HAL Id: inria-00073503
https://hal.inria.fr/inria-00073503
Submitted on 24 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Robust Location based Partial Correlation
Zhong-Dan Lan, Roger Mohr
To cite this version:
Zhong-Dan Lan, Roger Mohr. Robust Location based Partial Correlation. [Research Report] RR-3186, INRIA. 1997. �inria-00073503�
ISSN 0249-6399
a p p o r t
d e r e c h e r c h e
')(+*,-.0/1(2435.67(98 *:3;-<>= [email protected]#CEDF(+AA<GCH3I.67(98
JKLNMONPRQ;STMUV5WXSNMYZ>L[O]\_^a`cb;daZ
fe;gihkj7l
mNn!M\porqst]tvu
Bw"Bg
JKLNMONPRQ;STMUV5WXSNMYZ>L[O]\_^a`cb;daZ
*,+-
.0/1.2436587:9.<;>=@?<9ACBD7
+
BD/1/1.FEG/H=@?
+
AC7I.@J
AK/[email protected].0.0M0J?0BD7I7=@AKMMQ=@7I?0.0M
R
;QBTS.<9UWV"X5
Y
=[ZIZ$B@;9NI.\;Q.0?
+
.<;Q?
+
.]7^_2_`baDcd3fehgI7.i:Jj`bkDkhl43 `bkZ#[email protected]
monqp0rTsTtIuDrhv *"+
.xw_AKMg=@yz?0B@;Q;Q.0M{Z$BD7INI.07I?0.|ZI;QB@}yC.0/~ACM=H/H=bSB@;\AKMMgI.|AK7?0BD/Zg9.<;wACMACBD7$,B@;Q;Q.0yK=[9ACBD7ACM=
?0BD/1/HBD7|9B_BDyh B@;z9
+
ACM$ZI;>B@}yC.0/zUBDM{9q?0y=@MMAC?=@yD?0B@;;>.0yK=[9ACBD7|/1.<9
+
B:NIMz =@ACyD7.=[;j9
+
.NIAKM{Z#=[;QA9,NIACMQ?0BD79AC7_gIA9AK.0M0J
+ AC?
+
B_?0?0g;"=[99
+
.,}$BDgI7N=[;QAC.0MBD jB@}S.0?<9M05879
+
ACMZ#=[Z$.<;J=|@FDz>@F{D@9.0?
+
7AC:g.ACMZI;>[email protected]
MBDyCwh.P9
+
ACMjZI;>B@}yC.0/
Y
B@}gIM9qyCB_?=[9ACBD7/1.<9
+
B_NIM=[;Q.PgM.0N9BZ$.<;Q B@;Q/9
+
ACMzZ=[;9AK=@y_?0B@;;Q.0yK=[9ACBD7jBD/1Z#=[;QACMBD7M
=[;Q./H=@NI.
AC9
+
B@9
+
.<;"9.0?
+
7IAC_gI.0M,[email protected] Z$.<;QAC/1.07:9Q=@y;Q.0Mgy9Mw@=@yKACN=[9.\9
+
.=[ZZI;QBh=@?
+
¡d¢£¤¥¦ sT§qphv
w_AKMACBD7$J/H=[9?
+
AC7ILJ?0B@;Q;Q.0yK=[9ACBD7$J;>B@}gIM{97I.0MQM0JIyCB_?=[9ACBD7$JZ#=[;9A=@y$B:?0?0yCgMACBD7
¨C©ª
«¬G®
ª
«¯j°¬²±³0´
µ¶T·¹¸$º»>¼½"º¾F¸z¸¿[ÀTÀD»F¼Á²Â{Ã,ÄÅÆÇqÈÉDÊTËÇÌÈzËÍÊËÇÌÎÏо>Ñ[ÃÒÓQÔ
PÕDÖ×DØ
Ù"ÚDÙÛGÙFÜ{ÝDÙ>ÛÞÜ{ÝDÙ
c
Ý#ßà
Õ
ÙQá )âã
Ù>ä
l[åbåDæDç<è
Ù
Õ@é
ÙÚDÙ âêqé
ÛGà ã Ù
æig7j7g_gbë "pì "#!îí
Û
çbÕ
ÜQÙ<ï
4ØâØã Õ íg7g ëbñ|ò lflihåbóëbë,ô>ØâQØ ãhÖ íg7g ëñò l lih"åbó\åTó
¢ p ¢ v
=[ZZ#=[;QAC.0/1.07:9|.07:9{;Q.HNI.0g WAC/[email protected]{9gI7.?0y>O.HNI.7IBD/4}I;Q.0g WZI;>B@}y
-
.0/1.0M|NI.Z$.<;Q?0.<Z9ACBD7$Jz.07
Z#=[;Q9AC?0gIyCAC.<;"?0.0yKgIAjNI.4yK=Z$.<;Q?0.<ZI9ACBD7dM{9:O.<;_O.0BDM?0B@ZAC_gI.@
=1?0B@;;O.0yK=[9AKBD7.0M{9yK=MBDyKg9ACBD7yK=ZyCgIM?0BD/1/ gI7O.0/1.07:9
=@NIB@Z9:O.0.@
=?0B@;;_O.0yK=[9ACBD7'?0y=@MMAC_gI. ?0BD/1/1.x"=oNI.0MxNIA ?0gIy9:O.0.0M\ZI;
-
.0MxNI.0MxNIAKM?0BD79AK7:gIAC9:O.0Mx_gIAM. ZI;O.FE
M.07:9.07:9
-
=dyK=yCAC/HA9.1NI.0MB@}S.<9MB:?0?0gIyC9Q=@79M0x=@7IM?0.;>=[ZIZ$B@;90J7IBDgMZI;>[email protected]?
+
7IAC_gI.H}#=[ZI9ACM_O.0.
>@>0D@@F²0"Z$BDg;,;_O.0MBDgIN;Q.?0.|ZI;QB@}#y
-
.0/1.@BDgIM,gI9ACyCACMBD7My=bQD¹0D@ { !<G"Z$BDg;9{;>BDgwh.<;yK=
Q@IZ#=[;9AC.\NI.0M .07".<9{;Q.0M
-
=?0B@;;O.0yK.<;#BDgIM"=0whBD7IM,?0BD/Z#=[;O.|7IB@9{;>./ .<9O
+
B:NI.|=wh.0?|yC.0M/ .<9O
+
B_NI.0M"M9Q=@7IN=[;QNIM
.<9,yK.0M"/ O.<9
+
B_NI.0M,7IBD7Z#=[;>=@/ .<9{;QAC_gI.0M0.0M.F _ZOO .<;>AC/1.07:9Q=[9ACBD7IM"[email protected]9yK=/ .<9O
+
B_NI.@
$¦ rTp ¤ u% ¢ v
w_AKMACBD7$JI=[ZIZ#=[;>AC.0/1.07:90JI?0B@;;_O.0yK=[9ACBD7$J;QB@}gIM{9.0MQM.@JyCB_?=@yCACMQ=[9AKBD7$JB:?0?0gIyC9Q=[9ACBD7Z#=[;9AK.0yCyC.
& )
rTs ¦ §quDr¦
H¢ %tr ¢ § ¥¦ s
` B@;;Q.0yK=[9ACBD7=@7NAC/H=@LD.\9{;>=@7M B@;>/1M||x||x||x|x|||x|x|||x|x||x
Y
=@7=@7IN?0.07IMgM9{;>=@7IM B@;Q/ |x||x||x||x|x|||x|x|||x|x||x i
2 Y
B@}gIM{9;>.0L@;Q.0MMACBD7o}[email protected]Z#=[;9A=@yj?0B@;;Q.0yK=[9AKBD7 ||x|x|||x|x|||x|x||x c
m t %£ pp ¦ v%¦ uhtr¦ "! ¢:¥$#¦ r %
H¦ nqp0r ¢ p0r tr¦
¦& %¦ uhtIr¦ t
§(§'p #¢ sTp¦ (
` )PM{9AC/H=[9AKBD7BD ÌyKB:?=[9ACBD7=@7INNIACM{Z$.<;QMQACBD7H||x||x|x|||x|x|||x|x||x a
*
BD/1.xNI.,+7IAC9ACBD7IM=@[email protected] I=@/ZyK. |x||x||x|x|||x|x|||x|x||x a
2 UdX-)x||x||x||x||x||x||x||x|x|||x|x|||x|x||x k
. /ZIZyKAC?=[9ACBD7oBD UX-) .0M{9AK/H=[9B@; ||x||x||x|x|||x|x|||x|x||x `D`
0 H¦ nqp0r %¦ uhtr¦ nztIp ¢ § # t#srt %Pu ¦ sTs ¢ %tr¦ &
i:` /7=@yMACMBD ?0B@;;Q.0y=[9ACBD7d=@7INyKB:?=[9ACBD7Ðx||x||x|x|||x|x|||x|x||x `D`
i:
\.=@yCAK7IL A9 + 9 +
.|Z#=[;9AK=@y$B_?0?0yCgIMAKBD7oZI;QB@}yC.0/]x||x|x|||x|x|||x|x||x `D`
i:2 R
;QB_?0.0NIg;Q.\BD ÌZ#=[;9AK=@y$?0B@;Q;Q.0yK=[9ACBD7 ||x||x||x|x|||x|x|||x|x||x `
i:. UW=@AC7Z$BDAC7:9?0BD7IM{9{;>=@AK79x||x||x||x||x|x|||x|x|||x|x||x `
i:¹i )PM{9AC/H=[9AK7IL?0B@;;Q.0yK=[9ACBD7}.<9
.0.07d9
BHAC/ =@LD.
AC7INB
MH|x|||x|x|||x|x||x `b2
1 243 #¢ s ¢ rbt %Ps ¢ p %KrTp 5
c_` BD7IM{9{;>=@AC7:9MgIM.0N x||x||x||x||x||x|x|||x|x|||x|x||x `
c_
*
9Q=[9ACM{9AC?=@y$;>.0MgIy9M x||x||x||x||x||x|x|||x|x|||x|x||x `Ti
% 4¦
u% zp¦ t § zr Ìs ¢¥ ¦ s %
!|1Ì $
*"+
.w_ACMQg=@y?0B@;;Q.0M{Z$BD7IN.07I?0.ZI;QB@}yK.0/ACM=x/H=bSB@;)9Q=@M:AC7?0BD/1Zg9.<;wACMACBD7¹`D`J
J:cJ_=@MPA9gI7IN.<;QyCAC.0MÌ/1B@9ACBD7
=@7IN M9.<;Q.0B /H=[9?
+
AC7IL\Awh.07 9 B'AC/ [email protected] "9
+
.M>=@/1.oM?0.07I.@J=WZA .0yPAC7 BD7I.AK/H=@LD.?0B@;;Q.0M{Z$BD7INM49B'=
ZA .0yÌAC7W9
+
.B@9
+
.<;|AC 9
+
.<W=[;Q. ZI;QBTS.0?<9ACBD7IM\BD )9
+
.MQ=@/1.Z
+
_MQAC?=@yqMQ?0.07I..0yC.0/1.07:9058 9
+
.<W=[;>.9.0/Z$B@;>=@yCy
?0BD7IMQ.0?0g9Awh.@JI9
+
.\?0B@;;>.0M{Z$BD7INI.07I?0.NI.<9.<;Q/1AK7I.0M/1B@9ACBD7$58 j9
+
.<o=[;Q.\M{Z#=[9AK=@yKyHM.<Z#=[;F=[9.0No}g9MAC/ gIy[email protected]J
9 +
.?0B@;;Q.0M{Z$BD7INI.07?0.xNI.<9.<;Q/1AC7.0MM{9.<;Q.0B1NIAKM{Z#=[;QA9D
*,+
.|gIM.BD Ì?0B@;Q;Q.0yK=[9ACBD7W=@M=1MQAC/1ACyK=[;>A9GH/H.=@Mg;Q.x}$.<9 .0.079 B MQACLD7=@yCMACM .0yKy _7B 7$P59ACM,?0BD/1/HBD7Iy
gIMQ.0NAC74M9.<;Q.0B\w_AKMACBD7 B@;9
+
.PwACMg#=@y:?0B@;;>.0M{Z$BD7INI.07I?0.ZI;QB@}#yC.0/[email protected] _9.07IMQAwh.?0BD/Z#=[;QACMQBD7IM
+
=0wh.}$.0.07/H=@NI.
.<wD=@yCg=[9AC7LNIA.<;Q.079"?0B@;Q;Q.0yK=[9ACBD7?<;>A9.<;QAK=¹`
/MAC7INIAK?=[9.0N AC7i_Jl JB_?0?0yCgIMQACBD7'Zy=0M =@7 AC/Z$B@;9Q=@7:9;>gIyC.1AC7 M{9.<;Q.0BÐ/H=[9?
+
AC7IL5Þ7!lJ5879ACyKyC.gIM.0M
;>BDgI7IN1BD7:9{;QBDy
R
BDAC79M#"$
R&%
9Bx?0BD7IM9{;QgI?<99
+
.}$.0M{9NIACM{Z#=[;>A9G|Z=[9 +
*,+
ACMÌ/1.<9
+
B_NLDAwh.0M9
+
.}$.0M{9Ì;Q.0MgIy9M
AC '|
R
=[;Q.Pwh.<;4?0yKBDM.)9B9
+
.B_?0?0yCgIMAKBD7}$BDgI7IN#=[;QAC.0Mz=@M9
+
.0MQ.=[;>.)9
+
.;Q.0LDAKBD7IMqAK7
+ AC?
+
=@yCy_?0yK=@MMAC?=@y/1.<9
+
B:NIM
=@ACy
(
.Ð .0.0y,9
+
=[9BD7IyC 9 +
.dgIM.BD \;QB@}gIM9 /1.<9
+
B_NIM ?=@7(Bbwh.<;Q?0BD/1.W9
+
.W/H=[9?
+
AC7IL'Z;QB@}yC.0/1M19
+
=[9.<wh.<;Q
/1.<9
+
B:N.07?0BDgI79.<;>M=[9"9
+
.B:?0?0yCgNIAC7IL?0BD7:9BDg;QM0P5Þ7)k*J= _@<hÌ>@>0D@/1.<9
+
B_NACM"N.0M?<;QA}$.0N$J_}#[email protected]
BD7 8 !FP{$D FDIP5Þ9ZI;QBbw_AKNI.0M"LDB_B:N/ =[9?
+
.0M=[9B_?0?0yCgIMACBD7o?0BD7:9BDg;QM0
587Ð9
+
ACM\Z=[Z.<;bJz=@7IB@9
+
.<;|Z#=[;Q9AK=@y?0B@;;Q.0yK=[9ACBD7 /1.<9
+
B_N ACMxNI.<wh.0yCB@Z$.0N$J$}#[email protected] BD7 8 !FbQD@I15Þ9xgIM.0M
=H/1AC7IAC/ gI/ whBDyCg/1.|.0M{9AC/ =[9B@;+"²UdX ) %
J=@7INACM,=@yCMQB1?[email protected]
+
.UdX ) ?0B@;;Q.0y=[9ACBD7$59,?=@7}$.x9{;Q.=[9.0NÐ=@M=
;QB@}#gIM{9A+?=[9AKBD7BD q9
+ . \, ?0B@;;Q.0yK=[9AKBD7,".<;QBHU.=@7W;QBDMM,B@;;>.0yK=[9ACBD7
%
JIAC79
+
=[9,M.07IMQ.xA9"ACM9
+ . \"
BD7d9
+
.-|Z#=[;Q9M"BD Ì9
+
.|MACLD7#=@yCM0
*,+
ACMZ#=[Z$.<;"ACM"B@;>Lh=@7IA/.0.0Nd=@M BDyCyKB M0)=@ 9.<;=@7dAC79{;>B:NIg?<9ACBD7o9B;Q.0yK=[9.0N B@; J .|LDAwh.=M
+
B@;9,=@7=@yMACM
BD ?0yK=@MMQAC?=@y x"(?0B@;;Q.0yK=[9AKBD7 K;>BD/ 9 +
.xyCB_?=[9ACBD7wAC. Z$BDAC7:90
*"+
.07MBD/1. gI7N=@/1.07:9Q=@y$yCB:?=[9AKBD77IB@9ACBD7IM
=[;Q.NIAKM?0gIMM.0NjJ# BDyCyCB .0NW}:W=H;>B@}gIM{9\yCB_?=[9ACBD7W}#[email protected]?0B@;;Q.0yK=[9ACBD7jJ + AC?
+
ACMx=@yCMB?[email protected]ÐUdX )?0B@;Q;Q.0yK=[9ACBD7$
/9x9
+
..07INjJMQBD/1..F _Z$.<;QAC/H.079Q=@yq;Q.0MgIyC9M=[;Q.M + B
7$J$AC7?0yCgINIAC7L?0BD/Z#=[;QACMQBD7IM
A9 + 9
B?0yK=@MMAC?=@y/1.<9
+
B:NIM
x" =@7IN * *
1" $.<;QBUd.=@7
*
gI/ BD
*
_g=[;Q.0N NA.<;>.07I?0.0M
%
J$=[}A +
M;>=@7W=@7NW?0.07MgIM2¹`bcJz=@7N
=@7 M
/1.<9
+
B:N Y * *
|3"
Y
B@}gIM{9
*&*
A9
+
.07:9.<;"Z$BDAC7:9"?0BD7IM{9{;>=@AC7:9
% k
4 )% 65 !7
5879
+
ACM;>.<ZB@;Q90J . ACyCy_NI.0M?<;QAC}.= { !</H=[9?
+
AC7IL|=@yCLDB@;>A9
+
/
*"+
.<;Q."=[;>."=@yCMBxB@9
+
.<;QM)Z#=[Z$.<;QMBD71/H=[9?
+
AC7IL
ZI;>B@}yC.0/ AK7 + AK?
+
MBD/1.48 !F/1.<9
+
B_NIM=[;>.gIMQ.0N$98IB@;.F =@/Z#yC.@J_AK7H9
+
.,Z#=[Z$.<;BD zVxNIB@}$..=@7IN;:BDg9
+
.0/4
¹`b2*Jh=< F>=1D²@jACMqgIM.0Nx9B?0BD/Zg9.<;Ì9
+
.B@Z9AC?=@yE?B
Jh=@7INAC7@¹`bc*:=@7IN;2*Jh=\#@A_@{*=H0F9{;>=@7IMQ B@;Q/
ACM,gIM.0No}$.0 B@;>.\9
+
.|?0B@;;Q.0y=[9ACBD7$587Bk*J= _DFDQ@F{h@/1.<9
+
B:NAKMNI.0M?<;QA}$.0N$
CED$F G@HJIKIKLNM$O'PQ$HRSORUTVQ$WXOZYJL,PIKORU[ \]HJIWX[
*"+
.|wACMg=@yj?0B@;Q;Q.0M{Z$BD7INI.07I?0.4BD 9 BHwh.0?<9B@;>MxACM,gIMQg=@yCyNI.,+7I.0Nd}:?0B@;;>.0yK=[9ACBD7$
*"+
.<;Q.4=[;Q.4/H=@7:dNIA.<;Q.07:9
wh.<;QMQACBD7IMBD $9
+
ACM^"M.0.,9 + . *
=[}yC.`AC7<¹` B@;AC7IM{9Q=@7I?0.
%
) _Z$.<;QAK/1.079Q=@y?0BD/Z#=[;QACMQBD7IM=[;>.,;Q.<Z$B@;9.0N AC7_¹``=@7IN
//1BD7IL4.F AKM{9AC7IL|/H.<9
+
B:NM0J
. +
=wh.?
+
[email protected].<;QB /1.=@7H7IB@;>/H=@yCA/.0.0N ?<;QBDMM)?0B@;Q;Q.0yK=[9ACBD7_"x"
%
}$.0?=@gIM.
BD A9M"LDB_B_N.F _Z$.<;QAK/1.079Q=@yZ$.<;Q B@;Q/H=@7?0. A9
+
;Q.0M{Z$.0?<9"BD ÌwD=[;_AK7ILyCACL
+
9AC7IL?0BD7INIA9AKBD7IM0
"
%
" "
%
"
% %
"!$#&%(') )*#+%-,/.%-'
AC9
+
"012
%
J
"340562
% 9 +
.NIACM{Z=[;QA9GDJ
7
"
%8:9<;>=?
#+ ! #&%(,
;
'@ ! #+%-'A'B#+ * #&%(,/.%-,
;
'@ * #+%-,/.%('B'
C @
"9
+
.?0Bbw@=[;>AK=@7I?0.}$.<9 .0.07
"
%
=@7IND
"EF
% % J !
G 9 ;=?
#+"!#&%(,
;
'H@ "!#+%-'A'
*
C @
"9
+
.M{9Q=@7IN=[;QNÐNI.FE
wAK=[9ACBD7 BD ("
% % J
)*
IG 9<;>=?
#+ * #&%(,/.J%(,
;
'@ * #+%-,/.%('B'
*
C @
"9
+
.HM9Q=@7IN=[;QN NI.<w_A=[9ACBD7'BD ("IK4
% % J
L
"M5
%ONKP IQ
"M1
%SR TVU
M
UW
=@7N
TYX U
UWX[Z
J `"
%
=@7IN *"
% 9 +
.H/1.=@7
L@;>=yC.<[email protected]"BD Ì9
+
.\9
B AC/H=@LD.
AC7INIB
M( "
%
=@7N\"Y
%
=@7IN\]
"
(`
% " X ` %
ACM9 + .
AK7INIB
NAC/1.07IMAKBD7$
*,+
. * * ?0B@;;>.0yK=[9ACBD7 gI7I?<9ACBD7
+
=@M,=@yCMB}$.0.07gIMQ.0N0
_^^a`
"bY
%
! #&%('H@c* #+%-,/.%-'
AC9
+
"
T
% " L %d
"e L %fT
"eK4I L %
J=@7IN ! #&%('H@c * #+%-,/.%-' 9 +
.HM{9Q=@7IN#=[;QN NI.<wAK=[9ACBD7 BD
"
%aT
"Y
%
B@9AC?0.9
+
=[9
A9
+ * * J
.|/HAC7IAC/1A.0.,9
+
.?0B@;;Q.0yK=[9ACBD7d gI7?<9ACBD7
+ ACyK.
AC9
+
\,J
./H=T AK/1A/.0.\A90
/y9.<;Q7#=[9Awh.,=[ZIZI;>Bh=@?
+
.0M
+
=0wh."}$.0.07ZI;>[email protected]}#[email protected]{;F=@7IM B@;Q/1.0Nwh.<;QMAKBD7IMBD 9
+
.B@;QACLDAC7#=@y_MQACLD7=@y'/
?0BD/1/HBD74M{9Q=@7N=[;QN|9{;F=@7IM B@;Q/H=[9ACBD7|AKMj9 + .
=[ZyK=@?0AK=@74BD _9
+
.)AC/ =@LD.@
*"+
ACM
+
=@M=MAC/1AKyK=[;j. .0?<9Ì9B,7B@;Q/H=@yCA/.0AK7IL
9 +
.AC/H=@LD.=@M"AK7o9
+ . * * /1.<9
+
B:N$
/yCyBD 9
+
.0M./1.<9
+
B_NIM=[;Q.owh.<; M.07IMQA9Awh.9B BDg9yCAC.<;>M0J.@L M{9{;QBD7IL 7IBDAKM. B@;H9
+
.ZI;Q.0M.07I?0.BD ,Z#=[;9AK=@y
B_?0?0yCgIMAKBD7IM0Ì5Þ7 B@;QNI.<;)9B;Q.0NIgI?0.MgI?
+ . .0?<9M0JMBD/1.=@gI9
+
B@;QM)yKAC/1A99
+
.AC7 ?g.07I?0.BD 9
+
.,7_gI/1.<;QAK?=@yw@[email protected]}_
BD7IyC9Q=AC7ILHAC79B =@?0?0BDgI799
+
.MQACLD7BD Ì9 + .
=[ZyK=@?0AK=@7 ¹`Tl_Jj`
JB@;,9
+
.|B@;>AC.079Q=[9AKBD7BD Ì9
+
.|L@;[email protected]:9 ¹`Ti*5Þ7
9 +
.0MQ.4?[email protected]J$9
+
.Z$.<;9gI;}#=[9ACBD7IM,BD 9
+
.?0B@;Q;Q.0yK=[9ACBD7 =[;Q. AC7dNA;Q.0?<9,;F=[9ACBH9B 9 + . MA/.0.4BD 9
+
.4Z.<;Q9g;}$.0Nd=[;Q.=_
/MAC/1AKyK=[;AKNI.= ACM.F _Z#yCBDA9.0NAC79
+
.\;>=@7 =@7IN?0.07IMQgIM9{;>=@7IM B@;Q/ ¹`bc*$Z;Q.0M.07:9.0NAC7oM.0?<9ACBD7 *
.0?<9ACBD7 2
ZI;>.0M.079M=@7IB@9
+
.<;,AKNI.= 9BZ$.<;Q B@;Q/ _@<hÌ>@>0D@=@ K9.<;
+
=0wAC7IL;>.S{.0?<9.0Nd9
+
.H> FD Z#=[;Q9+k
CED C g<OR8h OZRUTji LNRU[lkU[ PIKOR[\]HJI W
$=[}#A
+
¹`bc*
+
=@MzNI.<wh.0yCB@Z$.0N=@7=[ZIZ;QBh=@?
+ 9 +
=[9j;Q.0yCAK.0M$BD7yCB_?=@y@9{;>=@7IM B@;Q/1Mj}#[email protected]|BD7H#@A_@{*=H0F/H.=@Mg;Q.0M0J
NI.0MQACLD7I.0N9B 9BDyK.<;>=[9. Z#=[;9AK=@yÌB:?0?0yKgIMACBD7$J
+ AK?
+W+
.?[email protected]hb@D = BD7E²Z#=[;>=@/H.<9{;QAC?M{9Q=[9ACM{9AK?0M ¹`oÌACM
NIAKM{9AC7ILDgIAKM
+
.0N}_19
+
.\gIM.BD zB@;>NI.<;QAC7IL4AC7I B@;Q/H=[9ACBD7o=@/1BD7ILN=[9Q=_J;>=[9
+
.<;9
+
=@79
+
.\N=[9Q=4w@[email protected]
+
.0/1M.0ywh.0M0
* ByCB:?=@yP7IBD7E²Z#=[;F=@/1.<9{;QAC?H9{;>=@7IM B@;Q/1M .<;Q. AC7:9{;QB:NgI?0.0N$
*"+
.{@ pW8@ Am0@ = ACM4=d7IBD7E²Z#=[;>=@/1.<9{;QAK?
/1.=@MQg;Q.BD yKB:?=@yqAC79.07MA9GDJ + ACyK.\9
+
.M.0?0BD7NBD7I.@J#9
+ . F< !48D Am0@ = JjAKM=17BD7E²Z#=[;>=@/1.<9{;>AC?|MgI/H/H=[;
BD yCB_?=@yjM{Z#=[9A=@y$M{9{;QgI?<9g;>.@
.<9Oq }$.=WZA .0yJa'"Hq
%
A9MAC7:9.07IMA9DJ=@7IN
"Hq
% 9 +
.M.<9BD "ZA .0yCMAC7 A9M7I.0AKL
+
}$B@;
+
B_B_N$ /yCy7IBD7E
Z#=[;F=@/1.<9{;QAC?9{;>=@7IMQ B@;Q/HM4NI.<Z$.07IN BD7 9 +
.o?0BD/Z#=[;>ACMBD7 BD - "Hq
%
=@7IN 9 + . AK79.07IMQA9G BD ,9
+
.ZA .0yCM AC7
"Hq
%
\.,+7I.sr "Hq-Jqut
%
9BÐ}$.`AC ( "Hqvt
%nw
"Hq
%
=@7NxoWB@9
+
.<; ACM.@ BD7E²Z#=[;>=@/1.<9{;>AC?oyCB:?=@yP9{;>=@7IMQ B@;Q/HM N.<Z.07N
MBDyK.0yBD79
+
.|M.<9,BD qZA .0y$?0BD/Z#=[;>ACMBD7IM0J
+ AC?
+
AKM9
+
.|M.<9"BD B@;QNI.<;>.0NZ#=@A;QM0
y z
{/|
=}
#{ '
"Hq
t
r "Hq~Jq t% %
*,+
.o8D p9{;>=@7IM B@;Q/ ACM|N.,+7I.0N'=@M|9
+
.7:gI/4}$.<;|BD ZA .0yCM\AC7Ð9
+
.1yCB_?=@y;Q.0LDACBD7 +
BDM.1AK79.07IMQA9GWACM|yC.0MM
9 +
=@79
+
.|AC7:9.07IMA9BD q9
+
.|?0.07:9.<;,Z#A .0y² 8B@;Q/H=@yCyDJ9
+
.\;>=@79{;>=@7IM B@;Q/ "Hq
%
ACM
"Hq
%
A"
Q q t N
"Hq
%SR
"Hq
t
%(w
'"Hq
%Z %
8B@;,9
+
.\;>=@7 H9{;>=@7IMQ B@;Q/J-"?0B@;;Q.0yK=[9AKBD7 "/1AC7IAC/HA/.=[9ACBD7oBD z9
+
.xMgI/ BD =[}MBDyCg[email protected] ÌNIA.<;Q.07I?0.0M
%
BD z9
+
.9{;>=@7IM B@;Q/1.0NAC/ [email protected] B@;>NI.<;9B ZI;Q.0M.<;wh.x= ;Q.0MZBD7M."9
+
=[9"N.<Z.07NIMPyKAC7I.=[;QyBD7o9
+
.\7_gI/4}.<;
BD BDg9yCAK.<;QM0
*,+
.4F< 9{;F=@7IM B@;Q/ "Hq
%
ACM=/H=[ZIZAK7IL| ;QBD/ 9 +
.yCB_?[email protected]
+
}B@;
+
B:B_N1BD j=ZA .0y q 9B=|}AC9)M{9{;QAC7L
;Q.<Z;Q.0M.07:9AC7IL9
+
.4M.<9,BD 7I.0AKL
+
}$B@;QAC7L4Z#A .0yKM
+
BDM.AC79.07MA9GACM,AC79.<;>ACB@;"9B19
+
=[9BD q
.<9
"Hq
%f
q
`
+
.<;Q.ACM9
+
.xUdAK7hB M _A$Mg/ =@7N
`
ACM=M.<9"BD qNIACM{Zy=@?0.0/1.079MJ=@7NyC.<[email protected]?0BD7I?=[9.07=[9AKBD7$
*"+
.
?0.07IMQgIM9{;>=@7IMQ B@;Q/ ?=@79
+
.07}$.\M{Z$.0?0A+.0N=@M 0
"Hq
%
=
r "Hq-JqKSJK
%S
R A .0yCMBD j?0.07IMgIM9{;>=@7M B@;>/1.0NoAC/[email protected]"=[;Q.\?0BD/Z#=[;>.0N B@;"MAC/HACyK=[;QA9gIMAC7IL9
+
.=@/1/HAC7IL NIACM{9Q=@7I?0. "9 + .
7_gI/4}.<;BD Ì}A9M"9
+
=[9NIA.<;
%
}$.<9 .0.079
+
.|9 B }A9"M9{;QAC7ILDM0
*"+
.4=@yCLDB@;QA9
+
/~?0BD/Zg9.0M9
+
.4?0B@;;Q.0M{Z$BD7INI.07?0.
}_/1AC7IAC/HA/.0AC7ILx9
+
.=@/H/1AC7ILNIAKM{9Q=@7I?0.|=@ 9.<;=[ZIZyAC7IL 9 +
.x?0.07MgIM9{;>=@7M B@;>/
CED g<H fk[ PIKLNYJIKLN[*[*QHR!UOZ[LNT#"OZIPKQOM_i HJIKIKLNM$O'PQ$HR
BD7MACNI.<;9
+
.o9
B AC7INIB
MBD ^8ACLDg;>.Ð`D
*,+
.< /H=[9?
+
.F ?0.<ZI9H B@;9
+
.gZE²;>ACL
+
9?0B@;Q7.<;QM0
( .
=@799B
;Q.0?0Bbwh.<;9
+
ACMx?0B@;;QgIZI9.0N Z#=[;90
( . =@MMg/1.9
+
=[9yKB:?=@yCyCd9
+
.1MQACLD7=@yB@}$.<_M4=@7 = H7I.19{;>=@7IM B@;Q/H=[9ACBD7 ;QBD/
BD7I.AC/[email protected]
+
.xB@9
+
.<;+"M.0. 8ÌAKLDg;Q.
% 0
*"%$4&
% ('
`"%$f405&52
%
*) ,+
+
.<;Q. "30162
%
ACM9
+
.|NIACM{Z=[;QA9GBD Ì9
+
.\Z$BDAC79-"012
%
J-$
0 M1&
2u
=@7IN.$
N Q 0 T SS
0 Z
&
N
Q 2
TX SS
25 XjZ
J
o`=@7IN X
o`=[;Q.9 + .
ACN9
+
=@7IN 9 + . + .0ACL
+
9ÌBD 9
+
.)9.0/Z#yK=[9.
AC7INIB
;>.0M{Z$.0?<9Awh.0y
=@7IN.+ACM,= |=@gIMMAK=@7dAC/H=@LD.|7IBDAKM.@
8ACLDgI;Q.`0V\?0?0yCgMACBD7$
.0 K90 0JI;QAKL
+
90
*
gI?
+
=@7= H7I.x;Q.0y=[9ACBD7ACM9{;Qg.xAC7o9 + .
AK7INIB
.F ?0.<ZI9 B@;"9
+ . Qb |Z#=[;Q9M0
/ MAK/ZyC.<;/1B_NI.0y
BDgIyCN=@yCyCB
SgIM{9,=M
+
AC K9AC7AK79.07IMQA9GDJIA.@)9
+
.xM?=@yC.| =@?<9B@;
'
ACMM.<9"9B`0
"%$4&
%8
"%$>Y05&542
%
*) ,+
58 P9
+
.?0.079.<; BD )9 + .
AK7INIB
AC9
+
?0B:B@;QNAC7=[9.0M "052
%
ACM#hB:?0?0yCgNI.0N$Jz=@7 =@NINA9ACBD7=@yÌ?0BD7IM{9{;>=@AK79|?=@7
}$.M{9Q=[9.0N$J + AC?
+ ACyCyq}.?[email protected]Ð9
+
.0/b>0G<\_D\QD F{@21d}$.0yCB 5Þ9SgIM{9xM{9Q=[9.0M|9
+
=[9x9
+
.?0.079.<; BD
9 + . AC7INIB +
=@M9B1M>=[9ACM 9 +
.\Z;Q.<w_AKBDgIM"= H7I.|B@;,9{;>=@7IMyK=[9AKBD7=@y$?0BD7IM{9{;>=@AK790
I2
I1 50
100 150 200 250
50 100 150 200 250
8ACLDg;>.
0/ 7I.;Q.0yK=[9AKBD71}$.<9
.0.07 =@7INO[J:9
+
.,M/ =@yCyI?0A;Q?0yK.0M;Q.<ZI;>.0M.079BDg9yCAC.<;>M0JA.*0ÌZ#=@A;QMPBD L@;Q.<1yC.<wh.0yCM
BD ÌZA .0yCM/1ACMQ/H=[9?
+
.0Nj
"012
% '
"0 Y052d42
%
,)U"²= 7I.|?=@M.
% J
"012
%
"0 Y052v42
%
,)U"9{;F=@7IMyK=[9ACBD7d?=@M.
%7
*,+
.PB_?0?0yCgIN.0N4Z=[;9M=[;Q. BDgI7N4gIMQAC7IL|{ < <²D< > \=0wAC7IL BDg7IN9
+
.0M.xQb Z#=[;9MJ@?0B@;Q;Q.0yK=[9ACBD7
ACM4;Q.0M{9{;>AC?<9.0N'9B9
+ . ;Q.0/H=@AC7IAK7ILoZ#=[;9M BD 9
+ . 9 B AK7INIB M0
( . ?=@yKy)9
+
AKM|9.0?
+
7IAC_gI._DFDQ@F0DDI
587Ð =@?<9_@<D)QD>0D@ACMx= M{Z$.0?0AK=@yÌ?[email protected]
<*_G WQ@F{h@JÌ=@M
. yK.=0wh. "M5
% ` "AC7IyKAC.<;
%
B@;
"M1
%
o "BDg9yCAC.<;>M
%
8B@; .F I=@/ZyC.@JqNI.,+7IAC7L
` % "L %
" L %T
"IK L %
J=@7IN
` %
9
#;
' #; '
9
#
; ' Jz9
+ .
.0ACL
+
9.0N * * ?=@7d}.x.F ZI;Q.0MQM.0Nd=@M" BDyCyCB
M0
^8^ `
" 64
%f
9 ;>=?
"` % "L
%aT ` % % " L %
9 ;=?
" L
%aT
`
"%4Q & & %| $495 >
*"+
.x gI7?<9ACBD7 x" BD q9 BO]
T
NIAK/1.07IMACBD7#=@y#yCAKM{9M(
"0
%)C
=@7N
"2
%)C
JACMNI.,+#7I.0Nd=@M0
9 C
# "!H@#%-'A#%$&!3@'#
( '
C @
) 9 C
#% "!3@ %('#
*
C @ 9 C
#%$&!3@ #
( '*
C @
+ AC?
+
?=@7(=@yKMB }$..F ZI;Q.0MQM.0N(=@M9
+
.;>=@NIAKB'BD x9
+
.W?0BwD=[;QAK=@7I?0.
" &
%
}$.<9 .0.07 =@7IN* =@7N 9 + .
LD.0BD/1.<9{;>AC?=@yz/1.=@7BD z9
+
.0A;"wD=[;QAK=@7I?0.0M
=@7IN
$
58
.?<;Q.=[9.1=oyCAKM{9BD )9
B[EGNIAC/1.07MACBD7=@yÌZ$BDAC7:9M_q
"Hq
% C
" "0
2
%,+% C
J$9
+
.07 A9Mx?0.07:9.<;ACM\LDAwh.07
}_.-q
"/- 0- % + "
921
!43 ! "!
C
951
!43 !
$&!
C % +
JIA9M,NIACM{Z$.<;QMAKBD7 AC7o9.<;>/1M"BD Ì9
+
.x?0BwD=[;QAK=@7?0./H=[9{;QA oACMLDACwh.07}_ 0
9 C
"Hq
T -q %
"Hq
T -q
%,+
] T `
76
"012
%
7
"052
%
$ 8
AC9
+ G
951
!3 !
# "!@
#
%-'
*
C @ J $ G
921
!43 !
#%$&!@
#
( '*
C @ J
"012
%u 921
!43 !
#% "!@
#
%-'A#$&!H@
#
( '
C @
JÌ=@7IN
<d
" &
%
#
$7'
!*
!H!
*H*
J + .<;>.
ACM9
+
.|.0yK.0/1.079BD
AC7o9 + .
+
;QB =@7N9 + .
+
?0BDyCgI/17j
(
.xMQ.0.,9
+
=[99
+ . x" gI7?<9ACBD7HBD
"0
%
=@7IN
"2
%
?=@7o}$.,?0BD/ZgI9.0N ;QBD/]9
+
.NIACM{Z$.<;QMAKBD71BD
q
"Hq
%8
"0
2
%,+
*,+
ACMZI;QB_?0.0NIg;>.AKM"?0BD/Z$BDM.0NBD q9
+
;Q.0.|M9.<ZM0
`D ))M{9AK/H=[9ACBD7BD Ì9
+
.xN=[9Q=?0.07:9.<;
))M{9AK/H=[9ACBD7BD Ì9
+
.x?0BwD=[;QAK=@7?0./H=[9{;QA
2_ ))M{9AK/H=[9ACBD7ÐBD 9
+
.?0B@;Q;Q.0yK=[9ACBD7 ?0B_. H?0AC.07:9x}:W?0BD/Zg9AK7ILH9
+
.;>=[9ACBoBD )9
+
.49 Bw@=[;QA=[}yC. ?0BwD=[;QAK=@7I?0.
=@7INo9
+
.|LD.0BD/H.<9{;QAC?=@yz/H.=@7BD Ì9
+
.0A;wD=[;QAK=@7I?0.0M0
5879
+
.|Z#=[;Q9AK=@yzB_?0?0yCgIMAKBD7d?=@M.@J9
+
.<;Q. =[;Q. MBD/1.BDgI9yCAC.<;QM0
*,+
..0M{9AK/H=[9ACBD7dBD 9
+
.?0.07:9.<;|=@7INd9
+
.?0BwD=TE
;QA=@7I?0.|/H=[9{;QA M
+
BDgIyCN}$.x;>B@}gIM{9,MQB19
+
=[99
+
..0M{9AK/H=[9ACBD7dBD Ì9
+
.4?0B@;;Q.0yK=[9AKBD7d?0B_. H?0AC.07:9ACM,;Q.0yCAK=[}#yC.@)5879 + .
7I.F 9,M.0?<9AKBD7$J
.|NIACM?0gMM9
+
.x;QB@}gM{9.0M{9AC/H=[9ACBD7dBD ÌyKB:?=[9ACBD7=@7INNIACM{Z$.<;QMQACBD7$
Q %Q| Q"1Q )!
587 9 +
ACMM.0?<9AKBD7$J .AC7:9{;QB_NIgI?0.9
+
.7IB@9AKBD7 BD = yKB:?=[9ACBD7 .0M9AC/H=[9B@;J?0BD7IMACN.<;ZI;QB@Z$.<;9AC.0M1=@M9{;F=@7IMyK=[9ACBD7
AC7:wD=[;QAK=@7I?0.@J#Z.<;>/4g9Q=[9AKBD7dAC7:w@=[;>AK=@7I?0.=@7INÐ= H7I.4AC7:w@=[;QA=@7I?0.@J=@7INNIACM?0gMM9
+ . 7IB@9ACBD7BD }I;Q.=NIB 7Z$BDAC7:90
/9o9
+
..07IN$J
.dAC7:9{;QB_NIgI?0.MBD/1.;QB@}gIM{9 .0M{9AC/H=[9B@;QMJ=@7IN LDAwh./HB@;Q.N.<9Q=@ACyCMBD |UX-)"J
+ AC?
+
ACMo= H7I.
AC7:wD=[;QAK=@7:90
*"+
ACMo.0M{9AC/ =[9B@;ACM=[ZIZ#yCAC.0N(9B NI.0NIg?0.'= ?0B@;;Q.0yK=[9AKBD7 g7I?<9ACBD7$J + AC?
+
ACMo;QB@}gM{9o9B Z#=[;9AK=@y
B_?0?0yCgIMAKBD7IM "M.0?<9ACBD7Ði
%
D$F [ PKQ$WXO'PKQHR!H\+MH i O'PKQHR ORT TUQ$[ "#LAI [*QHER
(
.NIAKM?0gIMM9
+
.o.0M{9AC/H=[9ACBD7 BD 9
+ . /?0.07:9.<; 1 BD ,=WM.<9BD Z$BDAC7:9M0J +
.<;Q.=@yKy)wD=[;QAK=[}yK.0M =[;Q.9{;Q.=[9.0N AK7 9 + .
MQ=@/H.
=D
(
.=[;>.=@yCMBAC7:9.<;Q.0M{9.0NAC79
+
.|N=[9Q=1NIACM{Z$.<;QMAKBD7=[;QBDg7INo9
+
.x?0.07:9.<;J
+ AC?
+
ACM.F ZI;Q.0MQM.0N}_ 9 + .
?0Bbw@=[;QA=@7I?0./H=[9{;>A oBD q9
+
ACMMQ.<9,BD ÌZBDAK79M0
*
gZIZ$BDM.9
+
=[9 . +
=wh.=M.<9,BD N=[9Q= 0
Q 3
SS
3 C Z
Q "3
3
)l
SS
3
% +
SS
"3 C 3 C l
SS
3 C % + Z
BD ]EGNIAC/1.07MACBD7=@y$Z$BDAC7:9M0
( . +
=wh.9B.0M{9AC/H=[9. A9M?0.079.<;bJ$=@7N=@yCMBA9MNIACM{Z$.<;QMAKBD7=[;QBDgI7INd9
+
.4?0.07:9.<;J
.F ZI;Q.0MMQ.0N}:=?0Bbw@=[;QA=@7I?0./H=[9{;>A
UB@;Q.x.F I=@?<9yDJ .x=[;Q.\yCB:B AC7IL B@;=@7.0M9AC/H=[9B@;HJMQgI?
+ 9 +
=[9
¢ 3 T "
%
BDyCyKB M=
+
_Z$B@9
+
.0MA/.0.0N
NIAKM{9{;QA}gI9ACBD7=@7N
+
=@M"MQBD/1.|gIM.0 gIyZI;>B@Z.<;Q9AC.0M058 qMQBD/1.|BDg9yCAK.<;QM"=@/1BD7ILH9 + . ¢
NB7IB@9"Z$.<;9g;}9
+
.x;Q.0MgIyC90J
9 +
..0M{9AC/H=[9B@;ACMMQ=@ACNo9BH}. { !<
DC HWXL TL&RUQPKQHER[ ORT OR L9OW "UM$L
(
.AC79{;>B:NIg?0.MQBD/1.ZI;QB@Z$.<;9AK.0MBD #;>B@}gIM{9.0M{9AK/H=[9B@;QM0J_AC7I?0yCgINAC7IL"9
+
.7IB@9ACBD7IMBD AC7:wD=[;QAK=@7I?0."=@7IN1}I;Q.=NIB 7
Z$BDAC7:90