQFH 3'

= = ccc

s 3)$

5

e\Q I \Q G\Q w s

= = ccc

s m

:H DOVR NQRZ WKDW 3'  b 30 3'

 b n UHVXOWLQJ LQ  b n w s m

5

I \Q G\Q



1RQ3DUDPHWULF DQG 6HTXHQWLDO /LNHOLKRRG 5DWLR 'HWHFWRUV

KHQFH s ›

= =

6LPLODUO\ 30

ccc

n

5

n › s

= = ccc

7KLV OHDGV WR D ERXQG RQ s

= = ccc

e\Q I \Q G\Q

5

e\Q › s

5

I \Q G\Q

s w

7KH 6/57 DW VWHS , EHFRPHV e\,

m

I \Q G\Q

7R GHFODUH D PLVV KHQFH

bn

+ ! s  s +

s

 b m

n

bm

WDNH DQRWKHU VDPSOH

7KLV LV LQWHUSUHWHG DV LI e\, ! s  ZH GHFODUH + WR EH WUXH DQG LI e\,  ZH GHFODUH + WR EH WUXH ,I e\, ! s  EXW VPDOOHU WKDQ s  ZH WDNH DQRWKHU VDPSOH JRLQJ WR e\,  DQG UHSHDW WKH WHVW 1RWH ZH DVVXPHG LQ DOO WKH GHULYDWLRQV WKDW WKH VDPSOHV DUH LLG DQG WKDW WKH SUREDELOLWLHV 30 n DQG 3)$ m UHPDLQ FRQVWDQW WKURXJKRXW WKH ZKROH WHVW 7KLV W\SH RI WHVW DOORZV WKH XVHU WR XVH FRPSXWDWLRQDO UHVRXUFHV PRUH HmHF WLYHO\ VLQFH WKH WHVW W\SLFDOO\ FRXOG EH WHUPLQDWHG HDUOLHU ZLWK WKH PRGLnHG 1H\PDQ3HDUVRQ FRQVWUDLQWV  ,Q RWKHU ZRUGV RQFH WKH SUHVHQFH RU DEVHQFH RI D WDUJHW KDV EHHQ GHWHUPLQHG ZLWK DQ DFFHSWDEOH OHYHO RI HUURU 3)$ RU 30  RWKHU SRVVLEOH WDUJHWV FDQ EH LQWHUURJDWHG ZKLOH GDWD UHODWHG WR WKH MXVW GHFODUHG GHWHFWLRQ RU QRQGHWHFWLRQ LV GHOHJDWHG WR D WUDFNLQJ RU FODVVLnFDWLRQ QHWZRUN

s 

([DPSOH 

8QGHU 8QGHU

+ +

 

\ Q \ Q

QQ P  QQ

IRU Q   c c c

7KH DGGLWLYH QRLVH LV *DXVVLDQ LLG ZLWK ]HUR PHDQ DQG YDULDQFH ZDQW WR WHUPLQDWH WKH WHVW 6357 ZKHQ

30 › m

RU

3)$ › m 

} 

:H



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

/57 YHUVXV WKH QXPEHU RI VDPSOHV WHVW WHUPLQDWHG DW 3

:H NQRZ WKDW

s › DQG WKDW

s w

 b 30

 b m

30  b 3)$

m  b m

3)$

 

m

7KH 6/57 DW VWHS , EHFRPHV

e\,

b3  }

H



, Q



3 \> 

, Q

Q



c \ >P 

Q

+ ! s  s +

WDNH DQRWKHU VDPSOH

6LPSOLI\LQJ WKH DERYH H[SUHVVLRQ OHDGV WR

]

;, Q

Q 

\

+ !  +

} P

OQs  PQ 

WDNH DQRWKHU VDPSOH } P

OQs  PQ 

)LJXUH  GHPRQVWUDWHV WKH RXWFRPH RI D SDUWLFXODU WHVW ZKHUH IRU

   

,

DQG  DQRWKHU VDPSOH QHHGV WR EH WDNHQ VLQFH QHLWKHU RI WKH WZR

WKUHVKROGV DUH FURVVHG :KHQ

,

 WKH ORZHU WKUHVKROG LV FURVVHG DOORZLQJ

WKH GHFODUDWLRQ RI HYHQW K\SRWKHVLV ]HUR WR EH WUXH

1RQ3DUDPHWULF DQG 6HTXHQWLDO /LNHOLKRRG 5DWLR 'HWHFWRUV





6800$5
@ +HOVWURP &: (OHPHQWV RI 6LJQDO 'HWHFWLRQ DQG (VWLPDWLRQ (QJOH ZRRG &OLmV 1- 3UHQWLFH+DOO 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

>@ .D]DNRV ' DQG .D]DNRV 3 'HWHFWLRQ DQG (VWLPDWLRQ 1HZ @ *LEVRQ -' DQG 0HOVD -/ ,QWURGXFWLRQ WR 1RQ3DUDPHWULF 'HWHF WLRQ ZLWK $SSOLFDWLRQV 1HZ @ *LEERQV -' 1RQ3DUDPHWULF 6WDWLVWLFDO ,QIHUHQFH 1HZ @ 0HOVD -/ DQG &RKQ '/ 'HFLVLRQ DQG (VWLPDWLRQ 7KHRU\ 1HZ @ $NWDV 8 DQG +LSSHQVWLHO 5 ?/RFDOL]DWLRQ RI *60 6LJQDOV 8VLQJ :DYHOHW 'HQRLVLQJ %DVHG RQ WKH )RXUWK 2UGHU 0RPHQW UG $VLORPDU &RQI RQ 6LJQDOV 6\VWHPV DQG &RPSXWHUV 3DFLnF *URYH &$ 2FWREHU  >@ 6ULQDWK 0' 5DMDVHNDUDQ 3. DQG 9LVZDQDWKDQ 5 ,QWURGXFWLRQ WR 6WDWLVWLFDO 6LJQDO 3URFHVVLQJ ZLWK $SSOLFDWLRQV (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ :DOG $ 6HTXHQWLDO $QDO\VLV 1HZ @ 1LNLDV &/ DQG 3HWURSXOX $3 +LJKHU2UGHU 6SHFWUD $QDO\VLV _ $ 1RQOLQHDU 3URFHVVLQJ )UDPHZRUN  (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO 

&KDSWHU 

'HWHFWLRQ RI '\QDPLF 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH 

,1752'8&7,21

,QLWLDOO\ ZH ZLOO FRYHU WKH GHWHFWLRQ RI NQRZQ VLJQDOV HPEHGGHG LQ ZKLWH *DXVVLDQ QRLVH 7KDW LV ZH DVVXPH WKDW ZH NQRZ DOO RI WKH VLJQDO DQG QRLVH SDUDPHWHUV

7KLV LV DQ LGHDO FDVH

/DWHU LQ WKLV FKDSWHU ZH ZLOO UHPRYH

VRPH RI WKH OLPLWDWLRQV LQ WKDW ZH ZLOO DOORZ WKH SDUDPHWHUV WR EH UDQGRP RU XQNQRZQ ,Q &KDSWHU  ZH ZLOO DOORZ WKH *DXVVLDQ QRLVH WR EH FRORUHG LH WKH QRLVH SURFHVV LV QRW GHOWD FRUUHODWHG  7KH ELODWHUDO ZKLWH QRLVH VSHFWUDO GHQVLW\ LV JLYHQ E\ LV JLYHQ E\

51 ~

61  1  DQG WKH FRUUHVSRQGLQJ FRUUHODWLRQ IXQFWLRQ 1  p~  1RWH DV ORQJ DV WKH QRLVH LV VSHFWUDOO\ oDW

RYHU WKH UHJLRQ RI LQWHUHVW LW FDQ EH LQWHUSUHWHG DV EDQGOLPLWHG ZKLWH QRLVH :H PLQLPL]H WKH XVH RI FRPSOH[ YDOXHG IXQFWLRQV ZKHQ SRVVLEOH DQG SRLQW RXW WKH FKDQJHV ZKHQ GHDOLQJ ZLWK FRPSOH[ YDOXHG IXQFWLRQV 7\SLFDOO\ WKH H[WHQVLRQV WR FRPSOH[ YDOXHG IXQFWLRQV UHTXLUH WKH FRPSOH[ FRQMXJDWLRQ RI WKH VHFRQG WHUP LQ WKH GHnQLWLRQ RI WKH LQQHU SURGXFW 7KH VHFRQG HGLWLRQ RI

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH E\ 0F'RQRXJK DQG :KDOHQ >@ KDV DPSOH

H[DPSOHV XVLQJ FRPSOH[ YDOXHG HQWLWLHV

:H ZLOO OHW WKH VLJQDOV EH FRQWLQXRXV IXQFWLRQV RI WLPH DQG VDPSOH WKHP SURSHUO\ UDWKHU WKDQ VWDUWLQJ ZLWK n[HG TXDQWLWLHV DV LQ &KDSWHU 

2QH

FRXOG KDYH LQWURGXFHG VLJQDOV ZLWK D WLPH YDU\LQJ QDWXUH LQ &KDSWHU  EXW IRU FOHDUQHVV RI SUHVHQWDWLRQ WKLV ZDV QRW DWWHPSWHG

7KLV SDUWLFXODU DS

SURDFK LQWURGXFHV WKH SURFHVVLQJ RI GDWD LQ D QDWXUDO IDVKLRQ 7KDW LV WKH





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

%LQDU\ GHWHFWLRQ

VHWWLQJ XS RI WKH FRQWLQXRXV WLPH SUREOHP DV D GLVFUHWH WLPH SUREOHP DOORZV WKH DSSOLFDWLRQ RI WKH WHFKQLTXHV OHDUQHG LQ &KDSWHU 

,W DOVR DOORZV WKH

GLUHFW GHULYDWLRQ RI WKH GLVFUHWH WLPH YHUVLRQ RI WKH DSSURSULDWH GHWHFWRUV 7KLV FKDSWHU GHULYHV WKH GLVFUHWH WLPH FRUUHODWRU DQG KHQFH WKH GLVFUHWH WLPH PDWFKHG nOWHU ,Q DGGLWLRQ ZH DUH H[SRVHG WR WKH SURMHFWLRQ RI WKH GDWD RQWR D VHW RI EDVLV IXQFWLRQV 7KH GDWD LV SURMHFWHG RQWR D UHSOLFD RI WKH VLJQDO LH WKH VLJQDOV VHUYH DV WKH EDVLV IXQFWLRQ  7KLV LV RXU nUVW H[SRVXUH WR WKH FRQFHSW RI DQ LQQHU SURGXFW LQ D +LOEHUW VSDFH VHWWLQJ 6HFWLRQ  ZLOO DGGUHVV WKH GLJLWDO DQG DQDORJ FRUUHODWRU VROXWLRQ ZKLOH 6HFWLRQV  DQG  DGGUHVV WKH PDWFKHG nOWHU DSSURDFK ,Q 6HFWLRQ  ZH ZLOO H[WHQG WKH ELQDU\ FDVH WR WKH PRUH JHQHUDO 0DU\ FDVH 7KH QRWLRQ RI SDUDPHWHU XQFHUWDLQW\ ZLOO EH WDNHQ XS LQ 6HFWLRQ  ,Q 6HFWLRQV  DQG  ZH ZLOO H[DPLQH WKH PXOWLSXOVH VFHQDULR DV ZHOO DV WKH QRWLRQ RI FRKHUHQW DQG QRQFRKHUHQW DYHUDJLQJ



7+( %,1$5< '(7(&7,21 352%/(0

6XSSRVH WKDW RQH RI WZR SRVVLEOH FRPPXQLFDWLRQ VLJQDOV LV WUDQVPLWWHG 7KDW LV ZKHQ K\SRWKHVLV VLJQDO

V W

+

LV WUXH VLJQDO

V W

DQG ZKHQ K\SRWKHVLV

+

LV WUXH

LV VHQW 7KLV LV D ELQDU\ K\SRWKHVLV SUREOHP ZKHUH WKH DGGLWLYH

QRLVH DV VSHFLnHG DERYH PDNHV WKH GHWHFWLRQ GLpFXOW

7KH ELQDU\ GHWHF

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nUVW ]HUR FURVVLQJ RI WKH QRLVH FRUUHODWLRQ IXQFWLRQ 6LQFH WKH QRLVH LV *DXVVLDQ DQG WKH VDPSOHV GXH WR WKH VDPSOLQJ DSSURDFK DUH XQFRUUHODWHG WKH VDPSOHV DUH DOVR LQGHSHQGHQW XQGHU HDFK K\SRWKHVLV 7KH FRQWLQXRXV ZDYHIRUP

\W

LV VDPSOHG DQG RQFH

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH



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

+



+



7

KDYH WKH IROORZLQJ UHSUHVHQWDWLRQ

\W

V W  QW

\W

V W  QW



IRU 

›W›7

6DPSOLQJ WKHVH FRPSRQHQWV ZH DVVXPH WKDW WKH VLJQDO LV VDPSOHG DW OHDVW DW D UDWH FRUUHVSRQGLQJ WR WKH 1\TXLVW UDWH OHDGV WR WKH GHVFULSWLRQ

\WP



\P

WKDW LV XQGHU HDFK K\SRWKHVLV

VLP  QP IRU  › P › N N

DQG

L



 



VDPSOHV DUH DYDLODEOH 8VLQJ WKH H[SUHVVLRQ

IURP &KDSWHU  ZH FDQ HDVLO\ ZULWH DQ H[SUHVVLRQ IRU WKH /57 DV

\

e

ZKHUH

+ I \  \  c c c  \N ! w I \  \  c c c  \N  +



w GHSHQGV RQ WKH FULWHULRQ WKDW LV EHLQJ HPSOR\HG )RU WKH WLPH EHLQJ w NQRZLQJ WKDW DW DQ\ WLPH

DQG ZLWKRXW ORVV RI JHQHUDOLW\ ZH OHDYH LW DV

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

61 

|

1 



IRU

MM  l

HOVH



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



%DQGOLPLWHG ZKLWH QRLVH VSHFWUDO GHQVLW\

)LJXUH 

)LJXUH 

%DQGOLPLWHG ZKLWH QRLVH FRUUHODWLRQ IXQFWLRQ

DQG

1 { 1 {

51 ~

l

~

VLQl

~

l

~

l VLQF l



7KH nUVW ]HUR FURVVLQJ RI WKH FRUUHODWLRQ IXQFWLRQ LV DW WKH FRUUHODWLRQ ODJ ZKHUH

~

W

c

{

l ,I ZH VDPSOH XVLQJ WKLV VDPSOLQJ LQWHUYDO WKHQ DOO

QRLVH VDPSOHV DUH XQFRUUHODWHG DQG WKH MRLQW GHQVLW\ OLNHOLKRRG IXQFWLRQ FDQ EH H[SUHVVHG DV WKH SURGXFW RI LQGHSHQGHQW WHUPV 7KLV PDNHV WKH GHULYDWLRQ RI WKH RSWLPDO GHWHFWRU YHU\ VLPSOH 7R KDYH D SURSHU /5 IXQFWLRQ H[SUHVVLRQ DOO ZH QHHG WR nQG LV WKH PHDQ XQGHU HDFK K\SRWKHVLV DQG WKH YDULDQFH 7KH YDULDQFH LQ WKLV FDVH LV LQGHSHQGHQW RI ZKLFK K\SRWKHVLV LV WUXH 7KH PHDQ XQGHU HDFK K\SRWKHVLV LV

( >\P @

( >VLP  QP @

VLP  P

IRU

L

 cc c  N





  

7KH FRPSXWDWLRQ RI WKH YDULDQFH LV MXVW VOLJKWO\ PRUH FRPSOLFDWHG )RU H[ DPSOH XQGHU

+L

ZH KDYH

}\P

( >\P b (\P @

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH



( >VLP  QP b VLP @

d e

( QP

1 l  { ZKLFK LV LQGHSHQGHQW RI WKH K\SRWKHVLV L 8VLQJ N LQGHSHQGHQW VDPSOHV cW XQLWV DSDUW WDNHQ RYHU WKH WLPH VHJPHQW  7  ZH REWDLQ WKH OLNHOLKRRG }1

51 

IXQFWLRQV DV JLYHQ E\



H[S

b }



H[S

b } 



I \

{}1 N





I \

{}1 N



N ;

1P  N  ; 1P



\P b VP 





\P b VP 





7KH /57  EHFRPHV

\

e

H[S

N b ;

b } 

1P



+

  ec ! \P VP b \P VP b VP b VP 

d



+

$V DOZD\V WKHVH W\SHV RI H[SUHVVLRQV DUH UHGXFHG WR WKH VLPSOHVW IRUP 7DNLQJ WKH QDWXUDO ORJDULWKP DQG VLPSOLI\LQJ OHDGV WR

b

N \ V ; P P P



}

1

+

 N \ V ; P ! P 

P

}

1



 +

w

OQ

b }

N b ;

1P



VP b VP

c



,I ZH PXOWLSO\ WKURXJK E\ WKH YDULDQFH RI WKH QRLVH ZH REWDLQ WKH GLVFUHWH DOVR FDOOHG GLJLWDO FRUUHODWRU IRUP 6R LI ZH DUH GHDOLQJ ZLWK GLVFUHWH VDP SOHV ZH ZRXOG VWRS KHUH DQG LPSOHPHQW WKH GLJLWDO FRUUHODWRU 7KLV FRUUHODWRU LV RI WKH IRUP

N ; P



\P VP

b

+ \P VP !  s P  +

N ;



7KH WKUHVKROG LV HDVLO\ REWDLQHG IURP WKH GLVFXVVLRQ OHDGLQJ WR WKH GLJLWDO FRUUHODWRU IRUP 7KH GLVFUHWH GLJLWDO FRUUHODWRU LV VKRZQ LQ )LJXUH  7KH nQDO SDUW RI WKLV VHFWLRQ GHDOV ZLWK WKH DQDORJ YHUVLRQ RI WKH GHWHFWRU LPSOHPHQWDWLRQ VR ZH QHHG WR DSSO\ VRPH OLPLWLQJ DUJXPHQWV 7R GR WKLV FRQVLGHU WKH H[SUHVVLRQ IRU WKH YDULDQFH RI WKH QRLVH VDPSOHV )RU D EDQGOLP LWHG ZKLWH QRLVH SURFHVV ZLWK VSHFWUDO KHLJKW

1

 WKH YDULDQFH LV JLYHQ E\

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



)LJXUH 

}1

1 l

'LVFUHWH WLPH FRUUHODWRU

{

  $ EDQGOLPLWHG SURFHVV QHHGV RQO\ WR EH VDPSOHG DW WZLFH

WKH KLJKHVW VSHFWUDO IUHTXHQF\ ,Q WKLV FDVH WKH VDPSOLQJ IUHTXHQF\ LQ UDGL

IV l { cW KHQFH l { cW 7KH QRLVH YDULDQFH EHFRPHV 1 l { 1 cW  7KLV DOORZV  WR EH ZULWWHQ DV

DQVVHF LV

}1

N

OLP

  c W N

cW

7

+  N N \ V ; \P VP ; ! P P b cW  cW  1 1

P

b1





N b ; P

P







VP

b





w

+



c

OQ

VP cW



,Q WKH OLPLW WKH VXPPDWLRQV EHFRPH LQWHJUDOV VR WKDW ZH REWDLQ WKH IROORZLQJ H[SUHVVLRQ

=7 

+ =7  \W V W GWb \W V W GW !  1  1 

OQ

=7

w

+



1



b

c

V W b V W GW 

2QH FDQ QRUPDOL]H WKLV H[SUHVVLRQ WR REWDLQ

=7 

+ =7 1 \W V W GWb \W V W GW !   

+

OQ

w

 

= 7b 

c

V W b V W GW

7

 7KLV GHWHFWRU LV VKRZQ LQ )LJXUH  ,Q WKH ULJKWKDQG VLGH WKH WKUHVKROG KDV EHHQ VLPSOLnHG WR EH UHSUHVHQWHG E\ WKH VLQJOH V\PERO

7 

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH

)LJXUH 



&RUUHODWLRQ UHFHLYHU

:H QRWH WKDW WKLV LV D FRUUHODWLRQ UHFHLYHU 7KDW LV WKH LQSXW LV FRUUHODWHG ZLWK WKH VWRUHG UHSOLFDV RI WKH VLJQDOV

V W DQG V W IRU D WLPH GHOD\ RI ]HUR

,I D PD[LPXP OLNHOLKRRG DSSURDFK LV XVHG DQG ERWK VLJQDOV KDYH WKH VDPH HQHUJ\ WKHQ WKH WHVW UHGXFHV WR WKH GHWHUPLQDWLRQ RI ZKLFK FRUUHODWRU KDV WKH GRPLQDQW RXWSXW )RU WKH 1H\PDQ3HDUVRQ W\SH FULWHULRQ ZH ZLOO GLVUHJDUG WKH YDOXH RI

7

DOWRJHWKHU 7KH DSSURSULDWH WKUHVKROG LV REWDLQHG E\ VROYLQJ

IRU WKH YDOXH XQGHU WKH ]HUR K\SRWKHVLV WR JXDUDQWHH WKH GHVLUHG

3)$ 

:H

DOVR QRWH WKDW LI RXU SUREOHP FDOOV IRU WKH LPSOHPHQWDWLRQ DV D FRPPXQLFDWLRQ UHFHLYHU WKHQ VXEVWLWXWLQJ WKH NQRZQ YDOXHV LQWR WKH H[SUHVVLRQ IRU

3  &

3

&



&

&

 OHDGV WR D

w

w

LH

RI RQH /HW XV QRZ

GHnQH D FRQYHQLHQW WHVW VWDWLVWLF IRU WKH UHFHLYHU DQG REWDLQ WKH SHUIRUPDQFH RI WKH UHFHLYHU 7KH GHVLUHG WHVW TXDQWLW\ LV

* DQG ZLWK OQ w

 ZH REWDLQ WKH IROORZLQJ

H[SUHVVLRQ

* 6LQFH

=7 

\W V W b V W GW 

 

= 7b 

V W

b

c

V W

+ GW !  +





\W KDV D *DXVVLDQ 3') D VFDOHG DQG LQWHJUDWHG YHUVLRQ RI LW DOVR KDV * DV GHnQHG LQ  LV QRUPDOO\ GLVWULEXWHG 7R

D *DXVVLDQ 3') +HQFH

REWDLQ YDOXHV IRU WKH SHUIRUPDQFH RI WKH GHWHFWRU RQO\ WKH PHDQV DQG WKH YDULDQFH RI WKH *DXVVLDQ UDQGRP YDULDEOH



* KDYH WR EH REWDLQHG

3HUIRUPDQFH $QDO\VLV

7KLV LV DOVR D YHU\ FRQYHQLHQW SODFH WR HYDOXDWH WKH SHUIRUPDQFH RI VRPH VWDQGDUG GLJLWDO FRPPXQLFDWLRQ PRGXODWLRQ WHFKQLTXHV :H XVH VRPH VLP SOLI\LQJ DVVXPSWLRQV WR HQKDQFH WKH FRPSDULVRQ :H DVVXPH WKDW

3

3 



&

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ &

 DQG

&

&

 7KLV LV WKH PLQLPXP SUREDELOLW\ RI HUURU

FULWHULRQ LQWURGXFHG LQ &KDSWHU  UHVXOWLQJ LQWR D WKUHVKROG RI

w

 +HQFH

WKH UHFHLYHU GHWHFWRU DQG WKUHVKROG FDQ EH PDGH DV VKRZQ LQ   7R REWDLQ WKH RSWLPDO UHFHLYHU GHWHFWRU IRUP DQG WKUHVKROG LV QRW VXp FLHQW 2QH DOVR QHHGV WR GHWHUPLQH WKH SHUIRUPDQFH 7KLV LV PRVW HDVLO\ GRQH XVLQJ WKH SUREDELOLW\ RI HUURU ZKLFK GXH WR WKH FRVWV DQG SULRU SUREDELOLWLHV

+ M+

LV QXPHULFDOO\ WKH VDPH DV 3UFKRRVH

LV WUXH  :H UHFDOO WKDW D VFDOHG

*DXVVLDQ UDQGRP YDULDEOH LV VWLOO D *DXVVLDQ UDQGRP YDULDEOH UDQGRP YDULDEOH

*

+HQFH WKH

LV *DXVVLDQ XQGHU HDFK K\SRWKHVLV 7R FRPSXWH DQ\ RI

WKH SHUIRUPDQFH LQGLFDWRUV ZH QHHG WR nQG WKH PHDQV RI

*

XQGHU HDFK K\

+ RU +  WKH * XQGHU WKH ]HUR

SRWKHVLV DQG LWV YDULDQFH 8QGHU HDFK K\SRWKHVLV PD\ LW EH YDULDQFH LV LGHQWLFDO IRU WKLV W\SH RI GHWHFWRU 7KH PHDQ RI K\SRWKHVLV LV GHQRWHG E\

( *M+

( *



'LVUHJDUGLQJ WKH QRLVH GHSHQGHQW WHUPV WKDW HTXDWH WR ]HUR LH WKH nUVW RUGHU PRPHQWV RI WKH QRLVH DUH ]HUR ZH REWDLQ

( *

( (

= 7 

V W V W b V W GW b

= 7 b

V W V W b



7KLV FDQ EH VLPSOLnHG WR

( *

b 

V W

=7 

b

V W

 



= 7b





V W

c

b

V W

c

  V W  V W

b V W  GW

 GW

 GW





7KH PHDQ XQGHU WKH RQH K\SRWKHVLV LV FRPSXWHG VLPLODUO\ DQG LV JLYHQ E\

( *M+

( *

b( *



=7





V W

 

b V W  GW 

7KH YDULDQFH LV D OLWWOH PRUH GLpFXOW WR FRPSXWH :H UHFDOO WKDW WKH YDULDQFH FRUUHVSRQGV WR WKH $& SRZHU LQ HOHFWURQLF FLUFXLWV RU LQ WHUPV RI VWDWLVWLFV LW GHVFULEHV WKH VSUHDG RI WKH SUREDELOLW\ GHQVLW\ IXQFWLRQ DERXW LWV PHDQ 7KDW LV LW LV LQGHSHQGHQW RI WKH PHDQ LH WKH '& WHUP DQG LQ WKLV FDVH FDQ EH FRPSXWHG HLWKHU XQGHU WKH ]HUR RU WKH RQH K\SRWKHVLV 8VXDOO\ ZH FKHFN ZKLFK ZD\ WKH UHVXOW FDQ EH REWDLQHG PRUH HDVLO\ ,Q PDQ\ FDVHV WKH ]HUR K\SRWKHVLV UHVXOWV LQ D ]HUR PHDQ 8QGHU WKLV FRQGLWLRQ ZH SUHIHU WR FRPSXWH WKH YDULDQFH XQGHU WKH

+

K\SRWKHVLV VLQFH WKH VHFRQG PRPHQW FRUUHVSRQGV

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH



WR WKH YDULDQFH DQG QR VSHFLDO FRQVLGHUDWLRQ IRU DQ\ PHDQ YDOXH KDV WR EH WDNHQ ,Q RXU FDVH ZH DUELWUDULO\ FKRRVH WR ZRUN ZLWK WKH ]HUR K\SRWKHVLV WKH RQH K\SRWKHVLV UHTXLUHV WKH VDPH DPRXQW RI FRPSXWDWLRQ  :H GHnQH WKH YDULDQFH RI WKH *DXVVLDQ WHVW VWDWLVWLF DV

( >*M+ b ( *M+ @

*

YDU

( >*M+ b ( *M+ @

*

YDU 



1RZ ZH nUVW H[DPLQH WKH H[SUHVVLRQ

=7

*M+ b ( *

 

 

=7 

V W

  

= 7b 

QW V W b V W GW

b

V W

c

V W

GW 



=7





V W

> 

b V W @ GW

QW >V W b V W @ GW



8VLQJ   ZH QRWH WKDW WKH YDULDQFH H[SUHVVLRQ EHFRPHV

(

*

YDU

‚= 7 



QW >V W b V W @ GW

= 7= 7 

1

=7







( >QW Q} @ >V W b V W @ >V } b V } @ GW G}

V W

> 

b V W @ GW

*

YDU 



5DWKHU WKDQ FDUU\LQJ WKLV VRPHZKDW FXPEHUVRPH H[SUHVVLRQ IXUWKHU DORQJ ZH XVH VRPH VLPSOLI\LQJ WHUPV /HW XV GHnQH WKH IROORZLQJ WHUPV

:H QRWH WKDW





|





= 7d 

=7



e

V W  V W GW



V W V W GW





 DQG | UHSUHVHQW WKH DYHUDJH HQHUJ\ DQG WKH QRUPDOL]HG FURVV | LV ERXQGHG DERYH E\

FRUUHODWLRQ FRHpFLHQW UHVSHFWLYHO\ 7KLV PHDQV WKDW  DQG EHORZ E\ FRPSDFW IRUP

=7 

V W

> 

b

:H FDQ QRZ H[SUHVV WKH IROORZLQJ LQWHJUDOV LQ D PRUH

h V W @ GW

= 7d 

e

V W  V W GW h 

 h |

=7 

V W V W GW





 

h | w 





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

8VLQJ WKHVH H[SUHVVLRQV WKH PHDQV DQG WKH YDULDQFH RI WKH *DXVVLDQ UDQGRP YDULDEOH

* EHFRPH

b b |  ( *  b |  YDU* YDU* 1  b |   7KH 3')V RI * XQGHU WKH ]HUR DQG RQH K\SRWKHVLV DUH z 1 b b |  1  b |  DQG z 1  b |  1  b |  UHVSHFWLYHO\ 7KH WKUHVKROG LV JLYHQ E\ ( *

 WKH KDOIZD\ PDUN EHWZHHQ WKH PHDQV ZKHUH HLWKHU

+

SUREDELOLW\ RI FKRRVLQJ

LQ WKH UDGDU DQG VRQDU

+

ZKLOH

WHUPLQRORJ\ LV JLYHQ E\

=

3)$



+ LV WUXH LH 3)$

RU

+

LV IUHH 7KH

I J GJ

=  H>J>|  1 >| S GJ {1  b |   ‚  b | 4 S 1  b |  V   b | $ 4 #



1

7KH SUREDELOLW\ RI HUURU LV 

3(



3)$  30  

3)$ KHQFH

3(

4

tU

  b | 1



u

:H QRWLFH WKDW WKH SUREDELOLW\ RI HUURU GHSHQGV RQO\ RQ



|  DQG 1

DQG LV

WKHUHIRUH LQGHSHQGHQW RI WKH VLJQDO VKDSH LH RQO\ PRPHQWV UHODWHG WR WKH VHFRQG RUGHU DUH LPSRUWDQW  :H FDQ QRZ HYDOXDWH WKLV H[SUHVVLRQ IRU YDULRXV YDOXHV RI WKH YDOXHV 

b DQG  DUH WKH PRVW LQWHUHVWLQJ RQHV ,I | = 3(



I J GJ

|

(VSHFLDOO\

 WKHQ

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH =  >\  1  {1  

‚› w    



4 ,I

|

b WKHQ



H

G\

1

=

S{1 H>J  1  GJ   t  u tU  u 4 S 4 1 1 

3(





,I ZH FRPSDUH WKH FRPSOHPHQWDU\ HUURU IXQFWLRQ LQWHJUDOV IRU DQG



|

HTXDO WR 

b ZH QRWH WKDW WKH ODWWHU H[SUHVVLRQ KDV KDOI WKH YDULDQFH RI WKH nUVW

RQH ,Q RWKHU ZRUGV WKH QRLVH SRZHU KDV EHHQ KDOYHG RU WKH HmHFWLYH DYHUDJH VLJQDO HQHUJ\ KDV GRXEOHG 7KLV VKRZV WKDW LI VLJQDOV KDYLQJ D QRUPDOL]HG FURVVFRUUHODWLRQ FRHpFLHQW RI

b DUH XVHG WKHQ WKLV PRGXODWLRQ VFKHPH KDV

D  G% DGYDQWDJH RYHU WKH RQH WKDW XVHV VLJQDOV ZLWK D QRUPDOL]HG FURVV FRUUHODWLRQ FRHpFLHQW RI  ,I ZH OHW

|

V W HTXDOV V W DQG =  H>J  S GJ 3( s { 

 WKHQ

 

7KLV LQGLFDWHV WKDW RQO\ KDOI RI WKH WLPH ZH ZRXOG PDNH WKH FRUUHFW GHFL VLRQ 7KLV FRQVWLWXWHV YHU\ SRRU UHFHSWLRQ EXW WKDW VKRXOG EH REYLRXV ZKHQ ZH UHFRJQL]H WKDW ZH ZRXOG XVH WKH VDPH V\PERO WR UHSUHVHQW WZR GLmHUHQW PHVVDJHV

([DPSOH 

&RKHUHQW 3KDVH 6KLIW .H\LQJ &36.  8QGHU + +

V W V W

 

$ VLQ W b$ VLQ W

IRU 

›W›7

DQG  LV VXFK WKDW LQ 7 VHFRQGV DQ LQWHJHU QXPEHU RI SHULRGV DUH JHQHUDWHG 7KH WZR PHVVDJHV DUH WZR VLQXVRLGV  GHJUHHV RXW RI SKDVH 'XULQJ WUDQV PLVVLRQ ZKLWH *DXVVLDQ QRLVH ZLWK YDULDQFH HTXDO WR 1  LV DGGHG WR WKH VLJQDO +HQFH = 7b c  $ 7     

DQG



$



|

VLQ





 W  $

=7

b



VLQ

 W GW

b$ VLQ  WGW





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

7KH SUREDELOLW\ RI HUURU LV JLYHQ E\ =  H>\  1  S{1 G\ 3( s   tU   u 4

1

7KLV LV WKH RSWLPXP V\VWHP VLQFH WKH QRUPDOL]HG FURVVFRUUHODWLRQ FRHpFLHQW HTXDOV b 7\SLFDO HUURU SUREDELOLWLHV DUH VKRZQ LQ )LJXUH  ([DPSOH  &RKHUHQW )UHTXHQF\ 6KLIW .H\LQJ &)6.  7KH WZR PHVVDJHV DUH WUDQVPLWWHG DW WZR GLmHUHQW IUHTXHQFLHV + +

V W V W

 

$ VLQ W $ VLQ W

IRU 

›W›7

:H DVVXPH WKDW  DQG  DUH RUWKRJRQDO WKDW LV RYHU 7 VHFRQGV RI REVHUYD WLRQ WKH SURMHFWLRQ RI RQH RI WKH VLQXVRLGV RQWR WKH RWKHU LV ]HUR LH |   7KH FRPPXQLFDWLRQ FKDQQHO DGGV ZKLWH *DXVVLDQ QRLVH ZLWK YDULDQFH HTXDO WR 1  6LQFH ERWK VLJQDOV KDYH WKH VDPH DPSOLWXGH LH DPSOLWXGH $  WKH\ KDYH WKH VDPH HQHUJ\ $ 7   +HQFH WKH DYHUDJH HQHUJ\ = 7b = 7b c     c   q

 



V W  V W GW



= 7b







V W  V W GW

c

$ VLQ  W  $ VLQ>  W GW

$ 7 

LV WKH VDPH DV LQ ([DPSOH  8VLQJ  WKH SUREDELOLW\ RI HUURU LV JLYHQ E\ tU  u 3(

4

1

7KLV SHUIRUPDQFH LV  G% ZRUVH WKDQ WKH RQH IRU &36. $ W\SLFDO SHUIRU PDQFH FXUYH LV VKRZQ LQ )LJXUH  ([DPSOH  2Q2m .H\LQJ 22. RU DPSOLWXGH VKLIW NH\LQJ $6.  + +

 

V W V W



% FRV W

IRU 

›W›7

7KH QRUPDOL]HG FURVVFRUUHODWLRQ FRHpFLHQW LV E\ LQVSHFWLRQ ]HUR |   7KH FRPPXQLFDWLRQ FKDQQHO DGGV ZKLWH *DXVVLDQ QRLVH ZLWK YDULDQFH HTXDO WR 1  2QO\ RQH FRUUHODWRU LV QHHGHG WR DFFRXQW IRU WKH VLJQDO V W  %\ GHnQLWLRQ VLQFH WKHUH LV QR HQHUJ\ XVHG IRU VLJQDO V W  WKH DYHUDJH HQHUJ\

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH



0

10

CPSK CFSK OOK

−1

10

−2

Probability of Error, Pe

10

−3

10

−4

10

−5

10

−6

10

−7

10

−8

10

0

2

4

6

)LJXUH 

8

10 Eb/No (dB)

12

14

16

18

20

'HWHFWDELOLW\ RI &36. &)6. DQG 22.

LV KDOI RI WKH HQHUJ\ RI VLJQDO V W  6RPHWLPHV ZH UHIHU WR WKLV W\SH RI PRG XODWLRQ DV DPSOLWXGH VKLIW NH\LQJ $6.  7KH HQHUJ\ RI VLJQDO V W LV JLYHQ E\ =7  %7 





%  FRV  WGW



:H GHQRWH WKLV DYHUDJH HQHUJ\ E\   ZKHUH  UHSUHVHQWV KDOI RI WKH HQHUJ\ LQ VLJQDO V W  7KH SUREDELOLW\ RI HUURU EHFRPHV tU  u  3( s 4 1 

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

,Q VXPPDU\ WKHVH WKUHH H[DPSOHV VKRZ WKH SHUIRUPDQFH RI WKUHH YHU\ W\SLFDO GLJLWDO PRGXODWLRQ VFKHPHV

,I WKH VLJQDO DPSOLWXGHV DUH LGHQWLFDO



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

WKHQ &36. ZLOO RXWSHUIRUP &)6. ZKLFK LQ WXUQ RXWSHUIRUPV 22. ,I WKH DPSOLWXGHV DUH NHSW LGHQWLFDO WKH SURFHVVLQJ ORVV LV  G% ZKHQ JRLQJ IURP &36. WR &)6. DQG DQ DGGLWLRQDO ORVV RI  G% ZKHQ JRLQJ IURP &)6. WR 22.

5DGDU6RQDU 5HFHSWLRQ :H FDQ LQWHUSUHW WKH UDGDU DQG DF WLYH VRQDU GHWHFWLRQ VFKHPH DV D UHDOL]DWLRQ RI 22. 7KH UHFHLYHG GDWD LV JLYHQ E\ ([DPSOH 

+ +

 

\W \W

QW $ FRV W  QW

IRU 

›W›7

7KH FKDQQHO PHGLXP DQG HOHFWURQLFV DGGV ZKLWH *DXVVLDQ QRLVH ZLWK YDUL DQFH HTXDO WR 1  7KH RSWLPDO GHWHFWLRQ VWDWLVWLF PRGLI\LQJ   LV JLYHQ E\ + =7 ! /

\W VW GW  s + WKH + K\SRWKHVLV DQG s



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z 1  } DQG I C z 1   } 7KH YDULDQFH XQGHU WKH ?]HUR DV ZHOO DV XQGHU WKH ?RQH DVVXPSWLRQ LV } 1   7KH SUREDELOLW\ RI IDOVH DODUP 3)$ LV JLYHQ E\ = =  H>C 1  S I C GC GC 3)$ s {1   s s tU  u 4 s

 1

,Q WKH UDGDUVRQDU SUREOHP ZH XVH WKH 1H\PDQ3HDUVRQ FULWHULRQ ZKLFK n[HV WKH FRQVWDQW IDOVH DODUP UDWH &)$5  7KH DFWXDO WKUHVKROG s LV REWDLQHG E\

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH

)LJXUH 



5DGDUVRQDU GHWHFWRU

)LJXUH 

&RUUHODWRU

H[DPLQLQJ WKH WHVW VWDWLVWLF LH GHWHFWRU RXWSXW XQGHU WKH + K\SRWKHVLV DQG FRPSXWLQJ WKH WKUHVKROG s IURP WKH VWDWLVWLF DQG WKH n[HG DODUP UDWH YDOXH 7KH SUREDELOLW\ RI GHWHFWLRQ LV JLYHQ E\ =  H>C>  } S  GC 3' s {} s U u t  4 s b  1 



7KH nQDO GHWHFWRU DQG LWV WKUHVKROG IRU DQ DUELWUDU\ VWDUW WLPH W  DUH JLYHQ LQ )LJXUH  7\SLFDOO\ WKH IDOVH DODUP UDWH LV NHSW DW D YHU\ VPDOO OHYHO LQ WKH RUGHU RI > WR >  

0$7&+(' ),/7(56

,Q 6HFWLRQV  DQG  WR PDNH WKH QRWDWLRQ VLPSOH ZH UHODEHO WKH LQSXW DV

[W 

7KLV IROORZV WKH JHQHUDO V\VWHPV QRWDWLRQ ODEHOLQJ WKH LQSXWV DV

DQG WKH RXWSXW DV

\W 

[W

,I ZH UHH[DPLQH WKH GHWHFWRU RI WKH ODVW VHFWLRQ WKH

FRUUHODWRU  ZH UHDOL]H WKDW WKH PXOWLSOLFDWLRQ E\ WKH UHSOLFD DQG WKH IROORZRQ LQWHJUDWLRQ FDQ EH UHSODFHG E\ D VWDQGDUG FRQYROXWLRQDO nOWHU 7KH FRUUHODWRU LV UHSURGXFHG LQ )LJXUH  ZKLOH WKH FRQYROXWLRQDO nOWHU LV VKRZQ LQ )LJXUH 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

&RQYROXWLRQDO nOWHU

)LJXUH 

7KH FRUUHODWRU RXWSXW IRU DQ DUELWUDU\ LQSXW

=7

\& 7 ZKLOH WKH nOWHU RXWSXW DW WLPH



7

LV JLYHQ E\

[W V W GW



LV GHVFULEHG E\

=7

\) 7

[W  DW WLPH 7



[7 b ~ K ~ G~



:H QRWH WKDW WKH nOWHU RXWSXW LV ZULWWHQ DV EHLQJ SURGXFHG E\ D FDXVDO LQSXW RSHUDWLQJ RQ D FDXVDO V\VWHP 6XSSRVH

K W ZKHUH

K W

K W LV UHODWHG WR V W E\

V 7 b W 

IRU 

›W›7



LV D WLPH UHYHUVHG LH D WLPH PLUURUHG YHUVLRQ RI

V W 

7KHQ

WKH RXWSXW RI WKH nOWHU LV JLYHQ E\

\) 7 :LWK WKH FKDQJH LQ YDULDEOHV

=7 

[7 b ~ V 7 b ~ G~

7 b~

\) 7

= 7 =7 

}

DQG

bG~



G} WKH RXWSXW EHFRPHV

[} V } b G} [} V } G}



:H QRWH WKLV HTXDOV WKH RXWSXW RI WKH FRUUHODWRU LI LW LV HYDOXDWHG DW WLPH

7

6LQFH WKH UHVSRQVH RI WKH nOWHU LV PDWFKHG WR WKH VLJQDO WR EH GHWHFWHG WKLV WHFKQLTXH LV FDOOHG WKH ?PDWFKHG nOWHU )RU ELQDU\ K\SRWKHVLV WHVWLQJ ZH KDYH WZR FKDQQHOV DV VKRZQ LQ )LJXUH 

1RWH WKH FRQFHSW HDVLO\ H[WHQGV WR WKH PDQ\ 0DU\ K\SRWKHVLV E\

DOORZLQJ D PDWFKHG nOWHU IRU HDFK VLJQDO WR EH GHWHFWHG

$ W\SLFDO VHW XS

LV VKRZQ LQ )LJXUH  7KLV SDUDOOHO nOWHU DUUDQJHPHQW LV DOVR NQRZQ DV D PDWFKHG nOWHU EDQN :H FDQ DOVR H[SUHVV WKH UHVSRQVH RI WKH PDWFKHG nOWHU LQ WKH IUHTXHQF\ GRPDLQ /HW

6 M

=7 

VW H>MW GW



'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH

)LJXUH 

7ZR FKDQQHO GHWHFWLRQ

)LJXUH 

0DWFKHG nOWHU EDQN



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



=7

DQG

+ M 1RZ

KRSW W

KW H>MW GW



V7 b W DQG DVVXPLQJ WKDW VW LV UHDO YDOXHG KHQFH

=7

+RSW M

V7 b W H>MW GW



A

‚= 7

VW H>MW GW

 RU

+RSW M





H>M~

H>M~ 6 A M



0$7&+(' ),/7(5 $3352$&+ PD[LPL]LQJ WKH RXWSXW 615

:KHQ D NQRZQ VLJQDO PLJKW EH HPEHGGHG LQ ZKLWH *DXVVLDQ QRLVH WKH PDWFK HG nOWHU FDQ EH GHULYHG XVLQJ D FULWHULRQ ZKHUH RQH PD[LPL]HV WKH RXWSXW 615 7KLV REWDLQV DQ RSWLPDO UHFHLYHU VWUXFWXUH E\ PD[LPL]LQJ WKH RXWSXW 615 UDWKHU WKDQ PLQLPL]LQJ WKH %D\HV FRVW )RU WKH *DXVVLDQ QRLVH FDVH WKH RSWLPDO GHWHFWRU UHFHLYHU ZLOO EH WKH VDPH LQGHSHQGHQW RI WKH FULWHULRQ XVHG

$ EULHI GHVFULSWLRQ IROORZV

6XSSRVH WKHUH LV D GHWHUPLQLVWLF VLJQDO

HPEHGGHG LQ ZKLWH *DXVVLDQ QRLVH WKHQ WKH LQSXW WR nOWHU FDQ EH ZULWWHQ

VW  QW  7KH RXWSXW RI WKH nOWHU FDQ EH GHVFULEHG DV \W DV [W \V W  \Q W  7KH VXEVFULSW GHQRWHV WKH RXWSXW GXH WR WKH VLJQDO DQG WKH QRLVH LQSXW 7KH RXWSXW 615 FDQ EH H[SUHVVHG DV

 >\ 7 @ 615 A V YDU\1 7 ZKHUH

\V 7 DQG

\1 7

=7 

=7 

\1 W @

>

= 7= 7 



7



QRLVH SRZHU

K~ V7 b ~ G~ K~ Q7 b ~ G~

7KH QRLVH FRUUHODWLRQ IXQFWLRQ LV JLYHQ E\ 7KH QRLVH SRZHU DW WLPH

VLJQDO HQHUJ\

( IQW QW J

1



pW

  

LV REWDLQHG E\ nUVW FRPSXWLQJ

K~ K~ Q7 b ~ Q7 b ~ G~ G~

b W  

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH



DQG WKHQ DSSO\LQJ WKH H[SHFWDWLRQ RSHUDWRU

= 7= 7

( >\1 W @





=7

1 



K~ K~

1 

p~ b ~ G~ G~

K ~ G~



7KH VHFRQG PRPHQW LV QXPHULFDOO\ HTXDO WR WKH YDULDQFH ZKHQ QR '& FRP SRQHQW LV SUHVHQW ,Q RXU FDVH WKHQ WKH ODVW H[SUHVVLRQ UHSUHVHQWV WKH RXWSXW QRLVH YDULDQFH DW WLPH

7

7KH RXWSXW 615 EHFRPHV

= 7

615 A





1

K~ V7 b ~ G~

=7 



K ~ G~

8VLQJ WKH 6FKZDU] LQHTXDOLW\ WKH ODVW H[SUHVVLRQ EHFRPHV

= 7

615

A



1



K~ V7 b ~ G~

=7 



@



K ~ G~

›



1

=7 

7KH HTXDOLW\ RQO\ RFFXUV ZKHQ WKH LPSXOVH UHVSRQVH WKH WLPH PLUURUHG VLJQDO

V7

b W 

V 7 b ~ G~ KW



LV SURSRUWLRQDO WR

8QGHU WKHVH FRQGLWLRQV WKH WKHRUHWLFDO

PD[LPXP RI WKH RXWSXW 615 LV DFKLHYHG WKDW LV

615



=7

1



V 7 b ~ G~



7KH RSWLPXP GHWHFWRU XVLQJ WKH FULWHULRQ WKDW PD[LPL]H WKH RXWSXW 615 LV FDOOHG WKH PDWFKHG nOWHU IRU WKH REYLRXV UHDVRQ WKDW WKH LPSXOVH UHVSRQVH DQG WKH WLPH VLJQDO DUH UHODWHG LH PDWFKHG  1RWH ZH REWDLQ IRU WKLV VFHQDULR NQRZQ VLJQDO LQ ZKLWH *DXVVLDQ QRLVH WKH VDPH VROXWLRQ DV LI ZH PLQLPL]HG D FRVW IXQFWLRQ DV GHnQHG LQ &KDSWHU 



0$5< &20081,&$7,21 6 1

5 7 >\W @ GW

ZKLOH WKH RQH K\SRWKHVLV OHDGV WR

I \Mt ZKLFK EHFRPHV

I \Mt

57  .H> 1 >\W >$ VLQF Wt @ GW

57   .H> 1 >\ W > $ VLQF Wt D VLQ F Wt @ GW

([SDQGLQJ WKH VHFRQG TXDGUDWLF WHUP RI WKH H[SRQHQWLDO LQ WKH ODVW H[ SUHVVLRQ OHDGV WR

=7 

$ VLQ F W  t GW

= 7 $ 



GW b

= 7 $ 



F W  t GW

FRV

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH

 {

7

,I WKH LQWHJUDWLRQ WLPH

F



LV PXFK ORQJHU WKDQ WKH SHULRG RI RQH

F\FOH RI WKH VLQXVRLG WKHQ WKH VHFRQG WHUP LQ WKLV HTXDWLRQ FDQ EH QHJOHFWHG DQG WKH LQWHJUDO LV DSSUR[LPDWHO\ HTXDO WR RI WKH VLQXVRLG LV VXFK WKDW LQ

7

$ 7

 2I FRXUVH LI WKH SHULRG

VHFRQGV DQ LQWHJHU QXPEHU RI KDOI F\FOHV DUH

GHVFULEHG WKHQ WKH VHFRQG WHUP LV DXWRPDWLFDOO\ HTXDO WR ]HUR +HQFH WKH /57 EHFRPHV

= {

57  H> 1 \ W >\W $ VLQF Wt GW Gt {  5 7 \W GW e\  1 >

H 7KLV FDQ EH IXUWKHU UHGXFHG XVLQJ WKH WULJRQRPHWULF LGHQWLW\ VLQF W  t VLQF W FRVt  FRVF W VLQt WR = {  Gt H>$ 7 1 H$T 1 FRVt>t  e\ {   H>$ 7 1

+HUH D VLPSOLnFDWLRQ RI WKH IRUP

=7

\W VLQF W GW

T FRVt

\W FRVF W GW

T VLQt



=7 

t FRVt VLQt VLQt

DQG WKH WULJRQRPHWULF LGHQWLW\ FRV 

t

FRV 

b t ZHUH

XVHG 1RZ WKH IROORZLQJ LQWHJUDO ZLOO VLPSOLI\ WKH H[SUHVVLRQ IRU WKH /57 

{

 ZKHUH

m

= { 

H[ FRVtm Gt

FDQ EH DQ\ UHDO QXPEHU DQG

, [

,  LV WKH PRGLnHG ]HURRUGHU %HVVHO

IXQFWLRQ RI WKH nUVW NLQG 1RWH WKH SDUDPHWHUV FDQ DOVR EH WUDQVODWHG LQWR WKH ORZHU DQG XSSHU OLPLWV RI WKH LQWHJUDO LH FKDQJH RI YDULDEOHV  :KDW

{

PDWWHUV LV WKDW WKH VXSSRUW RI WKH LQWHJUDO LV H[DFWO\ RYHU   EHFRPHV

H> $ 7

1 ,

  

\

e

t $T u + 1

! w   +

7KH /57



7KLV FDQ EH IXUWKHU VLPSOLnHG

,

t $T u + 1

! w H$7 1   +



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



7KH PRGLnHG %HVVHO IXQFWLRQ WLRQ RI

T

, $T 1 LV D PRQRWRQLFDOO\ LQFUHDVLQJ IXQF T LWVHOI RU T FDQ EH XVHG DV D

VHH $SSHQGL[ '  7KHUHIRUH HLWKHU

GHFLVLRQ UXOH 7KH GHWHFWLRQ UXOH EHFRPHV

+ T !  s +



RU

+  s  T !  s + )RU D VSHFLnHG w  WKH WKUHVKROG s IRU WKH GHWHFWLRQ UXOH LQ  LV REWDLQHG E\ VROYLQJ

w

 H> $ 7 1 ,

t $s u  



1

7KH FRUUHODWRU DQG PDWFKHG nOWHU EDVHG GHWHFWRUV DUH VKRZQ LQ )LJXUHV  DQG  UHVSHFWLYHO\

2QH FDQ XVH

T

RU HTXLYDOHQWO\

T

DV WKH GHWHFWLRQ

 VWDWLVWLF 7KH TXDQWLW\ T LV SURSRUWLRQDO WR WKH SRZHU VSHFWUDO GHQVLW\ DW IUHTXHQF\

F 

$ OLWWOH WKRXJKW ZLOO UHYHDO WKDW WKH LPSOHPHQWDWLRQ LQ )LJXUH

 FRUUHVSRQGV WR D SRZHU VSHFWUXP DQDO\]HU

%DVHG RQ WKH GLVFXVVLRQV

LQ &KDSWHUV  DQG  ZH UHDOL]H WKDW XQGHU WKH QRLVH RQO\ FRQGLWLRQ WKH TXDQWLW\

T

ZLOO KDYH DQ H[SRQHQWLDO 3') LH FKLVTXDUHG ZLWK WZR GHJUHHV

RI IUHHGRP  ,I WKH VLJQDO LV SUHVHQW LH

T

+

LV WUXH  WKH UDQGRP YDULDEOH

ZLOO KDYH D QRQFHQWUDO FKLVTXDUHG 3') :H QRWLFH WKDW WKH RXWSXW RI

WKH FRUUHODWRUV RI )LJXUH  FRUUHVSRQG WR D nQLWH WLPH )RXULHU WUDQVIRUP RI

\W  ZKHUH WKH WRS OHJ DQG ERWWRP OHJV FRUUHVSRQG WR WKH LPDJLQDU\ DQG

UHDO SDUW RI WKH WUDQVIRUP UHVSHFWLYHO\ 7KH TXDQWLW\ WUXH

T

T

FRUUHVSRQGV WR WKH HQYHORSH :KHQ WKH K\SRWKHVLV

ZLOO KDYH D 5D\OHLJK 3') &RQYHUVHO\ LI WKH K\SRWKHVLV

+

+

LV WUXH

LV

T

ZLOO EH D 5LFLDQ YDULDWH



5DQGRP $PSOLWXGH

7KH VDPH SURFHGXUH DV IRU WKH UDQGRP SKDVH FDVH LV IROORZHG 7KDW LV DQ /57 RI WKH IRUP

= \

e

LV IRUPHG

$

I \MD I D GD + ! w   I \ +



)URP SK\VLFDO LQVLJKW RU D ZRUVWFDVH VFHQDULR WKH SUREDELOLW\

GHQVLW\ RI WKH UDQGRP YDULDEOH

D FDQ EH HVWLPDWHG LI WKH 3') LV QRW DYDLODEOH

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH

)LJXUH 

)LJXUH 

&RUUHODWRU LPSOHPHQWDWLRQ

0DWFKHG nOWHU LPSOHPHQWDWLRQ





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



5DQGRP $PSOLWXGH DQG 3KDVH

7KH /57 LV RI WKH IRUP

=

\

e

=

ZKHUH

d

,I

I \MJ I J G J + ! w d   I \ +

= =

I \MJ I J G J

$

d



I \MD t I D t GD Gt

D DQG t DUH VWDWLVWLFDOO\ LQGHSHQGHQW WKHQ I D t

I D I t 

1RWH LI

WKH ]HUR K\SRWKHVLV FRQWDLQV D UDQGRP FRPSRQHQW WKHQ WKH GHQRPLQDWRU RI WKH /57 ZLOO DOVR EH DYHUDJHG WR UHPRYH WKH SHUWXUEDWLRQ GXH WR WKH UDQGRP FRPSRQHQW



5DQGRP )UHTXHQF\

$JDLQ WKH /57 LV VHW XS WR DOORZ WKH DYHUDJLQJ RXW RI WKH UDQGRP FRPSRQHQW 7KH /57 LV RI WKH IRUP

=

\

l

e

7KH 3')

I \M I  G + ! w   I \ +



I  FDQ EH DSSUR[LPDWHG DV I 

7KH VSDFLQJ EHWZHHQ

L

DQG

0 ; L L

L p b L

3U





LV VXLWDEO\ FKRVHQ LH

0

HYHQO\ VSDFHG

VSHFWUDO ORFDWLRQV  )LJXUH  VKRZV D GHWHFWRU PDWFKHG nOWHU LPSOHPHQ WDWLRQ IRU D FRQVWDQW DPSOLWXGH XQLIRUP SKDVH GLVWULEXWLRQ DQG D XQLIRUP IUHTXHQF\ GLVWULEXWLRQ

7KH EORFN GHVLJQDWHG E\

DQG WKH HQYHORSH GHWHFWRU SURGXFHV XUH 

KL W

IRU

L

  c c c  0

 

LQ WKH VDPH ZD\ DV LOOXVWUDWHG LQ )LJ

:H UHDOL]H WKDW HDFK FKDQQHO WR SHUIRUP WKH HQYHORSH GHWHFWLRQ

SURSHUO\ PXVW KDYH DQ WKH SKDVH LV XQNQRZQ 



TL

K, W DQG D FRUUHVSRQGLQJ K4 W ,

DQG

4 FKDQQHO LI

5DQGRP $UULYDO 7LPH

7KH EDVLF GHWHFWRU LH FRUUHODWRU RU PDWFKHG nOWHU FDQ EH XVHG WR H[DPLQH DGMDFHQW FRQWLJXRXV RU RYHUODSSLQJ GDWD VHJPHQWV LQ WLPH ,I RQH ORRNV IRU

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH



)LJXUH  0DWFKHG nOWHU NQRZQ DPSOLWXGH UDQGRP XQLIRUP SKDVH DQG IUH TXHQF\ HPEHGGHG LQ DGGLWLYH ZKLWH *DXVVLDQ QRLVH

D VLJQDO RI OHQJWK

7

ZKLOH WKH VHJPHQWV DUH FRQWLJXRXV DQG RI VL]H

7

D

PLVPDWFK RI  G% UHODWLYH WR D SHUIHFW OLQH XS LV SRVVLEOH LQ WKH ZRUVWFDVH VFHQDULR 7KLV LV EHFDXVH KDOI RI WKH HQHUJ\ ZRXOG IDOO LQWR HDFK RQH RI WZR FRQWLJXRXV VHJPHQWV $ VLPSOH PHWKRG WR UHGXFH WKH ORVV LV WR XVH RYHUODS SURFHVVLQJ WKDW LV VHOHFW GDWD VHJPHQWV RI GXUDWLRQ

7

WKDW RYHUODS $W  RYHUODS DOVR FDOOHG

 RYHUODS WKH ZRUVWFDVH SURFHVVLQJ ORVV LV OLPLWHG WR  G% LH WKH ZRUVWORVV LV  RI WKH HQHUJ\  ,I RQH JRHV WR D  RYHUODS DOVR FDOOHG D  RYHUODS WKH ZRUVWFDVH SURFHVVLQJ ORVV LV OLPLWHG WR  G% LH WKH ZRUVW ORVV LV  RI WKH HQHUJ\  ,Q WKH H[WUHPH FDVH RQH FDQ JR WR DQ DUELWUDULO\ GHQVH RYHUODS ZKLFK LV ZKDW ZDYHOHW SURFHVVLQJ VHH &KDSWHU  DFKLHYHV DW WKH KLJKHVW VSHFWUDO EDQG SULRU WR UHVDPSOLQJ +RZHYHU LQ FRQYHQWLRQDO SURFHVVLQJ LH FRUUHODWRU RU PDWFKHG nOWHUV  WKH SURFHVVLQJ ORDG PD\ EHFRPH WRR ODUJH ZKLOH QRW SURYLGLQJ D PHDVXUDEOH JDLQ LQ SHUIRUPDQFH



6XPPDU\

,W LV REYLRXV WKDW WKH RSWLPXP UHFHLYHU LQ WKH %D\HV VHQVH RU E\ PD[LPL]LQJ WKH RXWSXW 615 IRU QDUURZEDQG VLJQDOV WRQDOV VLQXVRLGV ZLWK XQNQRZQ IUHTXHQF\ XQNQRZQ SKDVH DQG XQNQRZQ DUULYDO WLPH LV WKH SRZHU VSHFWUDO GHQVLW\ 8VXDOO\ WKLV LV LPSOHPHQWHG XVLQJ WKH SHULRGRJUDP ZKLFK LQ WXUQ LV WKH VFDOHG PDJQLWXGH VTXDUHG RXWSXW RI WKH ))7 $V ZH UHFDOO IURP WKH GLVFXVVLRQV LQ &KDSWHU  WKH ))7 LV DQ H[WUHPHO\ HpFLHQW SURFHVVLQJ WRRO

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



&OHYHU FKRLFH RI WKH ZLQGRZ IXQFWLRQ QXPEHU RI SDGGHG RQ ]HURV DQG RYHUODS IDFWRU ZLOO SURYLGH D JRRG PDWFK LQ IUHTXHQF\ DQG WLPH >@



08/7,3/( 38/6( '(7(&7,21

:KHQ PXOWLSOH SXOVHV DUH DYDLODEOH WZR SURFHVVLQJ WHFKQLTXHV OHQG WKHPVHOYHV IRU XVH ZKHQ WU\LQJ WR XVH WKH LQIRUPDWLRQ IURP PRUH WKDQ RQH SXOVH 7KH WZR FKRLFHV DUH D ,QFRKHUHQW SURFHVVLQJ RU LQWHJUDWLRQ DOVR FDOOHG SRZHU DYHUDJLQJ 7KLV XVXDOO\ UHGXFHV WKH YDULDQFH RI WKH RXWSXW VWDWLVWLF DQG HQKDQFHV WKH SURFHVVLQJ JDLQ E\ FRPELQLQJ WKH SURMHFWLRQV E &RKHUHQW SURFHVVLQJ RU LQWHJUDWLRQ 7KLV XVXDOO\ WDNHV WKH LQQHU SURGXFW REWDLQHG E\ SHUIRUPLQJ WKH FRUUHOD WLRQ RU PDWFKHG nOWHULQJ DQG SURYLGHV WKH SURFHVVLQJ JDLQ E\ H[WHQGLQJ WKH HmHFWLYH LQWHJUDWLRQ WLPH



,QFRKHUHQW $YHUDJLQJ

:H UHFDOO WKDW WKH H[DFW H[SUHVVLRQ IRU D VLQJOH NQRZQ SXOVH LV JLYHQ E\   7KH %HVVHO IXQFWLRQ

, [ KDV H[SDQVLRQV WKDW DUH YDOLG IRU VPDOO DUJXPHQWV

LH ORZ 615V DQG IRU ODUJH DUJXPHQWV LH ODUJH 615  7KH DSSUR[LPDWLRQ RI WKH ]HURRUGHU PRGLnHG %HVVHO IXQFWLRQ IRU VPDOO DUJXPHQWV LV JLYHQ E\

, [ {  

[ 

t $T u $ T {  ,  1 1

KHQFH



7DNLQJ WKH QDWXUDO ORJ RI  

OQ 

[

+HQFH WKH QDWXUDO ORJ RI WKH





[  IRU VPDOO [ OHDGV WR [b

[ 



[

b [ c c c  

, $T 1 DW VPDOO 615V LV DSSUR[LPDWHG E\

t $ T u $ T t $T u { OQ   1  { 1  OQ ,  1 







)LJXUH  VKRZV WKH JHQHUDO LH WKH H[DFW UDGDUVRQDU GHWHFWRU LPSOH PHQWDWLRQ IRU

0

SXOVHV ZKLOH )LJXUH  VKRZV WKH UHFHLYHU EDVHG RQ D ORZ

615 DSSUR[LPDWLRQ

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH

)LJXUH 

)LJXUH 



5DGDUVRQDU  SXOVH DYHUDJLQJ H[DFW VFKHPH

5DGDUVRQDU  SXOVH DSSUR[LPDWLRQ DW ORZ 615 DYHUDJLQJ VFKHPH

$W D KLJK 615 ZH FDQ XVH D GLmHUHQW DSSUR[LPDWLRQ 7KH DSSUR[LPDWLRQ RI WKH ]HURRUGHU PRGLnHG %HVVHO IXQFWLRQ LV JLYHQ E\

, [ {

[ {[

SH

,Q RXU FDVH WKH DSSUR[LPDWLRQ RI WKH %HVVHO IXQFWLRQ KLJK 615 EHFRPHV

t $T u H$T 1 {U ,  1 

{



7KH QDWXUDO ORJ EHFRPHV

t t $T uu { OQ ,  1 

{

$T  1 

$T 1



$T 1

t $T u b  OQ { 1 





7KH ODVW DSSUR[LPDWLRQ WDNHV DGYDQWDJH RI WKH IDFW WKDW WKH ORJDULWKP RI D ODUJH SRVLWLYH QXPEHU LV PXFK VPDOOHU WKDQ WKH SRVLWLYH QXPEHU LWVHOI $Q LPSOHPHQWDWLRQ XVLQJ WKH KLJK 615 DSSUR[LPDWLRQ LV VKRZQ LQ )LJXUH 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

7KH RSHUDWLRQ RI VXPPLQJ SRVW GHWHFWLRQ LQWHJUDWLRQ >@

+LJK 615 DSSUR[LPDWLRQ 0 VXFFHVVLYH T

RU

T

RXWSXWV LV DOVR FDOOHG

)URP DQ DQDO\VLV SRLQW RI YLHZ VTXDUH ODZ

GHWHFWLRQ LH

T

WLYHO\ VLPSOH

7KLV LV GXH WR WKH IDFW WKDW DYHUDJHG FKLVTXDUH SUREDELOLW\

LV SUHIHUUHG VLQFH WKH PDWKHPDWLFDO H[SUHVVLRQV DUH UHOD

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

T  GLmHU E\ OHVV WKDQ  G% DW D W\SLFDO 3

)$   YDOXH RI >  RYHU D 3' UDQJH RI  WR  IRU YDOXHV RI 0 IURP  WR  WLRQ RXWSXWV SRZHU ZLVH LH

>@

7KLV LV ZK\ WKH SHUIRUPDQFH RI DQ HQYHORSH EDVHG V\VWHP FDQ EH DSSUR[ LPDWHG XVLQJ HQYHORSH VTXDUHG TXDQWLWLHV



&RKHUHQW 9HUVXV ,QFRKHUHQW ,QWHJUDWLRQ $YHUDJLQJ

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nUVW VSHFWUDO GHQVLW\ HVWLPDWH DQG WKHQ DGG WKH

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH

)LJXUH 

)LJXUH 



&RKHUHQW DYHUDJLQJ H[DPSOH

,QFRKHUHQW DYHUDJLQJ H[DPSOH

FRPSRQHQWV ZH KDYH LPSOHPHQWHG SRZHU RU LQFRKHUHQW DYHUDJLQJ 6HH )LJ XUH  IRU DQ H[DPSOH >@



6800$5


 LH 1H\PDQ3HDUVRQ FULWHULRQ  nQG

WKH WKUHVKROG

G )RU WKH SUREOHP LQ SDUW F  FRPSXWH WKH

3' 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

 8QGHU

+ +



UL

V 7 3 U+

VL

L



    

     

QL LLG >   b b b@ V 7 

UL

VL 

3 U+

F

F

z 1   >b b b   @ F



F



D 'HULYH WKH RSWLPDO GHWHFWRU E\ PLQLPL]LQJ WKH SUREDELOLW\ RI HUURU E :KDW LV WKH

3H  WKH SUREDELOLW\ RI HUURU"

 7KH ELQDU\ VLJQDO  DQG  UHSUHVHQWHG E\  DQG  YROWV UHVSHFWLYHO\ DUH WUDQVPLWWHG RYHU D V\PPHWULF FKDQQHO 7KH SULRU SUREDELOLWLHV DUH WKH VDPH 7KHUH LV ]HUR FRVW IRU FRUUHFW GHFLVLRQV DQG D JLYHQ FRQVWDQW FRVW

&F 

IRU PDNLQJ D PLVWDNH

7KH FKDQQHO DGGV ]HUR PHDQ ZKLWH

*DXVVLDQ QRLVH ZLWK D SUHVFULEHG YDULDQFH

+



+



\ z 1  } \ z 1  }

} 

6R ZH KDYH

VLQJOH RU PXOWLSOH VDPSOHV VLQJOH RU PXOWLSOH VDPSOHV

D 'HVLJQ WKH DSSURSULDWH GHWHFWRUV DQG SURYLGH IRU HDFK RQH WKH DS

3)$ DQG 3'  1RWH 52& 3)$  ZKHUH 3' DQG 3)$ DUH SORWWHG RQ

SURSULDWH GHWHFWRU IRUP WKUHVKROG DQG FXUYHV GLVSOD\

3'

YHUVXV

WKH YHUWLFDO DQG KRUL]RQWDO D[LV UHVSHFWLYHO\

L &RPSXWHU ([HUFLVH 8VLQJ RQH GDWD VDPSOH REWDLQ DQG SORW WKHRUHWLFDO 52& FXUYHV IRU

}

   

     DQG  7KLV SDUW

VKRZV WKH WKHRUHWLFDO GHJUDGDWLRQ DV D IXQFWLRQ RI 615 RQH SORW ZLWK nYH WKHRUHWLFDO 52& FXUYHV  LL 3LFN WKH ZRUVW GHWHFWLRQ VFHQDULR RI L DERYH LQFUHDVH WKH QXPEHU RI GDWD VDPSOHV IURP  WR  DQG SORW WKH FRUUHVSRQGLQJ 52& FXUYH VXSHULPSRVHG RQ WKH UHVSHFWLYH ZRUVWFDVH GHWHFWRU FXUYH ZKLFK LV XVHG RQ VDPSOH  5HSHDW WKH FRPSXWDWLRQ XV LQJ

1

 VDPSOHV DQG VXSHULPSRVH RQ WKH RWKHU WZR FXUYHV

2I FRXUVH WKH QRLVH ZLOO EH VWDWLVWLFDOO\ LQGHSHQGHQW IRU DOO DQG DQ\ VDPSOH 7KLV SDUW RI WKH H[HUFLVH GHPRQVWUDWHV WKH SHUIRUPDQFH HQKDQFHPHQW DV ZH DUH DOORZHG WR XVH PRUH DQG PRUH VDPSOHV LQ WKH GHWHFWLRQ SURFHVV 8VH PDQ\ FRPSXWD WLRQ SRLQWV VXpFLHQWO\ WR REWDLQ D IDLUO\ VPRRWK VHW RI FXUYHV *LYH D EULHI H[SODQDWLRQ RI \RXU UHVXOWV RQH SORW ZLWK WKUHH WKHRUHWLFDO 52& FXUYHV  LLL 3HUIRUP D QXPHULFDO VLPXODWLRQ 8VH WKH WKHRUHWLFDO SORWV IURP SDUW L IRU

}



  DQG IRU

}

 DQG FRPSDUH WR H[SHU

LPHQWDO UHVXOWV REWDLQHG IURP D VLPXODWLRQ

7KH QXPEHU RI

'HWHFWLRQ RI 6LJQDOV LQ :KLWH *DXVVLDQ 1RLVH



H[SHULPHQWV LQ WKH VLPXODWLRQ VKRXOG EH VXpFLHQWO\ ODUJH WR LOOXVWUDWH WKDW WKH WKHRUHWLFDO DQG H[SHULPHQWDO 52& FXUYHV DJUHH WZR SORWV RQH IRU HDFK

} VHOHFWHG HDFK SORW GLVSOD\LQJ

D WKHRUHWLFDO DQG RQH H[SHULPHQWDO 52& FXUYH  &RPPHQW RQ \RXU UHVXOWV  8QGHU

ZKHUH

+



UW

QW

QW LV D ZKLWH *DXVVLDQ QRLVH SURFHVV ZLWK YDULDQFH RI  +

UW



VW

VW  QW

W IRU  › W ›  + 3U+ &  & & 

3U

&

D 'HULYH WKH RSWLPDO GHWHFWRU E\ PLQLPL]LQJ WKH SUREDELOLW\ RI HUURU E :KDW LV WKH WKUHVKROG" F :KDW LV

3H  WKH SUREDELOLW\ RI HUURU"

G *LYH WKH H[SUHVVLRQ DQG GUDZ WKH LPSXOVH UHVSRQVH RI WKH DSSUR SULDWH PDWFKHG nOWHU  8QGHU

+ +

ZKHUH

1





QW LV D ZKLWH  { 7  7

UW QW   › W › 7 UW VW  QW   › W › 7 

*DXVVLDQ QRLVH SURFHVV ZLWK ELVSHFWUDO KHLJKW  PLOOLVHFRQG DQG

VW

VLQ{ W 

D 'HVLJQ WKH RSWLPDO GHWHFWRU UHFHLYHU IRU WKHVH FRQGLWLRQV 3UR YLGH D GLDJUDP RI WKH GHWHFWRU DQG WKH QXPHULFDO YDOXH RI WKH WKUHVKROG E :KDW LV GHWDLO

3 U' M+

DQG

3 U' M+ "

6KRZ DOO ZRUN LQ VXpFLHQW

F 6XSSRVH LQVWHDG RI WKH UHTXLUHG UHSOLFD ZH FDQ RQO\ XVH D KDUG FOLSSHG YHUVLRQ RI WKH VLQXVRLG LH D VTXDUH ZDYH ZLWK WKH VDPH

3 U' M+ LI ZH DGMXVW WKH WKUHVK 3 U' M+ DV REWDLQHG LQ WKH 0/ DSSURDFK LQ SDUW

SHULRG  :KDW LV WKH FKDQJH LQ ROG WR SURYLGH D D DERYH"

G :KDW LV WKH ORVV RU JDLQ FRPSDULQJ WKH UHVXOWV LQ SDUW D DQG F "



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

5HIHUHQFHV >@ :KDOHQ $ 

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH 1HZ @ 5REHUWVRQ *0 ?3HUIRUPDQFH 'HJUDGDWLRQ E\ 3RVW 'HWHFWRU 1RQOLQ

%HOO 6\VW 7HFK - 9RO  1R   SS ^ .D\ 60 0RGHUQ 6SHFWUDO (VWLPDWLRQ 7KHRU\ DQG $SSOLFDWLRQ (Q

HDULWLHV >@

JOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ 0DUSOH 6/ +DOO 

'LJLWDO 6SHFWUDO $QDO\VLV (QJOHZRRG &OLmV 1- 3UHQWLFH

6LJQDO 7KHRU\ (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO  1REOH % DQG 'DQLHO -: $SSOLHG /LQHDU $OJHEUD (QJOHZRRG &OLmV

>@ )UDQNV /( >@

1- 3UHQWLFH+DOO  >@ 6WUDQJ *

/LQHDU $OJHEUD DQG ,WV $SSOLFDWLRQV

QG HG 1HZ @ 5REHUWVRQ *0 ?2SHUDWLQJ &KDUDFWHULVWLFV IRU D /LQHDU 'HWHFWRU RI &: 6LJQDOV LQ 1DUURZEDQG *DXVVLDQ 1RLVH  1R   SS ^

%HOO 6\VW 7HFK - 9RO

5DGDU 3ULQFLSOHV 1HZ @  $V GLVFXVVHG LQ &KDSWHU  WKH ')7 DQG ))7 DUH WKH WUDQVIRUPDWLRQ WKDW GRPLQDWH WKH PDMRULW\ RI VLJQDO SURFHVVLQJ DSSOLFDWLRQV )RU WKH )RXULHU EDVHG UHSUHVHQWDWLRQ ZH XVH D VHW RI RVFLOODWRU\ W\SH EDVLV IXQFWLRQV ZKHUH WKH EDVLV IXQFWLRQV DUH ZHLJKWHG DQG VXPPHG DV JLYHQ E\ (TXDWLRQ  ,Q JHQHUDO DQ H[SDQVLRQ UHSUHVHQWDWLRQ LQ ZKLFK WKH EDVLV IXQFWLRQV DUH RUWKRJRQDO LV FKRVHQ 7KLV PDNHV WKH H[SDQVLRQ FRHpFLHQWV RI GDWD LH N WK

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



VLJQDOV REVHUYDWLRQ SURFHVVHV WKDW KDYH D *DXVVLDQ W\SH 3') VWDWLVWLFDOO\ LQGHSHQGHQW +HQFH WKH DOJRULWKPV LH WKH VLPSOH OLNHOLKRRG UDWLR WHVW GHYHORSHG IRU LQGHSHQGHQW VDPSOHV DSSO\ GLUHFWO\ ZLWK KLJK RUGHU MRLQW 3')V EHFRPLQJ WKH SURGXFW RI RQHGLPHQVLRQDO 3')V 7KH H[SDQVLRQV IRU nQLWH GXUDWLRQ GDWD FDQ EH EDVHG RQ WKH VLJQDO VHW RU RQ WKH QRLVH VSDFH ,I ZH XVH DQ H[SDQVLRQ EDVHG RQ WKH VLJQDO VHW ZH FDQ XWLOL]H RQH RI WZR W\SHV RI UHSUHVHQWDWLRQV 6KRXOG ZH XVH DQ H[SDQVLRQ EDVHG RQ WKH QRLVH FKDUDFWHULVWLFV WKHQ ZH GHSHQG RQ DQ HLJHQYDOXH W\SH GHFRPSRVLWLRQ D 7KH EDVLV IXQFWLRQ VHW VSDQV WKH VLJQDO VSDFH 7\SLFDO FDQGLGDWHV DUH )RXULHU NHUQHO >@ RU DQ\ RWKHU RUWKRJRQDO SRO\QRPLDO IXQFWLRQ >@ LH +HUPLWH /DJXHUUH &KHEE\FKHm HWF  8VXDOO\ D FRPSOHWH RUWKRQRUPDO &21 VHW LV SUHIHUHG IRU WKH DQDO \VLV )RU VLJQDO VSDFH GHSHQGHQW H[SDQVLRQV ZH FDQ XVH DQ H[SDQVLRQ DORQJ D SUHIHUUHG VLJQDO FRRUGLQDWH D[LV :H VHOHFW DV WKH nUVW EDVLV IXQFWLRQ RQH SDUWLFXODU VLJQDO LH WKH VLJQDO XQGHU WKH + FRQGLWLRQ DQG FKRRVH DOO RWKHU EDVLV IXQFWLRQV LQ D VHTXHQWLDO IDVKLRQ RUWKRJ RQDO WR WKH nUVW DQG DOO SUHYLRXVO\ VHOHFWHG EDVLV IXQFWLRQV $OO EDVLV IXQFWLRQV DUH QRUPDOL]HG SURYLGLQJ D VHW RI RUWKRQRUPDO EDVLV IXQFWLRQV LH *UDP6FKPLGW SURFHGXUH >@ DQG &KDSWHU   E 7KH HLJHQIXQFWLRQV RU HLJHQYHFWRUV VHW VSDQV WKH QRLVH VSDFH :H XVH WKH FRUUHODWLRQ IXQFWLRQ RI WKH QRLVH SURFHVV WR JHQHUDWH WKH EDVLV IXQFWLRQV LH WKH .DUKXQHQ/RtHYH H[SDQVLRQ  ,Q WKH FRQWLQXRXV WLPH FDVH LH VHH 6HFWLRQ   WKH HLJHQIXQFWLRQV RI WKH DXWRFRUUHODWLRQ IXQFWLRQ RI WKH QRLVH VHUYH DV EDVLV IXQFWLRQV >@ ,Q WKH GLVFUHWHWLPH FDVH ZH XVH WKH DXWRFRUUHODWLRQ PDWUL[ RI WKH QRLVH DQG WKH FRUUH VSRQGLQJ HLJHQYHFWRUV >@ ,QGHSHQGHQW RI ZKLFK EDVLV IXQFWLRQ VHW LV FKRVHQ LW LV HVVHQWLDO FULWLFDO WKDW WKLV VHW LV D FRPSOHWH RQH 7KDW LV QR PDWWHU ZKDW WKH VLJQDO IRUP LV LW PXVW EH H[SUHVVLEOH DV D FRXQWDEOH VXP RI ZHLJKWHG EDVLV IXQFWLRQV DV JLYHQ E\  \ W \N JN W ›W›7 

;

N 

ZKHUH IJN W J LV D VHW RI FRXQWDEOH EDVLV IXQFWLRQV DQG \N LV WKH H[SDQVLRQ FRHpFLHQW WKDW LV WR EH SDLUHG ZLWK WKH NWK EDVLV IXQFWLRQ )RU QRWDWLRQDO FRQYHQLHQFH WKH LQWHUYDO LV FKRVHQ WR EH > 7 @ EXW LQ JHQHUDO WKH PHWKRG FDQ EH XVHG IRU DQ\ DUELWUDU\ LQWHUYDO >W  W  7 @ 1RWH WKDW N FDQ UDQJH IURP RQH WR LQnQLW\ RU IURP RQH WR . D nQLWH QXPEHU  ,Q VRPH SUREOHPV WKH H[SDQVLRQ FRXQW N FDQ VWDUW DW ]HUR LH VHH WKH ')7 W\SH UHSUHVHQWDWLRQ LQ &KDSWHU   ,Q VRPH H[SDQVLRQV WKH FRXQW YDULDEOH N FDQ DOVR WDNH RQ QHJDWLYH YDOXHV VXFK DV LQ WKH FRPSOH[ )RXULHU VHULHV UHSUHVHQWDWLRQ :H



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

QRWH WKDW LQ PRVW GLVFXVVLRQV ZH DVVXPH WKDW WKH EDVLV IXQFWLRQV DUH UHDO YDOXHG ,I FRPSOH[ YDOXHG EDVLV IXQFWLRQV DUH WR EH FRQVLGHUHG WKH GHnQLWLRQ RI WKH LQQHU SURGXFW PXVW EH JHQHUDOL]HG E\ FRQMXJDWLQJ RQH RI WKH IXQFWLRQV LQYROYHG 7KLV LQGHHG JXDUDQWHHV WKDW WKH SURMHFWLRQ RI RQH EDVLV IXQFWLRQ RQWR LWVHOI LH LWV HQHUJ\ ZLOO DOZD\V EH UHDO YDOXHG 7KH NWK H[SDQVLRQ FRHpFLHQW \N  LV REWDLQHG E\ SURMHFWLQJ \W RQWR WKH WK N EDVLV IXQFWLRQ

=

K\W  JN W L

\N

7



\ W JN W GW



ZKHUH WKH DQJXODU EUDFNHWV GHQRWH WKH LQQHU SURGXFW 1RWH WKH XVDJH RI WKH DQJXODU EUDFNHWV WR GHQRWH WKH LQQHU SURGXFW ,Q &KDSWHU  WKH DQJXODU EUDFNHWV ZLOO EH XVHG WR LQGLFDWH D WLPH DYHUDJLQJ RSHUDWLRQ 7KH EDVLV VHW IJN W J FRQVLVWV RI D &21 VHW WKDW LV

KJL W  JN W L

ZKHUH

|

=



7

JL W JN W GW

pLN



L N  HOVH  LH WKH .URQHFNHU GHOWD DQG IJN W J ZLOO VSDQ WKH VLJQDO VSDFH ,I QHFHVVDU\ H[SUHVVLRQV  ^ FDQ HDVLO\ EH PRGLnHG WR DOORZ WKH UHSUHVHQWDWLRQ RI FRPSOH[ YDOXHG VLJQDOV RU WR UHSUHVHQW UHDO YDOXHG VLJQDOV XVLQJ FRPSOH[ YDOXHG EDVLV IXQFWLRQV $V WKH UHDGHU ZLOO UHFDOO WKH EDVLV IXQFWLRQV RI WKH ')7 DQG ))7 DUH FRPSOH[ YDOXHG H[SRQHQWLDOV )RU VRPH FRPSOH[ EDVLV IXQFWLRQ H[DPSOHV VHH &KDSWHU  ,Q PRVW ZRUN GLVFXVVHG LQ &KDSWHU  DV HDUOLHU VWDWHG ZH VKDOO GHDO ZLWK UHDO YDOXHG SURFHVVHV :H GHQRWH WKH PWK DSSUR[LPDWLRQ WR WKH IXQFWLRQ I W LH WKH NQRZQ VLJQDO  E\ IP W  WKDW LV pLN

IP W

VXFK WKDW

OLP  IP W

P

; I J W P

N 

N

; I J W N 

N



N

N

I W

:H ZLOO UHH[DPLQH WKH SUREOHP RI D NQRZQ VLJQDO HPEHGGHG LQ ZKLWH *DXV VLDQ QRLVH WR LOOXVWUDWH WKH XVH RI VHULHV UHSUHVHQWDWLRQV DQG WKH PDQLSXODWLRQV RI WKH UHVXOWLQJ WHUPV

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH 



'(5,9$7,21 2) 7+( &255(/$725 6758&785( 86,1* $1 $5%,75$5< &203/(7( 257+21250$/ &21 6(7

7KLV VHFWLRQ HPSOR\V WKH VHULHV H[SDQVLRQ WR GHULYH WKH VWDQGDUG FRUUHODWRU UHVXOW ,W SULPDULO\ VHUYHV WKH SXUSRVH RI H[SRVLQJ WKH UHDGHU WR WKH DSSOLFD WLRQ RI VHULHV H[SUHVVLRQV E\ DOORZLQJ WKH GHULYDWLRQ RI D NQRZQ UHVXOW XVLQJ DQ DOWHUQDWLYH WKH VHULHV EDVHG DSSURDFK :H VKDOO LQYHVWLJDWH WKH GHWHFWLRQ RI D NQRZQ VLJQDO LQ DQ DGGLWLYH ZKLWH *DXVVLDQ VFHQDULR DV LV DGGUHVVHG LQ &KDSWHU  6HFWLRQ  8QGHU +L  ZH KDYH L    +L  \W VL W  QW  ›W›7 ZKHUH QW LV ZKLWH *DXVVLDQ QRLVH 7KH NWK SURMHFWLRQ FRPSRQHQW DORQJ WKH N WK EDVLV IXQFWLRQ LV JLYHQ E\ L    \N VLN  QN ZKHUH \N

K\W  JN W L

VN

KV W  JN W L

VN

KV W  JN W L

QN

KQW  JN W L

= \W J W GW = V W J W GW = V W J W GW = QW J W GW 7

N



7





N



N

7



7

N



DQG IJN W J IRUPV D &21 DUELWUDU\ EDVLV VHW

([DPSOH  $Q H[DPSOH RI D VHFRQG RUGHU DSSUR[LPDWLRQ IRU WKH UHFHLYHG VLJQDO ZKHQ WKH + K\SRWKHVLV LV WUXH LH V W LV SUHVHQW  LV LOOXVWUDWHG LQ )LJXUH  +HUH XQGHU +  ZH KDYH \W V W  QW ZKLOH WKH VHFRQG RUGHU DSSUR[LPDWLRQ LV JLYHQ E\ \ W V W  Q W

;V 

N 

N

JN W 

; Q J W 

N 

N

N



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

6HFRQG RUGHU DSSUR[LPDWLRQ RI \W XQGHU WKH + FRQGLWLRQ

1RWH \ LV WKH SURMHFWLRQ RI \W RQWR WKH VHFRQG EDVLV IXQFWLRQ ZKLOH \ W LV WKH VHFRQG RUGHU DSSUR[LPDWLRQ RI \W  1RWH DOVR WKDW ZH XVH )LJXUH  WR UHSUHVHQW ERWK WKH SURMHFWLRQV RI WKH FRPSRQHQWV RQWR WKH EDVLV D[HV DV ZHOO DV WKH UHVXOWDQW WLPH GHSHQGHQW FRPSRQHQWV )RU H[DPSOH V DQG V UHSUHVHQW WKH SURMHFWLRQ RI V W RQWR J W DQG J W  UHVSHFWLYHO\ 7KH H[SUHVVLRQ V W UHSUHVHQWV WKH VXP RI V J W  V J W  ,Q JHQHUDO WKH H[SDQVLRQ FRHpFLHQWV \  \  c c c  DUH UDQGRP YDULDEOHV ZKLFK DUH GHnQHG E\ XQGHU +  \N K\W  JN W L KV W  QW  JN W L VN  QN  +  \N KV W  QW  JN W L VN  QN  :H QRWH WKDW QN KQW  JN W L LV D OLQHDU WUDQVIRUPDWLRQ RI QW DQG VLQFH QW LV *DXVVLDQ XQGHU HDFK K\SRWKHVLV QN  DQG KHQFH \N ZLOO EH *DXVVLDQ $OO WKDW LV UHTXLUHG LV NQRZOHGJH RI WKH PHDQV DQG WKH YDULDQFH 7KH PHDQV DUH JLYHQ E\ ( IVN  QN J VN ( I\N M+ J  ( IVN  QN J VN ( I\N M+ J 

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



ZKLOH WKH YDULDQFH LV GHVFULEHG E\ YDU I\N M+ J

YDU IVN  QN J ( IVN  QN J b (  IVN J ( IVN J  ( IQN J  ( IVN QN J b ( IVN J  WKH YDULDQFH EHFRPHV

DQG VLQFH ( IVN QN J YDU I\N M+ J

( IQN J ( IVN  QN J b (  IVN J YDU I\N M+ J



= = QW QV J W J V GW GV = = 1 pW b V J W J V GW GV  = J W GW 1 p 1 1

,Q WKH ZKLWH QRLVH FDVH ZH ZRXOG KDYH ( IQN J

7

(

7



7



N



7



N



N

N

7

      N  NN  :H FDQ IRUP WKH OLNHOLKRRG UDWLR WHVW XVLQJ WKH RUWKRJRQDO FRPSRQHQWV \ J W  \ J W  c cc  \N JN W DV JLYHQ E\ e \W OLP e \N W N 

I \ J W  \ J W  c cc  \N JN W OLP N I \ J W  \ J W  c cc  \ J W 

 

 

N N

:H UHFDOO WKDW LI D *DXVVLDQ KDV SDUDPHWHUV [ z 1 P[  }[  WKHQ \ m[ z 1 mP[  m }[ WKHUHIRUH WKH QWK WHUP IRU Q   c c c  LV JLYHQ E\ I \Q JQ W I \Q I \Q JQ W I \Q VR WKH /57 EHFRPHV I \  \  c cc  \N e \W OLP N I \  \  c cc  \N



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



< I \ I \ < Hb OLP  < b H N

OLP N



Q

Q  

Q

N

Q  N

N

b

 1  \Q VQ 

+ !  w +

b

 1  \Q VQ 

Q 



7DNLQJ WKH QDWXUDO ORJDULWKP RI WKH NWK DSSUR[LPDWLRQ RI WKH /57 OHDGV WR OQ e \N W



; \ V N

1 Q



Q

 Q b VQ  1

; bV N

 Q 

 Q

b

c +!



VQ

OQ w 

+

:H ZDQW WR REWDLQ DQ HTXLYDOHQW H[SUHVVLRQ IRU WKH nUVW DQG VHFRQG VXPPD WLRQV LQ   7R GR WKLV ZH ZLOO VWDUW ZLWK WKH IROORZLQJ LQWHJUDO

=



7

\N W VLN W GW

= ; \ J W  ; V N

7



N

Q

Q 

Q

U 

LU



JU W GW



ZKHUH \N W LV WKH NWK DSSUR[LPDWLRQ RI \W DQG VLN W LV WKH NWK DSSUR[LPD WLRQ RI VL W IRU L   DQG JQ W LV WKH QWK EDVLV IXQFWLRQ IRU Q  c c c  N %\ UHDUUDQJLQJ WKH RUGHU RI WKH OLQHDU RSHUDWRUV RI WKH ULJKWKDQG VLGH RI   ZH REWDLQ

; ; \ V = J W J W GW ; ; \ _Q ^] ` ; \ V  IRU L   N

N

Q U 

N

7

Q

LU

Q



U

Q

 

HOVH

N

Q U 

U

Q

VLU pQU

pQU

N

Q 

Q

LQ



5HYHUVLQJ WKH RUGHU RI WKH HTXDWLRQ ZH REWDLQ

;\ N

Q 

Q

VLQ

=

7



\N W VLN W GW



%\ DQDORJ\ LW LV HDVLO\ VKRZQ WKDW

;V = N

Q 



LQ



7

VLN W GW



'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



7KHUHIRUH WKH /57 FDQ EH ZULWWHQ DV  7 \ W >V W b V W @ GW OQ e \N W N N N 1

=





 1

= dV 7



 N



W b

+ e W GW ! 



VN

OQ w



+

7DNLQJ WKH OLPLW DV N   OHDGV WR OLP OQ e \N W OQ e \W N 

1

=

7



 1



\ W >V W b V W @ GW

= dV W b V W e GW +! 



7

 



 

OQ w 

+

ZKHUH w LV D IXQFWLRQ RI GHWHFWRU SKLORVRSK\ LH 0/ LPSOLHV w LV XQLW\  )RU %D\HV IRUPXODWLRQ ZH KDYH w

3 & b & 3 & b &

,Q JHQHUDO WKH GHWHFWRU EHFRPHV

=

7



ZKHUH

= \ W V W GW b 



7

+ ! \ W V W GW  o +

= b

c



 7 V W b V W GW OQ w       ,I D 1H\PDQ3HDUVRQ FULWHULRQ LV XVHG WKHQ o LV VHOHFWHG WR PHHW WKH GHVLUHG 3) $  1RWHV  $JDLQ ZH GHULYHG WKH FRUUHODWLRQ VWUXFWXUH WKH VDPH RQH WKDW ZDV nUVW GHYHORSHG LQ &KDSWHU  6HFWLRQ  7KLV WLPH UDWKHU WKDQ XVLQJ D GDWD VDPSOLQJ DSSURDFK ZH XVHG D GHFRPSRVLWLRQ EDVHG RQ DQ DUELWUDU\ &21 EDVLV VHW o

1

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



 ,I RQO\ ZKLWH QRLVH LV SUHVHQW DQ\ &21 VHW ZLOO EH DSSURSULDWH WR EH XVHG DV EDVLV IXQFWLRQ VHW IJQ W J LQGHSHQGHQW RI ZKHWKHU RU QRW WKH REVHUYDWLRQ LQWHUYDO 7 LV nQLWH RU QRW 'XH WR LWV QDWXUDO LQWHUSUHWDWLRQ DQG WKH HDVH RI LWV LPSOHPHQWDWLRQ >@ WKH )RXULHU EDVLV VHW LV XVXDOO\ SUHIHUUHG  ,I WKH VLJQDO GXUDWLRQ 7 EHFRPHV YHU\ ODUJH LH UDQJLQJ IURP b WR   WKH FRHpFLHQWV RI WKH )RXULHU H[SDQVLRQ RI DQ\ ZVV UDQGRP SURFHVV RI DQ\ VSHFWUDO VKDSH WHQG WR EH XQFRUUHODWHG >@  $ SUREOHP RFFXUV ZKHQ GHDOLQJ ZLWK D FRORUHG UDQGRP SURFHVV LI WKH VLJQDO LV RI nQLWH GXUDWLRQ LH 7    :H FDQQRW XVH WKH )RXULHU H[SDQVLRQ VLQFH WKH FRHpFLHQWV EHFRPH XQFRUUHODWHG RQO\ DV 7   >@ 7KH .DUKXQHQ/RtHYH H[SDQVLRQ >@ WR EH GLVFXVVHG LQ 6HFWLRQ  DOORZV GHFRPSRVLWLRQ RI WKH VLJQDOV RI LQWHUHVW VXFK WKDW WKH FRORUHG QRLVH H[SDQVLRQ FRHpFLHQWV DUH XQFRUUHODWHG 

*5$06&+0,'7 352&('85(

7KLV VHFWLRQ VHUYHV DV DQ LQWURGXFWLRQ WR WKH *UDP6FKPLGW SURFHGXUH D WHFKQLTXH IUHTXHQWO\ XVHG LQ VWDWLVWLFDO GLJLWDO VLJQDO SURFHVVLQJ 6XSSRVH WKDW ZH KDYH D +LOEHUW VSDFH LH D YHFWRU VSDFH ZLWK DQ LQQHU SURGXFW DQG D QRUP GHULYHG IURP WKH LQQHU SURGXFW  DQG DOORZLQJ IRU FRPSOH[ YDOXHG IXQFWLRQV RU YHFWRUV ZH FDQ GHnQH WKH LQQHU SURGXFW GHSHQGLQJ RQ WKH VFHQDULR DV [L \Le  IRU GLVFUHWH WLPH YDULDEOHV

! ; !!! = e !!! [W \ W GW ! ( [ \e L

K[ \L

 IRU FRQWLQXRXV WLPH YDULDEOHV



 IRU UDQGRP YDULDEOHV ZKHUH GHQRWHV FRQMXJDWLRQ 7KH QRUP LV GHnQHG DV  QRUP K[ [L  MM[MM *LYHQ D YHFWRU VSDFH LV VSDQQHG E\ WKH YHFWRUV Im  m  c c cJ ZH ZDQW WR VSDQ WKH VDPH YHFWRU VSDFH ZLWK D VHW RI FRPSOHWH RUWKRQRUPDO &21 YHFWRUV IT  T  c c c  J :H GR WKLV E\ VHTXHQWLDOO\ IRUPLQJ RUWKRJRQDO YHFWRUV IY  Y  c c c  J DQG SHUIRUPLQJ D QRUPDOL]DWLRQ WR REWDLQ IT  T  c c c  J ImJ b IYJ b ITJ DUELWU RUWKRJRQDO RUWKR YHFWRUV YHFWRUV QRUPDO YHFWRUV

_^]`

_^]`

_^]`

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



)LJXUH  )LUVW WZR YHFWRU UHODWLRQVKLSV ZKHQ XVLQJ WKH *UDP6FKPLGW SURFHGXUH )LUVW ZH VHOHFW D VWDUWLQJ YHFWRU :LWKRXW ORVV RI JHQHUDOLW\ ZH VHOHFW m DV WKH VWDUWLQJ YHFWRU DQG WKHQ VHTXHQWLDOO\ GHWHUPLQH WKH IROORZRQ YHFWRU Y Y m T MMYY MM KY  YY L   





Y m b N T ZKHUH N LV VHOHFWHG VXFK WKDW Y DQG T DUH RUWKRJRQDO Y DQG Y DUH RUWKRJRQDO  :KHQ YHFWRU Y " T UHDG Y RUWKRJRQDO WR T  WKHQ ZH KDYH  KT  Y L KT  m b N T L KT  m L b N KT  T L KT  m L b N  :H UHDOL]H WKDW KT  T L  KHQFH WKH FRHpFLHQW N LV REWDLQHG DV  N KT  m L Km  T L ZKLOH WKH RUWKRJRQDO YHFWRU Y LV JLYHQ E\ Y m b Km  T L T m b KYKmY YLL  KY  YY L       DQG RUWKRQRUPDO YHFWRU T LV REWDLQHG E\ T MMYY MM  

7KH nUVW WZR UHODWLRQVKLSV DUH LOOXVWUDWHG LQ )LJXUH  ,Q JHQHUDO WKH RUWKR QRUPDOL]DWLRQ SURFHGXUH LV REWDLQHG IURP D VHTXHQWLDO GHWHUPLQDWLRQ RI RU WKRJRQDO YHFWRUV IRU N      DV GHnQHG E\ N b RUWKRJRQDO  mN b KYKmNY YLL L KY  YYL L  YN VHW L L L L L 

;

t

u



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

t

u

YN YN RUWKRQRUPDO TN    VHW MMYN MM KYN  YN L ZKHUH WKH QRUPDOL]DWLRQ ZDV REWDLQHG E\ GLYLGLQJ HDFK YHFWRU E\ LWV QRUP ([DPSOH  *LYHQ Im  m  c c c  J IW  W  W  c c c  J ZKHUH b › W ›  )LQG DQ RUWKRQRUPDO VHW RYHU >b @ :H QRWH WKDW

=  c  GW   =b  c W GW W nnn  nb b 

Km  m L



     +HQFH WKH VHW Im  m  c c c  J GRHV QRW IRUP DQ RUWKRJRQDO EDVLV VHW IRU MWM ›  /HW Y m Y  S T  MMY MM   c GW 

Km  m L

 



V=

b

 ~  S S W b W   =   Wb W S GW S m b Km  T L T

Y



b

V=

MMY MM



b



W c W GW

U

Y MMY MM

T Y



nnn  nb W U

W

Wb

V W nn n  nb





 





W

U  U ~    W b W  W  W b W  S S = =   W GW b W GW W b W m b Km  T L T b Km  T L T 



  W b 







b





 b



'HWHFWLRQ RI 6LJQDOV LQ 1RLVH

MMY MM

nnnnnn  nnnnnn V= t  u nnW b  nn b W b  GW V t W W W unn   n  b  c  c    nb  S t u Y  S W b  



















T





MMY MM     7KH RUWKRQRUPDO VHW IT  T  T  c c cJ LV QRZ RI WKH IRUP S IT  T  T  c c c  J S   W   W b   c c c

 U

t u

 ZKLFK LV WKH VHW RI RUWKRQRUPDO /HJHQGUH SRO\QRPLDOV RYHU WKH LQWHUYDO b   :H OHDYH LW IRU WKH UHDGHU WR YHULI\ WKDW LQGHHG KT  T L KT  T L KT  T L  DQG WKDW KT  T L KT  T L KT  T L  

'(7(&7,21 2) $ .12:1 6,*1$/ ,1 $'',7,9( :+,7( *$866,$1 12,6( 86,1* 7+( *5$06&+0,'7 352&('85(

,Q WKLV VHFWLRQ ZH ZDQW WR DSSO\ WKH *UDP6FKPLGW RUWKRQRUPDOL]DWLRQ SUR FHGXUH WR WKH SUREOHP RI GHWHFWLQJ D NQRZQ VLJQDO HPEHGGHG LQ ZKLWH *DXV VLDQ QRLVH 7KLV HQKDQFHV WKH XQGHUVWDQGLQJ RI VHULHV UHSUHVHQWDWLRQV DQG OHDGV WR WKH JHQHUDO FRUUHODWRU UHVXOW XVLQJ FRRUGLQDWHV D[HV WKDW DUH VOLJKWO\ GLmHUHQW FRPSDUHG WR UHVXOWV GHULYHG HDUOLHU 6XSSRVH ZH ZDQW WR GHWHUPLQH ZKHWKHU RU QRW WKH VLJQDO V W RU V W LV HPEHGGHG LQ ZKLWH *DXVVLDQ QRLVH DQG WKDW ZH KDYH V W  QW XQGHU +  \W ›W›7  +  \ W V W  QW  :H ZDQW WR nQG WKH RSWLPDO GHWHFWRU XVLQJ WKH *UDP6FKPLGW SURFHGXUH /HW 7  KV W  V W L V W GW

=





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

EH WKH HQHUJ\ RI V W

EH WKH HQHUJ\ RI V W  DQG U

=

KV W  V W L





=

KV W  V W L

7

7



V W GW

V W V W GW

EH WKH XQQRUPDOL]HG FURVVFRUUHODWLRQ FRHpFLHQW :H DUELWUDULO\ KDYH FKRVHQ V W WR EH WKH nUVW EDVLV IXQFWLRQ QRWH WKDW ZH FRXOG KDYH XVHG V W MXVW DV ZHOO  KHQFH V W Y W V W V W T W S   KV W  V W L  Y W

V W b

KV W  V W L V W   KV W  V W L KV W  V W L 

U V W V W b S S   T W

V W b

U V W  

Y W

MMY W MM

V~t U u t U u V W b V W  V W b V W   V U U  b    V  b U

7KH QRUP RI Y W LV JLYHQ E\

MMY W MM















  



 







KHQFH T W

U V W S     b U

V W b

S





'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



)RU WKH ELQDU\ FDVH QR DGGLWLRQDO EDVLV IXQFWLRQV DUH QHHGHG VLQFH ZH DOUHDG\ KDYH D FRPSOHWH RUWKRQRUPDO VHW WR GHVFULEH WKH VLJQDOV V W DQG V W  ,I DQ\ ZHUH WR EH FKRVHQ WKH\ ZRXOG EH RUWKRJRQDO WR T W DQG T W  ,Q JHQHUDO \W

; \ T W 

›W›7

C C

C 



ZKHUH \C K\W  TO W L ,Q RXU FDVH \ W

; \ T W 

 › W › 7

C O

C 

:H UHDOL]H D WKDW \C LV WKH SURMHFWLRQ RI \W RQWR WKH EDVLV IXQFWLRQV TC W IRU C   DQG E GXH WR WKH OLQHDULW\ RI WKH LQQHU SURGXFW WKH YDULDEOHV \C  IRU C   ZLOO EH LQGHSHQGHQW *DXVVLDQ UDQGRP YDULDEOHV :H QHHG WKH PHDQV DQG YDULDQFHV RI WKHVH *DXVVLDQ UDQGRP YDULDEOHV %HFDXVH WKH QRLVH KDV ]HUR PHDQ ZH KDYH ( I\C M+ J

= =

7



7



\ WM+ TC W GW V W TC W GW

VC



C

  

6LPLODUO\ WKH PHDQ XQGHU WKH + K\SRWKHVLV DQG WKH YDULDQFH LQGH SHQGHQW RI WKH K\SRWKHVLV DUH JLYHQ E\ ( I\C M+ J

YDU I\C M+L J

= V W T W GW V C    = = (IQW QV J T W T V GW GV = T W GW 1  C    1 7

C





7





C

7

C



7





C

L     C  UHVSHFWLYHO\ ZKHUH C LV EDVLV IXQFWLRQ DQG L FRUUHVSRQGV WR WKH K\SRWK HVLV :H RUWKRQRUPDOL]HG WKH WZR RULJLQDO IXQFWLRQV DQG QHHG RQO\ WKH WZR RUWKRQRUPDO IXQFWLRQV DQG WKHLU H[SDQVLRQ FRHpFLHQWV LH e\W e\ W 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH  C

2SWLPDO GHWHFWRU XVLQJ *UDP6FKPLGW SURFHGXUH VLC  IRU L   GHQRWHV WKH CWK FRHpFLHQW RI WKH LWK VLJQDO 

  DQG

,I PRUH EDVLV IXQFWLRQV VD\ 0  DUH UHTXLUHG VXFK DV LQ DQ 0DU\ SUREOHP WKHQ e\W e\0 W  )RU RXU ELQDU\ SUREOHP WKH /57 EHFRPHV

<S H b {1 <S H b 

e\ W

OQ e\ W



; \ V 

N 

N

; \ V 

1 N

N





N 

RU

RU

N 

N



b VN

b

  1 \N VN 

{1

b

  1 \N VN 

 N b VN  1

;V 

 N 

;

 N

+ !  w +



+ !  +

OQ w 

b VN

+   V b V ! 1 OQ w b    N  N  N +

w



7KLV GHWHFWRU LV HDVLO\ LPSOHPHQWHG DV VKRZQ LQ )LJXUH  $V ZH VHH WKH VWUXFWXUH LV WKH VDPH DV WKH FRUUHODWRU VWUXFWXUH DGYRFDWHG LQ &KDSWHU  6HFWLRQ  DQG 6HFWLRQ  :H FDQ LQWHUSUHW WKH GHWHFWRU RSHUDWLRQ DV D SURMHFWLRQ RI WKH GDWD RQWR EDVLV IXQFWLRQV YHFWRUV WR GHWHUPLQH ZKLFK SUR MHFWLRQ FRHpFLHQW GRPLQDWHV 2I FRXUVH ZH DFFRXQW IRU SULRU LQIRUPDWLRQ FRVW DQG IRU GLmHUHQWLDO HQHUJ\ ,Q VXPPDU\ ZH KDYH DGGUHVVHG WKUHH ZD\V IRU GHWHFWLQJ NQRZQ VLJQDOV LQ ZKLWH *DXVVLDQ QRLVH (DFK DSSURDFK OHG WR WKH FRUUHODWRU VWUXFWXUH ZKLFK

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



LV D SURMHFWLRQ RI WKH GDWD RQWR D VHW RI EDVLV IXQFWLRQV IRU FRQWLQXRXV WLPH VLJQDOV RU YHFWRUV IRU GLVFUHWHWLPH VLJQDOV LQ D +LOEHUW VSDFH VHWWLQJ 



6(5,(6 (;3$16,21 )25 &217,18286 7,0( '(7(&7,21 )25 &2/25(' *$866,$1 12,6( ,QWURGXFWLRQ

7KLV VHFWLRQ LQWURGXFHV VRPH FRQFHSWV WKDW DUH HVVHQWLDO LQ VROYLQJ WKH SURE OHP RI GHWHFWLQJ VLJQDOV HPEHGGHG LQ FRORUHG *DXVVLDQ QRLVH :H ZLOO nUVW UH YLHZ 0HUFHU V WKHRUHP DQG WKH .DUKXQHQ/RtHYH H[SDQVLRQ ZKLFK DUH QHHGHG LQ WKH GHULYDWLRQ RI WKH GHWHFWRU ,I WKH DGGLWLYH *DXVVLDQ QRLVH SURFHVV LV FRORUHG DQG WKH VLJQDO GXUDWLRQ LV nQLWH WKHQ WKH XVH RI WUDGLWLRQDO RUWKRJRQDO RU RUWKRQRUPDO EDVLV IXQFWLRQV OHDGV WR FRUUHODWHG FRHpFLHQWV 7KLV GRHV QRW OHQG LWVHOI WR DQ DSSURDFK EDVHG RQ D VLPSOH /57 $ ZRUNDEOH VROXWLRQ LV DYDLODEOH LI ZH UHVRUW WR WKH .DUKXQHQ /RtHYH H[SDQVLRQ >@ 7KH .DUKXQHQ/RtHYH H[SDQVLRQ LV EDVHG RQ WKH FRUUHODWLRQ SURSHUWLHV RI WKH UDQGRP SURFHVV ,W SURYLGHV EDVLV IXQFWLRQV WKDW JXDUDQWHH WKH SURMHFWLRQV RI WKH QRLVH RQWR WKH EDVLV IXQFWLRQV WR EH XQFRUUHODWHG 7KH .DUKXQHQ /RtHYH H[SDQVLRQ LV EDVHG RQ WKH HLJHQIXQFWLRQ RI WKH QRLVH DXWRFRUUHODWLRQ IXQFWLRQ ZKHUH WKH DXWRFRUUHODWLRQ IXQFWLRQ LV V\PPHWULF LQ WKH DUJXPHQWV QRQQHJDWLYH DQG FRQWLQXRXV 1RWH ZH ZLOO XVH WKH ODEHO TW UDWKHU WKDQ J W WR GLVWLQJXLVK EDVLV IXQFWLRQ GHULYHG YLD WKH QRLVH VSDFH FKDUDFWHULVWLFV IURP WKH VLJQDO VSDFH EDVHG IXQFWLRQV 

0HUFHU V 7KHRUHP

$Q\ UHDO YDOXHG IXQFWLRQ 5W V  ZKLFK LV V\PPHWULF SRVLWLYH VHPLGHnQLWH DQG FRQWLQXRXV RQ WKH VTXDUH  › W V › 7 PD\ EH H[SDQGHG LQWR DQ DEVROXWHO\ DQG XQLIRUPO\ FRQYHUJHQW VHULHV >@ 5W V

; w T W T V  N 

N

N

N

 › W V › 7



7KH VHW IwN  TN W J FRQWDLQV WKH SDLUV RI HLJHQYDOXHV DQG HLJHQIXQFWLRQV RE WDLQHG IURP WKH LQWHJUDO HTXDWLRQ wN TN W

=



7

5W V TN V GV



ZKHUH ITN W J IRUPV D &21 VHW RYHU WKH LQWHUYDO  › W › 7  (TXDWLRQ  LV DOVR FDOOHG WKH KRPRJHQHRXV LQWHJUDO HTXDWLRQ >@



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



.DUKXQHQ/Rt HYH ([SDQVLRQ

7KH H[SDQVLRQ EDVHG RQ WKH HLJHQIXQFWLRQV RI WKH DXWRFRUUHODWLRQ IXQFWLRQ LV FDOOHG WKH .DUKXQHQ/RtHYH H[SDQVLRQ 7KH HLJHQIXQFWLRQV ITN W J IRUP D FRPSOHWH RUWKRQRUPDO VHW WKDW LV

=

7

TQ W TP W GW



|

pQP

Q



HOVH

P



7KH SURMHFWLRQ RI WKH QRLVH RQWR WKH NWK EDVLV IXQFWLRQ SURGXFHV

=

KQW  TN W L

QN

7



QW TN W GW



7KH QRLVH IXQFWLRQ LV JLYHQ E\

; Q T W  OLP 1

QW

1

 N

N



›W›7

N



ZKHUH OLP1  GHQRWHV WKH OLPLW LQ WKH PHDQ VTXDUH VHQVH OLP  (



; QW b Q T W

1



1

N 

,I ( IQW J  WKHQ

=

( IQN J



7

N



N

( QW JN W GW



7KH VHFRQG RUGHU PRPHQW LV JLYHQ E\

= = ( QW Q} T W T } GW G} ( IQ Q J   = %%%= 5 W } T } G}&&& T W GW #_ ^] `$ = ( IQ Q J w T W T W GW w p 7

N

P



7

N



7

P

7





1

P

N



wP TP W 

KHQFH

7

N

KHQFH LI N P

P



P P

( IQN QP J

N



P PN

 

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH ,I N P WKHQ

( IQN QP J

:H QRWH WKDW WKH YDULDQFH RI WKH REWDLQHG DV

= = 7



7 

N WK



YDU QN wN  WHUP RI WKH QRLVH H[SDQVLRQ FDQ DOVR EH

=

TN W 51 W V TN V GW GV

7 

TN W wN TN W GW

wN



DQG WKDW WKH VXP RI WKH HLJHQYDOXHV FRUUHVSRQGV WR WKH LQWHJUDO RI WKH FRUUH ODWLRQ IXQFWLRQ SRWHQWLDOO\ WKH WLPHYDU\LQJ LQVWDQWDQHRXV SRZHU DV VKRZQ E\ (

=

7



Q W GW

= 5 W W GW = ; w J W J W GW ; w 7

1



7



N 



N 

N

N

N

N



'(7(&7,21 2) .12:1 6,*1$/6 ,1 $'',7,9( &2/25(' *$866,$1 12,6(

7KLV VHFWLRQ IRFXVHV RQ GHULYLQJ WKH FRUUHODWRU VWUXFWXUH IRU NQRZQ VLJQDOV HPEHGGHG LQ FRORUHG *DXVVLDQ QRLVH 7KH ELODWHUDO /DSODFH WUDQVIRUP LV UH YLHZHG DQG WKH LQWHJUDO HTXDWLRQ IRU WKH OHXFRJHQLF QRLVH FDVH LV FRQYHUWHG LQWR D GLmHUHQWLDO HTXDWLRQ ZKLFK WKHQ FDQ EH VROYHG :H DOVR ORRN DW WKH SHU IRUPDQFH DVVHVVPHQW 6HFWLRQ  RI WKH GHWHFWRU DQG GLVFXVV DQ DSSURDFK EDVHG RQ ZKLWHQLQJ WKH DGGLWLYH QRLVH 6HFWLRQ   6XSSRVH WKDW WKH VLJQDO VL W LV UHFHLYHG LQ WKH SUHVHQFH RI DGGLWLYH ]HUR PHDQ FRORUHG *DXVVLDQ QRLVH 8QGHU +L  ZH KDYH L    \ W VL W  QW  ›W›7 /RRNLQJ DW WKH SURMHFWLRQ RI WKH QRLVH RQWR WKH M WK EDVLV IXQFWLRQ ZH KDYH D VHTXHQFH RI *DXVVLDQ UDQGRP YDULDEOHV QM

=



7

QW TM W GW 

M

  c c c



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



ZKHUH ( QM  VLQFH ( QW  7KH VWDWLVWLFDO GHSHQGHQF\ RI WKH LWK DQG N WK QRLVH FRPSRQHQW LV REWDLQHG DV VKRZQ LQ 

=

7

|w 

N M N  HOVH  6LQFH WKH FRHpFLHQWV DUH *DXVVLDQ DQG XQFRUUHODWHG LW IROORZV WKDW WKH\ DUH LQGHSHQGHQW UDQGRP YDULDEOHV :H VHH WKDW ( IQM QN J



= =

XQGHU +  \M



=

\M

7



VLPLODUO\ XQGHU + 

7

7 

wN TN W TM W GW

wN pNM

\ WM+ TM W GW

 V W  QW TM W GW

VM  QM

V W  QW TM W GW VM  QM 

M



M

  c c c

  c c c 

8QGHU HDFK K\SRWKHVLV WKH LQQHU SURGXFW JLYHQ E\ \M LV *DXVVLDQ KHQFH ZH FDQ VHW XS WKH /57 DV LQ WKH SUHYLRXV VHFWLRQV RQFH ZH REWDLQ PHDQV DQG YDULDQFHV 7KH PHDQV RI WKH UDQGRP YDULDEOHV \M DUH JLYHQ E\  ( I\M M+ J ( VRM  QM VM DQG ( I\M M+ J VM  7KH YDULDQFH LV LQGHSHQGHQW RI WKH PHDQ KHQFH YDU \M M+L YDU \M YDU QM  L    8VLQJ  DQG  YDU \M wM  WK $W WKLV SRLQW ZH FDQ IRUPXODWH WKH N DSSUR[LPDWLRQ RI WKH /57 N  Hb  \M bVM  wM +  M  {wM ! OQ w  e \N W N  Hb  \M bVM  wM  + M  {wM &DQFHOLQJ FRPPRQ WHUPV WDNLQJ WKH QDWXUDO ORJDULWKP DQG OHWWLQJ N   UHVXOWV LQ

<S <S

OQ e \W

; \ V M

;b

c+

  VM b VM ! OQ w M b VM    w  M  wM  M M



+



'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



%\ UHSODFLQJ WKH .DUKXQHQ/RtHYH FRHpFLHQWV ZLWK WKHLU FRUUHVSRQGLQJ LQQHU SURGXFWV WKLV FDQ EH ZULWWHQ DV

=

\M

7



\ W TM W GW

=

DQG QM

7



QW TM W GW

HWF

KHQFH  EHFRPHV

;  = \W T W GW = V W b V W T W GW w   = = ;      V W V } T W T } GW G} 7

M

M 

7

M







7

 M  wM

= = b 7



7













M







7







M





M

+ ! V W V } TM W TM } GW G}  +

OQ w

5HRUGHULQJ  OHDGV WR

= = \W V W b V W ;  T W T W GW GW w = = ;   >V W V } b V W V } @ w T W T } GW G} ! _ 7KLV LV LQGHSHQGHQW^]RI WKH UHFHLYHG GDWD `  7

7



7













M 

M

M

M









7













M

M 

M

M

KHQFH LW LV HTXDO WR D FRQVWDQW .

KHQFH ZH JHW

=

= \ W 

7





RU

= 

7



7

  T W ; V W b V W T W GW  GW 



;  = \ W 

M 

wM





7



M

M 



wM

M







OQ w

 + !  +

>V W b V W @ TM W GW TM W GW

OQ w b . 7  + !  7 +



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



ZKLFK EHFRPHV

= 

7

;  V b V T W GW \ W _ w ^] ` 

M

M

M 

M

M





KW

+ !  7 +



,I ZH ODEHO WKH WHUPV LQ WKH LQWHJUDQG RI   H[FOXGLQJ \W  DV KW DQG XVH W DV WKH GXPP\ YDULDEOH RI LQWHJUDWLRQ ZH FDQ ZULWH

=



7

+ ! \ W KW GW  7 +



:H KDYH D FRUUHODWLRQ UHFHLYHU ZKHUH WKH VWDQGDUG UHSOLFD LV UHSODFHG E\ KW ZKLFK LV JLYHQ E\ KW

; V ; V

M

M 

M

M 

wM

b VM T W M

wM

V ; T W b

M

M

M 

wM

TM W

K W b K W



:H QRWH LI WKH wM V HTXDO D FRQVWDQW IRU DOO M  VXFK DV LW LV WKH FDVH ZLWK ZKLWH QRLVH WKHQ  ZRXOG EHFRPH WKH VFDOHG GLmHUHQFH EHWZHHQ V W DQG V W  ,I GHVLUHG ZH FDQ VSOLW KW LQWR K W DQG K W  DV LQGLFDWHG LQ   7KLV ZLOO SURYLGH WKH WZR OHJV RI WKH FRUUHODWRU VWUXFWXUH RQH IRU HDFK VLJQDO DV VKRZQ LQ )LJXUH  7R XVH   ZH ZRXOG KDYH WR REWDLQ IwM J DQG ITM W J IRU DOO M  7KLV DSSURDFK LV YHU\ WHGLRXV 2I FRXUVH RQH FRXOG WU\ WR DSSUR[LPDWH KW RI  ZLWK D nQLWH QXPEHU RI WHUPV %XW WKHQ WKH TXHVWLRQ UHPDLQV ZKLFK WHUPV VKRXOG EH XVHG DQG KRZ PDQ\" 5DWKHU WKDQ VROYLQJ IRU KW LQ WKLV IDVKLRQ ZH SURSRVH WKH IROORZLQJ >@ 7R VROYH IRU KW  ZH WDNH KW DQG LWV GHnQLQJ HTXDWLRQ PXOWLSO\ E\ WKH QRLVH NHUQHO DQG LQWHJUDWH RYHU WKH VLJQDO GXUDWLRQ $VVXPLQJ WKDW WKH QRLVH LV ZVV ZH KDYH

=



7

51 W b ~ K~ G~

; t V

M

M 

b VM

wM

u= _



7

51 W b ~ TM ~ G~

^]

YLD  wM TM W

`

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH

)LJXUH 



&RUUHODWRU VWUXFWXUH FRORUHG *DXVVLDQ QRLVH 

; V b V w T W  w ; V b V T W V W b V W A VW M

M

M

M 

M

M 

M

M

M

M





+HQFH ZH REWDLQ D )UHGKROP LQWHJUDO HTXDWLRQ RI WKH nUVW NLQG >@

=

7



51 W b ~ K~ G~

VW 

›W›7



1RWH LI 51 ~ 1  p~ LH WKH ZKLWH QRLVH FDVH WKHQ

=

7



1

 pW b ~ K~ GW VW 

EHFRPHV

›W›7

1

RU

KW

 VW 1 

 KW VW D VFDOHG YHUVLRQ RI VW

7KHUHIRUH LQ WKH ZKLWH QRLVH FDVH ZH ZRXOG REWDLQ RXU RULJLQDO UHVXOW WKDW LV ZH FRUUHODWH WKH UHFHLYHG GDWD ZLWK D UHSOLFD RI WKH VLJQDO WKDW ZH DUH WHVWLQJ IRU 7KLV SDUDSKUDVHV WKH QRWH WKDW IROORZV   :H FDQ VSOLW WKH KW DQG VW IXQFWLRQV LQWR WKHLU ? DQG ? FRPSRQHQWV DV JLYHQ E\

=



7

51 W b ~ K ~ G~

V W

IRU  › W › 7



 DQG

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

=

7

51 W b ~ K W GW



V W

IRU  › W › 7



:H VHH WKDW WKH RSWLPXP UHFHLYHU GHWHFWRU LV WKH FRUUHODWRU ZKHUH WKH IXQFWLRQV KL W  IRU L   KDYH WR EH FRPSXWHG E\ VROYLQJ D )UHGKROP LQWHJUDO HTXDWLRQ RI WKH nUVW NLQG >@ ,Q JHQHUDO ZH QHHG WR VROYH WKH IROORZLQJ HTXDWLRQ IRU WKH TXDQWLW\ KW

=



7

51 W X KX GX

VW 

›W›7



ZKHUH 51 W X LV WKH FRUUHODWLRQ IXQFWLRQ RI WKH QRLVH DQG VW V W b V W  ,Q WKH VWDWLRQDU\ FDVH 51 W X EHFRPHV 51 W b X  2QH FDQ DOVR VROYH  IRU K W DQG K W E\ XVLQJ V W DQG V W DV GULYHUV DOVR FDOOHG H[FLWDWLRQ RU LQSXW  UHVSHFWLYHO\ 7R EH DEOH WR VROYH WKH LQWHJUDO HTXDWLRQV ZH ZLOO nUVW UHYLHZ VRPH RI WKH SURSHUWLHV RI WKH ELODWHUDO /DSODFH WUDQVIRUP 

%LODWHUDO /DSODFH 7UDQVIRUP

7KLV VHFWLRQ VHUYHV DV D TXLFN UHYLHZ RQ D WRSLF WKDW LV XVHG WR FRQYHUW WKH LQWHJUDO WR DQ HTXLYDOHQW GLmHUHQWLDO HTXDWLRQ ,W PD\ DOVR VHUYH DV DQ LQWUR GXFWLRQ WR WKH JHQHUDO /DSODFH WUDQVIRUPDWLRQ DOORZLQJ DQDO\VLV RI WZRVLGHG IXQFWLRQV VXFK DV FRUUHODWLRQ SRZHU VSHFWUDO GHQVLW\ RU SUREDELOLW\ GHQVLW\ IXQFWLRQV 7KH ELODWHUDO /DSODFH WUDQVIRUP RI WKH IXQFWLRQ I W LV JLYHQ E\ >@  /% II W J A ) V I W HbVW GW 

=

b

ZKHUH }  5H V  } OLHV LQ WKH UHJLRQ RI FRQYHUJHQFH 52& VHH )LJ XUH   7KH UHJLRQ RI FRQYHUJHQFH JXDUDQWHHV WKH H[LVWHQFH RI ) V  7KH WUDQVIRUP YDULDEOH ) V FDQ EH GHFRPSRVHG LQWR WZR SDUWV  ) V ) V  )b V ZKHUH  I W HbVW GW  ) V DQG

)b V

= =





b

I W HbVW GW



/HW / I J GHQRWH D FRQYHQWLRQDO RQHVLGHG /DSODFH WUDQVIRUP WKHQ ) V

A / II W J

=  I W Hb 

VW

GW



'HWHFWLRQ RI 6LJQDOV LQ 1RLVH

)LJXUH  DQG



5HJLRQ RI FRQYHUJHQFH IRU WKH ELODWHUDO /DSODFH WUDQVIRUP

)b V

/ IIb bW JMVbV

 ZKHUH Ib W LV I W IRU QHJDWLYH WLPH DUJXPHQWV DQG Ib bW LV I W IRU QHJ DWLYH WLPH DUJXPHQWV ZLWK W EHLQJ UHSODFHG E\ W  bW 7R YHULI\ WKDW WKH /DSODFH WUDQVIRUP RI WKH DQWLFDXVDO SDUW QHJDWLYH WLPH IXQFWLRQ LV GRQH SURSHUO\ ZH WDNH D ORRN DW WKH H[SUHVVLRQ

/ IIb bW JMVbV

=  I bW Hb GWnnn b n b =  I bW Hb b GW b = I W Hb GW VW



V

V

V W





b

VW

ZKLFK LV WKH GHVLUHG UHVXOW 7R REWDLQ WKH LQYHUVH WUDQVIRUP LQ D ELODWHUDO VHQVH DQ RYHUODSSLQJ UHJLRQ RI FRQYHUJHQFH IRU ERWK SRVLWLYH DQG QHJDWLYH WLPH VHJPHQWV PXVW H[LVW VHH WKH FURVVKDWFKHG VWULS LQ )LJXUH   :H SHUIRUP WKH LQYHUVH LQ IRXU VWHSV  )RU 5H V ! }  WDNH WKH RQHVLGHG FRQYHQWLRQDO LQYHUVH DQG FDOO LW I W XW   )RU 5H V  }  UHSODFH V  bV WDNH WKH FRQYHQWLRQDO RQHVLGHG LQYHUVH DQG FDOO WKH UHVXOW Ib bW 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

 5HSODFH W  bW LQ  DQG FDOO LW Ib W   I W I W XW  Ib W XbW  7R LOOXVWUDWH WKH SURFHGXUH ZH ZLOO ORRN DW D IHZ VLPSOH H[DPSOHV

([DPSOH  )RUZDUG WUDQVIRUP o! I W Hbo MWM RU HboW XW  HoW XbW I W I W XW  Ib W XbW

/ II W J / HboW XW

) V

j

 Vo )b V

k =  Hb 

oW

HbVW GW

 IRU 5H V ! bo

52&

/ IIb bW JMVbV  oW H XbW HbVW GW  WbW VbV  boW bVW  H H GW V  o VbV  VbV

= |=

nnn n



}nnn n

nnn nn

nnn n

bV  o IRU 5H V  o 52& VR o    bV  o bV  o  IRU 5H V ! bo IRU 5H V  o  52& M5H V M ! o  ) V

Vo

,QYHUVH WUDQVIRUP *LYHQ ) V

o b  o V

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



52& M5H V M  o ZKHUH o !  %\ SDUWLDO IUDFWLRQ H[SDQVLRQ ZH KDYH    ) V V  o bV  o 5H V ! bo 5H V  o  I W /b V  o HboW XW /b V  o HboW  Ib bW /b bV o VbV  Ib W HoW  I W HboW XW  HoW XbW HbMo MW  IRU DOO W ([DPSOH   ) V  bV  V   52&  b  5H V   1RWH WKHUH LV RQO\ IRU D YDOLG SROH DVVLJQPHQW

| }  nn nn

| }

WKH FDXVDO SROH UHVSRQVLEOH IRU WKH SRVLWLYH WLPH IXQFWLRQ LV DW V b WKH DQWLFDXVDO SROH UHVSRQVLEOH IRU WKH QHJDWLYH WLPH IXQFWLRQ

LV DW V  8VLQJ SDUWLDO IUDFWLRQ H[SDQVLRQ ZH KDYH   ) V  bV   V 52& 52& 5H V ! b 5H V  

|  } Hb XW V  b|  } n n  b / V / bV   nn b

I W

/b

 I bW





) V  )b V

W



V

V

 Ib W HW  I W HbW XW  HW XbW 





HbW



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



,QWHJUDO (TXDWLRQV

:KHQ WKH QRLVH LV D VHFRQG RUGHU UDQGRP SURFHVV DQG FDQ EH WKRXJKW RI DV EHLQJ REWDLQHG DV WKH RXWSXW RI D SK\VLFDOO\ UHDOL]DEOH nOWHU WKDW LV GULYHQ E\ ZKLWH QRLVH WKHQ ZH FDQ FRQYHUW WKH LQWHJUDO  LQWR D GLmHUHQWLDO HTXDWLRQ 7KLV VHFWLRQ GHDOV ZLWK WKH )UHGKROP LQWHJUDO HTXDWLRQ RI WKH nUVW DQG VHFRQG NLQG >@ :H FKDQJHG WKH QRWDWLRQ IRU WKH ULJKWKDQG VLGH RI WKH LQWHJUDO HTXDWLRQ E\ OHWWLQJ [W UHSODFH VW  7KLV LV WR DYRLG FRQIXVLRQ VLQFH WKH FDSLWDO OHWWHU 6 VWDQGV IRU SRZHU VSHFWUDO GHQVLW\ ZKHQ ZH HYDOXDWH WKH H[SUHVVLRQV LQ WKH VGRPDLQ 7KH HTXDWLRQ RI WKH IRUP

=

WI W

51 W X KX GX

[W

W › W › WI



LV WKH )UHGKROP HTXDWLRQ RI WKH nUVW NLQG ZKHUH 51 W X DQG [W DUH NQRZQ IXQFWLRQV DQG KW LV WKH XQNQRZQ TXDQWLW\ )RU 6HFWLRQ  ZH SUHIHU WKH PRUH JHQHUDO OLPLWV W DQG WI DQG HYDOXDWH WKHP DW W  DQG WI 7  ZKHQ FRQYHQLHQW ,I WKH FRORUHG QRLVH KDV DQ XQGHUO\LQJ VWDWLRQDU\ ZKLWH QRLVH FRPSRQHQW WKHQ WKH QRLVH DXWRFRUUHODWLRQ IXQFWLRQ KDV DQ LPSXOVHOLNH FRPSRQHQW 7KH QRLVH FRUUHODWLRQ IXQFWLRQ FDQ WKHQ EH ZULWWHQ DV 1 51 W X   pW b X  51F W X ZKHUH 51 W X LV WKH FRUUHODWLRQ IXQFWLRQ RI WKH QRLVH SURFHVV 1  pW b X LV WKH FRUUHODWLRQ IXQFWLRQ RI WKH ZKLWH QRLVH FRPSRQHQW DQG 51F W X LV WKH FRUUHODWLRQ IXQFWLRQ GXH WR WKH FRORUHG QRLVH FRPSRQHQW 7KHQ WKH LQWHJUDO HTXDWLRQ EHFRPHV 1

=

WI

51 W X KX GX [W  W › W › WI   W 7KLV LV FDOOHG WKH )UHGKROP LQWHJUDO HTXDWLRQ RI WKH VHFRQG NLQG 7KHUH LV DOVR WKH KRPRJHQHRXV LQWHJUDO  ZKLFK ZDV GLVFXVVHG HDUOLHU 8VXDOO\ VROXWLRQV WR LQWHJUDO HTXDWLRQV DUH WHGLRXV XQGHUWDNLQJV +RZ HYHU XQGHU VRPH UHDVRQDEOH FRQGLWLRQV ZH FDQ REWDLQ VROXWLRQV E\ WUDQVODWLQJ WR DQG VROYLQJ DQ HTXLYDOHQW GLmHUHQWLDO HTXDWLRQ 7KLV DSSURDFK LV DSSOLFD EOH ZKHQ RQH FDQ LQWHUSUHW WKH FRORUHG QRLVH DV EHLQJ WKH RXWSXW RI D OLQHDU WLPH LQYDULDQW V\VWHP WKDW LV GULYHQ E\ VWDWLRQDU\ ZKLWH QRLVH LH OHXFR JHQLF QRLVH DV FRLQHG LQ >@  ,Q WKLV FDVH WKH FRORUHG QRLVH LV VWDWLRQDU\ DQG LWV SRZHU VSHFWUDO GHQVLW\ LQ WKH VGRPDLQ LV WKH UDWLR RI WZR HYHQ RUGHUHG SRO\QRPLDOV 7KH GHJUHH RI WKH GHQRPLQDWRU LV JUHDWHU WKDQ WKH GHJUHH RI WKH QXPHUDWRU JXDUDQWHHLQJ WKDW WKH SRZHU RI WKH QRLVH SURFHVV LV nQLWH :H FDQ H[SUHVV WKH /DSODFH WUDQVIRUP RI WKH FRUUHODWLRQ IXQFWLRQ DV KW 

61 V

=  5 ~ Hb b

1

V~

G~

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



1 V 'V

:H UHFDOO WKDW KHQFH

/ IpW J / IpW b ~ J

ZLWK DQ LQYHUVH JLYHQ E\

=  pW Hb GW  b =  pW b ~ Hb GW Hb



VW

VW

b

V~



j k =  Hb H GV b =  H b GV p W b ~

/b HbV~ DQG



V~ VW

b

1RZ G p W b ~ GW

VW ~

=  G H b GV =b VGWH b GV



=V H



V W ~

VW ~

b

+HQFH LQ JHQHUDO GN p W b ~ GWN

8VLQJ WKH QRWDWLRQ S A G GW 1 S p W b ~

b

N

b

VW ~

=  1 V H b

b

GV

V W ~



GV



ZKHUH 1 S LV DQ HYHQ RUGHUHG SRO\QRPLDO RI GLmHUHQWLDO RSHUDWLRQV DQG 1 V LV D FRUUHVSRQGLQJ SRO\QRPLDO LQ HYHQ SRZHUV RI V LH LI 1 S D S  D S WKHQ 1 V D  D V  :H DOVR UHFDOO 51 X

=  6 V H b

1

VX



GV

ZKHUH 51 X LV WKH FRUUHODWLRQ IXQFWLRQ DQG 61 V LV WKH VSHFWUDO GHQVLW\ LQ WKH VGRPDLQ $ VKLIWHG YHUVLRQ WKHQ ORRNV OLNH LH OHW X W b ~ 51 W b ~

=  6 V H b

1

b

VW ~

GV



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



=  'V 6 V H

$SSOLFDWLRQ RI 'S WR ERWK VLGHV RI  OHDGV WR 'S 51 W b ~

5HDOL]LQJ WKDW

1 V

ZH REWDLQ

'S 51 W b ~ 

b

VW ~

1

GV



1 V 'V

61 V

RU



b

'V 61 V

=  1 V H b



b

VW ~



GV



7KLV H[SUHVVLRQ EHFRPHV XVLQJ WKH UHVXOWV IURP  RQ WKH ULJKWKDQG VLGH 'S 51 W b ~ 1 S p W b ~  ZKHUH S G GW :H FDQ XVH WKLV UHVXOW WR VROYH  IRU VWDWLRQDU\ QRLVH NHUQHOV ZKLFK LV UHSURGXFHG KHUH IRU FRQYHQLHQFH

=

=

WI

W

51 W b ~ K~ G~

W › W › WI

[W



+HQFH ZKHQ ZH RSHUDWH RQ ERWK VLGHV RI  ZLWK 'S  ZH REWDLQ WI

W

=

'S 51 W b ~ K~ G~

'S [W 

W › W › WI



8VLQJ  WKLV EHFRPHV RU

WI

W

1 S p W b ~ K~ G~ 1 S KW

'S [W 

'S [W 

W › W › WI

W › W › WI



 (TXDWLRQ  LV D GLmHUHQWLDO HTXDWLRQ (VVHQWLDOO\ ZH FRQYHUWHG DQ LQ WHJUDO HTXDWLRQ SUREOHP LQWR D GLmHUHQWLDO HTXDWLRQ SUREOHP ZKLFK FDQ EH VROYHG XVLQJ VWDQGDUG WHFKQLTXHV 7KH VROXWLRQ RI WKH GLmHUHQWLDO HTXDWLRQ  FRQVLVWV RI WKUHH SDUWV KW KK W  KS W  KH W  ZKHUH KK W LV WKH KRPRJHQHRXV VROXWLRQ LH QR H[FLWDWLRQ  KS W LV WKH SDUWLFXODU VROXWLRQ LH GXH WR WKH LQSXW [W  DQG KH W LV WKH HQG SRLQW VROXWLRQ >@ 7KH VROXWLRQ WR  LV REWDLQHG E\ VROYLQJ   :H ZLOO VNHWFK RXW WKH VWHSV DQG SURYLGH DQ H[DPSOH *LYHQ WKDW WKH GLmHUHQWLDO HTXDWLRQ LV RI WKH IRUP W › W › WI 1 S KW 'S [W 

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH ZH VROYH D 7KH KRPRJHQHRXV SDUW

1 S KW

KK W







VROXWLRQ RI  E 1 S KW 'S [W FDQ EH VROYHG XVLQJ /DSODFH WUDQVIRUP WHFK QLTXHV 1 V + V 'V ; V 'V + V ; V 1 V 61 V ; V KS W /b I6 V ; V J  F 7KH HQG SRLQW VROXWLRQ LV JLYHQ E\ KH W

;b b KD p

S T 

L 

L

L

W b W  EL pL W b WI

L



ZKHUH pL W LV WKH LWK GHULYDWLYH RI WKH GHOWD IXQFWLRQ S LV WKH RUGHU RI ' V  DQG T LV WKH RUGHU RI 1 V 

1XPHULFDO YDOXHV RI WKH FRHpFLHQWV FRQWDLQHG LQ WKH VHW IDL  EL J DUH RE WDLQHG E\ HYDOXDWLQJ  DQG XVLQJ KW KK W  KS W  KH W ([DPSOH  $ FRUUHODWLRQ EDVHG UHFHLYHU IRU D NQRZQ VLJQDO LQ DGGLWLYH FRORUHG *DXVVLDQ QRLVH  › W › 7 LV GHVLUHG :H DUH JLYHQ \ W [W  QW $ o 51 ~ $Hbo M~ M b 61 V bV  o 

=



7

51 W b ~ K~ G~

[W



›W›7

7KLV LV D )UHGKROP LQWHJUDO HTXDWLRQ RI WKH nUVW NLQG bV  o  V S  RUGHU RI V 'V 1 V $oV T  RUGHU RI V 6ROXWLRQ ›W›7 1 S KW 'S [W 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

ZH KDYH D 1 S KW  KRPRJHQHRXV SDUW  RU  $ o KW  7KLV LV QRW D GLmHUHQWLDO HTXDWLRQ MXVW DQ DOJHEUDLF RQH +HQFH D  KK W LV RI WKH IRUP KK W  E

_

G

^]

]`_^

1 S 

`

b GW [W  o  [W $o KW ' S  [ W

SDUWLFXODU VROXWLRQ

LV DQ DOJHEUDLF HTXDWLRQ LQ KW D GLmHUHQWLDO HTXDWLRQ LQ [W  +HQFH E  KS W LV RI WKH IRUP   KS W $o o [W b [ƒW

d

e

F (QG SRLQWV L KDV D UDQJH IURP

 WR SbTb 

;D p 

KH W

L 

L

L



p L W

p  W

p W

W  EL pL W b 7

D p W  E p W b 7

7KH WRWDO VROXWLRQ LV JLYHQ E\ KS W  KK W  KH W KW   KW $o o [W b [ƒW  D pW  E pW b 7 :H VWLOO QHHG WR VROYH IRU D DQG E  +HUH [W LV WKH NQRZQ VLJQDO \W LV WKH UHFHLYHG VLJQDO DQG QW LV WKH DGGLWLYH FRORUHG *DXVVLDQ QRLVH )URP   ZH KDYH

c

b

[W

= =

7

>

7

>

51 W b ~ K~ G~ $Hbo MWb~ M

LQFOXGHV HQG SRLQWV

|  bo [~ b [ƒ~ c  D p~  E p~ b 7 } G~ $o







'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



7KH LQWHJUDO LV VSOLW LQWR WZR UHJLRQV WR HOLPLQDWH WKH DEVROXWH YDOXH V\PERO W  o  [~ b [ƒ~  D p~ G~ [W $Hbo Wb~   $o >

| t u } | } = b c  b  $H $o o [~ b [ƒ~  E p~ b 7 G~ =  bo [~ b [ƒ~ c G~ b b D $H  E $ H  $Hb b $o = $H b  bo [~ b [ƒ~ c G~  =

7

 o W ~





W

oW



7

=

1RZ

W

>

Ho~ [ƒ~ G~

HoW [B W

= 

DQG

7 W

Hbo~ [ƒ~ G~

W

[W

b [B  b o HoW [W b o [

o  Ho

~

[~ G~

Hbo7 [B 7 b HboW [B W  oHbo7 [7 b oHboW [W

= 

W

VR ZH REWDLQ





$o



=

o W ~

>

 o W ~

W

W

o W 7



7

o  Hbo~ [~ G~

w › | [B  [ [W  H D $  o b  w} › [B 7 [7 b b H H E $b oW

oW

o7





o

7

IRU  › W › 7 %XW [W s [W FXUO\ EUDFNHW WHUPV PXVW EH ]HUR IRU DOO  › W › 7  FDQ VSOLW DQG VHW HDFK b [ D $  [B WHUP WR ]HUR VLQFH o   VHFRQG WHUP LQ WKH ODVW [B 7 [7 H[SUHVVLRQ KDV D WLPH E $ b  YDU\LQJ WHUP LH HoW DQG HboW o b 

! ! !!

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



RU

 o[ b [B  $o

D

 $o o[7  [B 7 +HQFH WKH GHVLUHG UHIHUHQFH FRUUHODWLRQ LH UHSOLFD IXQFWLRQ LV JLYHQ E\  o[ b [B  pW  o[7  [B 7 pW b 7  o  [W b [ƒW KW $o IRU  › W › 7  E

k

j



3HUIRUPDQFH RI WKH 2SWLPXP 5HFHLYHU IRU .QRZQ 6LJQDOV LQ &RORUHG *DXVVLDQ 1RLVH

7KLV VHFWLRQ VKRZV KRZ RQH FDQ VROYH LQ SULQFLSOH IRU 3) $ DQG 3' WR WHVW IRU WKH SUHVHQFH RI D NQRZQ VLJQDO LQ DGGLWLYH FRORUHG *DXVVLDQ QRLVH )URP )LJXUH  ZH FDQ DVFHUWDLQ WKDW WKH RXWSXW RI WKH RSWLPXP GHWHFWRU ZLOO EH D *DXVVLDQ UDQGRP YDULDEOH 7KLV LV WUXH XQGHU HDFK K\SRWKHVLV VLQFH D OLQHDU RSHUDWLRQ RQ D *DXVVLDQ UDQGRP YDULDEOH UHVXOWV LQ D *DXVVLDQ UDQ GRP YDULDEOH 5HSOLFDWLQJ   IRU FRQYHQLHQFH DQG GHQRWLQJ WKH RXWSXW *DXVVLDQ UDQGRP YDULDEOH DV ] ZH KDYH ]

=

+ ! \ W KW GW  7 +

7





7KH PHDQ RI ] XQGHU WKH + K\SRWKHVLV LV JLYHQ E\

= =

( ]

7 

7 

V W  QW KW GW V W KW GW



ZKLOH WKH PHDQ RI ] WKH + K\SRWKHVLV LV JLYHQ E\ ( ]

=

7 

V W KW GW

7KH YDULDQFH LQYDULDQW XQGHU HDFK K\SRWKHVLV LV JLYHQ E\ }]

}] M+

}] M+



'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



= V W  QW KW GW b = V W KW GW@ J = ( I> QW KW GW@ J = = 5 W V KW KV GW GV = = 5 W b V ;  w V b V J W ;  w V b V J V GW GV ; ;  w w V b V V b V = w J W J W GW ;  w V b V  (L I>

7

7

L





7

L

7

1



7







7





L

7

1



M

M

L

L

M

M

L M

L

L

L

L

M

7

L

L

M

L

L

L

M

M



M M

M

L

6R WKH IDOVH DODUP UDWH FDQ EH H[SUHVVHG DV 3) $ 47 b ( ] }]  ZKLOH WKH SUREDELOLW\ RI GHWHFWLRQ LV JLYHQ E\ 3' 47 b ( ] }]  ZKHUH WKH PHDQV DQG WKH YDULDQFH DUH GHnQHG DERYH 7R REWDLQ DFWXDO YDOXHV RQH KDV WR VROYH WKH KRPRJHQHRXV LQWHJUDO HTXDWLRQ ZKLFK DW EHVW LV D WHGLRXV XQGHUWDNLQJ 

:KLWHQLQJ )LOWHU

:KLWHQLQJ ZKLFK LV DOVR VRPHWLPHV FDOOHG VSHFWUDO VKDSLQJ RU SUHZKLWHQLQJ LV D VLPSOH WHFKQLTXH ZKLFK DV WKH QDPH LPSOLHV FRQYHUWV WKH FRORUHG QRLVH WR ZKLWH QRLVH ,I SRVVLEOH WKH ZKLWHQHU ZLOO FRUUHFW WKH VSHFWUDO VKDSH RI WKH QRLVH DQG FRQYHUW WKH SUREOHP LQWR RQH WKDW FDQ WKHQ EH VROYHG ZLWK WKH VWDQGDUG PDWFKHG nOWHU RU FRUUHODWRU 2QH FDQ VROYH IRU D ZKLWHQLQJ nOWHU E\ DVVXPLQJ WKDW WKH LPSXOVH UHVSRQVH RI WKH ZKLWHQHU LV RI WKH IRUP >@ K: W X

; K T W T X M

M

M

M



ZKHUH ITM W J LV D &21 VHW EDVHG RQ WKH QRLVH NHUQHO HLJHQGHFRPSRVLWLRQ 1RWH WKDW WKLV LPSXOVH UHVSRQVH DGPLWV D QRQVWDWLRQDU\ EHKDYLRU :LWK VRPH



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

ZRUN >@ RQH FDQ VKRZ WKDW IKM J K: W X

;Iww JT +HQFH W T X M

M

M

M

M



7R REWDLQ WKH HLJHQYDOXHV DQG HLJHQIXQFWLRQV WKH KRPRJHQHRXV LQWHJUDO HTXD WLRQ KDV WR EH VROYHG $ SUDFWLFDO VROXWLRQ H[LVWV ZKHQ WKH QRLVH LV VWDWLRQDU\ DQG WKH VLJQDO GXUDWLRQ 7 PXFK ODUJHU WKDQ WKH FRUUHODWLRQ ZLGWK RI WKH QRLVH RU WKH RE VHUYDWLRQ LQWHUYDO !! 7  DQG WKH QRLVH VSHFWUXP LV RI D UDWLRQDO OHXFRJHQLF IRUP ,I ZH OHW WKH VWDUW WLPH DSSURDFK b WKHQ WKH ZKLWHQLQJ nOWHU RI  FDQ EH PDGH WLPHLQYDULDQW WKDW LV K: ~ K: W b X  ZKHUH ~ W b X +HQFH JLYHQ WKH QRLVH FRUUHODWLRQ IXQFWLRQ 51 ~  WKH VSHFWUDO GHQVLW\ LV JLYHQ E\ 61 V 61 V 61b V  ZKHUH 61 V KDV DOO LWV SROHV DQG ]HURV LQ WKH OHIWKDQG VSODQH LH WKH FDXVDO FRQWULEXWLRQ DQG 61b V KDV DOO LWV SROHV DQG ]HURV LQ WKH ULJKWKDQG VSODQH LH WKH DQWLFDXVDO FRQWULEXWLRQ  WKHQ WKH WUDQVIHU IXQFWLRQ RI WKH ZKLWHQLQJ nOWHU LV JLYHQ E\ +: V  61 V  DQG WKH FRUUHVSRQGLQJ LPSXOVH UHVSRQVH LV GHnQHG E\ K: W /b I 61 V J  ,I WKLV nOWHU LV XVHG WR ZKLWHQ WKH UHFHLYHG GDWD LW PXVW DOVR EH DSSOLHG WR WKH UHSOLFD VLJQDOV VHH )LJXUH  WR DOORZ WKH VLJQDOV WR FRUUHODWH ZKHQ WKH\ DUH SUHVHQW 



',6&5(7(7,0( '(7(&7,21 _ .12:1 6,*1$/6 (0%(''(' ,1 &2/25(' *$866,$1 12,6( ,QWURGXFWLRQ

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

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



)LJXUH  7ZR FKDQQHO UHDOL]DWLRQ RI WKH DQDORJ ZKLWHQHU D 7KH VSHFWUDO UHJLRQ RI WKH VLJQDOV RI LQWHUHVW LH WKH EDQGZLGWK LV QDUURZ UHODWLYH WR WKH FHQWHU IUHTXHQF\ DQG WKH 36' RI WKH QRLVH LV DOPRVW FRQVWDQW RYHU WKH EDQG ,Q WKLV SDUWLFXODU FDVH DV IDU DV WKH UHFHLYHU LV FRQFHUQHG WKH QRLVH DIWHU SURSHU nOWHULQJ FDQ EH WUHDWHG DV D EDQGOLPLWHG ZKLWH QRLVH SURFHVV ZKHUH WKH 36' RXWVLGH WKH EDQG WDNHV RQ YHU\ VPDOO YDOXHV +HQFH WKH QRLVH FRUUHODWLRQ IXQFWLRQ LV DSSUR[LPDWHO\ D VLQFIXQFWLRQ ZKRVH nUVW ]HUR FURVVLQJ RFFXUV DW ODJ RQH 7KLV VD\V WKDW 51 P z  IRU MPM w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nOWHUHG VLJQDO HPEHGGHG LQ ZKLWH *DXVVLDQ QRLVH +HQFH WKH VWDQGDUG DSSURDFK RI UHSOLFD FRUUHODWLRQ DSSOLHV DV ZDV GHULYHG LQ &KDSWHU  6HFWLRQ  7KH PDMRU GLmHUHQFH LV WKDW QRZ WKH UHSOLFD IXQFWLRQ ZLOO EH UHSODFHG ZLWK D nOWHUHG YHUVLRQ RI WKH RULJLQDO VLJQDO 7KLV WHFKQLTXH LV LOOXVWUDWHG LQ )LJXUH  :H QRWH WKDW RQH FDQ FRPELQH WKH ZKLWHQLQJ DQG nOWHULQJ RSHUDWLRQV LQWR D RQHVWHS RSHUDWLRQ )RU FODULW\ ZH W\SLFDOO\ GR QRW LPSOHPHQW WKLV W\SH RI



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

'LVFUHWHWLPH FRORUHG QRLVH GHWHFWRU

VWUHDPOLQHG RSHUDWLRQ 5DWKHU WKDQ WKDW ZH XVXDOO\ SUHIHU WR NHHS WKH EXLOG LQJ EORFNV DQ H[SOLFLW ZKLWHQHU DQG WKH FRUUHODWRU VHSDUDWH :H DOVR ZDQW WR SRLQW RXW WKDW WKH ZDYHOHW WUDQVIRUPDWLRQ WHQGV WR ZKLWHQ WKH QRLVH >@ EXW WKH GLmHUHQW nOWHUV ZLOO KDYH D GLmHUHQW EDQGZLGWK DQG RXWSXW VDPSOLQJ UDWH LH VHH &KDSWHU  DQG $SSHQGL[ (  

:KLWHQLQJ YLD 6SHFWUDO )DFWRUL]DWLRQ

7KHUH DUH VHYHUDO ZD\V WR DFFRPSOLVK WKH ZKLWHQLQJ RI WKH QRLVH 0DQ\ RI WKHP DUH FKDPSLRQHG LQ WKH GLJLWDO VLJQDO SURFHVVLQJ OLWHUDWXUH >^@ .QRZOHGJH RI WKH QRLVH DXWRFRUUHODWLRQ IXQFWLRQ LV HTXLYDOHQW WR NQRZLQJ WKH SRZHU VSHFWUDO GHQVLW\ RI WKH QRLVH 0RVW RI WKH WLPH WKH 36' FDQ EH IDFWRUHG LQWR LWV SROH DQG ]HUR FRQVWLWXHQWV 7KLV DOORZV IRU WKH VHSDUDWLRQ RI WKH SROHV DQG ]HURV WKDW OLH LQVLGH WKH XQLW FLUFOH IURP WKRVH WKDW GR QRW :H DVVXPH WKDW WKH 36' RI WKH FRORUHG QRLVH GRHV QRW WDNH RQ D ]HUR YDOXH KHQFH WKHUH DUH QR ]HURV RQ WKH XQLW FLUFOH 7KHUH FDQQRW EH DQ\ SROHV RQ WKH XQLW FLUFOH VLQFH WKDW ZRXOG UHVXOW LQ VSHFWUDO UHVRQDQFH LH D UHVSRQVH GXH WR D VLQXVRLG  6R LQ WKLV GLVFXVVLRQ RQO\ SROHV DQG ]HURV LQVLGH DQG RXWVLGH WKH XQLW FLUFOH ZLOO EH FRQVLGHUHG :H IXUWKHU DVVXPH WKDW WKH REVHUYDWLRQ RU SURFHVVLQJ WLPH LV PXFK ODUJHU WKDQ WKH VLJQDO GXUDWLRQ XQOHVV LW LV FDOOHG IRU RWKHUZLVH 7KH 36' FDQ EH ZULWWHQ DV  61 ] =I51 P J ZKLFK FDQ EH IDFWRUHG DV  61 ] . 61 ] 61b ]  ZKHUH 61 ] FRQWDLQV DOO WKH SROHV DQG ]HURV WKDW UHVLGH LQVLGH WKH XQLW FLUFOH 61b ] FRQWDLQV DOO SROHV DQG ]HURV RXWVLGH WKH XQLW FLUFOH DQG . LV D SRVLWLYH

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



)LJXUH  *HQHUDWLRQ RI WKH FRORUHG QRLVH VHTXHQFH QQ IURP WKH ZKLWH QRLVH VHTXHQFH ZQ 

)LJXUH 

:KLWHQHU VSHFWUDO HTXDOL]DWLRQ 

YDOXHG FRQVWDQW ,I ZH OHW + ] 61 ]  WKHQ ZH FDQ nOWHU WKH QRLVH SURFHVV XVLQJ WKH LQYHUVH RI WKH nOWHU WUDQVIHU IXQFWLRQ LH + b ] WR REWDLQ D FRQVWDQW SRZHU VSHFWUDO GHQVLW\ 7KLV LV HTXLYDOHQW WR WKH OHXFRJHQLF SURFHVV LQ DQDORJ V\VWHPV >@ DQG LWV JHQHUDWLQJ PHFKDQLVP LV VKRZQ LQ )LJXUH  2QH DVVXPHV WKDW WKH FRORUHG QRLVH LV JHQHUDWHG E\ SURFHVVLQJ VWDWLRQDU\ ZKLWH QRLVH ZLWK D V\VWHP WKDW KDV D WUDQVIHU IXQFWLRQ ZLWK D nQLWH QXPEHU RI SROHV DQG ]HURV $OO SROHV DQG ]HURV PXVW OLH ZLWKLQ WKH XQLW FLUFOH LH WKH VWDEOH UHJLRQ  7KH UHTXLUHPHQW RQ WKH RUGHU RI WKH SROHV UHODWLYH WR WKH ]HURV LV UHOD[HG ,W LV SHUPLVVDEOH WR KDYH PRUH ]HURV WKDQ SROHV LH ),5 nOWHU WUDQVIHU IXQFWLRQV KDYH D QRQWULYLDO QXPHUDWRU SRO\QRPLDO VLQFH WKH UHJLRQ RI VXSSRUW H[WHQGV RQO\ IURP b{ WR { UDGLDQV 7KH ZRUGLQJ QRQWULYLDO SRO\QRPLDO UHIHUV WR KDYLQJ URRWV QRW ORFDWHG DW WKH RULJLQ $ YDOLG PRGHO IRU WKH QRLVH SURFHVV LV JLYHQ E\ )LJXUH  7KH ]HURV RI WKH QXPHUDWRU $] DQG GHQRPLQDWRU % ] SRO\QRPLDOV DUH DOO LQVLGH WKH XQLW FLUFOH 7KH ZKLWHQLQJ LV DFFRPSOLVKHG DV WKH GDWD VHTXHQFH QQ LV HTXDOL]HG VHH )LJXUH   WKDW LV WKH RXWSXW EHFRPHV VSHFWUDOO\ oDW 7KH nOWHU UHVSRQVH LV REWDLQHG E\ XVLQJ WKH IDFWRUHG SRZHU VSHFWUDO GHQ VLW\ $]  + ] 61 ] b % ] 1RZ WKH ZKLWHQHG QRLVH LV JLYHQ E\ Q: Q

QQ

e K Q



KHQFH 61: ]

61 ] M+ ] M

FRQVWDQW

M% ] $] M M$] % ] M 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

)LJXUH 

([DPSOH RI D ZKLWHQHU EDVHG RQ VSHFWUDO IDFWRUL]DWLRQ

7KHUHIRUH WKH QRLVH KDV EHHQ ZKLWHQHG 2I FRXUVH WKH VLJQDO ZLOO DOVR KDYH EHHQ nOWHUHG WKDW LV WKH RXWSXW VLJQDO LV WKH SUHYDLOLQJ LQSXW VLJQDO FRQ YROYHG ZLWK WKH LPSXOVH UHVSRQVH K Q  7KLV DSSURDFK OHDGV WR D ZKLWHQHU WKDW RSHUDWHV LQ UHDO WLPH LH RQ OLQH LQ D FRQWLJXRXV IDVKLRQ LH VDPSOH E\ VDPSOH  1RZ WKDW WKH DGGLWLYH QRLVH FRPSRQHQW LV ZKLWH WKH VWDQGDUG DSSURDFK VHH &KDSWHU  DSSOLHV WR WKH nOWHUHG VLJQDOV HPEHGGHG LQ ZKLWH *DXVVLDQ QRLVH ([DPSOH  ] b       ] b 61 ] ] b       ] b RU  b   ] b  b   ] 61 ]  b   ] b  b   ]

KHQFH

 b   ] b ] b    b   ] b ] b   7KH WUDQVIHU IXQFWLRQ RI WKH ZKLWHQHU LV  ]b  + ]  61 ] ] b   7KH GLmHUHQFH HTXDWLRQ LV HDVLO\ GHULYHG YLD 61

< ]

UHVXOWLQJ LQ

 b  ]b ; ]  b  ] b

  \Q b   [Q b   [Q b  7KH nOWHU WKDW XVHV WKLV WUDQVIHU IXQFWLRQ LV VKRZQ LQ GHWDLO LQ )LJXUH  6LQFH WKLV nOWHU VKDSHV WKH HPEHGGHG VLJQDO WKH nOWHU PXVW DOVR EH DSSOLHG WR WKH VWRUHG UHSOLFD WR DOORZ HmHFWLYH FRUUHODWLRQ \Q

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH 



:KLWHQLQJ 8VLQJ &RUUHODWLRQ 'RPDLQ ,QIRUPDWLRQ

,I WKH REVHUYDWLRQ WLPH LV UHODWLYHO\ VKRUW LH 7REVHUY { 7VLJQDO 7  WKHQ DQ DSSURDFK EDVHG RQ WKH GHFRPSRVLWLRQ RI WKH QRLVH DXWRFRUUHODWLRQ IXQFWLRQ FDQ EH XVHG ,Q WKLV FDVH ZH FDQ IDFWRUL]H WKH FRUUHODWLRQ PDWUL[ DQG XVH WKH UHVXOWDQW PDWUL[ WR nOWHU D nQLWH GDWD VHJPHQW $ JRRG FDQGLGDWH LV WKH / '/ /+ GHFRPSRVLWLRQ 7KH / '/ /+ GHFRPSRVLWLRQ LV WKH VSHFLDO FDVH RI WKH / '/ 8+ GHFRPSRVLWLRQ ZKHQ WKH PDWUL[ WR EH IDFWRUHG LV V\PPHWULF 8VLQJ WKH PDWUL[ REWDLQHG YLD WKH IDFWRUL]DWLRQ RQH FDQ WUDQVIRUP nOWHU WKH GDWD RI OHQJWK / LH / d / LV WKH VL]H RI WKH FRUUHODWLRQ PDWUL[ LQWR / XQFRUUHODWHG VDPSOHV >@ 7KH WUDQVIRUPDWLRQ LV REWDLQHG E\ IDFWRULQJ WKH FRUUHODWLRQ PDWUL[ DV JLYHQ E\ 51 / '/ /+  ZKHUH + GHQRWHV WKH +HUPLWLDQ WUDQVSRVH LH FRPSOH[ FRQMXJDWH WUDQVSRVH  7KH PDWUL[ / LV D XQLW ORZHU WULDQJXODU PDWUL[ ZLWK XQLW HOHPHQWV RQ WKH GLDJRQDO 3UHPXOWLSO\LQJ E\ /b DQG SRVWPXOWLSO\LQJ E\ /+ b  ZH REWDLQ  /b 51 /+ b /b / '/ /+ /+ b '/ +HQFH '/ LV D FRUUHODWLRQ PDWUL[ RI WKH UDQGRP YHFWRU JLYHQ E\  Z /b Q 6LQFH '/ LV D GLDJRQDO PDWUL[ WKH FRPSRQHQWV RI WKH YHFWRU Z DUH RUWKRJRQDO RUWKRQRUPDO LI '/ LV WKH LGHQWLW\ PDWUL[  6LQFH / LV D ORZHU WULDQJXODU PDWUL[ /b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mHFW ([DPSOH  6XSSRVH WKH QRLVH FRUUHODWLRQ IXQFWLRQ LV JLYHQ E\ WKH H[SR QHQWLDO H[SUHVVLRQ 51 L HbMLM IRU L     7KH RQHVLGHG FRUUHODWLRQ YHFWRU LV JLYHQ E\ 51 >   @7



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

 

 

7KH / '/ /+ GHFRPSRVLWLRQ SURYLGHV DQ / PDWUL[ RI WKH IRUP         /        

 

6R WKH GHVLUHG WUDQVIRUPDWLRQ PDWUL[ LV JLYHQ E\    b     b /  b    b

   

 

ZKLFK FDQ EH XVHG LQ   ,I ZH ZHUH WR XVH WKH &KROHVN\ GHFRPSRVLWLRQ WKDW LV 51 & &7  WKHQ & LV JLYHQ E\         &        

 

 

7KHQ WKH GHVLUHG WUDQVIRUPDWLRQ PDWUL[ LV JLYHQ E\    b     b &  b    b

 

   

 

ZKLFK FDQ EH XVHG LQ   ([DPSOH  6XSSRVH WKH QRLVH FRUUHODWLRQ IXQFWLRQ LV JLYHQ E\ WKH H[SUHV VLRQ 51 L MLM IRU L     7KH RQHVLGHG FRUUHODWLRQ YHFWRU LV JLYHQ E\ 51 >   @7 7KH / '/ /+ GHFRPSRVLWLRQ SURYLGHV DQ / PDWUL[ RI WKH IRUP         /        

 

 

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH

 

6R WKH GHVLUHG WUDQVIRUPDWLRQ PDWUL[ LV JLYHQ E\    b     b /  b    b

    

 

ZKLFK FDQ EH XVHG LQ   ,I ZH ZHUH WR XVH WKH &KROHVN\ GHFRPSRVLWLRQ WKDW LV 51 & &7  WKHQ & LV JLYHQ E\         &        

 

 

7KHQ WKH GHVLUHG WUDQVIRUPDWLRQ PDWUL[ LV JLYHQ E\    b     b &  b    b

 

   

 

ZKLFK FDQ EH XVHG LQ   ([DPSOH  6XSSRVH WKH QRLVH FRUUHODWLRQ IXQFWLRQ LV JLYHQ DV WKH OLQHDU FRPELQDWLRQ RI WZR H[SRQHQWLDO W\SH WHUPV 51 L b MLM  MLM IRU L     7KH RQHVLGHG FRUUHODWLRQ YHFWRU LV JLYHQ E\ 51 > b  b@7 7KH / '/ /+ GHFRPSRVLWLRQ SURYLGHV DQ / PDWUL[ RI WKH IRUP     b     /  b   b  b 

 

 

 

6R WKH GHVLUHG WUDQVIRUPDWLRQ PDWUL[ LV JLYHQ E\        b / b    b 

   

 



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

ZKLFK FDQ EH XVHG LQ   ,I ZH ZHUH WR XVH WKH &KROHVN\ GHFRPSRVLWLRQ WKDW LV 51 & &7  WKHQ & LV JLYHQ E\     b     &  b   b  b 

 

 

 

7KHQ WKH GHVLUHG WUDQVIRUPDWLRQ PDWUL[ LV JLYHQ E\        b & b    b 

   

 

ZKLFK FDQ EH XVHG LQ   7KH PDWUL[ GHFRPSRVLWLRQ PHWKRG ZLOO ZRUN IRU VHJPHQWV RI OHQJWK / LH WKH GLPHQVLRQ RI WKH QRLVH FRUUHODWLRQ PDWUL[ ZKLFK HTXDOV WKH VLJQDO GXUD WLRQ  7KLV LPSOLHV WKDW WKH VWDUW DQG VWRS WLPHV RI WKH VLJQDO VKRXOG OLQH XS ZLWK WKH VHJPHQW XQGHU FRQVLGHUDWLRQ LH NQRZQ VLJQDO WLPLQJ SDUDPHWHUV  ,I ZH VHJPHQW WKH GDWD DQG WKH H[SHFWHG VLJQDO GXUDWLRQ LV RI OHQJWK / WKHQ LW LV SRVVLEOH WR EH PLVDOLJQHG DV PXFK DV E\ O RI WKH OHQJWK RI WKH GDWD SRLQWV ,I WKH VLJQDO HQHUJ\ LV KRPRJHQHRXV LQ WLPH SRWHQWLDOO\ KDOI RI WKH HQHUJ\ FRXOG EH ORVW LH D  G% ORVV  7KLV ORVV FDQ EH UHGXFHG E\ SHUIRUPLQJ RYHUODS SURFHVVLQJ FKRRVLQJ RYHUODSSLQJ VHJPHQWV RI GDWD IRU SURFHVVLQJ  2I FRXUVH WKH SULFH SDLG LV WKH DGGLWLRQDO SURFHVVLQJ FRVW LH IRU D  SHUFHQW RYHUODS IDFWRU WKH SURFHVVLQJ FRVW GRXEOHV UHODWLYH WR DQ DSSURDFK LQ ZKLFK FRQWLJXRXV QRQRYHUODSSLQJ VHJPHQWV DUH XVHG 7KH QRQRYHUODS VHJPHQW SUR FHVVLQJ OHQGV LWVHOI WR VFHQDULRV LQ ZKLFK VLJQDO V\QFKURQL]DWLRQ LV HQVXUHG LH ELQDU\ FRPPXQLFDWLRQ VLJQDOV n[HG SXOVH UHSHWLWLRQ IUHTXHQF\ 35) UDGDU VLJQDOV VRQDU VLJQDOV HWF  1RLVH VDPSOHV IURP VHJPHQW WR VHJPHQW ZLOO EH FRUUHODWHG EXW ZLWKLQ RQH VHJPHQW WKH / VDPSOHV ZLOO EH LQGHSHQGHQW :H QRWH WKDW LI RQH ZHUH WR OHW WKLV nOWHU RSHUDWH LQ D FRQWLJXRXV PRGH LH VKLIW RQH GDWD SRLQW WR GHnQH D QHZ VHJPHQW  RXWSXW VDPSOHV VHSDUDWHG E\ D QXPEHU RI VDPSOHV VPDOOHU WKDQ WKH nOWHU OHQJWK ZLOO EH FRUUHODWHG KHQFH EH FRORUHG 

:KLWHQLQJ YLD $XWR5HJUHVVLYH 0RGHOLQJ

$Q LPSRUWDQW DVSHFW RI GLJLWDO VLJQDO SURFHVVLQJ LV DXWRUHJUHVVLYH PRGHOLQJ +HUH RQH DVVXPHV WKDW WKH GDWD LV JHQHUDWHG XVLQJ DQ DXWRUHJUHVVLYH PRGHO LH DQ ,,5 nOWHU GULYHQ E\ D ZKLWH QRLVH VRXUFH  $V LQGLFDWHG LQ )LJXUH  ZH VHH WKDW WKH REVHUYDWLRQ QRLVH LV REWDLQHG E\ SURFHVVLQJ ZKLWH QRLVH

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH

)LJXUH 

)LJXUH 



$5 PRGHO

$5 JHQHUDWLRQ ZKLWHQHU DQG RSWLPDO GHWHFWRU

ZLWK DQ ,,5 nOWHU 7KLV QRLVH LV DOVR FDOOHG DQ DXWRUHJUHVVLYH SURFHVV DQG WKH VKDSLQJ nOWHU LV FDOOHG DQ DOO SROH PRGHO GLVUHJDUGLQJ ]HURV GXH WR WKH WULYLDO QXPHUDWRU SRO\QRPLDO ZLWK URRWV RQO\ DW WKH RULJLQ  :H DOUHDG\ NQRZ IURP WKH HDUOLHU GLVFXVVLRQ LQ WKLV VHFWLRQ WKDW SURSHU nOWHULQJ RI WKH FRORUHG QRLVH VHTXHQFH SURGXFHV D ZKLWH QRLVH VHTXHQFH )LJXUH  VKRZV WKH SURFHVVLQJ VWUHDP WR SHUIRUP GHWHFWLRQ RI WKH VLJQDO FRPSR QHQWV HPEHGGHG LQ FRORUHG *DXVVLDQ QRLVH +HUH QQ LV D FRORUHG QRLVH VHTXHQFH DQG VL Q LV WKH nOWHUHG VLJQDO FRPSRQHQW 7R REWDLQ WKH RSWLPDO nOWHU ZHLJKWV RQH KDV WR VROYH WKH :LHQHU <XOH:DONHU 1RUPDO HTXDWLRQV 7KHVH HTXDWLRQV DUH LQWLPDWHO\ UHODWHG VHH IRU H[DPSOH >@ :H FDQ VROYH IRU WKH RSWLPDO nOWHU E\ VWXG\LQJ WKH PRGHO LQ )LJXUH  :H KDYH QQ KQ e Z Q ZKHUH KQ LV WKH JHQHUDO LPSXOVH UHVSRQVH ZQ LV WKH ZKLWH QRLVH LQSXW VHTXHQFH DQG QQ LV WKH FRORUHG QRLVH RXWSXW VHTXHQFH :H FDQ H[SUHVV WKH V\VWHP GLmHUHQFH HTXDWLRQ DV

; D ZQ b L QQ b S

L

L 

Z Q



ZKHUH DL GHQRWHV WKH YDULDEOH KL  IRU DOO UHTXLUHG YDOXHV RI L 3RVWPXOWLSO\LQJ E\ QQ b O DQG WDNLQJ H[SHFWDWLRQ OHDGV WR 51 O b

; D 5 O b L S

L 

L

1

5: 1 O





'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

1RZ ( IZ Q QQ b O J

5: 1 O

( IZ Q

;} S



:

L 

; D ZL KQ b O b L J S

L

L 

p Q b L KQ b O b L

 }: KbO

6R  EHFRPHV

; D 5 bL }  ; b D 5 bL     ; b D 5 bL     b

51  51 b 51 b

S

L  S

L  S L 

  51 bS b

L



1

L

1

L

1

:

 

; D 5 bL  S S

L 

L



1

IRU O  IRU O  IRU O    IRU O S

,Q PDWUL[ IRUP ZH FDQ ZULWH

51 ZLWK DQG

  

51 D U 51 b 51  51 

51  51  51 

 

51 3

b 

 

51 3

b 



51 b 51 b 51 

 

51 3

ccc ccc ccc

b  c c c

51 b3 51 b3 51 b3

     

  5Q 

  

D7 >D  D  c c c  DS @

U7 >51  51  cc c 51 3 @ 7KLV DSSURDFK ZRUNV ZLWK VHTXHQWLDO GDWD VDPSOHV LH UHDO WLPH  KHQFH WKHUH LV QR QHHG WR VHJPHQW WKH GDWD $OO QRLVH RXWSXW VDPSOHV RI WKH ),5

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



ZKLWHQHU ZLOO EH VWDWLVWLFDOO\ LQGHSHQGHQW RQFH WKH ),5 nOWHU KDV SURFHVVHG / GDWD VDPSOHV LH KDV SURFHVVHG VDPSOH / b   7KH ZKLWHQHG RXWSXW FDQ WKHQ EH FRUUHODWHG ZLWK WKH nOWHUHG VL Q FRPSRQHQWV LH VL Q VL Q e KQ  WKH ),5 nOWHU ZHLJKWV  :H DOVR QRWH WKDW WKH DXWRUHJUHVVLYH PRGHOLQJ FDQ EH LQWHUSUHWHG DV D VSHFWUDO IDFWRUL]DWLRQ ZKHUH RQO\ WKH GHQRPLQDWRU $] LV D QRQWULYLDO SRO\QRPLDO ([DPSOH  6XSSRVH ZH KDYH WKH RQHVLGHG FRUUHODWLRQ YHFWRU DV JLYHQ E\ 51 >  @7 :H FDQ HDVLO\ YHULI\ WKDW WKH FRUUHVSRQGLQJ FRUUHODWLRQ IXQFWLRQ JLYHQ E\ IRU L    51 L MLM 6XSSRVH ZH OHW WKH PRGHO RUGHU EH 3  WKHQ ZH KDYH 51  D 51  KHQFH D 51  51     6XSSRVH ZH OHW WKH PRGHO RUGHU EH 3  WKHQ ZH KDYH 51  D  51  D 51  51  D  51  D 51  KHQFH   D   D  D   D  6ROYLQJ WKHVH WZR VLPXOWDQHRXV HTXDWLRQV OHDGV WR D  DQG D  ,I ZH ZRXOG LQFUHDVH WKH PRGHO RUGHU 3  ZH ZLOO VHH WKDW DOO FRHpFLHQWV SDVW WKH nUVW RQH LH D ZLOO EH ]HUR  6R ZH VHH ZH FDQ HDVLO\ H[WUDFW WKH SURSHU SROH ORFDWLRQ HYHQ ZKHQ WKH DVVXPHG PRGHO RUGHU H[FHHGV WKH WUXH PRGHO RUGHU 

:KLWHQLQJ E\ 'LVFUHWL]LQJ WKH &RQWLQXRXV 7LPH .DUKXQHQ/Rt HYH (TXDWLRQV

:H FDQ DOVR DSSURDFK WKH FRORUHG QRLVH SUREOHP XVLQJ WKH UHVXOWV REWDLQHG LQ 6HFWLRQ  6XSSRVH WKDW ZH GR KDYH FRQWLQXRXV WLPH SURFHVVHV VR WKDW

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



 LV DSSOLFDEOH )XUWKHU DVVXPH WKDW ZH VDPSOH WKH GDWD LQ WKH 1\TXLVW VHQVH WKHQ VQ  DQ\ VLJQDO FRPSRQHQW FDQ EH DFFXUDWHO\ UHSUHVHQWHG E\ WKH ')7 UHSUHVHQWDWLRQ LH WKH GLVFUHWH )RXULHU V\QWKHVLV HTXDWLRQ VQ



1

;b 6N H

1 

M { 1 NQ

N 



ZKHUH 6 N LV WKH FRPSOH[ DPSOLWXGH VSHFWUXP DQG {N 1 N  7KH KLJKHVW IUHTXHQF\ FRPSRQHQW LH PD[  KDV D FRUUHODWLRQ IXQFWLRQ FRPSRQHQW RI WKH IRUP 5P $ FRV P ZKHUH  FRUUHVSRQGV WR WKH KLJKHVW IUHTXHQF\ SUHVHQW 2QH FDQ VDPSOH WKH FRUUHODWLRQ IXQFWLRQ DW WKH VDPH UDWH DV ZRXOG EH DSSURSULDWH IRU WKH DQDORJ FRQWLQXRXV WLPH GDWD +HQFH  FDQ EH UHSUHVHQWHG E\

;b 5 O b P KP

/ 

P 

1

VO

V O b V O

›O ›/b



,Q PDWUL[ IRUP ZH KDYH XVLQJ V V b V DV WKH VLJQDO GLmHUHQFH VHTXHQFH  51 K V ZKHUH ZH UHSODFHG WKH FRQWLQXRXV YHUVLRQ RI VW LQ  ZLWK WKH DSSURSULDWH VDPSOHG YHUVLRQ VO V O b V O  7KH PDWUL[ HTXDWLRQ LV VLPLODU WR WKH PDWUL[ HTXDWLRQ RI WKH DXWRUHJUHVVLYH DSSURDFK LH VHH   EXW ZLWK K DQG U UHSUHVHQWHG E\ D7 K7 >K  b K   K  b K   c c c  K / b  b K / b  @ DQG U7 V7 >V  b V   V  b V   c c c  V / b  b V / b  @ 2QH FDQ DOVR VWDUW GLUHFWO\ ZLWK WKH GLVFUHWHWLPH VHTXHQFHV DQG SURGXFH WKH GLVFUHWHWLPH HTXLYDOHQW RI WKH .DUKXQHQ/RtHYH H[SDQVLRQ :LWK VRPH ZRUN RQH FDQ VKRZ WKH UHVXOW LV HTXLYDOHQW WR WKH ODVW PDWUL[ HTXDWLRQ :H QRWH LI WKH QRLVH SURFHVV LV ZKLWH WKHQ  FDQ EH VROYHG DV JLYHQ E\ V L b V L KL 51  UHVXOWLQJ LQ WKH FRUUHODWRU VROXWLRQ :H DOVR QRWH WKDW PDQ\ WLPHV WKH DQDO\WLF IRUP RI WKH QRLVH FRUUHODWLRQ IXQFWLRQ RU WKH QRLVH 36' DUH QRW DYDLODEOH 2QH FDQ HVWLPDWH WKH FRUUHODWLRQ IXQFWLRQ QRWLQJ WKDW WKH VLJQDO PD\ EH SUHVHQW DQG WKDW LWV HmHFWV PXVW EH

'HWHFWLRQ RI 6LJQDOV LQ 1RLVH



QXOOLnHG 6RPH DSSOLFDWLRQ RI DGDSWLYH nOWHULQJ WHFKQLTXHV PLJKW SURYLGH ZD\V WR JHW D XVHIXO DSSUR[LPDWLRQ RI WKH FRUUHODWLRQ IXQFWLRQ 7KLV DSSURDFK WULHV WR WDNH DGYDQWDJH RI WKH WHPSRUDO DQG VSHFWUDO GLm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

0DWFKHG )LOWHULQJ

)RU WKH NQRZQ VLJQDO FDVH LH GHWHUPLQLVWLF VLJQDO LQ FRORUHG QRLVH UHIHU HQFH >@ GHULYHV WKH RSWLPDO ),5 nOWHU WR PD[LPL]H WKH RXWSXW 615 ,Q WKLV SDUWLFXODU FDVH WKH ZKLWHQLQJ DQG FRUUHODWLRQ DUH FRPELQHG WR RFFXU VLPXO WDQHRXVO\ ZKLOH SHUIRUPLQJ WKH ),5 nOWHU RSHUDWLRQ )RU UHDO YDOXHG VLJQDOV WKH RSWLPDO nOWHU ZHLJKWV DUH JLYHQ E\  K V7 5b1 V b  5b1 aV ZKHUH aV LV WKH WLPH UHYHUVHG YHFWRU V )RU ZKLWH QRLVH WKH ),5 nOWHU ZHLJKWV EHFRPH  K V7 V b  aV ZKLFK LV WKH VFDOHG WLPH UHYHUVHG YHUVLRQ RI WKH VLJQDO XQGHU FRQVLGHUDWLRQ 7KLV SDUWLFXODU DSSURDFK ZRUNV DOVR LI WKH QRLVH LV QRW *DXVVLDQ EXW QRWH WKH RSWLPXP UHFHLYHU LV EDVHG RQ PD[LPL]LQJ WKH RXWSXW 615 DQG QRW RQ PLQLPL]LQJ WKH %D\HV FRVW ,Q WKH QRQ*DXVVLDQ FDVH WKHVH WZR DSSURDFKHV GR QRW SURYLGH WKH VDPH VROXWLRQ 

6800$5
M W   

HM W



$SSHQGL[ % ) IFRV  WJ

" " " \

{ >p  b   p    @

+

) IFRV  WJJ

+I

M{> p b

" " > ") \

>

)  I+I) IFRV  WJJJ

  p    @

[W J

+I

FRV  W

+I

b

[ AW

J

 M b HM W  H>M W c   bHM W H>M W c VLQ  W  M ([DPSOH ,I DW LV EDQGOLPLWHG LH DW $  IRU  WKHQ DW VLQ  W DW FRV W DW VLQ W DW FRV  W b

b

J

)I

J

+I

+I

M

! : 

J

J

b

3URSHUWLHV     

M

[W

DQG [AW DUH RUWKRJRQDO [W [AW [W [ AW +LOEHUW WUDQVIRUP EHKDYHV DV D F SKDVH VKLIWHU IRU VLQXVRLGDO VLJQDOV ,I \W KW [W WKHQ AW [W KA W \AW KW [  ,I DW LV EDQGOLPLWHG WKHQ DW FRV  W DW VLQ  W DW VLQ  W DW FRV  W J

+I

+I

J

b

e

e

+I

J

+I

J

e

b



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

 7LPH DYHUDJHV

5;; ~

5;A ;A ~

= 7  OLP [ AW [W 7  7 >7 A;; ~ 5 A;; ~ 5 5;; A  ~ 5; ;A ~ 

5;; A ~

5; ;A ~

~ GW

b

5;; A ~

b

b

5; ; A b~

b

5; ;A 

([SHFWDWLRQV

b

AW (;W ;

5; ;A ~

>~ 5; ;A ~ 5;; A ~ 5;A ;A ~ 6;; 

A;; ~ 5

b

A;; ~ 5 5;; ~ 6;A ;A 

3UH(QYHORSHV $QDO\WLF 6LJQDO DQG &RPSOH[ (QYHORSHV :H GHnQH WKH SUHHQYHORSH RI [W [S W [S W [W  M [AW ZKHUH [AW +LOEHUW WUDQVIRUP RI [W [W UHDO [S W LPDJ [S W [ AW [W ;  ) [W ;S  ) [S W ;   M > M VJQ @;   !  ;  ;       J

I

+I

J

I

J

I

I

J

b

J

$SSHQGL[ %



7KH WZR VNHWFKHV GUDZQ EHORZ LOOXVWUDWH WKH LGHD EHKLQG WKH FRQFHSW RI WKH SUHHQYHORSH ,W VKRZV LQ WKH )RXULHU GRPDLQ KRZ WKH EDQGSDVV DQG '& FRPSRQHQWV DUH PDQLSXODWHG

:H FDQ JHW [S W WZR GLmHUHQW ZD\V  JHW [W [AW DQG XVH DQ HDUOLHU HTXDWLRQ [S W [W  M [AW RU  JHW ;  ) [W OHW  !  ;  ;    ;S     DQG LQYHUVH WUDQVIRUP  =  ;  HMW G [S W {  +I

J

I

J

:H DOVR FDOO WKH SUHHQYHORSH WKH DQDO\WLF VLJQDO ,I WKH FDUULHU IUHTXHQF\ LV UHPRYHG WKHQ WKH DQDO\WLF VLJQDO EHFRPHV WKH FRPSOH[ HQYHORSH GHQRWHG E\ [aW  7KLV LV VKRZQ ZLWK WKH QH[W H[DPSOH



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

([DPSOH %DQGSDVV VLJQDO

:H VHH WKDW

a  ;

RU

VKLIWHG YHUVLRQ RI ;S  [ aW

([DPSOH I W

IAW

IS W

>M W

[S W H

I W

D QDUURZEDQG VLJQDO [W FRV  W \W VLQ  W [W VLQ  W  \ W FRV  W [W  M [ AW [W FRV  W \W VLQ  W M [W VLQ  W  \W FRV  W [W  M\W FRV  W  M VLQ  W [W  M\W HM W IS W H>M W [W  M\ W b

b

IaW

$SSHQGL[ %



7KH HQYHORSH LV GHnQHG WR EH WKH PDJQLWXGH RI WKH FRPSOH[ HQYHORSH S I W [ W  \ W IaW ([DPSOH %DQGOLPLWHG ZKLWH QRLVH QW [W FRV  W \ W VLQ  W M

M

b

QW

LV ZVV ZLWK 5;; ~ 5< < ~ 6;  6<  VR

Q A W [W \ W

b

[W

DQG \W DUH ZVV DQG 5;  5<  } QDUURZEDQG SURFHVV

[W VLQ  W  \ W FRV  W QW FRV  W  Q AW VLQ  W Q A FRV  W b QW VLQ  W

,I QW LV *DXVVLDQ WKHQ QA W LV *DXVVLDQ WKHUHIRUH [W DQG \W DUH *DXV VLDQ 1RWH QW [W FRV  W \ W VLQ  W LQSKDVH TXDGUDWXUH FRPSRQHQW SKDVH FRPSRQHQW /HW S [ W  \ W 5D\OHLJK 3') HW €W WDQ> \W [W XQLIRUP 3') WKHQ QW HW FRV W  €W )RU PRUH GHWDLO RQ VLJQDOV FRQVXOW >@ b



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

')7 ,QWHUSUHWDWLRQ 6XSSRVH ZH KDYH D FRPSOH[ YDOXHG WLPH VLJQDO LH VLJQDO ZLWK D RQHVLGHG QDUURZEDQG VSHFWUXP  VD\ [Q DQ HM Q ZKHUH DQ LV ORZ SDVV QDUURZ EDQG SURFHVV Q 1 > ; ;Q HM [P H>MP  ZKHUH Q LV WKH WLPH LQGH[ P Q

DQG WKH ')7 RXWSXW UDWH HTXDOV WKH LQSXW UDWH Q 1 > ; DP H>MP> $Q   b

P Q

z

,I DQ GRHV QRW YDU\ PXFK RYHU WKH 1 GDWD SRLQWV WKHQ WKH WUDQVIRUPV FDQ EH DSSUR[LPDWHG E\ Q 1 > ; DQ H>MP> |

P Q

1 DQ







HOVH [Q VLQFH VLJQDO LV DOUHDG\ RQHVLGHG [S Q H>M Q [Q H>M Q DQ HM Q H>M Q

[S Q [ aQ

DQ

([DPSOH 2QHVLGHG QDUURZEDQG VSHFWUDO GDWD UHVLGLQJ LQ RQH VSHFWUDO ELQ  )RU WKH ')7 WKH RXWSXW DW WKH SDUWLFXODU ELQ FRUUHVSRQGV WR D GH VDPSOHG FRPSOH[ HQYHORSH ,I WKH PD[LPXP RYHUODS FKDQJH RQH SRLQW SHU WUDQVIRUP LV XVHG WKH RXWSXW UDWH HTXDOV WKH LQSXW 1RWH 7KH FRPSOH[ WLPH GDWD RXW RI WKDW VSHFWUDO ELQ FRUUHVSRQGV WR WKH FRPSOH[ HQYHORSH ,I UHDO YDOXHG QDUURZEDQG GDWD LV LQSXWHG UHVLGHV DW D JLYHQ VSHFWUDO N ORFDWLRQ WKHQ WKH RXWSXW DW VSHFWUDO ORFDWLRQ FRUUHVSRQGV WR WKH FRPSOH[ HQYHORSH VFDOHG E\ D IDFWRU RI 1  DQG UHVDPSOHG  h

5HIHUHQFHV

>@ 2SSHQKHLP $9 :LOOVN\ $6 DQG @ /XGHPDQ /& )XQGDPHQWDOV RI 'LJLWDO 6LJQDO 3URFHVVLQJ 1HZ @ 2SSHQKHLP $9 DQG 6FKDIHU 5: 'LJLWDO 6LJQDO 3URFHVVLQJ (QJOH ZRRG &OLmV 1- 3UHQWLFH+DOO  >@ 0RKDQW\ 1 5DQGRP 6LJQDOV (VWLPDWLRQ DQG ,GHQWLnFDWLRQ $QDO\VLV DQG $SSOLFDWLRQV 1HZ @ 6KDQPXJDQ .6 DQG %UHLSRKO $0 5DQGRP 6LJQDOV 'HWHFWLRQ (VWLPDWLRQ DQG 'DWD $QDO\VLV 1HZ @ 5HIHUHQFHV

>@ +RmPDQ . DQG .XQ]H 5 /LQHDU $OJHEUD (QJOHZRRG &OLmV 1- 3UHQWLFH+DOO  >@ +RmPDQ . $QDO\VLV LQ (XFOLGHDQ 6SDFH (QJOHZRRG &OLmV 1- 3UHQWLFH +DOO 

$33(1',; ' 6RPH 0DWKHPDWLFDO ([SUHVVLRQV DQG 0RPHQWV RI 3UREDELOLW\ 'HQVLW\ )XQFWLRQ

, [ , , [

= {

 {



H[ VLQ t Gt

PRGLnHG %HVVHO IXQFWLRQ VW NLQG RI RUGHU ]HUR  [Q ; [ [ [ Q 

>

M Q -Q M[

, [

- M[

, [



= {

{



= {

{





= {

{





 c 

 c  c 

>

b

[ 

[





c



b

[  c  c 

H[S[ FRVt  € Gt

{[

t

,

{



[





u  c   ccc  [ [ 

IRU ODUJH [

ccc

H M Q{  -Q M[

DV\PSWRWLF H[SDQVLRQ ODUJH DUJXPHQWV

H[S [

S



HM[ VLQ t Gt

IRU VPDOO [

, [



FRV[ VLQ t Gt







Q Q 

,Q [

- [

>@



, [

H[S[

{ S

{

S



[



ccc

>@

'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



HUI [ 'HnQH 

HUI [

S

HUI F[

S

{

 {

= [

> > =   H>\  G\

 H \  G\

[

 b HUI [

4[

1RWH  

[  

1RWH 



%HZDUH RI

HUI F[

ZKHUH

:H VHH

z

D QHJDWLYH QXPEHU

HUI M[ M ‚ D SRVLWLYH QXPEHU HUI FM[ M  b HUI b[

HUI F[ HUI [ HUI [

GHnQLWLRQ WDEOH RI WKH HUURU IXQFWLRQ

r3 

s

IRU [ ! 

>

[  L L  DL W H

D D D D D

>

DFFXUDF\  HUURU   c  

 b

W

   H[

H



 b



6HH $EUDPRZLW] DQG 6WHJXQ >@ :H FDQ DOVR VROYH IRU [ LWHUDWLYHO\ ZKHQ   3) $   

[

>

HUI F  3 ) $

$SSHQGL[ '



*XHVV D UHDVRQDEOH [

› [

t

 OQ

› [

t

 OQ

5W 3) $ 5W

uw



uw



W

ZKHUH

  H[





W

3) $

  H[

   UHSHDW XQWLO

[L

b

>

[L  

ZKHUH 5W LV IURP ODVW SDJH

5W

1RWH

)RU 1

z

 ;

L 

DL WL

P }  RYHU b [

S

 { }

= [

> 

S

{

t 4

> >

  H \ P } G\

= [>P }

>

[bP }

u

>

 H W  GW

t HUI

[bP }

u



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

&RPSOHPHQWDU\ (UURU )XQFWLRQ 7DEOH =  (5) & [



S

{

[

>

 H X  GX

7KH YDOXHV DUH FRPSXWHG XVLQJ WKH 0$7/$%  RU  YHUVLRQ 7KH 0DWKZRUNV ,QF DV JLYHQ EHORZ

(5) & [

 

t HUI F0$7/

%$[

S



u

$SSHQGL[ '



6RPH +DQG\ ([SUHVVLRQV 

>

1 ; L 



D  LG

ZKHUH H



1 D  H

D  1

b

 G 

ccc 1



1

     c c c  1



>

1 ; L  LI

D b U 1

DU L

b

 OQ  [

[b

S

{[

z

 ; DUL

WKHQ

L 

[ 



[ 

b

[ 

D bU

 ccc 

[

H[S[

 , [



b

[ [   ccc       c  c  c  [ IRU VPDOO [   

z

U

bU

U

 , [

1 1  

H[S[ S

{[

| 

 [



 c  ccc [ 

IRU ODUJH [

[

›



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ



'HQVLW\ 3RLVVRQ

)XQFWLRQDO )RUP

&KDUDFWHULVWLF )XQFWLRQ

0HDQ

}

3 U [ D[ HbD  [ [    D!

€ ; 

D

D

Q

Q

???

I ; [

&KLVTXDUH

€ ; 



rQ [Q

b

M HDH 

s

> 

Q 

b Hb[ 



>

 M Q 

IRU [ !  ([SRQHQWLDO

*DXVVLDQ

I ; [

DHbD[ IRU [ !  I ; [

5{}



€ ;  D D M

>

€ ;  HM[x Hb



D

D

[x

}

 } 

Hb[b[x

 }

5D\OHLJK

8QLIRUP

I ; [ [ b[ D H D IRU [ !  I ; [ E



>D

IRU D  [  E

D

€ ;  HME HMD ME D

T{

r





DE

E



> >

> {

s

D

> D  

>  D  E   0D[ZHOO

I ; [

T

   D [

{

IRU

[!

HbD [  D!  

T

 

{ D

r



> {

s



D

$SSHQGL[ ' )XQFWLRQDO )RUP

'HQVLW\

&KDUDFWHULVWLF )XQFWLRQ

I; [ DQ [Qb HbD[ Q   IRU [ !  Q    D!

(UODQJ

>



0HDQ

}

Q D

Q D

€ ;  DQ D M Q

>

???

/RJ QRUPDO

D  HE}

I; [

  HxbIOQ[bD bEJ }

 



HE} H}



> 

5{} [ > D

} !  [ ! D

>  D E   r  s

I; [

:HLEXOO

rD

DE[Eb HbD[

E

b 

IRU [ ! 

I; [ D {  D    [ E  D! IRU [

>

>



E

s

r  s

Q rD

b 

K r

b 

D E !  &DXFK\

E



E



s

E R sL 

E

€ ;  HMEbDMM



5HIHUHQFHV

6FKDXP V 2XWOLQH 6HULHV 7KHRU\ DQG 3UREOHPV 0DWK HPDWLFDO +DQGERRN RI )RUPOXODV DQG 7DEOHV 1HZ @ 6SLHJHO 05

%RRN &R 

+DQGERRN RI 0DWKHPDWLFDO )XQF WLRQV 1HZ @ DV LV EULHo\ LQGLFDWHG QH[W )RXULHU 7UDQVIRUP )7  :KHQ ERWK GRPDLQV DUH FRQWLQXRXV DQG QRQSHULRGLF WKH WUDQVIRUP LV FDOOHG WKH )RXULHU WUDQVIRUP )7   ,I WKH WLPH GRPDLQ LV GLVFUHWH WKH WUDQVIRUP LV FDOOHG WKH GLVFUHWHWLPH )RXULHU WUDQVIRUP '7)7  7KH IUHTXHQF\ GRPDLQ UHSUHVHQWDWLRQ ZLOO EH SHULRGLF  ,I WKH IXQFWLRQ LV FRQWLQXRXV DQG SHULRGLF LQ WLPH WKH WUDQVIRUP LV FDOOHG WKH )RXULHU VHULHV )6  7KH IUHTXHQF\ GRPDLQ UHSUHVHQWDWLRQ ZLOO EH GLVFUHWH  ,I WKH IXQFWLRQ LV GLVFUHWH LQ WLPH DQG n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nOWHU IUHTXHQF\ DQG EDQGZLGWK 7KH :7 FDQ EH FKDUDFWHUL]HG DV LQ WKH )7 FDVH  ,I WKH LQSXW LV FRQWLQXRXV DQG RXWSXWV WKDW DUH FRQWLQXRXV LQ ERWK GLPHQVLRQV GHOD\ DQG VFDOH  WKH WUDQVIRUP LV FDOOHG WKH FRQWLQXRXV ZDYHOHW WUDQVIRUP &:7  



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

 ,I WKH LQSXW LV FRQWLQXRXV DQG WKH RXWSXWV DUH GLVFUHWH VKLIW DQG VFDOH DUH HYDOXDWHG RQ D JULG  WKH WUDQVIRUPDWLRQ LV FDOOHG WKH GLVFUHWH ZDYHOHW WUDQVIRUP ':7   ,I WKH WLPH VLJQDO LV GLVFUHWH DQG WKH WUDQVIRUPHG GLPHQVLRQV DUH GLV FUHWH WKH WUDQVIRUP LV FDOOHG WKH GLVFUHWHWLPH ZDYHOHW WUDQVIRUP '7:7  7KH '7:7 LV WKH RQH WKDW LV XVXDOO\ SHUIRUPHG LQ :7 DSSOLFDWLRQV 8VLQJ WKH VRFDOOHG 0DOODW DOJRULWKP OHDGV WR D IDVW LP SOHPHQWDWLRQ VRPHWLPHV FDOOHG WKH IDVW ZDYHOHW WUDQVIRUP  ,I WKH LQSXW WLPH VLJQDO LV GLVFUHWH DQG WKH WUDQVIRUPHG GLPHQVLRQV DUH FRQWLQXRXV WKH WUDQVIRUP LV FDOOHG WKH GLVFUHWHWLPH FRQWLQXRXV ZDYHOHW WUDQVIRUP '7&:7  7KH '7&:7 LV PHQWLRQHG IRU FRPSOHWHQHVV VDNH 7KH DXWKRU LV QRW DZDUH RI DQ\ DSSOLFDWLRQV RI WKLV WUDQVIRUP :H ZLOO H[DPLQH WKH &7:7 DQG WKH '7:7 LQ VRPH GHWDLO 7KH GLV FXVVLRQ DQG QRWDWLRQ ZLOO IROORZ WKH RQHV DGYRFDWHG LQ >@ 7KLV DSSHQGL[ LV ZULWWHQ VR WKDW LW FDQ EH IROORZHG ZLWK OLWWOH GLpFXOW\ 7KH RQO\ RGGLW\ LV RXU FRQYHQWLRQ WR ODEHO RUGHU VFDOHV LQ WKH VDPH VSHFWUDO VHQVH DV WKH IUHTXHQFLHV )URP D YHFWRU VSDFH SRLQW RI YLHZ DQG IURP D VLPSOHU ODEHOLQJ DELOLW\ RXU RUGHULQJ LV PHDQLQJIXO :H QRWH VFDOH ODEHOLQJ LV WRWDOO\ DUELWUDU\ &RPPRQ VHQVH VXJJHVWV WKDW WKH ODEHOLQJ JRHV LQ WKH RSSRVLWH GLUHFWLRQ WKDW IUHTXHQF\ ODEHOLQJ JRHV ZKLFK LQGHHG PRVW DUWLFOHV DQG ERRNV DGKHUH WR :H ZLOO XVH WKH ZRUGV VKLIW DQG GHOD\ LQWHUFKDQJHDEO\ $ ZDYHOHW GHQRWHG E\ ‚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{ I W GI

>

ZKHUH ) I LV WKH ZHLJKW DQG IHM{W J DUH WKH EDVLV IXQFWLRQV ,Q WKH ZDYHOHW WUDQVIRUP UHSUHVHQWDWLRQ RQH LV QRW OLPLWHG WR VLQXVRLGV UDWKHU WKDQ WKDW DQ\ EDQG SDVV IXQFWLRQ FDQ EH XVHG 7KLV PHDQV WKH ZDYHOHW LH EDQG SDVV IXQFWLRQV GR QRW DOORZ D '& FRPSRQHQW DQG PXVW EH RVFLO ODWRU\ DQG RI nQLWH GXUDWLRQ LH FRPSDFW VXSSRUW  7KLV PHDQV WKDW RQO\

$SSHQGL[ (



EDQGSDVV SKHQRPHQD FDQ EH UHSUHVHQWHG ZLWK ZDYHOHWV 2QH H[WUHPH FDVH LV WKH VHW RI WKH +DDU EDVLV IXQFWLRQV DOVR FDOOHG 'DXEHFKLHV ZDYHOHWV RI RUGHU  7KH\ FRQVLVW RI IXQFWLRQV WKDW DUH SRVLWLYH IRU WKH n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‚W  ZH GHnQH WKH ZDYHOHW WUDQVIRUP DV :I D E

: 7 II W J K‚DE W  I W L

 ‚ W I W GW > DE

=

7KH LQYHUVH H[LVWV LI &€

DQG

=

 M‚l M l



k

Gl 



 ‚W GW  >

=

WKHQ WKH RULJLQDO WLPH IXQFWLRQ LV JLYHQ E\  =  =  : ) D E ‚ W GD GE I W I DE & D ‚

>



'LVFUHWH :DYHOHW 7UDQVIRUP ':7 7KH &:7 LV YHU\ UHGXQGDQW DQG QRW YHU\ SUDFWLFDO 7R HOLPLQDWH WKHVH SURE OHPV WKH VKLIW GHOD\ DQG VFDOH SDUDPHWHUV DUH DOORZHG WR WDNH RQ GLVFUHWH YDOXHV RQO\ LH VDPSOH WKH &:7 DW GLVFUHWH SRLQWV LQ LWV WUDQVIRUP GRPDLQ  *LYHQ DQ\ VSHFWUDO UHJLRQ bI%  I%  ZH FDQ SDUWLWLRQ WKH VSHFWUDO UHJLRQ LQWR WZR HTXDO VL]HG SDUWV LH D ORZ SDVV DQG D KLJK SDVV UHJLRQ  +HQFH ZH FDQ WDON DERXW WZR GLmHUHQW VHWV RI EDVLV IXQFWLRQV WKDW VSDQ WKHVH WZR VSHFWUDO UHJLRQV 7KH ORZ SDVV UHJLRQ LV VSDQQHG E\ WKH VFDOLQJ LH ORZ SDVV IXQF WLRQV ZKLOH WKH EDQG SDVV UHJLRQ LV VSDQQHG E\ WKH ZDYHOHW IXQFWLRQV /HW XV WDNH WKH H[DPSOH RI EDQG OLPLWHG VSHFWUDO UHJLRQ DV SORWWHG LQ WKH GUDZLQJ RQ WKH QH[W SDJH DQG LQWHUSUHW WKH GLmHUHQW VSHFWUDO UHJLRQV



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

7KH UHJLRQV :  :  DQG : DUH EDQG SDVV UHJLRQV ZKLOH 9  9  9  DQG 9 DUH ORZ SDVV UHJLRQV 7KH GLUHFW VXP RI 9 j : 9 DQG 9  HWF 6XSSRVH ZH KDYH D SURWRW\SH UHDO YDOXHG IXQFWLRQ ‚W  9 j : ZKLFK LV D EDQGSDVV ZDYHOHW IXQFWLRQ DQG D ORZ SDVV VFDOLQJ IXQFWLRQ €W  'HnQH €N W €W b N  IRU N  = DQG €  / VR WKDW 9

VSDQ I€N W

7KH VXEVSDFHV DV LQGLFDWHG LQ WKH DERYH GUDZLQJ DUH DUUDQJHG DV IJ

   | 9>

| 9> | 9 | 9 | 9

ZKHUH 9M j :M 9M  6XSSRVH WKDW I W  9M WKHQ I W VFDOLQJ DQG ZDYHOHW IXQFWLRQ LV JLYHQ E\ K€MN W  ‚MO W L

=



   /

9M  7KH LQQHU SURGXFW RI WKH

€MN W ‚MO W GW



ZKLFK VD\V WKDW IRU DOO VFDOHV ODEHOHG M DQG IRU DOO GHOD\V WKH VFDOLQJ DQG ZDYHOHW IXQFWLRQV DUH RUWKRJRQDO $OVR =

€[ €[ b P G[

.pP

ZKLFK VD\V WKDW WKH VFDOLQJ IXQFWLRQ LQ D JLYHQ VFDOH LV GHOWD FRUUHODWHG 7KH HQHUJ\ RI WKH VFDOLQJ IXQFWLRQ LV GHQRWHG E\ .  ,I . HTXDOV RQH WKHQ WKH EDVLV IXQFWLRQV DUH QRUPDOL]HG ,Q JHQHUDO DQ\ IXQFWLRQ I W FDQ EH ZULWWHQ DV

 ;

I W N

>

FN €W b N 

ZKHUH €PN W P  €P  W b N 

  ; ;

P N

>

GP N ‚PN W

$SSHQGL[ (



6LQFH €W b Q VSDQV 9 9 j :  €W b Q VSDQV 9  DQG ‚W b Q VSDQV : D ZHLJKWHG VXP ZLOO VSDQ 9  DV JLYHQ E\ €W

;

S

KQ  €W b N WKH VFDOLQJ HTXDWLRQ

Q

DQG D ZHLJKWHG VXP ZLOO VSDQ : DV JLYHQ E\ ‚W

;

S

K Q  €W b N WKH ZDYHOHW HTXDWLRQ

Q

$V LW WXUQV RXW K Q LV VXFK WKDW K Q b Q KO b Q  ,W LV D WLPH UHYHUVHG RGG RUGHUHG ORFDWLRQV VLJQ UHYHUVHG FRS\ RI WKH ),5 nOWHU GHQRWHG E\ KQ  'LVFUHWH7LPH :DYHOHW 7UDQVIRUP '7:7 :LWK VRPH ZRUN RQH FDQ REWDLQ D UHFXUVLYH UHODWLRQVKLS EHWZHHQ FP N DQG GP N  DOORZLQJ DGGUHVVLQJ GLVFUHWHWLPH SUREOHPV E\ UHSODFLQJ FP ZLWK I Q  WKH ORZ SDVV VLJQDO ZLWK WKH KLJKHVW VDPSOLQJ UDWH LH RULJLQDO VDPSOHG WLPH GDWD 

FP N

;

KO b N FP O

O

GP N

GP N

;

K O b N FP O

O

7KLV DUUDQJHPHQW LV VKRZQ VXSSUHVVLQJ WKH WLPH GHSHQGHQF\ LQ WKH IRO ORZLQJ GUDZLQJV



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

,Q WKLV GHFRPSRVLWLRQ 0DOODW V DOJRULWKP  GP UHSUHVHQWV WKH GHWDLO KLJK SDVV  ZKLOH FP VWDQGV IRU WKH ORZ SDVV RXWSXW DW VFDOH P 7KH GHFRPSRVLWLRQ FDQ EH VWRSSHG DQ\ZKHUH LQ WKH VFDOH VSDFH ,I LW LV DOORZHG WR FRQWLQXH XQWLO LW UXQV RXW RI GDWD WKHQ DW WKH EDQGSDVV nOWHU DQG WKH DVVRFLDWHG ORZ SDVV nOWHU DW WKH ODVW VFDOH ZLOO RXWSXW RQO\ RQH GDWD SRLQW 7\SLFDOO\ WKH GDWD OHQJWK LV FKRVHQ WR EH 1 U  ZKHUH U LV DQ LQWHJHU 7KHQ WKH QXPEHU RI VWDJHV LH VFDOHV LV OQ U U LI LQGHHG D WRWDO GHFRPSRVLWLRQ LV GHVLUHG 7KH SURFHVV FDQ HDVLO\ EH LQYHUWHG WR UHFRYHU WKH RULJLQDO VLJQDO I W  &KDSWHU  DQG $SSHQGL[ ( FRQVWLWXWH DQ DWWHPSW WR SURYLGH D VLPSOLnHG LQWURGXFWLRQ WR ZDYHOHW SURFHVVLQJ )RU PRUH GHWDLO VHH UHIHUHQFHV >@ 5HIHUHQFHV

>@ 2SSHQKHLP $9 :LOOVN\ $6 DQG 1DZDE 6+ 6LJQDOV DQG 6\VWHPV QG HG 8SSHU 6DGGOH 5LYHU 1- 3UHQWLFH+DOO  >@ %XUUXV &6 *RSLQDWK 5$ DQG *XR + ,QWURGXFWLRQ WR :DYHOHWV DQG :DYHOHW 7UDQVIRUPV $ 3ULPHU 8SSHU 6DGGOH 5LYHU 1- 3UHQWLFH +DOO  >@ 6WUDQJ * DQG 1JX\HQ 7 :DYHOHWV DQG )LOWHU %DQNV :HOOHVOH\ 0$ :HOOHVOH\&DPEULGJH 3UHVV 

,QGH[ $

ER[FDU DYHUDJHU  

D SRVWHULRUL SUREDELOLW\  D SULRUL SUREDELOLW\ 

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nOWHU   ZKLWHQHU  FRQWLQXRXV ZDYHOHW WUDQVIRUP &:7  

DGDSWLYH nOWHULQJ  DGDSWLYH OLQH HQKDQFHU $/(   DPELJXLW\ IXQFWLRQ  DPSOLWXGH VKLIW NH\LQJ $6.   DQDO\WLF VLJQDO  DSSUR[LPDWLRQ  DV\PSWRWLFDOO\ XQELDVHG HVWLPDWH  DXWRFRUUHODWLRQ  DXWRUHJUHVVLYH $5 PRGHOLQJ  DYHUDJLQJ FRKHUHQW  LQFRKHUHQW  SRZHU  

% EDQGOLPLWHG ZKLWH QRLVH    EDQGSDVV nOWHU     %D\HV GHWHFWLRQ  %D\HV WKHRUHP   ELDVHG HVWLPDWH   ELER VWDELOLW\   ELODWHULDO WUDQVIRUP /DSODFH  UHJLRQ RI FRQYHUJHQFH 52&   ELQDU\ GHWHFWLRQ SUREOHP  ELVSHFWUXP  



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

FRQYROXWLRQ  FRRUGLQDWH WUDQVIRUPDWLRQ  FRUUHODWLRQ LQVWDQWDQHRXV  VSHFWUDO  FRUUHODWLRQ UHFHLYHU  FRUUHODWRU  FRUUHODWRU EDQN  FRVW  FRYDULDQFH  &UDPHU5DR &5 ERXQG   FURVV SRZHU VSHFWUDO GHQVLW\  FURVV FRUUHODWLRQ  FXPXODQW VSHFWUXP  FXPXODQWV  F\FORVWDWLRQDU\ SURFHVVLQJ ))7 DFFXPXODWLRQ PHWKRG )$0   PHWKRG  VWULS VSHFWUDO FRUUHODWLRQ PHWKRG 66&$   VPRRWKHG F\FOLF SHULRGRJUDP 

' GHFLPDWLRQ  GHFRPSRVLWLRQ  GHWDLO IXQFWLRQ   GLVFUHWH )RXULHU WUDQVIRUP ')7     GLVFUHWH WLPH FRQYROXWLRQ  FRUUHODWLRQ  ')7   GLVFUHWH WLPH FRQWLQXRXV ZDYHOHW WUDQV IRUP '7&:7   GLVFUHWH WLPH FRUUHODWRU  GLVFUHWH WLPH )RXULHU WUDQVIRUP '7)7     GLVFUHWH ZDYHOHW WUDQVIRUP ':7  

( HpFLHQW HVWLPDWH  HLJHQIXQFWLRQ  HLJHQYDOXH  HQVHPEOH DYHUDJH  HQYHORSH GHWHFWRU  HUJRGLFLW\  HUURU IXQFWLRQ   HUURU RI WKH nUVW NLQG  HUURU RI WKH VHFRQG NLQG  HVWLPDWLRQ 

) IDVW FRUUHODWLRQ  IDVW )RXULHU WUDQVIRUP ))7     IDVW ZDYHOHW WUDQVIRUP  nOWHULQJ  nQLWH LPSXOVH UHVSRQVH ),5 nO WHU    )RXULHU WUDQVIRUP )7   )RXULHU VHULHV   )UHGKROP HTXDWLRQ RI WKH nUVW NLQG  )UHGKROP HTXDWLRQ RI WKH VHFRQG NLQG  IXQGDPHQWDO WKHRUHP RI WUDQVIRU PDWLRQ 

* *DERU WUDQVIRUP  *DXVVLDQ GHQVLW\   *UDP6FKPLGW SURFHGXUH  

+ KLJKHU RUGHU PRPHQWV  +LOEHUW WUDQVIRUP  KRPRJHQHRXV LQWHJUDO HTXDWLRQ  K\SRWKHVLV WHVWLQJ 



,QGH[

, LQSKDVH DQG TXDGUDWXUH ,4 WHUPV  LQFRKHUHQW LQWHJUDWLRQ   LQGHSHQGHQW  LQQHU SURGXFW  LQVWDQWDQHRXV FRUUHODWLRQ IXQFWLRQ  LQWHJUDO HTXDWLRQ 

. .DUKXQHQ/RtHYH H[SDQVLRQ    /DJUDQJH PXOWLSOLHU   /DSODFH WUDQVIRUP  /'/% IDFWRUL]DWLRQ  /HLEQLW] UXOH  OHXFRJHQLF QRLVH   OLNHOLKRRG UDWLR /5   OLQHDULW\  ORZ SDVV nOWHU  ORZ SDVV VLJQDO 

0 PDJQLWXGH VTXDUHG FRKHUHQFH 06& IXQFWLRQ  0DOODW V DOJRULWKP   0$3 HVWLPDWLRQ  0DU\ FRPPXQLFDWLRQ V\VWHPV  PDWFKHG nOWHU    PDWFKHG nOWHU EDQN   PD[LPXP D SULRUL 0$3 GHWHF WLRQ  PD[LPXP OLNHOLKRRG 0/ GHWHF WRU  0HUFHU V WKHRUHP  0LQ0D[ FULWHULRQ  PLQLPXP PHDQ VTXDUHG HUURU 006(    PLQLPXP SUREDELOLW\ RI HUURU  PLQLPXP YDULDQFH HVWLPDWH  0/ HVWLPDWLRQ 

PRPHQWV   PRWKHU ZDYHOHW VFDOLQJ IXQFWLRQ  PXOWLSOH K\SRWKHVLV WHVWLQJ  PXOWLSOH SXOVH GHWHFWLRQ  PXOWLUDWH nOWHU 

1 1H\PDQ3HDUVRQ FULWHULRQ    QRQFHQWUDO FKLVTXDUHG GHQVLW\   QRQSDUDPHWULF GHWHFWLRQ  QRQVWDWLRQDU\  QRUPDO HTXDWLRQ  QXLVDQFH SDUDPHWHUV 

2 RQRm NH\LQJ 22. $6.   RUWKRJRQDO  RUWKRJRQDOLW\ SULQFLSOH  RXWSXW 36' 

3 SHULRGRJUDP    SRO\VSHFWUD  SRZHU DYHUDJLQJ  SRZHU RI WKH WHVW  SRZHU VSHFWUDO GHQVLW\ 36' IXQF WLRQ  SRZHU VSHFWUXP DQDO\]HU  SUHGLFWLQJ   SUREDELOLW\ GHQVLW\ IXQFWLRQ 3')   &DXFK\   FKLVTXDUHG  FKLVTXDUHG 1 GHJUHHV RI IUHH GRP   H[SRQHQWLDO   *DXVVLDQ   ORJQRUPDO  QRQFHQWUDO FKLVTXDUHG   5D\OHLJK  



'HWHFWLRQ 7KHRU\ $SSOLFDWLRQV DQG 'LJLWDO 6LJQDO 3URFHVVLQJ

5LFLDQ   XQLIRUP   SUREDELOLW\ GLVWULEXWLRQ  SUREDELOLW\ RI GHWHFWLRQ  SUREDELOLW\ RI HUURU  SUREDELOLW\ RI IDOVH DODUP 

4

GHFLVLRQ  HYHQW  REVHUYDWLRQ  VLJQDO  VSHFWUDO IDFWRUL]DWLRQ  VSHFWURJUDP  VWDQGDUG GHYLDWLRQ  VXPPHU ER[FDU DYHUDJHU  

4IXQFWLRQ 

7

5 UDGDU  UDQGRP SDUDPHWHUV DPSOLWXGH  IUHTXHQF\  SKDVH  WLPH RI DUULYDO  UDQGRP SURFHVV  5D\OHLJK GHQVLW\   UHFHLYHU RSHUDWLQJ FKDUDFWHULVWLF 52& FXUYH  UHFWLnHU  UHJLRQ RI FRQYHUJHQFH 52&  

6 VFDODU SURGXFW  VFDOH  VFDOLQJ HTXDWLRQ  6FKZDU] LQHTXDOLW\ VHH DOVR &DXFK\ 6FKZDU] LQHTXDOLW\   VHTXHQWLDO OLNHOLKRRG UDWLR GHWHFWLRQ  VHTXHQWLDO OLNHOLKRRG UDWLR WHVW 6/57   VHULHV UHSUHVHQWDWLRQ  VLJQ GHWHFWRU  VLJQDOWRQRLVH UDWLR 615   VLQF VLQ [ [ IXQFWLRQ  VL]H RI WKH WHVW  VPRRWKLQJ  VRQDU  VSDFHV

WLPH DYHUDJH   WLPH GLmHUHQFH RI DUULYDO 7'2$   WLPH IUHTXHQF\ GLVWULEXWLRQV  WUDQVIRUPV )RXULHU  +LOEHUW  /DSODFH  =  WUHOOLV  WULVSHFWUXP 

8 XQELDVHG HVWLPDWH  XQFRUUHODWHG  XQLIRUP GHQVLW\   XQLIRUP PRVW SRZHUIXO 803 WHVW  

: :DOG V VHTXHQWLDO /57  ZDYHOHW GHWHFWRU  HTXDWLRQ  SURFHVVLQJ  WUDQVIRUP  ZDYHOHWV   ZKLWHQLQJ nOWHU FRQWLQXRXV WLPH   ZKLWHQLQJ nOWHU GLVFUHWH WLPH   ZLGH VHQVH VWDWLRQDU\ :66   :LHQHU nOWHULQJ  

,QGH[

:LJQHU9LOOH GLVWULEXWLRQ  :LOFR[RQ GHWHFWRU  <XOH:DONHU HTXDWLRQ 

= =WUDQVIRUP