IdentifiabilitÃ© et MÃ©thode des Moments dans les mÃ©langes ...

Thèse de doctorat de l’Université Paris 6 

Spécialité : Mathématiques 

Option : Statistiques 

Présentée par 

Emmanuel MONFRINI 

Sujet de la thèse 

Identifiabilité et Méthode des Moments 

dans les mélanges généralisés 

de distributions du système de Pearson 

TOME I 

Thèse soutenue le 4 janvier 2002, devant le jury composé de : 

M. 

M. 

M. 

M. 

M. 

M. 

M. 

Marc YOR 

Alain HILLION 

Christian ROBERT 

Michel BRONIATOWSKI 

Paul DEHEUVELS 

Wojciech PIECZYNSKI 

Daniel PIERRE-LOTI-VIAUD 

Université Paris 6 

ENST Bretagne 

Université Paris Dauphine 

Université de Reims 


INT 


Président 

Rapporteur 

Rapporteur 

Examinateur 

Examinateur 

Directeur 

Directeur 

Mme 

Annick VALIBOUZE 


Invitée 

Ces travaux on été, pour une large part, effectués au département CITI de l’Institut National des Télécommunications.

Thèse de doctorat de l’Université Paris 6 

Spécialité : Mathématiques 

Option : Statistiques 

Présentée par 

Emmanuel MONFRINI 

Sujet de la thèse 

Identifiabilité et Méthode des Moments 

dans les mélanges généralisés 

de distributions du système de Pearson 

TOME I 

Thèse soutenue le 4 janvier 2002, devant le jury composé de : 

M. 

M. 

M. 

M. 

M. 

M. 

M. 

Marc YOR 

Alain HILLION 

Christian ROBERT 

Michel BRONIATOWSKI 

Paul DEHEUVELS 

Wojciech PIECZYNSKI 

Daniel PIERRE-LOTI-VIAUD 


ENST Bretagne 

Université Paris Dauphine 

Université de Reims 


INT 


Président 

Rapporteur 

Rapporteur 

Examinateur 

Examinateur 

Directeur 

Directeur 

Mme 

Annick VALIBOUZE 


Invitée 

Ces travaux on été, pour une large part, effectués au département CITI de l’Institut National des Télécommunications.

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qrxv prgìolvrqv oh euxlwdjh sdu glìuhqwhv glvwulexwlrqv gh suredelolwì1 Oh sureoëph gh od vhjphqwdwlrq vwdwlv0 

wltxh g*lpdjhv vh gìfrpsrvh hq ghx{ skdvhv frpsoìphqwdluhv1 Od suhplëuh frqvlvwh ã hvwlphu ohv sdudpëwuhv ghv 

glvwulexwlrqv prgìolvdqw ohv euxlwdjhv ghv fodvvhv dlqvl txh ohv sdudpëwuhv fdudfwìulvdqw oh prgëoh/ od vhfrqgh frq0 

vlvwdqw hq xqh fodvvlfdwlrq ghv sl{hov1 Dlqvl/ od skdvh g*hvwlpdwlrq frqvlvwh ã hvwlphu ohv sdudpëwuhv g*xq pìodqjh 

gh glvwulexwlrqv euxlwdqwhv/ vxu gh odujhv ìfkdqwloorqv/ gh frxudpphqw 589 589 lqglylgxv1 Revhuyrqv/ fhshqgdqw/ 

tx*hq vhjphqwdwlrq g*lpdjhv rq fkhufkh ã sulyloìjlhu o*hvwlpdwlrq ghv glvwulexwlrqv frpsrvdqwhv gx pìodqjh/ ã 

o*hvwlpdwlrq gx pìodqjh hq wdqw txh glvwulexwlrq/ sxlvtxh ohv dojrulwkphv gh fodvvlfdwlrq/ txl shuphwwhqw od uhfrq0 

vwuxfwlrq ghv lpdjhv/ xwlolvhqw ohv glvwulexwlrqv frpsrvdqwhv srxu lghqwlhu ohv glìuhqwhv fodvvhv1 O*remhfwli g*xqh 

sduwlh gh qrwuh wudydlo d ìwì g*dqdo|vhu/ ã wudyhuv o*hvwlpdwlrq gh vhv sdudpëwuhv/ od frpsoh{lwì gh od glvwulexwlrq 

gh pìodqjh/ wurs vrxyhqw frqvlgìuìh frpph xqh vrpph srqgìuìh gh glvwulexwlrqv frxudqwhv1 Oh idlw g*lqvfuluh qrv 

wudydx{ gdqv oh fdguh jìqìudo gh od vhjphqwdwlrq vwdwlvwltxh g*lpdjhv/ qrxv shuphw gh frpsduhu o*dssurfkh sured0 

elolvwh gx pìodqjh hq wdqw txh glvwulexwlrq/ ã odtxhooh vhud frqvdfuìh od pdmrulwì gh fh grfxphqw/ dyhf fhooh xwlolvìh 

gdqv xq grpdlqh gh srlqwh gh od vwdwlvwltxh dssoltxìh1 Od vhjphqwdwlrq vwdwlvwltxh ghv lpdjhv idlw/ hq hhw/ sdu0 

wlh ghv judqgv wkëphv gh uhfkhufkh gx JHW +Whohfrp Sdulv/ Whohfrp Euhwdjqh hw LQW Hyu|,1 Xq fhuwdlq qrpeuh 

gh prgëohv hw gh wudlwhphqwv ruljlqdx{ rqw ìwì sursrvìv ghsxlv xqh gl}dlqh g*dqqìhv/ ghvtxhov lo uhvvruw txh ohv 

pìwkrghv vwdwlvwltxhv sduirlv wuëv orxughv hw vrxyhqw wuëv frpsoh{hv/ vrqw dxvvl wuëv sxlvvdqwhv +yrlu hq sduwlfxolhu 

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Suìvhqwdwlrq gx fdguh suredelolvwh1 

Srxu ghv udlvrqv dxwdqw klvwrultxhv txh sudwltxhv/ qrxv fkrlvlvvrqv gh flufrqvfuluh o*hqvhpeoh ghv glvwulexwlrqv 

grqw rq ìwxglhud od fodvvh ghv pìodqjhv/ ã od idplooh ghv glvwulexwlrqv gx v|vwëph gh Shduvrq1 Gdqv vd vìulh 

g*duwlfohv lqwlwxoìh rqwulexwlrq wr wkh Pdwkhpdwlfdo Wkhru| ri Hyroxwlrq +^Shduvrq4;

54 

Ghx{lëph sduwlh = Hvwlpdwlrq ghv sdudpëwuhv gh glvwulexwlrqv gh pìodqjhv sdu od pìwkrgh ghv 

prphqwv1 

Qrxv dyrqv vrxkdlwì gìyhorsshu/ gdqv fhwwh sduwlh gx wudydlo/ xqh pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv g*xq 

pìodqjh ql +yrlu ^Hyhulww;4`/ Ôlqgvd|

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Oruvtxh qrxv qrxv sodêrqv gdqv oh fdv wkìrultxh rü ohv prphqwv vrqw hhfwlyhphqw ohv prphqwv gx pìodqjh 

gh ghx{ glvwulexwlrqv qrupdohv/ o*h{lvwhqfh g*xqh vroxwlrq/ txh qrxv vdyrqv xqltxh sdu oh uìvxowdw g*lghqwldelolwì 

ghv pìodqjhv qlv gh glvwulexwlrqv qrupdohv/ qrxv dvvxuh/ txlwwh ã idluh hqwuhu hq mhx vx!vdpphqw g*ìtxdwlrqv/ 

gh o*xqlflwì gh od vroxwlrq dx sureoëph ghv prphqwv1 Oh fdv rü ohv sdudpëwuhv dssduwlhqqhqw ã Y 5 qh srvh/ gh 

fh idlw/ sdv gh uìho sureoëph gdqv o*ìwxgh wkìrultxh gh o*lqyhuvlelolwì/ sxlvtx*lo vx!w gh whvwhu o*dgìtxdwlrq ghv 

fdqglgdwv vroxwlrq vxu xq qrpeuh vx!vdqw g*ìtxdwlrqv gh prphqwv1 Qrxv srxyrqv fhshqgdqw vxssrvhu/ hw fhod vh 

frquph oruvtxh o*rq uhpsodfh ohv prphqwv wkìrultxhv sdu ohv prphqwv hpslultxhv/ txh o*hvwlpdwlrq gh sdudpëwuhv 

dssduwhqdqw ã xq yrlvlqdjh gh Y 5 vhud vrxufh g*xq grxeoh sureoëph1 G*xqh sduw o*devhqfh gh udflqh shuphwwdqw 

o*lqyhuvlrq vwdwlvwltxh gx v|vwëph/ g*dxwuh sduw/ o*rewhqwlrq gh ghx{ udflqhv grqqdqw ghx{ hqvhpeohv gh sdudpëwuhv 

vroxwlrqv soxv rx prlqv surfkhv/ txh o*lqfhuwlwxgh vxu ohv prphqwv hpslultxhv qh shuphwwud sdv gh vxssulphu gh 

idêrq deoh1 

Xqh lgìh qdwxuhooh/ phqwlrqqìh sdu Shduvrq/ hvw gh vh vhuylu gh od vl{lëph ìtxdwlrq gh prphqw srxu frqvwuxluh 

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prphqwv hpslultxhv1 Qrxv sursrvrqv g*doohu soxv orlq/ hq frqvlgìudqw ghx{ v|vwëphv ghv prphqwv/ oh suhplhu 

ìwdqw fhoxl gh Shduvrq hw oh vhfrqg ìwdqw rewhqx hq uhpsodfêdqw gdqv oh v|vwëph gh Shduvrq/ od flqtxlëph ìtxdwlrq 

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hq fdofxodqw vrq mdfrelhq/ qrwì M 5 1 Qrxv ìwdeolvvrqv/ gdqv xq suhplhu whpsv txh fh mdfrelhq v*dqqxoh vxu od vrxv0 

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pìodqjhdqfh gh 3=81 Qrxv prqwurqv dxvvl/ hw f*hvw fh txl qrxv shuphwwud/ sdu od vxlwh/ gh gìqlu xqh qrxyhooh 

pìwkrgh g*hvwlpdwlrq/ txh fh mdfrelhq qh v*dqqxoh sdv hq píph whpsv txh M 4 1 huìvxowdwqrxvdvvxuhghod 

frpsoìphqwdulwì hqwuh ohv ghx{ v|vwëphv ghv prphqwv1 Xqh xwlolvdwlrq frpelqìh/ txh qrxv glvfxwrqv gdqv oh 

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ghjuì/ shuphwwud xqh hvwlpdwlrq deoh hw vwdeoh ghv sdudpëwuhv sdu od pìwkrgh ghv prphqwv1 Qrwrqv txh o*lqyhuvlrq 

dojìeultxh gx vhfrqg v|vwëph ghv prphqwv shuphw gh uìlqyhvwlu srxu xqh odujh sduwlh ohv fdofxov hhfwxìv gdqv oh 

fdguh gh o*lqyhuvlrq gx suhplhu v|vwëph/ hw frqgxlw ã uhfkhufkhu ohv udflqhv srvlwlyhv g*xq sro|qrph gh ghjuì 451 

Hq hhw/ od pìwkrgh txh qrxv dyrqv dgrswìh srxu o*lqyhuvlrq gx suhplhu v|vwëph ghv prphqwv/ shuphw g*rewhqlu 

dlvìphqw o*lqyhuvlrq gx vhfrqg/ sxlvtx*lo qh uhvwh ã hhfwxhu txh od ghuqlëuh ìwdsh gh vxevwlwxwlrq ghv uìvxowdwv gdqv 

fhwwh od vl{lëph ìtxdwlrq gh prphqw1 Juåfh ã o*ìwxgh ghv ghx{ mdfrelhqv/ qrxv vdyrqv/ dxvvl/ txh qrxv srxyrqv 

olplwhu od pìwkrgh ã o*lqyhuvlrq gh ghx{ v|vwëphv1 

Od plvh dx srlqw g*xq surwrfroh g*hvwlpdwlrq ghv sdudpëwuhv edvì vxu fhv ghx{ v|vwëphv/ sdvvh doruv/ dxvvl/ 

sdu xqh ìydoxdwlrq gh o*ìsdlvvhxu gh od }rqh g*lqfhuwlwxgh dxwrxu ghv ydulìwìv vxu ohvtxhoohv ohv mdfrelhqv vrqw 

qxov1 Qrxv dyrqv fkrlvl g*hvwlphu khxulvwltxhphqw fhwwh ìsdlvvhxu hq surfìgdqw ã ghv whvwv g*lqyhuvlrq sdu ohv ghx{ 

v|vwëphv/ gh od idêrq od soxv frpsoëwh srvvleoh/ dx yrlvlqdjh ghv glìuhqwhv ydulìwìv vxvfhswleohv gh suryrtxhu ghv 

gìjudgdwlrqv/ yrlu ghv lpsrvvlelolwìv g*hvwlpdwlrq1 O*hqvhpeoh ghv uìvxowdwv gh fhv hvwlpdwlrqv/ qìfhvvdluhv srxu 

frpsuhqguh oh pìfdqlvph gh od pìwkrgh g*hvwlpdwlrq txh qrxv sursrvrqv/ vrqw suìvhqwìv hq dqqh{h/ hw frqvwlwxhqw/ 

sdu dloohxuv/ xqh edvh gh grqqìhv vxvfhswleohv g*lqvsluhu dx ohfwhxu xqh dssurfkh soxv wkìrultxh gh od gìwhuplqdwlrq 

ghv ìsdlvvhxuv ã ulvtxh1 Qrxv frqfoxrqv dlqvl/ fhwwh sduwlh gx wudydlo/ hq sursrvdqw/ gdqv oh fkdslwuh 45/ xqh 

pìwkrgh sudwltxh ruljlqdoh g*hvwlpdwlrq ghv sdudpëwuhv gx pìodqjh gh ghx{ glvwulexwlrqv qrupdohv/ edvìh vxu od 

pìwkrgh ghv prphqwv1 

Ohv ghx{ fkdslwuhv vxlydqwv shuphwwhqw g*ìwhqguh od pìwkrgh ghv prphqwv ã g*dxwuhv glvwulexwlrqv gx v|vwëph 

gh Shduvrq1 Oh suhplhu/ txl hvw frqvdfuì dx pìodqjh gh ghx{ glvwulexwlrqv h{srqhqwlhoohv +yrlu Ûlghu94` hw 

Ûlghu95`, qrxv shuphw g*derxwlu gluhfwhphqw ã ghv h{suhvvlrqv olwwìudohv ghv sdudpëwuhv hq irqfwlrq ghv prphqwv 

qrq fhqwuìv1 Oh mdfrelhq/ M 6 / qh v*dqqxodqw txh gdqv oh fdv gx pìodqjh gh ghx{ glvwulexwlrqv lghqwltxhv/ lo q*| 

d sdv gh glvfxvvlrq gdqv fh fdv1 Oh vhfrqg/ dssurirqglvvdqw oh fdv gx pìodqjh g*xqh glvwulexwlrq qrupdoh hw 

g*xqh glvwulexwlrq h{srqhqwlhooh/ sQ + 4 > 4 , . +4 s, H +p 5 ,/ looxvwuh ohv gl!fxowìv vxssoìphqwdluhv txh o*rq shxw 

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uhqfrqwuhu oruvtxh oh v|vwëph qh shxw íwuh vlpsolì hq xwlolvdqw od v|pìwulh ghv ìtxdwlrqv1 Lo qrxv d ìwì lpsrvvleoh 

gh grqqhu xqh idfwrulvdwlrq jìqìudoh gx mdfrelhq M 7 = 

dyhf = 

 

M 7 @ e 3 H > 6 2 n2e> 2 6 2 2 3 ef > 6 2 n2S6 e 2 f 

> S 3 2 > D 6 2 nSf> e 6 2 2 3 Sf > 6 2 n 222 > 2 6 e 2 3 2f > 6 D 2 n S62 

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6 

Qrxv sduyhqrqv/ fhshqgdqw dx sul{ gh pdqlsxodwlrqv dojìeultxhv frpsoh{hv hw gh fdofxov lqhqylvdjhdeohv vdqv 

oh vrxwlhq g*xq orjlflho gh fdofxo irupho/ ã grqqhu xqh pìwkrgh uljrxuhxvh g*lqyhuvlrq/ edvìh sulqflsdohphqw vxu od 

uhfkhufkh ghv udflqhv dssduwhqdqw ã `3> 4^ g*xq sro|q÷ph gh ghjuì 45 hq s1 O*devhqfh gh idfwrulvdwlrq gx mdfrelhq/ 

qh qrxv shuphw sdv gh suìvhqwhu xqh ìwxgh dxvvl frpsoëwh txh srxu oh pìodqjh gh ghx{ glvwulexwlrqv qrupdohv/ hw 

qrwdphqw gh fhuqhu ohv ydulìwìv ã ulvtxh1 Qrxv derxwlvvrqv dlqvl/ ã xqh pìwkrgh ghv prphqwv srxu xq pìodqjh 

qrupdoh0h{srqhqwlhooh/ ã pdqlsxohu dyhf od píph suìfdxwlrq txh od pìwkrgh lqlwldoh gh Shduvrq1 

Juåfh ã o*hqvhpeoh ghv pìwkrghv yxhv suìfìghpphqw/ qrxv frqfoxrqv fhwwh sduwlh gx wudydlo sdu xq fkdslwuh 

suìvhqwdqw xqh pìwkrgh ruljlqdoh g*hvwlpdwlrq ghv sdudpëwuhv +dlqvl txh ghv glvwulexwlrqv frpsrvdqwhv, g*xq 

pìodqjh jìqìudolvì gh ghx{ glvwulexwlrqv lvvxhv gh od idplooh uìxqlvvdqw ohv glvwulexwlrqv qrupdohv hw h{srqhqwlhoohv 

+vdqv lpsrvhu d sulrul oh w|sh ghv glvwulexwlrqv,1 hwwh ìwxgh idlw dssdudñwuh ohv qrxyhoohv gl!fxowìv olìhv dx fkrl{ 

gx w|sh ghv glvwulexwlrqv frpsrvdqw oh pìodqjh1 rpswh whqxgxprghghgìflvlrqfkrlvlsrxufhwwhìwdshgh 

o*hvwlpdwlrq/ edvìh vxu ohv ìtxdwlrqv gh prphqwv g*ruguh 8 rx 9/ qrxv phwwrqv hq ìylghqfh o*lpsruwdqfh gh od wdlooh 

ghv ìfkdqwloorqv frqvlgìuìv srxu oh fkrl{ gx w|sh gh pìodqjh ìwxglì1 

 

> 

Wurlvlëphsduwlh=Vhjphqwdwlrq vwdwlvwltxh g*lpdjhv1 

hwwh sduwlh gx wudydlo vh glvwlqjxh irqgdphqwdohphqw ghv ghx{ suhplëuhv/ pdlv q*hq hvw sdv prlqv ohxu frpsoì0 

phqw qdwxuho1 Qrxv dyrqv/ mxvtx*lfl/ wudlwì oh vxmhw vxu xq sodq wkìrultxh/ fh txl d prqwuì od qìfhvvlwì gh flufrqvfuluh 

oh vxmhw dx pd{lpxp/ srxu srxyrlu o*ìwxglhu gdqv ghv frqglwlrqv udlvrqqdeohv1 O*hvsulw gdqv ohtxho lo idxw derughu 

fh txl yd vxlyuh hvw udglfdohphqw glìuhqw1 Qrwuh dpelwlrq/ lfl/ hvw gh frqvwuxluh ghv surfìgxuhv gh vhjphqwd0 

wlrq g*lpdjhv/ ohv soxv suìflvhv hw ohv soxv udslghv srvvleoh/ txh qrxv ydolghurqv sdu oh eldlv gh o*h{sìulphqwdwlrq 

lqirupdwltxh1 hv surfìgxuhv qrxv shuphwwhqw/ dx sul{ gh frqfhvvlrqv vxu od frqqdlvvdqfh wkìrultxh gh ohxu frp0 

sruwhphqw/ gh gìsdvvhu odujhphqw oh fdguh ghv pìodqjhv txh qrxv qrxv vrpphv {ìv gdqv od Ghx{lhph Sduwlh1 

Qrxv hvwlprqv/ frxudpphqw/ ohv sdudpëwuhv gh pìodqjhv gh soxv gh ghx{ frpsrvdqwhv/ ohvtxhoohv srxuurqw íwuh 

fkhufkìhv gdqv xqh sdohwwh gh idploohv gh glvwulexwlrqv euxlwdqwhv txdvl loolplwìh1 D fh sursrv/ oh Wkìruëph g*lghq0 

wldelolwì/ txh qrxv dyrqv sursrvì gdqv od Suhplhuh Sduwlh/ shuphw gh mxvwlhu/ vrxv uìvhuyh ghv uhvwulfwlrqv txh 

qrxv | dyrqv vljqdoìhv/ gx fkrl{ gh glvwulexwlrqv euxlwdqwhv dssduwhqdqw ã od idplooh gh Shduvrq1 

Elhq txh od edvh gx wudydlo/ hq sduwlfxolhu/ od gìwhuplqdwlrq gx prgëoh hw oh gìyhorsshphqw ghv dojrulwkphv/ 

ghpdqghxqirqghphqwvwdwlvwltxhwkìrultxhwuëvfrpsoh{h/odsduwlhghplvhhqrhxyuhlqirupdwltxhruhxqhodujh 

sodfh ã o*h{suhvvlrq gh o*lqjìqlrvlwì gx surjudpphxu1 O*hqvhpeoh ghv fkrl{ gh surjudppdwlrq/ hw ohv glyhuvhv dvwxfhv 

lqirupdwltxhv/ vrqw/ hq hhw/ gìwhuplqdqwv srxu o*rswlplvdwlrq gh od udslglwì g*xq dojrulwkph gìql wkìrultxhphqw/ 

hw ohv soxv dmrxwìv dx frxuv gx gìyhorsshphqw/ shuphwwhqw g*dpìolruhu hpslultxhphqw od txdolwì ghv uìvxowdwv1 

Qrxv fkhufkhurqv/ grqf/ gdqv fhwwh wurlvlëph sduwlh gx wudydlo/ ã idluh ydorlu o*h!fdflwì gx prgëoh hw ghv dojrulwkphv 

txh qrxv sursrvrqv/ hw ã prqwuhu ã wudyhuv ohv h{hpsohv txh qrxv | wudlwrqv/ od glvwdqfh h{lvwdqw hqwuh ohv fdv 

ìwxgldeohv wkìrultxhphqw hw fhx{ wudlwìv hq Vwdwlvwltxh Dssoltxìh1 

Qrxv frpphqêrqv sdu suìvhqwhu ohv prgëohv ghv fkdpsv gh Pdunry fdfkìv hw ghv fkdñqhv gh Pdunry fdfkìhv 

xwlolvìv srxu od vhjphqwdwlrq g*lpdjh/ dlqvl txh ohv pìwkrghv gh fodvvlfdwlrq ed|ìvlhqqh Prgh ghv Pdu0 

jlqdohv d srvwhulrul +PSP, hw Pd{lpxp D Srvwhulrul +PDS, fruuhvsrqgdqwhv +yrlu Êhvdj:7`/ ^Jhpdq;7` 

hw ^Pduurtxlq;:`,1 Dsuëv txhotxhv udsshov gh wkìrulh ghv judskhv/ qrxv grqqrqv ohv ìtxlydohqwv ghv dojrulwkphv 

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suìfìghqwv srxu ohv dueuhv gh Pdunry fdfkìv1 Od vwuxfwxuh ghv dueuhv gh Pdunry ìwdqw surfkh gh fhooh ghv fkdñqhv 

gh Pdunry/ ohv dojrulwkphv gx PDS hw gx PSP | vrqw hqwlëuhphqw gìwhuplqlvwhv hw grqf wuëv udslghv/ fh txl 

mxvwlh tx*rq v*| lqwìuhvvh +yrlu Êdxp:3` hw ^kduglq33`,1 Oh fkdslwuh vxlydqw/ gdqv ohtxho qrxv udsshorqv ohv 

glìuhqwv dojrulwkphv g*hvwlpdwlrq ghv sdudpëwuhv gh w|sh HP hw LH/ dvvrflìv ã fhv glìuhqwv prgëohv/ shuphw 

g*dfkhyhu gh qrxv idploldulvhu dyhf ohv prgëohv pdunrylhqv1 

O*ìwdorqqdjh gx prgëoh gh gldueuh txh qrxv suìvhqwrqv/ hqvxlwh/ vh gìurxoh doruv hq wurlv ìwdshv1 Gdqv xq 

suhplhu whpsv/ lo v*djlw g*dvvrflhu dx prgëoh/ o*dojrulwkph g*hvwlpdwlrq ghv sdudpëwuhv shuphwwdqw g*rswlplvhu ohv 

uìvxowdwv gh od fodvvlfdwlrq sdu oh PSP vxu oh gldueuh1 Od frpsdudlvrq/ gdqv xq vhfrqg whpsv/ vxu ghv h{hpsohv/ 

ghv uìvxowdwv gh vhjphqwdwlrq qrq vxshuylvìh/ sdu oh gldueuh/ dyhf fhx{ lvvxv g*xq prgëoh gh fkdpsv gh Pdunry 

vlpsoh/ qrxv shuphw gh phwwuh hq ìylghqfh od frpsìwlwlylwì gx prgëoh txh qrxv sursrvrqv/ sdu udssruw ã xq prgëoh 

d|dqw idlw vhv suhxyhv1 Hqq/ xqh ìwxgh ghv olplwhv gh qrwuh prgëoh/ gdqv oh fdv gh glvwulexwlrqv euxlwdqwhv wuëv 

surfkhv/ qrxv dvvxuh gh od vwdelolwì gx prgëoh gx gldueuh gh Pdunry fdfkì/ hq pdwlëuh gh vhjphqwdwlrq g*lpdjhv1 

Qrxv lqwurgxlvrqv oh fkdslwuh vxlydqw hq sursrvdqw xqh qrxyhooh pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv/ vxu oh 

gldueuh/ gdqv oh fdguh ghv pìodqjhv jìqìudolvìv1 Elhq txh od pìwkrgh vrlw ydodeoh srxu xqh idplooh soxv judqgh/ qrxv 

qh frqvlgìurqv txh od idplooh ghv glvwulexwlrqv qrupdohv hw h{srqhqwlhoohv/ gìmã ìwxglìh gdqv od Ghx{lhph sduwlh1 

Od frpsdudlvrq/ vxu ghv lpdjhv/ ghv shuirupdqfhv gh od pìwkrgh ghv prphqwv srxu ohv pìodqjhv jìqìudolvìv/ dyhf 

ohv dojrulwkphv g*hvwlpdwlrq txh qrxv dyrqv sursrvìv/ prqwuh/ frpswh whqx gh od wdlooh +45; 45;, ghv lpdjhv gh 

v|qwkëvh frqvlgìuìhv/ od soxv judqgh h!fdflwì ghv pìwkrghv VHP jìqìudolvìhv srxu fhwwh wdlooh g*ìfkdqwloorq1 

Qrxv gìwdloorqv/ hqq/ xqh dssolfdwlrq ruljlqdoh gh od pìwkrgh ghv pìodqjhv jìqìudolvìv/ ã od vhjphqwdwlrq 

g*lpdjhv uìhoohv srxu od plvh dx srlqw g*xq orjlflho gh ohfwxuh g*xq fduwrq g*rujxh gh eduedulh1 hwwh dssolfdwlrq 

looxvwuh/ rxwuh o*lqwìuíw g*dyrlu ìwhqgx od pìwkrgh jdxvvlhqqh fodvvltxh dx{ pìodqjhv jìqìudolvìv/ od udslglwì ghv 

dojrulwkphv vxu ohv dueuhv gh Pdunry fdfkìv1 Hq hhw/ o*dojrulwkph plv hq rhxyuh dxwrulvh xqh frqyhuvlrq/ hq 

whpsv uìho gh o*lpdjh oxh sdu od fdpìud/ hq xq fklhu xwlolvdeoh sdu xq ruglqdwhxu/ txh fh vrlw srxu uhsurgxluh oh 

fduwrq/ rx jìqìuhu xq fklhu pxvlfdo gh w|sh plgl1 

Lqwurgxfwlrq1

Suhplëuh sduwlh1 

PÌODQJHV GH ORLV GH 

SUREDELOLWÌ/ LGHQWLILDELOLWÌ1

5: 

0kdslwuh40 

Lqwurgxfwlrq dx{ pìodqjhv1 

414 Gìqlwlrq1 

Vrlw ghx{ hqwlhuv qrq qxov p hw q1 rqvlgìurqv x @ iI +{> , >{5 U q > 5 p j xqh idplooh sdudpìwultxh gh 

irqfwlrqv gh uìsduwlwlrq vxu U q / sdudpìwuìh sdu @ + 4 >===> p , 5 p / p U p / whooh txh I +{> , hvw phvxudeoh 

vxu U q p 1 rqvlgìurqv dxvvl J/ xqh fodvvh gh irqfwlrqv gh uìsduwlwlrq p0glphqvlrqqhoohv1 Xq ìoìphqwv gh J hvw 

qrwì J hw od phvxuh dvvrflìh ã J hvw qrwìh J 1 Rq gìqlw/ doruv/ od fodvvh K/ ghv irqfwlrqv gh uìsduwlwlrq K vxu 

U q / sdu od fruuhvsrqgdqfh = 

P = J$K 

] 

J :$ K/ whoohtxhK +{, @ 

F I +{> , gJ +, / { 5 Uq 1 

O*ìoìphqw gh K dvvrflì ã J hvw dsshoì J0pìodqjh gh x rx soxv vlpsohphqw pìodqjh1 

^414` 

415 Pìodqjhv qlv1 

Vl J hvw xqh suredelolwì glvfuëwh txl qh fkdujh/ dyhf ghv srlgv srvlwliv/ tx*xq qrpeuh ql Q gh srlqwv/ qrwìv 

4 >===> Q /gh p / ohv ìoìphqwv gh K vrqw ghv pìodqjhv glwv qlv1 Gdqv fh fdv o*ìfulwxuh ghv pìodqjhv vh vlpsolh 

frqvlgìudeohphqw hw suhqg od iruph = 

Q[ 

K+{, @ f l I {> l Q[ 

/dyhf f l @4hw ;l> f l A 3= 

l@4 

l@4 

Lqwurgxfwlrq dx{ pìodqjhv1

5; 416 Gìqlwlrq suredelolvwh ghv pìodqjhv1 

416 Gìqlwlrq suredelolvwh ghv pìodqjhv1 

Qrxv uhsuhqrqv/ gdqv fh sdudjudskh/ ohv gìyhorsshphqwv gh K1 Ureelqv gdqv Ûreelqv7;`1 Lo v*djlw g*ìwdeolu 

o*h{lvwhqfh ghv pìodqjhv frqwlqxv ghv glvwulexwlrqv g*xqh idplooh gh orlv1 Oh fdv ghv pìodqjhv qlv hvw xq fdv 

sduwlfxolhu txl hq gìfrxoh1 

41614 Udssho vxu ohv irqfwlrqv gh uìsduwlwlrq1 

Vrlw { @ +{ 4 > ===> { q , xq yhfwhxu gh o*hvsdfh hxfolglhq U q hw vrlw E q od wulex hqjhqguìh sdu ohv Eruìolhqv gh U q 1 

Rq qrwhud V { oh sdyì vhpl0rxyhuw gh U q gìql sdu = 

V { @ i+z 4 >===>z q , 5 U q > z 4 { 4 >===>z q { q j = 

Rq udsshooh txh/ vl rq vh grqqh od suredelolwì vxu E q / od irqfwlrq gh uìsduwlwlrq I dvvrflìh ã hvw gìqlh sdu= 

I +{, @ +V { , / srxu wrxw { gh U q 1 

^415` 

hvw doruv o*xqltxh suredelolwì gh E q yìuldqw ^415`1 

41615 Wkìruëph irqgdphqwdo1 

Vrlw/ pdlqwhqdqw J xqh irqfwlrq gh uìsduwlwlrq xqh irqfwlrq gh uìsduwlwlrq gìqlh vxu p U q 1 Vrlw dxvvl 

xqh irqfwlrq I gìqlh vxu p U q yìuldqw srxu J suhvtxh wrxw |/ I +=> |, hvw xqh irqfwlrq gh uìsduwlwlrq/ hw 

srxu wrxw { gh U q / I +{> =, hvw xqh irqfwlrq eruìolhqqh1 Od suredelolwì vxu E q dvvrflìh ã od irqfwlrq gh uìsduwlwlrq 

I +=> |,/ vhudqrwìh | 1 

Wkìruëph 414= Odirqfwlrq 

] 

K +{, @ I +{> |, gJ +|, 

F ^416` 

hvw xqh irqfwlrq gh uìsduwlwlrq vxu U q 1 Gh soxv/ vl hvw od suredelolwì vxu E q dvvrflìh ã K/ doruv srxu wrxw V 5E q / 

| +V, hvw xqh irqfwlrq eruìolhqqh gh | hw 

] 

+V, @ 

F | +V, gJ +|, = 

^417` 

Gìprqvwudwlrq = yrlu Dsshqglfh D1[ 

hfl ìwdeolw o*h{lvwhqfh ghv pìodqjhv hq wdqw tx*remhw pdwkìpdwltxh hw ã wlwuh g*h{hpsoh rq shxw prqwuhu txh 

od frqyroxwlrq vxu U hvw xq pìodqjh1 Hq hhw/ J hw I ìwdqw ghv irqfwlrqv gh uìsduwlwlrq vxu U/ I +{ |, hvw xqh 

irqfwlrq gh +{> |, 5 U 5 yìuldqw ohv k|srwkëvhv gx wkìruëph 414 1 Sdu frqvìtxhqw = 

] 

K +{, @ 

U 

I +{ |, gJ +|, 

hvw od irqfwlrq gh uìsduwlwlrq g*xq pìodqjh vxu U txh o*rq qrwh kdelwxhoohphqw K +{, @ I +{, J +{,/ rü hvw 

o*rsìudwhxu gh frqyroxwlrq1 


417 Glvwulexwlrq g*xq ìfkdqwloorq1 5< 

417 Glvwulexwlrq g*xq ìfkdqwloorq1 

Vl o*rq vh sodfh pdlqwhqdqw vxu U/ txh o*rq | frqvlgëuh oh pìodqjh 

] 

K +x, @ I +x> y, gJ +y, srxu x 5 U 

U 

^418` 

hw txh o*rq gìqlw/ srxu { @ +{ 4 >===>{ q ,/ 

K +{, @ K +{ 4 , K +{ q , 

hw J +{, @ J +{ 4 , J +{ q , > 

doruv K hw J vrqw ghv irqfwlrqv gh uìsduwlwlrq vxu U q 1 Hq sduwlfxolhu K +{, hvw od irqfwlrq gh uìsduwlwlrq g*xq 

ìfkdqwloorq gh q yduldeohv lqgìshqgdqwhv d|dqw fkdfxqh srxu irqfwlrq gh uìsduwlwlrq K grqqìh hq ^418`1 

Vl o*rq srvh 

I +{> |, @ I +{ 4 >| 4 , I +{ q >| q , srxu { @ +{ 4 > ===> { q , 5 U q hw | @ +| 4 > ===> | q , 5 U q > 

doruv/ srxu J 

suhvtxh wrxw |/ I +=> |, hvw xqh irqfwlrq gh uìsduwlwlrq hw I +{> =, hvw xqh irqfwlrq eruìolhqqh/ srxu 

wrxw { 5 U q 1 Sdu oh wkìruëph gh Ixelql qrxv dyrqv doruv 

K +{, 

K +{, 

K +{, 

@ 

@ 

@ 

] 

] 

] 

U 

U 

] 

I +{ 4 >| 4 , gJ +| 4 , I +{ q >| q , gJ +| q , > 

U 

] 

I +{ 4 >| 4 , I +{ q >| q , gJ +| 4 , gJ +| q , > 

U 

I +{> |, gJ +|, > 

UG fh txl prqwuh txh K +{, hvw xqh irqfwlrq gh uìsduwlwlrq gh pìodqjh1 


64 

0kdslwuh50 

Lghqwldelolwì g*xqh fodvvh gh pìodqjhv1 

O*lghqwldelolwì gh od fodvvh gh wrxv ohv pìodqjhv g*xqh idplooh grqqìh gh glvwulexwlrqv/ dvvxuh o*xqlflwì gh 

od gìfrpsrvlwlrq g*xq pìodqjh vxu fhwwh fodvvh gh glvwulexwlrqv1 Od txhvwlrq gh o*lghqwldelolwì g*xqh fodvvh gh 

pìodqjhv gh glvwulexwlrqv hvw xq suìdodeoh qìfhvvdluh ã wrxwh ìwxgh vwdwlvwltxh idlvdqw lqwhuyhqlu fhwwh fodvvh1 Lo 

hvw/ hq hhw/ sulprugldo/ gdqv ohv sureoëphv vwdwlvwltxhv dìuhqwv dx{ pìodqjhv gh glvwulexwlrqv/ tx*xq ìfkdqwloorq 

vxlydqw xqh whooh orl qh sxlvvh sdv íwuh gìfrpsrvì gh soxvlhxuv idêrqv glìuhqwhv1 

Od frpsoh{lwì ghv uìvxowdwv suìvhqwìv gdqv fhwwh sduwlh prqwuh od gl!fxowì g*rewhqlu ghv uëjohv jìqìudohv vxu 

o*lghqwldelolwì ghv fodvvhv gh pìodqjhv1 


Uhsuhqrqv ohv qrwdwlrqv gx fkdslwuh 4 = x @ iI +{> , >{5 U q > 5 p j ìwdqw xqh idplooh sdudpìwultxh gh 

irqfwlrqv gh uìsduwlwlrq vxu U q / sdudpìwuìhv sdu 5 p / J ìwdqw xqh fodvvh gh irqfwlrqv gh uìsduwlwlrq p0 

glphqvlrqqhoohv/ od fodvvh K ghv pìodqjhv hvw gìqlh sdu od fruuhvsrqgdqfh = 

P = J$K 

] 

J :$ K/ whooh txh K +{, @ 

F I +{> , gJ +, srxu { 5 Uq 1 

^519` 

rq glud txh K hvw lghqwldeoh vl P hvw xqh fruuhvsrqgdqfh elmhfwlyh gh J vxu K1 

Gh píph wrxwh vrxv0fodvvh K 3 gh K hvw glwh lghqwldeoh vl P uìdolvh xqh elmhfwrq gh J 3 @ P 4 +K 3 , vxu K 3 1 

Uhpdutxh 514= Xqh frqvìtxhqfh lppìgldwh gh fhv gìqlwlrqv hvw txh vl K 3 K 4 K/ o*lghqwldelolwì gh K 4 

hqwudñqh fodluhphqw o*lghqwldelolwì gh K 3 1 

Ohv sdudjudskhv vxlydqwv suìvhqwhqw ghv uìvxowdwv vxu o*lghqwldelolwì gh fhuwdlqhv fodvvhv gh pìodqjhv1 

Lghqwldelolwìg*xqhfodvvhghpìodqjhv1

65 515 Lghqwldelolwì ghv pìodqjhv gh glvwulexwlrqv ã sdudpëwuhv dgglwliv hw pxowlsolfdwliv1 

515 Lghqwldelolwì ghv pìodqjhv gh glvwulexwlrqv ã sdudpëwuhv 

dgglwliv hw pxowlsolfdwliv1 

51514 Idploohv dgglwlyhphqw ihupìhv1 

Hq 4 , > 5 Gj/ hvwglwhdgglwlyhphqw 

ihupìh vl rq d = 

rü G hvw xq vhpl0jurxsh deìolhq dgglwli1 

I +=> 4 , I +=> 5 , @ I +=> 4 . 5 , / ; 4 > 5 5 G> 

51515 Wkìruëph g*lghqwldelolwì1 

Rq vdlw/ ghsxlv ^Whlfkhu87`/ txh ohv glvwulexwlrqv g*xqh idplooh sdudpìwultxh hq glphqvlrq 4/ dgglwlyhphqw 

ihupìh vxu G @ Q/ G @ T . rx G @ U . / srvvìghqw xqh irqfwlrq fdudfwìulvwltxh/ I +=> ,/ ghodiruph= 

I +=> , @ + I +=> 4,, / w 5 U/ 

rü I +=> 4, hvw xqh irqfwlrq fdudfwìulvwltxh lqgìshqgdqwh gh 1 Gdqv fh fdv/ wrxw ìoìphqw gh K G d xqh irqfwlrq 

fdudfwìulvwltxh K +w,/ ghodiruph= 

] 

K +w, @ 

G 

+ I +w> 4,, gJ +, / w 5 U1 

G*dsuëv o*ìwxgh gh o*lqwìjudoh L +}>J, @ U G } gJ +, vxu od erxoh rxyhuwh gh ud|rq 4/ sulyìh gh o*ruljlqh/ hw juåfh 

dx wkìruëph g*lghqwlwì ghv wudqvirupìhv gh Irxulhu/ rq ìwdeolw txh/ vl ghx{ irqfwlrqv gh uìsduwlwlrqv J 4 hw 

J 5 hqjhqguhqw oh píph pìodqjh gh K G / hoohv vrqw qìfhvvdluhphqw ìjdohv1 

*hvw oh wkìruëph = 

Wkìruëph 514= Od fodvvh K G ghv pìodqjhv g*xqh idplooh sdudpìwultxh vxu U dgglwlyhphqw ihupìh oruvtxh G hvw 

o*xq ghv wurlv Q/ T . rx U . / hvw lghqwldeoh1 

h wkìruëph wurxyh o*xqh gh vhv dssolfdwlrqv gdqv oh fdv rü xqh irqfwlrq gh uìsduwlwlrq I hqjhqguh od idplooh x G / 

sdu h{hpsoh srxu x 4 @ iI 4 +=> , @ I +=, > 5 Gj rx x 5 @ iI 5 +=> , @ I += , > 5 Gj1 Rq shxw dlqvl ìwdeolu od 

sursrvlwlrq 514 = 

Sursrvlwlrq 514= Vl od wudqvirupìh gh Irxulhu gh od irqfwlrq gh uìsduwlwlrq I +=, @ I +h = , q*hvw sdv lghqwltxhphqw 

qxooh vxu xq lqwhuydooh qrq gìjìqìuì gh U/ ohv fodvvhv K 4 / uhvshfwlyhphqw K 5 / ghv pìodqjhv gh glvwulexwlrqv ghv 

x 4 / uhvshfwlyhphqw x 5 / vrqw lghqwldeohv1 

Lghqwldelolwì g*xqh fodvvh gh pìodqjhv1

516 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh od idplooh h{srqhqwlhooh1 66 

51516 H{hpsohv g*dssolfdwlrqv1 

Oh wkìruëph 514 hw od sursrvlwlrq 514 v*dssoltxhqw dx fdv gh od glvwulexwlrq jdppd ã ghx{ sdudpëwuhv1 Vrlw/ hq 

hhw/ od idplooh gh glvwulexwlrqv = 

Qrwrqv srxu s 5 U . hw t 5 U . = 

; 

< 

 

]{ 

x @ 

= +=> s> t, @ 4 x t4 

+t, s t h{s x 

@ 

gx> s 5 U . >t 5 U . 

s 

> 1 

3 

+w> s> t, @ +4 lsw, t > 

od irqfwlrq fdudfwìulvwltxh gh od orl Jdppd gh irqfwlrq gh uìsduwlwlrq +=> s> t,1 D s {ì/ x hvw xqh idplooh 

dgglwlyhphqw ihupìh hw oh wkìruëph dvvxuh grqf o*lghqwldelolwì ghv J +t,0pìodqjhv gh x 1 Oruvtxh t hvw {ì/ x 

hvw hqjhqguìh sdu od irqfwlrq +=> 4>t,/ gh soxv/ od irqfwlrq fdudfwìulvwltxh gh I hvw +t.lw, 

+t, 

/ fh txl shuphw/ juåfh ã 

od sursrvlwlrq 514/ gh frqfoxuh ã o*lghqwldelolwì ghv J +s,0pìodqjhv gh x 1 

Uhpdutxrqv txh od qrq lghqwldelolwì/ hq jìqìudo/ gh od fodvvh K ghv J +s> t,0pìodqjhv txl d ìwì prqwuìh gdqv 

Ûreelqv7 > , @ 

4 

< 

4 

@ 

5 4 i>.j +x, gx> A 3> 5 U 

> / 

vrqw lghqwldeohv1 Lo q*hq q*hvw sdv gh píph/ elhq vþu/ srxu od fodvvh K gh wrxv ohv pìodqjhv gh x X / xqh glvwulexwlrq 

xqliruph ìwdqw hooh píph xq pìodqjh gh glvwulexwlrqv xqliruphv = 

Srxu fh txl hvw gh od fodvvh ghv glvwulexwlrqv elqrpldohv = 

X 

 

=> 4 5 > 4 

@ 4 

5 5 X => 4 7 > 4 

. 4 

7 5 X => 6 7 > 4 

1 

7 

; 

< 

 

x E @ 

= E +=> q> s, @ [ qm 

@ 

s m +4 s, qm > / 

m= 

rq rewlhqw/ sdu xq udlvrqqhphqw dqdorjxh/ o*lghqwldelolwì/ ã s {ì/ gh od fodvvh ghv J +q,0pìodqjhv gh x E 1 Sdu 

frqwuh/ ã q {ì/ od fodvvh ghv J +s,0pìodqjhv gh x E q*hvw sdv lghqwldeoh1 

516 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh od idplooh h{0 

srqhqwlhooh1 

Hq 4

67 517 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh glvwulexwlrqv jdxvvlhqqhv1 

Vrlw x @ iI +{> , >{5 U q > 5 p j/ rü/ srxu xqh phvxuh 0qlh vxu E q rq d = 

3 

4 

p[ 

gI +{> , @ d +, e +{, h{s C m k m +{, D g +{, > 

dyhf d +, A 3/ e +{, 3 hw d> e> k l > 4 l p/ phvxudeohv1 

Vl J 4 /J 5 5J/rqsrvh= 

Vrlw doruv = 

3 

i n +{, @ gK 

] 

g +{, @ e +{, F d +, h{s C 

hw hqq/ k> |l ìwdqw oh surgxlw vfdodluh gh hw gh |/ 

m@4 

4 

p[ 

m k m +{, D gJ n +, / n @4> 51 

m@4 

@ i{ = i 4 +{, @ i 5 +{, 9@ 3j 

hw @ i| @ +k 4 +{, >===>k p +{,, = { 5 j / 

] 

in +|, @ 

F d +, h{s ^k> |l` gJ n +, / n @4> 51 

Vrxv fhv qrwdwlrqv/ rq d i4 +|, @ i5 +|, vl | 5 1 Lo hvw doruv srvvleoh/ vl o*rq uhvwuhlqw / txh fhod lpsoltxh J 4 @ J 5 1 

Srxu fhod/ qrwrqv f +, o*hqyhorssh frqyh{h 4 gh hw glvwlqjxrqv ohv txdwuh fdv vxlydqwv = 

+l, hvw ql1 

Oh fdv +l, uhjurxsh ohv pìodqjhv qh suhqdqw tx*xq qrpeuh ql gh ydohxuv qrq qxoohv hw fhx{ ghv glvwulexwlrqv 

srxu ohvtxhoohv ohv irqfwlrqv k l vrqw ã ydohxuv gdqv xq hqvhpeoh ql1 *hvw w|sltxhphqw oh fdv gh od glvwulexwlrq 

Elqrpldoh txl qh suhqg tx*xq qrpeuh ql gh ydohxuv qrq qxoohv1 Od glvfxvvlrq gh Whlfkhu gdqv ^Whlfkhu96`/ srxu 

fhwwh glvwulexwlrq/ grqqh xqh lgìh ghv sureoëphv vrxohyìv gdqv fh fdv1 

+ll, hvw lqql/ f +, hvw eruqìh hw q*d sdv gh srlqw g*dffxpxodwlrq gdqv o*lqwìulhxu 5 gh f +,1 

Oh fdv +ll, hvw oh prlqv lqwìuhvvdqw g*xq srlqw gh yxh vwdwlvwltxh1 

+lll, frpph +ll, vdxi txh f +, hvw vxssrvì qrq eruqì1 

Oh fdv +lll, hvw fhoxl ghv glvwulexwlrqv glvfuëwhv + q*d sdv gh srlqw g*dffxpxodwlrq ,/ srxu ohvtxhoohv rq qh shxw 

sdv eruqhu o*xqh/ dx prlqv/ ghv irqfwlrqv k l 1 *hvw oh fdv gh od glvwulexwlrq gh Srlvvrq1 

+ly, hvw lqql hw d xq srlqw g*dffxpxodwlrq ã o*lqwìulhxu gh f +,1 

Oh fdv +ly, hvw hvvhqwlhoohphqw fhoxl gh glwulexwlrqv frqwlqxhv1 

Juåfh ã txhotxhv suìflvlrqv g*ruguh whfkqltxhv/ rq rewlhqw/ gdqv ohv fdv +lll, hw +ly, oìjëuhphqw prglìv/ o*ìjdolwì 

gh J 4 hw J 5 1 Sdu dssolfdwlrq gh fhv uìvxowdwv/ Eduqgru0Qlhovhq surxyh o*lghqwldelolwì gh od fodvvh ghv pìodqjhv gh 

glvwulexwlrqv gh Srlvvrq hw gh fhooh ghv pìodqjhv ghv surgxlwv gh qA4 glvwulexwlrqv qrupdohv lghqwltxhv1 Whlfkhu/ 

gdqv ^Whlfkhu93`/ d gdloohxuv prqwuì txh fh q*hvw sdv od fdv vl q @4hw gdqv ^Whlfkhu9:` txh fh q*hvw sdv qrq soxv oh 

fdv vl ohv glwulexwlrqv qh vrqw sdv lghqwltxhv1 

O*hqyhorssh frqyh{h g*xqh sduwlh g*xq hvsdfh d!qh . hvwodsoxvshwlwhsduwlhfrqyh{hgh. frqwhqdqw 1 


517 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh glvwulexwlrqv jdxvvlhqqhv1 68 

517 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh glvwulexwlrqv 

jdxvvlhqqhv1 

Gdqv xq duwlfoh/ wuëv frpsohw/ vxu o*lghqwldelolwì ghv pìodqjhv/ Euxql hw Nrfk +yrlu Êuxql;8`,/ ìqrqfhqw ghx{ 

wkìruëphv g*lghqwldelolwì gh od fodvvh ghv pìodqjhv gh glvwulexwlrqv qrupdohv1 Qrxv vdyrqv txh od fodvvh ghv 

pìodqjhv gh glvwulexwlrqv qrupdohv Q +{> +|,, >{5 U s > 

Q +{> +|,, @ 

 

4 

+5, s@5 m S +|,m h{s 4 4@5 5 +{ p [ 

4 +|,,W +|, +{ p +|,, 

q*hvw sdv lghqwldeoh hq jìqìudo/ f*hvw srxutxrl lo hvw qìfhvvdluh gh uìgxluh o*hqvhpeoh ghv sdudpëwuhv @ +p> S ,1 

51714 Glvwulexwlrq qrupdoh ã xqh glphqvlrq1 

Vrlw G xq frpsdfw gh U/ rq gìqlw oh vrxv0hqvhpeoh 4 gh 4 +G, 6 = 

4 @ 

@ +4 > 5 , 5 4 +G, = m 4 +|,m N 4 .4> 3 v 4 5 +|, v 5 .4> 

 

3 4 +|, . 3 5 +|, N 5 .4> ;| 5 G 

 

/ 

sxlv o*rq suìflvh fhw hqvhpeoh = 

 

5 @ 

vrxv hqvhpeoh gh 4 dyhf/ gh soxv/ 

;> 3 5 5 > ;|> | 3 5 G/ +|, @ 3 +| 3 , , | @ | 3 

 

> 

rü dxvvl = 

 

6 @ 

5 4 = +|, @ +| 3 , , | @ | 3 / ;|> | 3 5 G /hw 

srxu 4 > 5 prqrwrqhv > m 4 +|,m . m 5 +|,m A 3> ;| 5 G 

 

1 

Od wrsrorjlh vxu fhv hqvhpeohv hvw od wrsrorjlh qdwxuhoohphqw lqgxlwh sdu od qruph vxs1 Lo hvw lpsruwdqw gh 

uhpdutxhu tx*xq hqvhpeoh 6 qh frqwlhqw sdv qìfhvvdluhphqw g*hqvhpeoh 5 1 Lo q*hvw sdv qrq soxv qìfhvvdluhphqw 

frqwhqx gdqv xq hqvhpeoh 5 1 

51715 Wkìruëph g*lghqwldelolwì1 

Vlo*rqhqwhqgsdu0 +G, o*hqvhpeoh ghv phvxuhv vljqìhv gh vxssruw G/ ã yduldwlrqv eruqìhv/ pxql gh od wrsrorjlh 

ydjxh Êloolqjvoh|9;`/ rq qrwhud S +G, oh vrxv hqvhpeoh ghv suredelowìv gh 0 +G,1 Rq gìqlw o*rsìudwhxu 

W = 4 0 +G, $ O 4 +U, 7 sdu = 

] 

i @ W +> , = i +{, @ 

G 

Q +{> +|,, +g|, = 

Rq prqwuh dlvìphqw txh W hvw xqlirupìphqw frqwlqxh vìsduìphqw sdu udssruw ã hw hw hvw grqf frqwlqxh 

vxu 4 0 +G,1 

Euxql hw Nrfk ìwdeolvvhqw doruv ohv ghx{ wkìruëphv vxlydqwv = 

Hqvhpeoh ghv irqfwlrqv ã ydohxuv uìhoohv/ frqwlqxhv hw gìulydeohv gh (1 

e u " EU hvw o*hvsdfh ghv irqfwlrqv gh U gdqv U lqwìjudeohv srxu od qruph vxs1 


69 518 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh glvwulexwlrqv frqwlqxhv ã xq sdudpëwuh vxu U1 

Wkìruëph 515= Srxu> 3 5 5 / > 3 5S+G,/ vlW +> , @ W 3 > 3 /doruv 

txl prqwuh o*xqlflwì srxu > 3 5 5 1 

@ 3 hw @ 3 + ss=, > 

Wkìruëph 516= Srxu> 3 5 6 / > 3 5S+G,/ vlW +> , @ W 3 > 3 / lo h{lvwh/ doruv/ xqh irqfwlrq 4 / k = G $ 

G/ vwulfwhphqw prqrwrqh hw gìqlh gh pdqlëuh xqltxh vxu oh vxssruw gh who txh = 

3 +gk +|,, @ +g|, hw 3 +k +|,, @ +|, + ss=, > 

txl ìwdeolw xq uìvxowdw soxv iruw txh oh suìfìghqw/ vxu xq hqvhpeoh glìuhqw1 

hv uìvxowdwv frquphqw fhx{/ txh qrxv ìqrqfhurqv gdqv oh fkdsñwuh vxlydqw/ vxu o*lghqwldelolwì ghv fodvvhv gh 

pìodqjhv qlv srxu ohv glvwulexwlrqv qrupdohv1 Gdqv fh fdv/ hvw xqh suredelolwì glvfuëwh/ lo q*| d soxv gh sureoëph 

vxu od frqwlqxlwì gh hw ohv k|srwkëvhv gh prqrwrqlh gh 7 vrqw lqxwlohv1 

51716 H{whqvlrq dx fdv pxowlglphqvlrqqho1 

Ohv gìyhorsshphqwv hhfwxìv gdqv oh fdv ghv orlv vxu U shxyhqw íwuh ìwhqgxv dx{ orlv qrupdohv pxowlglphqvlrq0 

qhoohv1 Dq gh frpsuhqguh oh sulqflsh gh o*h{whqvlrq/ rq shxw uhjdughu oh fdv q @51 

Rq frqvlgëuh srxu fhod = 

hw o*rq gìqlw 5 4 sdu = 

; 

 

5 4 @ 

= 

p +|, @ + 4 +|, > 5 +|,, 

 

6 +|, 

hw +|, @ 

7 +|, 

7 +|, 8 +|, 

5 4 +G, = m l +|,m n 4 .4>l@4> ===> 8> 3 l +|, >l@6> 8> 

6 +|, 8 +|, 5 7 +|, 8S 

v4 A 3> 

l@4 

 

3 

4 +|, n5 .4> ;| 5 G 

Vl o*rq gìqlw 5 5 hw 5 6 frpph suìfìghpphqw/ pdlv hq irqfwlrq gh 5 4/ rq shxw ìwhqguh ohv wkìruëphv 515 hw 5161 

< 

@ 

> 1 

518 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh glvwulexwlrqv 

frqwlqxhv ã xq sdudpëwuh vxu U1 

Oh wkìruëph txl yd vxlyuh/ suìvhqwì sdu Wdoolv gdqv ^Wdoolv9 , ã xq sdudpëwuh xqlglphqvlrqqho1 *hvw 

xq wkìruëph wuëv shx sudwltxh ã phwwuh hq rhxyuh/ txh o*rq shxw fhshqgdqw dgdswhu gdqv fhuwdlqv fdv sduwlfxolhuv 

+yrlu ^Wdoolv9 , hvw frqwlqxh hq / rq shxw/ vdqv shuwh gh jìqìudolwì/ vxssrvhu txh 5 ^4> .4`1 

rqvlgìurqv/ grqf oh pìodqjh 

K +{, @ 

] .4 

4 

I +{> , gJ +, = 

^51:` 


519 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh phvxuhv surgxlwv1 6: 

Vrlw W +{> , @ CI+{>, 

C 

/ txh o*rq vxssrvhud frqwlqxh hq hw gh fduuì lqwìjudeoh vxu ^4> .4` ^4> .4`1 Oh pìodqjh 

^51:` shxw/ doruv/ dsuëv xqh lqwìjudwlrq sdu sduwlhv/ íwuh plv vrxv od iruph = 

O +{, @ 

] .4 

4 

W +{> , J +, g/ 

^51;` 

rü O +{, @ I +{> 4, K +{,1 Wdoolv ìwdeolw txh od fodvvh ghv pìodqjhv ^51:` hvw lghqwldeoh vl hw vhxohphqw v*lo h{lvwh 

xqh xqltxh vroxwlrq gh fduuì lqwìjudeoh ã ^51;`1 Hq srvdqw N +{> |, @ U .4 

W +{> }, W +|> }, g} rq d = 

4 

Wkìruëph 517= Od fodvvh ghv pìodqjhv ^51:` hvw lghqwldeoh vl hw vhxohphqw vl o*hqvhpeoh ghv irqfwlrqv sursuhv 

dvvrflìhv ã ghv ydohxuv sursuhv qrq qxoohv gh N hvw frpsohw1 

h uìvxowdw hvw od frqvìtxhqfh g*xqh dssolfdwlrq gx wkìruëph gh Kloehuw0Vfkplgw srxu ohv qr|dx{ qrq0v|pìwultxhv 

^Wulfrpl8:`1 

519 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh phvxuhv sur0 

gxlwv1 

Hq 4pq 

@ 

+ 

I {> h 

 

= I {> h 

@ 

q\ 

I {> l > rü I , 

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l@4 

srxu h @ 4 >===> q 5 qp @ 

p p ~} 

p 

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> 

q irlv 

rq gìqlw xqh fodvvhx q>pq sduwlfxolëuh/ vxu qp / grqw od glvwulexwlrq jìqìultxh hvw od phvxuh surgxlw I 

{> h 

1 

Hq idlw/ x q>pq hvw od idplooh ghv glvwulexwlrqv mrlqwhv ghv ìfkdqwloorqv gh q yduldeohv lqgìshqgdqwhv d|dqw fkdfxqhv 

xqh glvwulexwlrq gdqv x 4>p 1 Od mxvwlfdwlrq hw od gìqlwlrq ghv pìodqjhv gh glvwulexwlrqv gh x q>qp hvw grqf dvvxuìh 

sdu oh gìyhorsshphqw vxu oh sureoëph ghv ìfkdqwloorqv gx sdudjudskh L14171 Qrxv dyrqv doruv oh uìvxowdw vxlydqw = 

Wkìruëph 518= Vl od fodvvh gh wrxv ohv pìodqjhv gh x 4>p hvw lghqwldeoh/ doruv srxu wrxw qA4/ od fodvvh ghv 

pìodqjhv gh x q>pq hvw dxvvl lghqwldeoh1 Uìflsurtxhphqw/ vl srxu dx prlqv xq qA4/ od fodvvh gh wrxv ohv pìodqjhv 

gh x q>pq hvw lghqwldeoh/ lo hq hvw gh píph srxu x 4>p 1 

Gìprqvwudwlrq = yrlu Dsshqglfh E1 [ 

Juåfh ã fh wkìruëph/ rq shxw/ sdu h{hpsoh/ ìwdeolu txh od fodvvh gh wrxv ohv pìodqjhv gh orlv qrupdohv prqrgl0 

phqvlrqqhooh/ x @ x 4>5 q*ìwdqw sdv lghqwldeoh/ gdqv oh fdv jìqìudo/ lo hq vhud gh píph srxu od fodvvh gh wrxv ohv 

pìodqjhv gh glvwulexwlrqv gh x q>5q ghv surgxlwv gh q glvwulexwlrqv qrupdohv1 h uìvxowdw hvw elhq vþu prgxoì sdu 

oh sdudjudskh L1516 txl dvvxudlw o*lghqwldelolwì ghv pìodqjhv ghv glvwulexwlrqv surgxlw gh q qrupdohv lghqwltxhv1 


6; 51: Lqgìshqgdqfh gh o*lghqwldelolwì sdu udssruw ã od sdudpìwulvdwlrq1 

51: Lqgìshqgdqfh gh o*lghqwldelolwì sdu udssruw ã od sdudpìwul0 

vdwlrq1 

Wdoolv hw khvvrq sursrvhqw/ hq 4 V,/ dlqvl txh g*xqh idplooh sdudpìwultxh gh suredelolwìv iS > 5 p j1 Vrlw/ dxvvl/ E/ 

xqh wulex vxu p / whooh txh S +V, vrlw E0phvxudeoh srxu wrxw V 5V1 Xq pìodqjh gh S hvw doruv xqh phvxuh/ 

] 

K +V, @ 

F S +V, g J +, / V 5V/ 

rü J hvw xqh phvxuh vxu +> E,1 Ohv pìodqjhv gh S vrqw doruv lghqwldeohv/ g*dsuëv od gìqlwlrq fodvvltxh/ vl od 

fruuhvsrqgdqfh = 

] 

J :$ 

F S g J +, > J ìwdqw xqh suredelolwì vxu + p > E, 

hvw elmhfwlyh1 

Srxu dyrlu xqh glvfxvvlrq soxv joredoh/ f*hvw ã gluh lqgìshqghpphqw gh / vxu o*lghqwldelolwì ghv pìodqjhv gh 

S / lo idxw sdudpìwuhu S / sdu oxl0píph1 O*hqvhpeoh ghv sdudpëwuhv hvw doruv xq hqvhpeoh gh suredelolwìv vxu 

+[> V,/vxu ohtxho rq gìqlw od wulex S1 Xq pìodqjh hvw doruv = 

] 

 

Sg+S , /rü hvw xqh suredelolwì vxu +> S, 1 

hw o*lghqwldelolwì gh whov pìodqjhv hvw ìtxlydohqwh ã od elmhfwlylwì gh od fruuhvsrqgdqfh 

] 

:$ 

 

Sg+S , > 

rü hvw xqh suredelolwì vxu +> S,1 Juåfh ã fhwwh gìqlwlrq rq shxw v*dudqfklu gh od gìshqgdqfh hq 1 

Uhpdutxh 515= hv ghx{ gìqlwlrqv ghv pìodqjhv vrqw ìtxlydohqwhv vl o*rq frqvlgìuh ã od irlv @iS > 5 p j 

hw ohv ghx{ wulexv V hw E ohv soxv shwlwhv srxu txh ohv fruuhvsrqgdqfhv uhvshfwlyhv/ :$ S +V, hw S :$ S +V,/ 

vrlhqw phvxudeohv srxu wrxw V 5V1hwwh sursulìwì dvvxuh/ hq hhw/ od elmhfwlylwì gx fkdqjhphqw gh yduldeoh srxu 

v*dudqfklu gh od sdudpìwulvdwlrq1 

Od frpsoh{lwì gh od plvh hq rhxyuh gh fhwwh gìqlwlrq hw sulqflsdohphqw gh o*ìwdeolvvhphqw gh o*h{suhvvlrq gh od 

phvxuh / q*d sdv shuplv/ mxvtx*lfl/ g*ìwhqguh wrxv ohv uìvxowdwv rewhqxv gdqv oh fdv fodvvltxh gh sdudpìwulvdwlrq1 

Lo shuphw fhshqgdqw dx{ dxwhxuv g*rewhqlu xq uìvxowdw wkìrultxh lqwìuhvvdqw vxu o*lghqwldelolwì ghv pìodqjhv grqw 

o*lqyhuvh hvw eruqìh/ dlqvl tx*xqh QV g*lghqwldelolwì ghv fodvvhv gh pìodqjhv vxu ghv idploohv qlhv gh glvwulexwlrqv1 


0kdslwuh60 

Lghqwldelolwì ghv pìodqjhv qlv1 

Od vlpsolfdwlrq gx prgëoh/ oruvtxh o*rq sdvvh ghv pìodqjhv frqwlqxv dx{ pìodqjhv qlv/ q*hvw sdv txh iruphooh1 

Lo hvw/ hq hhw/ ehdxfrxs soxv idfloh g*lqwhusuìwhu vwdwlvwltxhphqw fhv ghuqlhuv> Vl xqh srsxodwlrq [/ ghirqfwlrqgh 

uìsduwlwlrq I / shxw íwuh glylvìh hq q vrxv srsxodwlrqv [ l / gh irqfwlrq gh uìsduwlwlrq I l / grqw od sursruwlrq gdqv 

S 

od srsxodwlrq [ hvw/ uhvshfwlyhphqw/ f l / rq lpdjlqh dlvìphqw txh [ vxlw xqh orl gh pìodqjh = I @ 

q f l I l 1 Gh 

l@4 

qrpeuhx{ h{hpsohv g*ìwxghv sudwltxhv ghv pìodqjhv qlv vrqw grqqìv gdqv ohv ghx{ rxyudjhv gh uìiìuhqfh vxu oh 

vxmhw txh vrqw ^Hyhulww;4` hw ^Wlwwhulqjwrq;8`1 

Od qrwlrq g*lghqwldelowì/ hooh dxvvl/ suhqg xq vhqv soxv dffhvvleoh gdqv oh fdv ghv pìodqjhv qlv1 O*lghqwldelolwì 

gh od fodvvh ghv pìodqjhv qlv vxu od fodvvh I @ I +{> , >{5 U n > 5 p 

q[ 

K+=, @ f l I +=> l , dyhf 

l@4 

q[ 

f l @4hw ;l> f l A 3> 

l@4 

vh wudgxlw/ vl o*rq vxssrvh ohv l glìuhqwv ghx{ ã ghx{ 8 / sdu o*xqlflwì gx q dlqvl txh ghv frxsohv +f l > l , >l@4> >q/ 

uìdolvdqw od gìfrpsrvlwlrq gh K1 Sdu h{whqvlrq/ rq glw txh oh pìodqjh K hvw lghqwldeoh vxu I/ rx soxv vlpsohphqw 

lghqwldeoh/ v*lo q*| d sdv gh grxwh1 

Gdqv oh fdguh gh o*ìwxgh gh o*lghqwldelolwì ghv fodvvhv gh pìodqjhv qlv/ o*lgìdo hvw gh prqwuhu o*lghqwldelolwì 

gh K 3 txl hvw od fodvvh gh wrxv ohv pìodqjhv qlv hw txl frqwlhqw wrxv ohv K n / ;n 5 Q +K n hw K 3 vrqw gìqlv gdqv 

od sduwlh L1415,1 Píph vl ghv uìvxowdwv rqw ìwì rewhqxv gdqv ghv fdv vlpsolìv/ qrwdpphqw oruvtxh o*hqvhpeoh x 

hvw ql/ qrxv qh gìyhorsshurqv txh oh fdv jìqìudo ghv hqvhpeohv gh sdudpëwuhv frqwlqxv/ txl hvw fhoxl txh qrxv 

uhqfrqwurqv gdqv oh fdguh gh qrwuh ìwxgh sudwltxh1 

D h txh o*rq shxw wrxmrxuv idluh txlwwh ã uhjurxshu wrxv ohv whuphv frqwhqdqw oh píph w 1

73 614 rqglwlrq vx!vdqwh srxu o*lghqwldelolwì ghv fodvvhv gh pìodqjhv qlv1 

614 rqglwlrq vx!vdqwh srxu o*lghqwldelolwì ghv fodvvhv gh 

pìodqjhv qlv1 

Ohv suhplhuv wkìruëphv vxu o*lghqwldelolwì ghv pìodqjhv qlv rqw ìwì ìwdeolv sdu Whlfkhu txl prqwuh gëv 4 { 5 U q > l 5 p xqh idplooh gh irqfwlrq gh uìsduwlwlrq srxu 

ohvtxhoohv lo h{lvwh xqh wudqvirupdwlrq / gh grpdlqh gh gìqlwlrq V / whooh txh P = I :$ / vrlw olqìdluh hw 

elmhfwlyh1 Vxssrvrqv gh soxv tx*lo h{lvwh/ vxu x/ xq ruguh wrwdo/ qrwì Q/ who txh I 4 R I 5 lpsoltxh 

V " V # / 

< w 4 5 V 9 

" / +w 4 lqgìshqgdqw gh V # , who txh olp # +w, 

w$w" " +w, @31 

Doruv od fodvvh K 3 ghwrxvohvpìodqjhvqlvghx hvw lghqwldeoh1 

Gìprqvwudwlrq =yrlu Dsshqglfh 1 [ 

61415 Lghqwldelolwì gh od fodvvh ghv pìodqjhv qlv gh glvwulexwlrqv qru0 

pdohv1 

Od sursrvlwlrq txl vxlw gìfrxoh gx wkìruëph g*lghqwldelolwì gh Whlfkhu1 

Sursrvlwlrq 614= 

Od fodvvh gh wrxv ohv pìodqjhv qlv gh orlv qrupdohv hvw lghqwldeoh1 

Gìprqvwudwlrq = hwwh gìprqvwudwlrq hvw grqqìh ã wlwuh g*h{hpsoh1 Rq frqvlgìuh od fodvvh 

; 

 

x @ 

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4 

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s h{s +x ,5 

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5 5 5 gx> { 5 U>5 U> 5 5 U . > > 

ghv irqfwlrqv gh uìsduwlwlrq ghv orlv qrupdohv gh pr|hqqh hw gh yduldqfh 5 1 Suhqrqv srxu 

w> > 5 @ 

 

h{s 

# w # 

5 

w >w 5 U/ od wudqvirupìh gh Odsodfh dvvrflìh1 Rq shxw rugrqqhu x dxpr|hqg*xqruguhoh{l0 

frjudskltxh = vl Q 4 @ Q {> 4 >4 5 hw Q5 @ Q {> 5 >5 5 vrqw ghx{ ìoìphqwv glìuhqwv gh x/ rq d Q4 R 

Q 5 vl 5 4 5 5 / rx/ vl 5 4 @ 5 5 hw 4 5 1 Lo vx!w/ doruv/ srxu frqfoxuh/ g*dssoltxhu oh wkìruëph 614 dyhf 

V l @ `4>.4^ hw w 4 @.41[ 

61416 Lghqwldelolwì ghv fodvvhv gh pìodqjhv gh glvwulexwlrqv jdppdv grqw 

ohv ghqvlwìv rqw oh píph vxssruw1 

Whlfkhu ìwdeolw dxvvl od sursrvlwlrq vxlydqwh frqfhuqdqw ohv orlv jdppdv1 

S hflgìvljqhohsoxvshwlwhqvhpeohihupìfrqwhqdqw7 x" 

Lghqwldelolwì ghv pìodqjhv qlv1

614 rqglwlrq vx!vdqwh srxu o*lghqwldelolwì ghv fodvvhv gh pìodqjhv qlv1 74 

Sursrvlwlrq 615= Odfodvvhghwrxvohvpìodqjhvqlvghorlvjdppdvghodidplooh= 

; 

< 

 

]{ 

x @ 

= +{> s> t, @ 4 x t4 

+t, s t h{s x 

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gx> { 5 U . >s5 U . >t 5 U . 

s 

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3 

ghv irqfwlrqv gh uìsduwlwlrq ghv orlv jdppdv ã ghx{ sdudpëwuhv s hw t/ hvw lghqwldeoh1 

Lo hvw lpsruwdqw gh frqvwdwhu txh fhwwh sursrvlwlrq vh olplwh dx{ glvwulexwlrqv gìqlhv vxu U . 1 Srxu od vxlwh 

gh o*h{srvì/ lo hvw qìfhvvdluh srxu qrxv g*ìodujlu fhwwh sursrvlwlrq1 Qrxv suìvhqwrqv oh uìvxowdw vxlydqw = 

Sursrvlwlrq 616= Od fodvvh gh wrxv ohv pìodqjhv qlv gh orlv jdppdv/ grqw ohv ghqvlwìv rqw oh píph vxssruw/ 

ù 3 >.4^/ hvw lghqwldeoh1 

Od suhxyh hvw xqh dgdswdwlrq gh fhooh gh Whlfkhu1 

Gìprqvwudwlrq = Rq frqvlgìuh od fodvvh = 

; 

 

x u! @ 

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4 

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ghv irqfwlrqv gh uìsduwlwlrq ghv orlv jdppdv ã wurlv sdudpëwuhv s> t hw u 3 / grqw oh ghuqlhu hvw {ì1 rqvlgìurqv od 

wudqvirupdwlrq olqìdluh txl ã +=> s> t> u 3 , dvvrflh u! +=> s> t, whooh txh = 

k l 

u! +w> s> t, @ H h +[u !,w 

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h{s x +4.sw, 

L^3>.4^ +x, gx> 

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t 

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L^3>.4^ +x, gx> 

s 

s 

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~} € 

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@ +4.sw, t >wA 4 s = 

Rq yìulh txh/ u 3 ìwdqw {ì/ od fruuhvsrqgdqfh +=> s> t> u 3 , :$ u! +=> s> t, hvw elhq xqh elmhfwlrq1 

Rq shxw wrxmrxuv rugrqqhu x u! dx pr|hq g*xq ruguh oh{lfrjudskltxh = vl 4 @ +{> s 4 >t 4 >u 3 , hw 5 @ 

+{> s 5 >t 5 >u 3 , vrqw ghx{ ìoìphqwv glìuhqwv gh x u! /rqd 4 R 5 vl s 4 As 5 /rx/vls 4 @ s 5 hw t 4 At 5 1 Lo vx!w/ 

doruv/ srxu frqfoxuh/ g*dssoltxhu oh wkìruëph 614 dyhf 

V u 

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4 

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! 

s 4 s ># 4 @ 4 =[ 

5 s 4 


75 615 Xqh QV g*lghqwldelolwì gh od fodvvh ghv pìodqjhv qlv1 

61417 Lghqwldelolwì gh od fodvvh ghv pìodqjhv qlv gh glvwulexwlrqv lqyhuvh0 

jdppd grqw oh vxssruw ghv ghqvlwìv hvw oh píph1 

Dq g*ìodujlu od froohfwlrq gh idploohv lghqwldeohv gh Whlfkhu qrxv prqwurqv od sursrvlwlrq txl vxlw = 

Sursrvlwlrq 617= = Od fodvvh gh wrxv ohv pìodqjhv qlv gh orlv lqyhuvh0jdppdv/ grqw ohv ghqvlwìv vrqw gh píph 

vxssruw/ hvw lghqwldeoh1 

Gìprqvwudwlrq = Rq frqvlgìuh od fodvvh 

; 

 

x ! u ! 

@ 

= Lqy +{> s> t> u 3, @ 

] { 

4 

4 +x u 3 , t4 

 

5 

+t, s t h{s 

Lû! >.4^ +x, gx> 

s +x u 3 , 

 

{ 5 U> u 3 5 U> 

s 5 U . >t5 U . < 

@ 

> > 

ghv irqfwlrqv gh uìsduwlwlrq ghv orlv jdppdv ã wurlv sdudpëwuhv s> t hw u 3 / grqw oh ghuqlhu hvw {ì1 rqvlgìurqv od 

wudqvirupdwlrq olqìdluh txl ã Lqy +=> s> t> u 3 , dvvrflh u! +=> s> t, whooh txh = 

 

u! +w> s> t, @ H h{s 

5w 

@ 

[ u 3 

@ 

].4 

4 

4 

+t, 

x t4 

s t 

4 

].4 

4 

4 

+t, 

+x u 3 , t4 

s t 

 

h{s 

5 

s +x u 3 , 

5w 

Lû!>.4^ +x, gx> 

x u 3 

 

h{s 5 

sx 5w 

Lû!>.4^ +x . u 3 , gx/ dsuëv fkdqjhphqw gh yduldeoh / 

x 

].4 

t 

@ +4.sw, t 4 4.sw 

+t, xt4 h{s 5 4.sw 

L^3>.4^ +x, gx> 

s 

x s 

@ +4.sw, t >wA 4 s > 

frpph suìfìghpphqw1 u 3 ìwdqw {ì/ od fruuhvsrqgdqfh +=> s> t> u 3 , :$ u! +=> s> t, hvw elhq xqh elmhfwlrq1 

Rq shxw dxvvl rugrqqhu x ! u ! 

dx pr|hq gx píph ruguh oh{lfrjudskltxh txh fl0ghvvxv = vl Lqy 4 @ Lqy +{> s 4 >t 4 >u 3 , 

hw Lqy 5 @ Lqy +{> s 5 >t 5 >u 3 , vrqw ghx{ ìoìphqwv glìuhqwv gh x ! u ! 

/rqdLqy 4 R Lqy 5 vl s 4 As 

k 5 /rx/vls 

l 4 @ 

k 

s 5 

hw t 4 At 5 1 Lo vx!w/ doruv/ srxu frqfoxuh/ g*dssoltxhu oh wkìruëph 614 dyhf V @ u 

l 4 ! >" s " 

>.4 4 s # 

>.4 @ 

V hw w u 

! ># 4 @ 4 s " 

1[ 

615 Xqh QV g*lghqwldelolwì gh od fodvvh ghv pìodqjhv qlv1 

*hvw hq 4 l > l 5 p >{5 U q / gh glvwulexwlrq grqqìh1 

61514 Wkìruëph gh \dnrzlw} hw Vsudjlqv1 


615 Xqh QV g*lghqwldelolwì gh od fodvvh ghv pìodqjhv qlv1 76 

Wkìruëph 615= Od fodvvh gh wrxv ohv pìodqjhv qlv gh x hvw lghqwldeoh vl hw vhxohphqw vl x hvw xqh idplooh 

olqìdluhphqw lqgìshqgdqwh vxu U : 1 

Gìprqvwudwlrq = Dsshqglfh G1 [ 

Gìfrxodqw gx idlw txh x hvw olqìdluhphqw lqgìshqgdqwh vl hw vhxohphqw vl vrq lpdjh sdu xq lvrprusklvph hvw 

olqìdluhphqw lqgìshqgdqwh vxu o*hvsdfh lpdjh/ rq d oh fruroodluh = 

ruroodluh 614= Odfodvvhghwrxvohvpìodqjhvqlvghx hvw lghqwldeoh vl hw vhxohphqw vl o*lpdjh gh x sdu xq 

lvrprusklvph yhfwrulho hvw olqìdluhphqw lqgìshqgdqwh vxu o*hvsdfh lpdjh1 

h uìvxowdw shuphw gh prqwuhu o*lghqwldelolwìv ghv fodvvhv gh pìodqjhv qlv gh soxvlhxuv idploohv sdudpìwultxhv1 

61515 Dssolfdwlrqv gx wkìruëph gh \dnrzlw} hw Vsudjlqv1 

Juåfh dx wkëruëph 615 hw ã vrq fruroodluh/ lo hvw srvvleoh g*ìwdeolu ohv 8 sursrvlwlrqv vxlydqwhv/ grqw qrxv qh 

grqqhurqv sdv ohv gìprqvwudwlrqv1 

Sursrvlwlrq 618= Vl n hvw xq hqwlhu srvlwli hw x 4 od idplooh ghv surgxlwv gh n irqfwlrqv gh uìsduwlwlrq gh od orl 

h{srqhqwlhooh/ od fodvvh K 4 gh wrxv ohv pìodqjhv qlv gh x 4 hvw lghqwldeoh1 

Sursrvlwlrq 619= Odidploohx 5 ghv irqfwlrqv gh uìsduwlwlrq jdxvvlhqqhv n0glphqvlrqqhoohv jìqëuh xqh fodvvh K 5 

gh pìodqjhv qlv lghqwldeoh1 

Sursrvlwlrq 61:= Vlx 6 @ x 4 ^ x 5 / od fodvvh K 6 gh wrxv ohv pìodqjhv qlv g*ìoìphqwv gh x 6 hvw lghqwldeoh1 

Sursrvlwlrq 61;= Od idplooh x 7 ghv glvwulexwlrqv elqrpldohv qìjdwlyhv/ qrq gìjìqìuìhv/ hqjhqguh xqh fodvvh K 7 

gh pìodqjhv qlv lghqwldeoh1 

Sursrvlwlrq 61 , @ 

 

5 . +{ , 5>{5 U/ 5 U. / 5 U/ 

&S 

. x hvw xqh idplooh olqìdluhphqw lqgìshqgdqwh vxu -/ vl& M ( % M - (E@ c uuuc@ & M - & ( ' c 8 ' 8 c @ 8 E% 'fi 

E@ c uuuc@ & 'Efc uuuc f1 

' 


77 616 Pìodqjhv qlv gh phvxuhv surgxlwv1 

hw vd irqfwlrq fdufwìulvwltxh hvw = 

+w> > , @h{s+lw mwm, >w5 U/ 5 U . / 5 U1 

Od fruuhvsrqgdqfh 4 = i :$ hvw olqìdluh1 Od fruuhvsrqgdqfh 5 = :$ ! /dyhf 

! +w> > , @ +w> > , /vlw 3> 

@ 3 /vlqrq/ 

hvw hooh dxvvl olqìdluh1Vl 6 dvvrflh ã 3 vd wudqvirupìh gh Odsodfh/ 6 hvw dxvvl xqh dssolfdwlrq olqìdluh1 Rq d grqf/ 

;v 5 G ; /vle @ l/ 

6 5 4 +i +{> > ,, @ +i +{> > ,, @ +v . e, 4 >5 U . / 5 U/ 

shxw íwuh ìwhqgxh ã xq lvrprusklvph vxu x 9 1 

Gh soxv/ vl o*rq d 

q[ 

q_ 

f m +v . e m , 4 / v 5 G @ G m / 

m@4 

m@4 

^61

617 G*dxwuhvpìodqjhvqlvsduwlfxolhuv1 78 

617 G*dxwuhvpìodqjhvqlvsduwlfxolhuv1 

G*dxwuhv wkìruëphv g*lghqwldelolwì vrqw gìyhorssìv gdqv ghv fdv wuëv sduwlfxolhuv/ txh qrxv qh ihurqv txh flwhu 

lfl1 Gdqv Îudvhu;4`/ rq prqwuh o*lghqwldelolwì gh od fodvvh ghv pìodqjv qlv gh glvwulexwlrqv gh Yrq0Plvhv hw gdqv 

^Nhqw;6`/ lo hvw prqwuì txh gdqv xqh judqgh pdmrulwì ghv fdv/ od fodvvh ghv pìodqjhv qlv ghv glvwulexwlrq gh grqqìhv 

gluhfwlrqqhoohv hvw lghqwldeoh1 


7: 

0kdslwuh70 

Lqwurgxfwlrq ghv pìodqjhv jìqìudolvìv1 

Qrxv dyrqv mxvtx*lfl ìyrtxì ohv pìodqjhv gh glvwulexwlrqv wrxwhv lvvxhv gh od píph idplooh sdudpìwultxh1 Ohv 

pìodqjhv idlvdqw lqwhuyhqlu ghv idploohv sdudpìwultxhv glìuhqwhv vrqw dfwxhoohphqw ã o*ìwxgh hq vwdwlvwltxh ds0 

soltxìh/ qrwdpphqw gdqv oh wudlwhphqw vwdwlvwltxh ghv lpdjhv lvvxhv gh od wìoìgìwhfwlrq1 Qrxv whqwrqv lfl gh grqqhu 

xqfdguhwkìrultxhãfhw|shghpìodqjh1 

714 Gìqlwlrq jìqìudoh 

Vrlw +[> K,/ xq hvsdfh pìwultxh/ hw vrlw S/ o*hqvhpeoh ghv suredelolwìv/ S / vxu fhw hvsdfh1 Qrwrqv / xqhwulex 

gìqlh vxu S1 Vrlw/ doruv/ J +S,/ od fodvvh ghv suredelowìv/ S/ vxu+S> , hw vrlw S 3 / xq vrxv hqvhpeoh gh S1 Rq 

gìqlw/ doruv/ od fodvvh K +S 3 ,/ ghv suredelolwìv K vxu +[> K,/ sdu od fruuhvsrqgdqfh = 

P = J +S, $K+S 3 , 

] 

S :$ K/ whoohtxhK +D, @ S +D, gS +S , / D 5 K= 

S ! 

^7143` 

Xq ìoìphqw gh K hvw dsshoì S0pìodqjh gh S rx soxv vlpsohphqw pìodqjh jìqìudolvì1 

hwwh gìqlwlrq ghv pìodqjhv hvw ã udssurfkhu gh fhooh gh Wdoolv hw khvvrq h{srvìh gdqv ^Wdoolv;5`1 Qrxv qh 

o*xwlolvrqv/ srxuwdqw sdv gdqv oh píph exw/ hw q*h{sorlwrqv sdv ohv wkìruëphv g*lghqwldelolwì suìvhqwìv sdv Wdoolv 

hw khvvrq/ wurs frpsoh{hv ã phwwuh hq rhxyuh1 

Vl S 3 hvw xqh uìxqlrq qlh rx gìqrpeudeoh gh idploohv sdudpìwultxhv glvmrlqwhv/ S 3 @ .4 V 

S l / o*ìfulwxuh gh ^7143` 

l@4 

vh vlpsolh hq = 

K +D, 

@ 

Q[ 

l@4 

] 

S +S l , S +D, gS +S mS l , / 

S l 

Lqwurgxfwlrq ghv pìodqjhv jìqìudolvìv1

7; 715 Pìodqjhv qlv jìqìudolvìv1 

rx K +D, 

@ 

.4[ 

l@4 

] 

S +S l , S +D, gS +S mS l , / D 5 K 

S l 

hw vl o*rq qrwh S l @ S + mS l ,/ 

K +D, 

rx K +D, 

@ 

@ 

Q[ 

l@4 

.4[ 

l@4 

] 

S +S l , S +D, gS l +S , / 

S l 

] 

S +S l , S +D, gS l +S , / D 5 K> 

S l 

^7144` 

^7145` 

fh txl qrxv grqqh xqh vrpph gìqrpeudeoh srqgìuìh gh pìodqjhv fodvvltxhv/ rx hqfruh xq pìodqjh gìqrpeudeoh 

gh pìodqjhv1 

715 Pìodqjhv qlv jìqìudolvìv1 

Vl/ srxu xq pìodqjh gx w|sh ^7144` ohv S l qh fkdujhqw/ dyhf ghv srlgv srvlwliv/ tx*xq qrpeuh ql g*ìoìphqwv gh 

S l / qrxv vrpphv hq suìvhqfh g*xq pìodqjh ql jìqìudolvì1 

Vrlw/ srxu n 3 hw q/ hqwlhuvsrvlwliv/hwsrxuwrxwl/ 4 l n 3 / ghv hqwlhuv qrq qxov p l / dlqvl txh ghv idploohv gh 

glvwulexwlrqv sdudpìwultxhv glvmrlqwhv gh irqfwlrqv gh uìsduwlwlrq I l vxu U q / 

x l q @ I l {> l >{5 U q > l 5 pl 

> 

lqgh{ìhv uhvshfwlyhphqw sdu l 5 p l / vrxv hqvhpeoh eruìolhq gx p l0hvsdfh hxfolglhq U p l 1 I l {> l hvw phvxudeoh 

vxu U q p l 

1 Rq shxw doruv gìqlu od idplooh 

I 

q>n! @ 

txl hvw xqh idplooh gh glvwulexwlrqv gh U q 1 

Srxu Q/ xq hqwlhu srvlwli hw wrxw m/ 4 m Q/ rq gìqlw xqh vxlwh +n m , m 

g*hqwlhuv srvlwliv/ lqiìulhxuv rx ìjdx{ 

 

ã n 3 hw rq vh grqqh f m > nm 

m 5 U . p n m 1 Od irqfwlrq gh uìsduwlwlrq K vxu U q gìqlh sdu = 

^n ! 

l@4 

x l q> 

K +{, @ 

Q[ 

f m I n m 

m@4 

{> n m 

m 

 

/dyhf 

Q[ 

f m @41 

m@4 

^7146` 

hvw xq pìodqjh ql jìqìudolvì1 Rq yìulh dlvìphqw txh wrxw pìodqjh gh od iruph ^7146` shxw vh phwwuh vrxv od iruph 

^7144`/ hw ^7146` gìfulw grqf od fodvvh gh wrxv ohv pìodqjhv qlv jìqìudolvìv gh glwulexwlrqv gh I q>n! 1 

Lqwurgxfwlrq ghv pìodqjhv jìqìudolvìv1

0kdslwuh80 

V|vwëph ghv glvwulexwlrqv gh Shduvrq1 

Dq gh grqqhu xq fdguh suìflv ã qrwuh ìwxgh/ qrxv dyrqv fkrlvl gh qrxv frqvdfuhu ã o*ìwxgh ghv pìodqjhv qlv sulv 

gdqv od idplooh ghv glvwulexwlrqv gh Shduvrq1 h v|vwëph/ gìql sdu xqh ìtxdwlrq glìuhqwlhooh ã txdwuh sdudpëwuhv/ 

frqwlhqw ohv sulqflsdohv glvwulexwlrqv vwdwlvwltxhv hw vh suëwh sduidlwhphqw dx fdguh dssoltxì gh qrwuh ìwxgh1 


Xqh ghqvlwì i/ vxuU/ dssduwlhqw dx v|vwëph ghv ghqvlwìv gh Shduvrq/ vl hooh hvw vroxwlrq gh o*ìtxdwlrq glìuhqwlhooh 

ã txdwuh sdudpëwhv = 

4 gi +{, 

i+{, g{ 

@ 

{ . d 

 

f 3 . f 4 { . f 5 { 5 

rü d> f 3 >f 4 >f 5 vrqw ohv sdudpëwuhv1 

^8147` 

*hvw xqh ìtxdwlrq glìuhqwlhooh grqw od iruph ghv vroxwlrqv hvw iruwhphqw olìh ã o*h{lvwhqfh hw dx w|sh ghv 

vroxwlrqv gx sro|q÷ph S +{, @ f 3 . f 4 { . f 5 { 5 1 Rq qh shxw/ sdu frqvìtxhqw/ sdv grqqhu xqh iruph jìqìultxh 

ghv vroxwlrqv1 Rq qh v*lqwìuhvvh/ gh soxv/ tx*dx{ vroxwlrqv txl vrqw ghv ghqvlwìv1 Od glvfxvvlrq txl vxlw/ shuphw 

g*rugrqqhu wrxwhv fhv ghqvlwìv1

83 815 Ghqvlwìv dssduwhqdqw dx v|vwëph gh Shduvrq1 

815 Ghqvlwìv dssduwhqdqw dx v|vwëph gh Shduvrq1 

Rq shxw/ doruv/ uìsduwlu ohv ghqvlwìv vroxwlrqv i l>4l;> hq kxlw idploohv glwlqfwhv iI 4 > ===> I ; j1 Ohv qrwdwlrqv/ txl 

uhsuhqqhqw fhoohv gh Shduvrq/ qh vrqw sdv fhoohv xwlolvìhv frxuudphqw/ pdlv rqw ìwì lqwurgxlwhv vrxv fhwwh iruph sdv 

o*dxwhxu1 

Qrxv gìwdloorqv fhv idploohv hq idlvdqw oh olhq dyhf od glvfxvvlrq vxu ohv udflqhv gh S +[, shuphwwdqw od uìvroxwlrq 

gh o*ìtxdwlrq glìuhqwlhooh1 

Od idplooh I 7 fruuhvsrqg dx fdv rü S +{, q*d sdv gh udflqhv uìhoohv1 Vhv ìoìphqwv rqw ghv glwulexwlrqv grqw od 

ghqvlwì yìulh = 

 

k 

i 7 +{, @ N 7 3 . f 5 +{ . 4 , 5l " 

t 

#B # 

h{s s d" 

!f 

dufwdq f# 

# ! 

+{ . 4 , > 

dyhf 3 @ f 3 f# " 

7f # 

/ 4 @ f" 

5f # 

/ 

Ohv ìoìphqwv gh od idplooh I : vrqw ghv glvwulexwlrqv gh ghqvlwìv= 

 

 

i : +{, @ N : f3 . f 5 { 5 " 

 

t #B # h{s s d 

f!f # 

dufwdq f# 

Hoohv vrqw rewhqxhv gdqv oh vrxv fdv gh od idplooh I 7 srxu ohtxho f 4 @31 

N 7 hw N : vrqw ohv frqvwdqwhv gh qrupdolvdwlrq N l @ U LU i l +{, g{ / l @7> : 1 

hv ghx{ fdv vrqw ohv prlqv lqwìuhvvdqwv hq sudwltxh/ vl o*rq h{fhswh od orl N > ^8148` gh dxfk| hw od orl fhqwudoh 

gh Vwxghqw w > /txlvrqwghvorlvghw|shLY 1 

Ohv dxwuhv glvwulexwlrqv vrqw wuëv xwlolvìhv hq vwdwlvwltxh/ hw rqw xq uìho lqwìuíw sudwltxh1 Qrxv suìvhqwrqv fhv 

glvwulexwlrqv vrxv ohxu iruph kdelwxhooh/ hq h{sulpdqw ohv sdudpëwuhv fodvvltxhv hq irqfwlrq ghv sdudpëwuhv gh 

o*ìtxdwlrq glìuhqwlhooh1 

Od idplooh I 4 XI 5 hvw od idplooh ghv orlv ehwdv/ ghghqvlwì= 

f ! 

{ 

 

> 

i 4>5 +{, @ 4 +{e ", I" +e #{, J" 

E+s>t, +e #e ", I.J4 Lê" >e #`+{, > 

dyhf e 4 @ 4 

5f # 

f 4 . s 

/ e 5 @ 4 

5f # 

f 4 s 

/ s @ d.e" 

f # +e # e " , .4/ t @ d.e# 

f # +e # e " , .4/ 

rü @f 5 4 7f 3 f 5 1 

hv ghx{ w|shv fruuhvsrqghqw dx fdv rü S +{, d ghx{ udflqhv uìhoohv gh vljqhv rssrvìv/ I 5 ìwdqw oh fdv rü s @ t1 

Od idplooh I 9 / srxu odtxhooh S +{, d ghx{ udflqhv uìhoohv/ glvwlqfwhv hw gh píph vljqh/ hvw od idplooh ghv orlv 

ehwdv gx vhfrqg w|sh1 Ohxu ghqvlwì hvw gh od iruph = 

i 9 +{, @ 

vJ +{u, I" 

E+s>t, +{+uv,, 

Lû>.4^+{, > 

I.J 

dyhf v @ s 

f # 

/ u @ 4 

5f # 

f 4 s 

/ s @ d.u 

f # v .4/ t @ 4 f # 

4 1 

Od idplooh I 8 hvw od idplooh ghv orlv lqyhuvh0jdppdv/ rewhqxh gdqv oh fdv rü S +{, hvw xq fduuì sduidlw/ sursrvh 

ghv glvwulexwlrqv gh ghqvlwì = 

 

i 8 +{, @ 4 +{u, J" 

+t, s 

h{s 5 

J 

s+{u, 

Lû>.4^+{, > 

V|vwëph ghv glvwulexwlrqv gh Shduvrq1

816 Judskh gh Shduvrq1 84 

dyhf s @ f# 

d B " 

#B # 

/ t @ 4 f # 

4 / u @ f" 

5f # 

1 

Lo uhvwh/ hqvxlwh/ ohv fdv rü S +{, q*hvw sdv gx vhfrqg ghjuì1 

Vl f 5 @3pdlv txh f 4 9@3rq rewlhqw od idplooh I 6 hvw od idplooh ghv orlv jdppdv/ ghghqvlwì 

 

dyhf s @ f 4 / t @ 4 f! 

f " f " 

d .4/ u @ f ! 

f " 

1 

 

i 6 +{, @ 4 +{u, J" 

+t, s 

h{s 

J 

+{u, 

s 

 

Lû>.4^+{, > 

Hw srxu qlu/ vl f 5 @ f 4 @3rq dxud od idplooh I ; hvw od idplooh ghv orlv qrupdohv/ ghghqvlwì 

dyhf @ d / 5 @ f 3 1 

 

i ; +{, @ s 4 h{s 5 # 

+{,# 

5 # 

 

> 

Od gìqlwlrq ghv glvwulexwlrqv gx v|vwëph gh Shduvrq sdu od píph ìtxdwlrq glìuhqwlhooh shuphw gh gìwhuplqhu 

fhv glvwulexwlrqv xqltxhphqw ã sduwlu gh ohxu txdwuh suhplhuv prphqwv1 hwwh sursulìwì/ txh qrxv gìyhorssrqv 

gdqv oh sdudjudskh vxlydqw/ grqqh wrxw vrq lqwìuíw ã fhwwh idplooh/ srxu od sureoìpdwltxh txl qrxv lqwìuhvvh1 

816 JudskhghShduvrq1 

Vrlw [ xqh yduldeoh doìdwrluh grqw od ghqvlwì dssduwlhqw dx v|vwëph gh Shduvrq hw frqvlgìurqv vrq prphqw 

g*ruguh 4/ 

hw srxu t @5> 6> 7> ohv prphqwv fhqwuìv = 

4 @ H^[` 

juåfh dx{txhov rq gìqlw ohv frh!flhqwv/ 

t @ H^+[ H^[`, t`> 

4 @ + 6, 5 

+ 5 , 6 hw 5 @ 7 

+ 5 , 5 > 

rü s 4 hvw dsshoìh vnhzqhvv hw 5 nxuwrvlv1 

Ohv txdwuh frh!flhqwv gh o*ìtxdwlrq glìuhqwlhooh ^8147` vrqw doruv uholìv ã 4 hw 5 sdu ohv uhodwlrqv= 

d 

@ 

+ 5 .6, s 4 5 

43 5 45 4 4; 4> 

f 3 @ 5+7 5 6 4 , 4 + 5 .6, s 4 5 . + 4 , 5 +5 5 6 4 9, 

> 

43 5 45 4 4; 


85 817 rpsdudlvrq hqwuh od sdudpìwulvdwlrq fodvvltxh hw fhooh gx v|vwëph gh Shduvrq1 

f 4 @ + 5 .6, s 4 5 5 4 +5 5 6 4 9, 

43 5 45 4 4; 

hw f 5 @ +5 5 6 4 9, 

43 5 45 4 4; > 

fh txl shuphw gh glvfxwhu ohv vroxwlrqv gh ^8147` vxlydqw ohv ydohxuv gh 4 / 5 hw gx sdudpëwuh / shuphwwdqwgh 

glvfxwhu ohv glìuhqwhv vroxwlrqv gh f 3 . f 4 { . f 5 { 5 / 

@ f5 4 

 

@ 

4 + 5 .6, 5 

7f 3 f 5 7+7 5 6 4 ,+5 5 6 4 ,+5 5 6 4 9, = 

Rq uhwurxyh/ doruv/ ohv ghqvlwìv vroxwlrqv i l>4l;> hw ohv kxlw idploohv iI 4 > ===> I ; j/ gìqlhv suìfhghpphqw = 

ii 4 5 I 4 j +, i3j > 

ii 5 5 I 5 j +, i @3dyhf i 4 @3hw 5 6jj > 

ii 6 5 I 6 j +, i hvw 4 fdu i5 5 6 4 9@3jj > 

ii 7 5 I 7 j +, i3 4j > 

ii 8 5 I 8 j +, i @4j > 

ii 9 5 I 9 j +, iA4j > 

ii : 5 I : j +, i @3dyhf i 4 @3hw 5 A 6jj > 

ii ; 5 I ; j +, i @3dyhf i 4 @3hw 5 @6jj = 

Od frqqdlvvdqfh ghv prphqwv 4 > 5 > 6 hw 7 shuphw grqf gh gìwhuplqhu/ od vrxv idplooh I l ã odtxhooh dssduwlhqw od 

glvwulexwlrq/ dlqvl txh ohv sdudpëwuhv gh fhwwh glvwulexwlrq1 Qrwrqv/ gh soxv/ tx*rq qh shxw mdpdlv dyrlu 5 4 4 

31 

Oh judskh gh Shduvrq/ yrlu Ilj1 L14/ h{sulpdqw 5 hq irqfwlrq gh 4 prqwuh od uìsduwlwlrq ghv glìuhqwhv idploohv 

gh glvwulexwlrqv gdqv oh sodq + 4 > 5 ,1 

817 rpsdudlvrq hqwuh od sdudpìwulvdwlrq fodvvltxh hw fhooh 

gx v|vwëph gh Shduvrq1 

Gdqv od phvxuh rü qrxv vrxkdlwrqv ìwxglhu o*lghqwldelolwì gh pìodqjhv gh glvwulexwlrqv gx v|vwëph gh Shduvrq/ 

hw txh fhwwh lghqwldelolwì gìshqg gh od sdudpìwulvdwlrq fkrlvlh/ lo hvw qìfhvvdluh/ vl qrxv yrxorqv xwlolvhu ohv uìvxowdwv 

frqqxv gdqv oh fdv gh sdudpìwulvdwlrq fodvvltxh ghv glvwulexwlrqv/ gh frpsduhu ohv ghx{ sdudpìwulvdwlrqv1 Qrxv 

gìyhorssrqv lfl fhwwh frpsdudlvrq1 

Qrxv uhwhqrqv/ srxu I 7 ohv glvwulexwlrqv gh fdxfk| txl vrqw gh od iruph = 

 

N > +{, @ 

 

5 . +{ , 5>{5 U/ 5 U. / 5 U/ ^8148` 

txl fruuhvsrqg dx fdv rü f 5 @3=8/ f 4 @ / d @ f 4 hw f 3 @ 5 . # 

5 

1 Wrxwhv ohv glvwulexwlrqv gh dxfk| vrqw grqf 

frqwhqxhv xqh hw xqh vhxoh irlv gdqv oh v|vwëph gh Shduvrq vrxv od sdudpìwulvdwlrq = 

d 5 U/ f 5 @3=8/ f 4 @ d hw f 3 d5 

5 1 


817 rpsdudlvrq hqwuh od sdudpìwulvdwlrq fodvvltxh hw fhooh gx v|vwëph gh Shduvrq1 86 

Type_V 

15 

Type_IV 

Type_VII 

Type_III 

10 

Distribution_Normale 

Type_VI 

5 

Type_I 

0 

Type_II 

Limite d ’existence des 

distributions. 

0 1 2 3 4 5 6 

Ilj1L1410JudskhghShduvrq1 

Srxu ohv glvwulexwlrqv gh Vwxghqw/ txl dssduwlhqqhqw dxvvl ã I 7 /qrxvdyrqv= 

w > +{, @ # 

.4 

5 

s 

 

 

4. 

5 

$ 

." 

# 

+{ ,5 

> 

 

dyhf d @ / f 5 @ 4 

.4 / f 4 @5df 5 hw f 3 @4.f 5 

 

d 5 4 1 rpph srxu od idplooh gh glvwulexwlrqv suìfìghqwh/ wrxwh 

glvwulexwlrq gh Vwxghqw hvw frqwhqxh xqh hw xqh vhxoh irlv gdqv od idplooh ghv glvwulexwlrqv gh Shduvrq1 Qrwrqv txh 

fhuwdlqhv glvwulexwlrqv dssduwlhqqhqw ã od irlv ã od idplooh ghv glvwulexwlrqv gh dxfk| hw ã fhooh ghv glvwulexwlrqv 

gh Vwxghqw/ txl rqw xqh lqwhuvhfwlrq qrq ylgh/ elhq tx*dxfxqh qh frqwlhqqh o*dxwuh1 

Srxu ohv glvwulexwlrqv ehwdv gh I 4 XI 5 rq d = 

4 +{ e 4 , s4 +e 5 {, t4 

i 4>5 +{, @ 

E+s> t, +e 5 e 4 , s.t4 Lê" >e #`+{,> 

rü e 4 @ 4 

5f # 

f 4 . s 

/ e 5 @ 4 

5f # 

f 4 s 

/ s @ d.e" 

f # +e # e " , .4/ t @ d.e# 

f # +e # e " , .4hw @f5 4 7f 3 f 5 1 

Vljqdorqv txh/ srxu txh ohv udflqhv e 4 hw e 5 gh S +[, vrlhqw gh vljqhv rssrvìv/ lo hvw qìfhvvdluh txh f 3 hw f 5 

vrlhqw gh vljqhv rssrvìv1 


87 817 rpsdudlvrq hqwuh od sdudpìwulvdwlrq fodvvltxh hw fhooh gx v|vwëph gh Shduvrq1 

Sdu lqyhuvlrq ghv irupxohv gh sdudpìwulvdwlrq fl0ghvvxv rq rewlhqw = 

f 5 @ 

4 

5 +t . s, >f 4 @ e 4 . e 5 

5 +t . s, >f 3 @ 

e 4 e 5 

5 +t . s, hw d @ e 4 +t 4, . e 5 +s 4, 

/ 

5 +t . s, 

fh txl dvvxuh xqh fruuhvsrqgdqfh elmhfwlyh gh o*hqvhpeoh ghv glvwulexwlrqv ehwd gh w|sh 4 gx v|vwëph gh Shduvrq vxu 

o*hqvhpeoh ghv glvwulexwlrqv ehwd fodvvltxhv grqw ohv eruqhv gx vxssruw vrqw gh vljqhv rssrvìv hw grqw od vrpph 

ghv sdudpëwuhv s hw t q*hvw sdv ìjdoh ã 51 

Srxu ohv glvwulexwlrqv ehwd gh od idplooh I 9 /odghqvlwìhvwghodiruph= 

i 9 +{, @ 

vt +{ u, s4 

Lû>.4^+{,> 

E+s> t, +{ +u v,, 

s.t 

dyhf v @ s 

f # 

/ u @ 4 

5f # 

f 4 s 

/ s @ d.u 

f # v .4/ t @ 4 f # 

4 /rü @f 5 4 7f 3 f 5 1 

rqwudluhphqw dx fdv suìfìghqw/ lo idxw txh ohv udflqhv u hw u v gh S +[, vrlhqw gh píph vljqh/ +pdlv sdv 

ìjdohv,/ fh txl hvw oh fdv vl hw vhxohphqw vl f 5 hw f 3 vrqw gh píph vljqh1 

Sdu lqyhuvlrq ghv irupxohv rq rewlhqw = 

f 5 @ 4 

t .4 / f 4 @ 5u v 

t .4 / f u +u v, 

3 @ hw d @ 

sv 

t .4 t .4 u/ 

fh txl dvvxuh xqh fruuhvsrqgdqfh elmhfwlyh gh o*hqvhpeoh ghv glvwulexwlrqv ehwd gh w|sh 5 gx v|vwëph gh Shduvrq vxu 

o*hqvhpeoh ghv glvwulexwlrqv ehwd gìjìqìuìhv grqw ohv eruqhv gx vxssruw vrqw gh píph vljqh hw grqw oh sdudpëwuh 

t q*hvw sdv ìjdoh ã 41 

Oh ghuqlhu fdv rü oh sro|q÷ph S d ghv udflqhv hvw fhoxl rü ohv ghx{ udflqhv vrqw lghqwltxhv hw rü o*rq hvw hq 

suìvhqfh gh glvwulexwlrqv lqyhuvh jdppd gh ghqvlwì = 

dyhf s @ f # 

d.u / t @ 4 f # 

4 / u @ f " 

5f # 

1 

i 8 +{, @ 4 +{ u, t4 

5 

+t, s t h{s 

s +{ u, 

Lû>.4^+{,/ 

Qrwrqv/ srxu frpphqfhu/ txh f 3 q*lqwhuylhqw sdv gdqv od uhsdudpìwulvdwlrq1 rpph qrxv vrpphv gdqv oh fdv 

rü oh glvfulplqdqw gx elq÷ph hvw qxo rq d = 

fh txl grqqh/ srxu od uhsdudpìwulvdwlrq = 

f 5 4 7f 3 f 5 @3/ f 3 @ f5 4 

7f 5 

> 

f 5 @ 4 

t .4 / f 4 @ 5u 

t .4 / d @ 4 

s +t .4, u hw f 3 @ 

u5 

t .4 = 

Od fruuhvsrqgdqfh hvw elmhfwlyh fdu sA3 hw tA31 

Sdvvrqv pdlqwhqdqw dx{ glvwulexwlrqv jdppdv gx v|vwëph gh Shduvrq/ grqw od ghqvlwì hvw grqqìh sdu = 

i 6 +{, @ 4 +{ u, t4 

+{ u, 

+t, s t h{s Lû>.4^+{,/ 

s 


817 rpsdudlvrq hqwuh od sdudpìwulvdwlrq fodvvltxh hw fhooh gx v|vwëph gh Shduvrq1 88 

 

dyhf s @ f 4 / t @ 4 f! 

f " f " 

d .4/ u @ f! 

f " 

1 

Lo q*| d lfl txh wurlv sdudpëwuhv gdqv ohv ghx{ sdudpìwulvdwlrqv +sxlvtxh f 5 @3,1 Rq ìwdeolw/ frpph suìfìghp0 

phqw ohv fruuhvsrqgdqfhv = 

f 4 @ s/ f 3 @ su hw d @ s +t 4, u> 

fh txl prqwuh/ ã qrxyhdx o*ìtxlydohqfh hqwuh od fodvvh I 6 ghv glvwulexwlrqv gh Shduvrq hw od idplooh ghv glvwulexwlrqv 

jdppdv +ã wurlv sdudpëwuhv,1 

Hqq/ srxu od idplooh ghv orlv qrupdohv/ gh ghqvlwì= 

i ; +{, @ 

# $ 

4 

s h{s +{ ,5 

 

5 

5 5 5 / 

dyhf @ d / 5 @ f 3 1 Od fruuhvsrqgdqfh hvw fodluhphqw elmhfwlyh +sxlvtxh f 4 hw f 5 vrqw qxoohv,/ hqwuh od idplooh 

ghv glvwulexwlrqv qrupdohv fodvvltxhv hw fhoohv gx v|vwëph gh Shduvrq1 


8: 

0kdslwuh90 

Lghqwldelolwì ghv pìodqjhv qlv gh glvwulexwlrqv 

gx v|vwëph gh Shduvrq1 

rpswh whqx gh o*ìtxlydohqfh ghv sdudpìwulvdwlrqv txh qrxv yhqrqv gh prqwuhu/ qrxv srxyrqv lqglìuhpphqw 

ìwxglhu ohv sureoëphv srvìv sdu o*lghqwldelolwì vxu od sdudpìwulvdwlrq fodvvltxh rx vxu fhooh gh Shduvrq1 Qrxv 

fkrlvlvvrqv gh oh idluh dyhf ohv yduldeohv fodvvltxhv1 

O*lghqwldelolwì ghv fodvvhv ghv pìodqjhv qlv jìqìudolvìv vh wudgxlw sdu o*xqlflwì gx qrpeuh Q gh frpsrvdqwhv 

gx pìodqjh/ dlqvl txh o*xqlflwì gh o*hqvhpeoh ghv sdudpëwuhv gx pìodqjh = 

q 

f m > n m 

m 

 

r 

5 `3> 4^ nm / 4 m Q = 

1 

Gdqv oh fdv ghv pìodqjhv jìqìudolvìv/ od txhvwlrq gh o*lghqwldelolwì hvw oh soxv vrxyhqw uëjoìh sdu o*ìwxgh gx 

frpsruwhphqw dx{ eruqhv ghv vxssruwv ghv glìuhqwhv idploohv sdudpìwultxhv hq suìvhqfh1 

914 Xq fdv sduwlfxolhu1 

O*lghqwldelolwì gh wrxv ohv pìodqjhv qlv gh glvwulexwlrqv gx v|vwëph gh Shduvrq/ q*hvw sdv yìulìh1 *hvw oh 

fdv/ frpph qrxv o*ìwdeolvvrqv lfl/ srxu ohv glvwulexwlrqv gx w|sh I 4 XI 5 1Hqhhw/rqd= 

g*rü 

+4 {, 5 @4 5{ . { 5 > 

{ s4 +4 {, t4 @ { s4 +4 {, t6 +4 {, 5 > 

Lghqwldelolwì ghv pìodqjhv qlv gh glvwulexwlrqv gx v|vwëph gh Shduvrq1

8; 915 Gìfrpsrvlwlrq ghv pìodqjhv qlv jìqìudolvìv1 

hw 

{ s4 +4 {, t4 @ { s4 +4 {, t6 5{ s +4 {, t6 . { s.4 +4 {,> 

vrlw 

{ s4 +4 {, t4 .5{ s +4 {, t6 @ { s4 +4 {, t6 . { s.4 +4 {, t6 = 

Rq d grqf = 

E+s> t, {s4 +4 {, t4 

E+s> t, 

.5E+s .4>t 5, {s +4 {, t6 

E+s .4>t 5, 

@ 

E+s> t 5, {s4 +4 {, t6 

E+s> t 5, 

.E+s .5>t 5, {s.4 +4 {, t6 

E+s .5>t 5, > 

^9149` 

hw hq lqwìjudqw ^9149` vxu ^3> 4`/ 

E+s> t, .5E+s .4>t 5, @ E+s> t 5, . E+s .5>t 5, @ V= 

O*ìjdolwì ^9149` qrxv grqqh o*ìjdolwì/ srxu wrxw tA5/ gh ghx{ pìodqjhv glìuhqwv gh ghx{ orlv ehwdv = 

s 4 

{ s4 +4 {, t4 

E+s> t, 

. +4 s 4 , {s +4 {, t6 

E+s .4>t 5, @ s { s4 +4 {, t6 

5 

. +4 s 5 , {s.4 +4 {, t6 

E+s> t 5, 

E+s .5>t 5, > 

rü s 4 @ 4 V E+s> t,/ 4 s 4 @ 4 V 5E+s .4>t 5,/ s 5 @ 4 V E+s> t 5, hw 4 s 5 @ 4 V 

E+s .5>t 5,1 

hfl frqwuhglw od sursulìwì g*lghqwldelolwì gh od fodvvh ghv pìodqjhv gh orlv ehwdv gx suhplhu w|sh1 Vljqdorqv 

tx*xqh gìprqvwudwlrq lghqwltxh shxw íwuh idlwh srxu ohv glvwulexwlrqv ehwdv gx ghx{lëph w|sh1 

Sursrvlwlrq 914= Ohv fodvvhv ghv pìodqjhv qlv ghv glvwulexwlrqv ehwdv gx suhplhu hw gx ghx{lìph w|sh qh vrqw 

sdv lghqwldeohv1 

915 Gìfrpsrvlwlrq ghv pìodqjhv qlv jìqìudolvìv1 

Srxu prqwuhu ghv sursulìwìv g*lghqwldelolwì ghv fodvvhv gh pìodqjhv qlv jìqìudolvìv/ lo idxw vìsduhu ohv gli0 

iìuhqwv w|shv gh glvwulexwlrqv hq suìvhqfh/ dq gh phwwuh hq ìylghqfh ohv vsìflflwìv gh fhv w|shv txl shuphwwurqw/ 

ìyhqwxhoohphqw/ gh frqfoxuh ã o*lghqwldelolwì1 

Srxu od foduwì gx sursrv/ frqvlgìurqv xq pìodqjh gh ghx{ w|shv gh glvwulexwlrqv1 h fdv v*ìwhqg qdwxuhoohphqw 

ã xq pìodqjh txhotxrqtxh1 Vxssrvrqv txh ohv q suhplëuhv glvwulexwlrqv vrqw wrxwhv gx píph w|sh hw txh ohv p 

vxlydqwhv vrqw gx vhfrqg w|sh1 Rq dxud doruv vl q.p S 

s l @4= 

l@4 

q.p 

[ 

l@4 

s l I l +=, @ 

q[ 

s l I l +=, . 

l@4 

q.p 

[ 

l@q.4 

s l I l +=, 

^914:` 


916 Vxssruw ghv ghqvlwìv hw lghqwldelolwì1 8< 

hw hq qrwdqw = 

q[ 

s l @ V hw 

l@4 

q[ 

s l @ V 3 /rqdV . V 3 @4= 

l@4 

^914:` ghylhqw grqf = 

q.p 

[ 

l@4 

s l I l +=, @ V 

q[ 

l@4 

q.p 

s l 

V I [ 

l +=, . V 3 

l@q.4 

s l 

V 3 I l +=, = 

Hq srvdqw/ 

K +=, @ 

q[ 

l@4 

s l 

V I l +=, hw K 3 +=, @ 

qrxv phwwrqv hq ìylghqfh xq pìodqjh ghv pìodqjhv K hw K 3 = 

q[ 

l@4 

s l 

V I l +=, > 

q.p 

[ 

l@4 

s l I l +=, @ VK +=, . V 3 K 3 +=, / V . V 3 @41 

Od uìflsurtxh hvw lppìgldwh1 Od gìprqvwudwlrq gx fdv jìqìudo hvw lghqwltxh1 

Sursrvlwlrq 915= Od fodvvh gh wrxv ohv pìodqjhv qlv g*xqh idplooh gh glvwulexwlrqv uìxqlvvdqw n w|shv glìuhqwv 

gh glvwulexwlrqv/ hvw od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh uìxqlvvdqw ohv n fodvvhv ghv pìodqjhv qlv vxu 

ohv idploohv gh glvwulexwlrqv gh fkdfxq ghv w|shv vìsduìphqw1 

Rq shxw ìwhqguh fhwwh sursrvlwlrq dx fdv rü od idplooh gh glvwulexwlrqv hvw frpsrvìh g*xqh lqqlwì gh w|shv 

glìuhqwv gh glvwulexwlrqv1 

Sursrvlwlrq 916= Vrlw +x d , d5D 

xqh idplooh gh idploohv gh glvwulexwlrqv lqglfìh sdu xq hqvhpeoh D1 Vrlw y d! od 

fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh x d! hw y D od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh V x d 1 

d5D 

Doruv od fodvvh y D hvw od fodvvh/ y/ ghwrxvohvpìodqjhvqlvghodidplooh V y d 1 

Gìprqvwudwlrq = Rq d fodluhphqw y y D 1 

Prqwurqv/ doruv txh wrxw ìoìphqw K D! gh y D hvw xq ìoìphqw 

V 

gh y1 K D! ìwdqw xq pìodqjh ql/ K D! hvw xq 

ìoìphqw gh od fodvvh y D! gh wrxv ohv pìodqjhv qlv gh od idplooh x d rü D 3 shxw íwuh/ vdqv shuwh gh jìqìudolwì/ 

d5D ! 

frqvlgìuì frpph xq hqvhpeoh ql/ 

V 

sxlvtxh oh pìodqjh hvw ql1 G*dsuëv od sursrvlwlrq 915/ y D! hvwodfodvvhghwrxv 

ohv pìodqjhv qlv gh od idplooh y d V y d 1Rq d grqf K D! 5 y D! y1 [ 

d5D ! d5D 

d5D 

916 Vxssruw ghv ghqvlwìv hw lghqwldelolwì1 

Rq vh sodfh gdqv oh fdguh g*xqh idplooh I gh glvwulexwlrqv wrwdohphqw rugrqqìh dx vhqv gh o*lqfoxvlrq1 Vl rq 

qrwh od uhodwlrq g*ruguh/ rq dxud/ srxu wrxw frxsoh +i 4 >i 5 , gh glvwulexwlrqv gh I = 

i 4 i 5 / V i" V i# dx vhqv vwulfw/ 


93 916 Vxssruw ghv ghqvlwìv hw lghqwldelolwì1 

rü V il hvwohvxssruwghi l 1 

Gdqv oh fdguh gh fh w|sh gh idploohv gh glvwulexwlrqv/ rq shxw ìwdeolu od sursrvlwlrq vxlydqwh = 

Sursrvlwlrq 917= Vrlw I xqh idplooh gh irqfwlrqv/ srvlwlyhv vxu ohxu vxssruw/ I ìwdqw wrwdohphqw rugrqqìh/ dx 

vhqv vwulfw/ dx pr|hq gh o*ruguh gìql fl0ghvvxv1 I hvw xqh idplooh oleuh1 

Gìprqvwudwlrq = Srxu od gìprqvwudwlrq/ qrxv dsshoohurqv V q oh vxssruw gh 

V 

gh i l /rqdV q @ q V il 1 Prqwurqv/ doruv/ sdu uìfxuuhqfh/ txh = 

l@4 

q S 

l@4 

d l i l +{, hw vl V il 

5 

6 

7 ;q 5 Q/ ;{ 5 V qS 

q/ d l i l +{, @3> 

8 

l@4 

@, ^;l> 4 l q/ rqdd l @3` 1 

dyhf ;l> 4 l q/ d l 5 U hw i l 5 I 

hvw oh vxssruw 

Dx udqj 4/ rqd;{ 5 V i" / 

d 4 i 4 +{, @3/dyhfd 4 5 U hw i 4 5 I / 

^914;` 

i 4 ìwdqw xqh irqfwlrq srvlwlyh vxu vrq vxssruw/ rq d qìfhvvdluhphqw d 4 @31 

Vxssrvrqv txh od sursulìwì hvw yudlh dx udqj q/ hw prqwurqv tx*hooh o*hvw wrxmrxuv dx udqj q .41 

rqvlgìurqv srxu fhod = ;{ 5 V q.4 = 

q.4 

[ 

d l i l +{, @3> dyhf ;l> 4 l q .4/ d l 5 U hw i l 5 I= 

l@4 

^914 

fh txl lpsoltxh/ i n! ìwdqw srvlwlyh vxu vrq vxssruw/ txh d n! @31 Rq wluh doruv gh ^914

917 Olhq hqwuh lghqwldelolwì hw qìjoljhdelolwì1 94 

Rq shxw dxvvl ìwhqguh ohv sursrvlwlrqv 616 hw 6171 Qrxv gìprqwurqv juåfh ã od sursrvlwlrq 917 od sursrvlwlrq 

jìqìudoh g*lghqwldelolwì ghv pìodqjhv qlv gh od idplooh ghv glvwulexwlrqv jdppdv1 

Sursrvlwlrq 918= Odfodvvhy gh wrxv ohv pìodqjhv qlv gh od idplooh = 

; 

 

]{ 

x @ 

= +{> s> t> u, @ 

4 

ghv glvwulexwlrqv jdppd hvw lghqwldeoh1 

+x u, t4 

+t,s t 

 

+x u, 

h{s Lû>.4^ +x, gx> 

s 

 

{ 5 U> u 5 U> 

s 5 U . > t 5 U . < 

@ 

> > 

Gìprqvwudwlrq = Xq ìoìphqw gh y vh gìfrpsrvh/ g*dsuëv od sursrvlwlrq 915/ hq xq pìodqjh gh pìodqjhv qlv/ 

K 4 > ===> K n / ghv fodvvhv y u" > ===> y un gh pìodqjhv ghv idploohv x u" > ===> x un / gh od sursrvlwlrq 616/ ohv u l ìwdqw glìuhqwv 

ghx{ ã ghx{1 kdtxh glvwulexwlrq K l hvw xqh glvwulexwlrq sdudpìwultxh srvlwlyh vxu vrq vxssruw hw od fodvvh y ul 

hvw lghqwldeoh1 Od idplooh ghv glvwulexwlrqv K l hvw rugrqqìh dx vhqv gìql gdqv od sursrvlwlrq 9171 G*dsuëv oh 

fruroodluh 914/ y hvw grqf lghqwldeoh1 [ 

Od sursrvlwlrq dvvxudqw gh o*lghqwldelolwì gh od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh lqyhuvh0jdppd vh 

gìprqwuh gh od píph idêrq = 

Sursrvlwlrq 919= Odfodvvhghy 3 wrxvohvpìodqjhvqlvghodidplooh= 

; 

 

]{ 

x 3 @ 

= Lqy +{> s> t> u, @ +x u, t4 

+t,s t 

4 

 

h{s 

 

5 

Lû>.4^ +x, gx> 

s +x u, 

 

{ 5 U> u 5 U> 

s 5 U . > t 5 U . < 

@ 

> > 

ghv glvwulexwlrqv lqyhuvh0jdppd hvw lghqwldeoh1 

Gìprqvwudwlrq = Od gìprqvwudwlrq hvw lghqwltxh ã od suìfìghqwh1 [ 

917 Olhq hqwuh lghqwldelolwì hw qìjoljhdelolwì1 

huwdlqhv frqglwlrqv vxu oh frpsruwhphqw olplwh ghv ghqvlwìv ghv glvwulexwlrqv gh od idplooh frqvlgìuìh/ dvvxuhqw 

o*lghqwldelolwì gh od fodvvh gh vhv pìodqjhv qlv1 

Sursrvlwlrq 91:= rqvlgìurqv xqh idplooh I d / d 5 U/ gh glvwulexwlrqv/ grqw ohv vxssruwv ghv ghqvlwìv rqw wrxv 

xqh eruqh d +qlh rx lqqlh,/ dx prlqv/ hq frppxq1 Vxssrvrqv txh ohv frpsruwhphqwv olplwhv +sdu h{hpsoh od 

gìfurlvvdqfh yhuv 3, ghv ghqvlwìv hq fhwwh eruqh vrqw wrwdohphqw rugrqqìv dx vhqv gh od qìjoljhdelolwì hw ghx{ ã ghx{ 

qrq ìtxlydohqwv1 

Od fodvvh gh wrxv ohv pìodqjhv qlv gh glvwulexwlrqv gh fhwwh idplooh hvw lghqwldeoh1 

Gìprqvwudwlrq = Prqwurqv/ sdu uìfxuuhqfh txh od idplooh I d hvw oleuh1 rqvlgìurqv od frpelqdlvrq olqìdluh gh q 

ghqvlwìv grqw ohv ylwhvvhv gh gìfurlvvdqfh vrqw ìfkhorqqìhv1 Vxssrvrqv/ vdqv shuwh gh jìqìudolwì/ txh 

hw prqwurqv txh 

i l 

d 

i l.4 > 4 l q 4 

;{ 5 G> 

q[ 

s l i l +{, @3+, s l @3> 4 l q= 

l@4 

^9153` 


95 917 Olhq hqwuh lghqwldelolwì hw qìjoljhdelolwì1 

Vl q @4od sursulìwì hvw lppìgldwh1 Vxssrvrqv ^9153` yudlh dx udqj q 41 Rqd= 

;{ 5 G> 

q[ 

s l i l +{, @3 

l@4 

+, ;{ 5 G> i q +{, 

+, ;{ 5 G> 

l@4 

# q4 

$ 

[ i l +{, 

s l 

i 

l@4 q +{, . s q @3> 

q4 

[ i l +{, 

s l 

i q +{, . s q @3= 

^9154` 

Hq sdvvdqw ã od olplwh/ hq d/ gdqv ^9154` rq d s q @3hw rq vh udpëqh dx udqj q 4 fh txl shuphw gh frqfoxuh1[ 

hwwh sursrvlwlrq shuphw g*dydqfhu vxu o*lghqwldelolwì ghv fodvvhv gh pìodqjhv qlv gh glvwulexwlrqv gh Vwxghqw1 

Hq hhw/ vl o*rq frqvlgëuh xqh fodvvh gh whoohv glvwulexwlrqv/ ã pr|hqqh 3 {ìh/ rq d srxu ìtxlydohqw hq .4 hw 4 = 

+.", 

.4 

5 5 

w ! > +{, s 

4 

 

{ +.4, = 

5 

hv ìtxlydohqwv vrqw wrwdohphqw rugrqqìv dx vhqv gh od sursrvlwlrq 91: hw o*rq hq wluh od sursrvlwlrq = 

Sursrvlwlrq 91;= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh gh Vwxghqw = 

hvw lghqwldeoh1 

; 

 

I @ 

= w >/ w > +{, @ # 

.4 

5 

s 

 

 

4. 

5 

$ < 

." 

# 

+{ ,5 

@ 

/ 5 Q 

 

> 

Od idplooh ghv glvwulexwlrqv ehwdv gh vxssruw {ì/ ^3> d`/ dA3/ hqwuh dxvvl gdqv oh fdguh ghv dssolfdwlrqv gh 

fhwwh sursrvlwlrq1 Hq hhw gh whoohv glvwulexwlrqv rqw srxu ghqvlwì = 

hw rqw grqf xq ìtxlydohqw hq d = 

hw xq ìtxlydohqw hq 3 = 

i d>s>t +{, @ 

4 

E+s> t, 

{ s4 +d {, t4 

d s.t4 

4 +d {, t4 

i d>s>t +{, 

d E+s> t, d t > 

4 { s4 

i d>s>t +{, 

3 E+s> t, d s = 

L^3>d`+{,> 

hv ìtxlydohqwv vrqw wrwdohphqw rgrqqìv dx vhqv gh od qìjoljhdelolwì/ hw o*rq shxw idluh ydorlu od sursrvlwlrq 91: 

srxu ìwdeolu txh = 

Sursrvlwlrq 91s! @ ii d>s! >t+{,>t A3j hw fhooh gh wrxv 

ohv pìodqjhv qlv gh od idplooh I d>t! @ ii d>s>t! +{,>sA3j ghv glvwulexwlrqv ehwdv L gh vxssruw ^3> d`/ dA3/ vrqw 

lghqwldeohv1 

Gìprqvwudwlrq = fhwwh sursrvlwlrq hvw xqh frqvìtxhqfh gluhfwh gh od sursrvlwlrq 91:1[ 


917 Olhq hqwuh lghqwldelolwì hw qìjoljhdelolwì1 96 

hwwh ghuqlëuh sursrvlwlrq qrxv dpëqh ã qrxv lqwìuhvvhu ã o*lghqwldelolwì ghv fodvvhv gh wrxv ohv pìodqjhv qlv 

gh od idplooh ghv idploohv gh glvwulexwlrqv +I d>s! , d5U . hw +I d>t! , d5U .1 Hq hhw fhv ghx{ idploohv hqwuhqw gdqv 

od fdwìjrulh ghv idploohv grqw ohv vxssruwv ghv ghqvlwìv vrqw wrwdohphqw rugrqqìv dx vhqv gh o*lqfoxvlrq1 Sdu xqh 

gìprqvwudwlrq dqdorjxh ã fhooh frqgxlwh gdqv oh fdguh ghv glvwulexwlrqv jdppdv hw lqyhuvh0jdppdv rq gìprqwuh/ 

juåfh ã od sursrvlwlrq 917 txh = 

Sursrvlwlrq 9143= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh +I d>s! , d5U . hvw lghqwldeoh1 

Sursrvlwlrq 9144= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh +I d>t! , d5U . hvw lghqwldeoh1 

Hw/ srxu ohv píphv udlvrqv = 

Sursrvlwlrq 9145= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh ghv glvwulexwlrqv ehwd L/ grqw oh sdudpëwuh s 

hvw {ì ã s 3 hw grqw od eruqh vxsìulhxuh gx vxssruw hvw 3/ hvwlghqwldeoh1 

Sursrvlwlrq 9146= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh ghv glvwulexwlrqv ehwd L/ grqw oh sdudpëwuh t 

hvw {ì ã t 3 hw grqw od eruqh vxsìulhxuh gx vxssruw hvw 3/ hvw lghqwldeoh1 

hv sursrvlwlrqv rqw srxu froorudluh/ txl qrxv lqwìuhvvh gluhfwhphqw = 

ruroodluh 915= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh ghv glvwulexwlrqv gh w|sh IL ^ ILL gx v|vwëph 

gh Shduvrq/ uhvwuhlqwh dx fdv f 3 @3hw f 4 3/ dyhfs rx t {ì/ hvw lghqwldeoh1 +fdv fruuhvsrqgdqw dx{ sursrvlwlrqv 

9143 hw 9144,1 

hw gh píph = 

ruroodluh 916= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh ghv glvwulexwlrqv gh w|sh IL ^ ILL gx v|vwëph 

gh Shduvrq/ uhvwuhlqwh dx fdv f 3 @3hw f 4 A 3/ dyhfs rx t {ì/hvw lghqwldeoh1 +fdv fruuhvsrqgdqw dx{ sursrvlwlrqv 

9145 hw 9146,1 

Gdqv oh fdguh gh o*ìwxgh g*lpdjhv uìhoohv +udgdu/ vdwwholwhv 111,/ ohv ìfkdqwloorqv vrqw irxuqlv vrxv od iruph gh 

qlyhdx{ gh julv/ txh o*rq shxw wrxmrxuv frqvlgìuhu frpph srvlwliv1 Lo hvw grqf udlvrqqdeoh gh v*lqwìuhvvhu dx{ 

glvwulexwlrqv ghv sursrvlwlrqv 9143 rx 9144 gdqv xq who fdguh1 

Dq gh frpsoìwhu o*lghqwldeolwì ghv pìodqjhv qlv ghv glvwulexwlrqv ehwdv/ lqwìuhvvrqv qrxv dx{ glvwulexwlrqv 

gh od idplooh I 9 1 *hvw od idplooh ghv glvwulexwlrqv gh w|sh = 

i 9 +{, @ 

vt +{ u, s4 

E+s> t, +{ +u v,, 

s.t 

Lû>.4^+{,= 

Vl o*rq wudydlooh vxu gh whoohv glvwulexwlrqv ã vxssruw {ì +v 3 hw u 3 {ìv,/ hw txh o*rq suhqg od suìfdxwlrq gh {hu 

dxvvl s 3 rx t 3 / rq prqwuh frpph suìfìghpphqw txh 

Sursrvlwlrq 9147= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh gh glvwulexwlrqv = 

I 9>u! >v ! >s ! 

@ 

 

i 9 +{, @ 

hwfhoohghwrxvohvpìodqjhvqlvghodidplooh 

vrqw lghqwldeohv1 

I 9>u!>v !>t ! 

@ 

i 9 +{, @ vt! 3 

E+s> t 3 , 

v t 3 +{ u 3 , s 

!4 

E+s 3 >t, +{ +u 3 v 3 ,, s !.t Lû !>.4^+{, 

+{ u 3 , s4 

 

+{ +u 3 v 3 ,, s.t! Lû !>.4^+{, 


97 918 Dssolfdwlrq ã o*ìwxgh ghv pìodqjhv jìqìudolvìv1 

Gìprqvwudwlrq = hwwh sursrvlwlrq hvw xqh dssolfdwlrq gluhfwh gh od sursrvlwlrq 91:1 [ 

Vl o*rq frqvlgëuh pdlqwhqdqw ohv idploohv ghv glvwulexwlrqv = 

I 9>v! >s ! 

@ 

 

i 9 +{, @ 

v t 3 +{ u, s!4 

 

E+s 3 >t, +{ +u v 3 ,, s !.t Lû>.4^+{, / v 3 hw s 3 vrqw {ìv/ 

hw 

I 9>v! >t ! 

@ 

i 9 +{, @ vt! 3 

E+s> t 3 , 

+{ u, s4 

 

+{ +u v 3 ,, s.t Lû>.4^+{, / v 

! 3 hw t 3 vrqw {ìv/ 

rq v*dsshuêrlw txh fhv idploohv vrqw ghx{ idploohv gh idploohv gh glvwulexwlrqv grqw ohv vxssruwv vrqw wrwdohphqw 

rugrqqìv dx vhqv gh od sursrvlwlrq 9171 Hq uhf|fodqw/ ã qrxyhdx/ oh sulqflsh ghv gìprqvwudwlrqv gh o*lghqwldelolwì 

ghv fodvvhv gh wrxv ohv pìodqjhv ghv idploohv gh glvwulexwlrqv jdppd hw lqyhuvh0jdppdv/ rq frqfoxw txh = 

Sursrvlwlrq 9148= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh I 9>v! >s ! 

od idplooh I 9>v!>t ! 


hwfhoohghwrxvohvpìodqjhvqlvgh 

Vl o*rq vrxkdlwh frpsoìwhu od idplooh gh glvwulexwlrqv ehwd L hqylvdjìh soxv kdxw/ rq ìwdeolw oh fruroodluh vxlydqw= 

ruroodluh 917= Od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh 

I 9>s! @ 

 

u t +{ u, s!4 

i 9 +{, @ 

E+s 3 >t, { s Lû>.4^+{, / uA3 

!.t 

hw fhooh gh wrxv ohv pìodqjhv qlv gh od idplooh gh glvwulexwlrqv 


I 9>t! @ 

 

i 9 +{, @ ut ! 

+{ u, s4 

Lû>.4^+{, / uA3 

E+s> t 3 , { s.t! 

Gìprqvwudwlrq = h fruurodluh qh vh gìgxlw sdv/ ã sursuhphqw sduohu/ gh od sursrvlwlrq suìfìghqwh/ pdlv vd 

gìprqvwudwlrq hvw lghqwltxh1[ 

918 Dssolfdwlrq ã o*ìwxgh ghv pìodqjhv jìqìudolvìv1 

Rq shxw ìwhqguh ohv ghx{ sursrvlwlrqv 917 hw 91:/ dq gh ohv dssoltxhu dx{ pìodqjhv jìqìudolvìv = 

Sursrvlwlrq 9149= Vrlw +x D , DD 

xqh idplooh gh idploohv gh glvwulexwlrqv grqw oh vxssruw uhvshfwli ghv ghqvlwìv 

hvw D DU q 1 Vxssrvrqv/ gh soxv txh fhwwh idplooh hvw wrwdohphqw rugrqqìh dx vhqv vwulfw gh o*lqfoxvlrq1 Vrlw 

y D od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh x D 1 

Od fodvvh/ y/ gh wrxv ohv pìodqjhv qlv gh od idplooh x @ 

V x D hvw lghqwldeoh vl hw vhxohphqw vl ;D D> 

y D hvw lghqwldeoh1 

Gìprqvwudwlrq = G*dsuëv od sursrvlwlrq 916/ o*ìoìphqw K gh y v*ìfulw vrxv od iruph g*xq pìodqjh gh n glvwulex0 

wlrqv/ K D" >===>K Dn suryhqdqw uhvshfwlyhphqw ghv fodvvhv ghv pìodqjhv qlv gh I D" > ===> I Dn 1 Rqshxwvxssrvhuvdqv 

shuwhghjìqìudolwìtxhohvD l vrqw glìuhqwv ghx{ ã ghx{1 Ohv +K Dl , 4ln 

vrqw ghv glvwulexwlrqv grqw ohv vxssruwv 

ghv ghqvlwìv vrqw wrwdohphqw rugrqqìv1 Sdu dssolfdwlrq gh od sursrvlwlrq 917/ od fodvvh gh whov pìodqjhv hvw lghqwl0 

DD 


919 Vxssruw hw frpsruwhphqw dv|pswrwltxh ghv orlv gx v|vwëph gh Shduvrq1 98 

deoh vl hw vhxohphqw vl/ ;l/ 4 l n/ K Dl hvw xqltxhphqw gìwhuplqìh gdqv y D 1 O*lghqwldelolwì/ ;D D/ghvy D 

hvw xqh frqglwlrq vx!vdqwh srxu fhod1 

Od uìflsurtxh hvw lppìgldwh1 [ 

Rq wudlwh gh od píph idêrq/ od jìqìudolvdwlrq gh od sursrvlwlrq 91: = 

Sursrvlwlrq 914:= rqvlgìurqv xqh idplooh/ I @ V 

l5D 

I dl / uìxqlrq gh idploohv gh glvwulexwlrqv gh +I dl , l5D 

/ 

D U n 1 Ohv ìoìphqwv gh I rqw ghv ghqvlwìv grqw ohv vxssruwv rqw wrxv xqh h{wuìplwì d/ qlh rx lqqlh/ dx prlqv/ 

hq frppxq1 Rq vxssrvhud/ gh soxv/ txh 

;l> m 5 D/ ;i 5 I dl > ;j 5 I dm > i 

d 

j 1 ^9155` 

Doruv/ od fodvvh y D ghwrxvohvpìodqjhvqlvghI hvw lghqwldeoh vl hw vhxohphqw vl ohv fodvvhv y dl 

qlv gh fkdfxqhv ghv I dl / l 5 D oh vrqw1 

ghv pìodqjhv 

Gìprqvwudwlrq = G*dsuëv od sursrvlwlrq 916/ xq ìoìphqw gh od fodvvh gh wrxv ohv pìodqjhv qlv gh od idplooh gh 

glvwulexwlrqv I v*ìfulw vrxv od iruph g*xq pìodqjh gh n glvwulexwlrqv/ K d" > ===> K dn suryhqdqw uhvshfwlyhphqw ghv 

fodvvhv ghv pìodqjhv qlv gh I d" > ===> I dn / txh o*rq shxw vxssrvhu/ vdqv shuwh gh jìqìudolwì/ glìuhqwhv ghx{ ã ghx{1 

G*dsuëv ^9155` rq d/ txlwwh ã uìlqglfhu o*hqvhpeoh/ 

;l> m 5 D/ K dl 

d 

K dm 1 

Sdu dssolfdwlrq gh od sursrvlwlrq 91:/ od fodvvh gh wrxv ohv pìodqjhv qlv gh I hvw lghqwldeoh vl hw vhxohphqw vl 

fkdfxq ghv K d" >===>K dn hvw xqltxhphqw gìwhuplqì gdqv y D 1 O*lghqwldelolwì gh fkdfxqh ghv y dl hvw xqh frqglwlrq 

vx!vdqwh srxu fhod1 

Od uìflsurtxh hvw lppìgldwh1 [ 

Hqq/ srxu srxyrlu dssoltxhu dx v|vwëph gh Shduvrq ohv glìuhqwv wkìruëphv txh qrxv yhqrqv g*ìwdeolu/ qrxv 

dxurqv ehvrlq gh od sursrvlwlrq vxlydqwh = 

Sursrvlwlrq 914;= rqvlgìurqv wurlv idploohv gh glvwulexwlrqv/ v 4 >v 5 hw v 6 1 Ohv glvwulexwlrqv gh v 4 vrqw ghv 

glvwulexwlrqv grqw oh vxssruw ghv ghqvlwìv hvw gh od iruph 

n V 

l@4 

^d l > e l`/ d l e l / +d l > e l , 5 U 5 1 Ohv glvwulexwlrqv gh v 5 

V 

vrqw ghv glvwulexwlrqv grqw oh vxssruw ghv ghqvlwìv hvw gh od iruph n ^d l >.4^/ d l 5 U1 Ohv glvwulexwlrqv gh v 6 vrqw 

l@4 

ghv glvwulexwlrqv grqw oh vxssruw ghv ghqvlwìv hvw U1 

Od fodvvh gh wrxv ohv pìodqjhv qlv gh glvwulexwlrqv gh od idplooh v 4 ^ v 5 ^ v 6 hvw lghqwldeoh vl hw vhxohphqw vl 

ohv fodvvhv gh wrxv ohv pìodqjhv qlv gh fkdfxqhv ghv idploohv v 4 >v 5 hw v 6 oh vrqw1 

Gìprqvwudwlrq = G*dsuëv od sursrvlwlrq 915/ xq ìoìphqw gh od fodvvh y gh wrxv ohv pìodqjhv qlv gh od idplooh 

gh glvwulexwlrqv v 4 ^ v 5 ^ v 6 v*ìfulw vrxv od iruph g*xq pìodqjh gh wurlv glvwulexwlrqv/ K 4 >K 5 hw K 6 suryhqdqw 

uhvshfwlyhphqw ghv fodvvhv ghv pìodqjhv qlv y 4 / y 5 hw y 6 gh v 4 >v 5 hw v 6 1 Prqwurqv tx*xqh whooh idplooh hvw oleuh1 

Vl rq d = 

;{ 5 U/ f 4 K 4 +{, . f 5 K 5 +{, . f 6 K 6 +{, @3/ 

rq d qìfhvvdluhphqw f 6 @3/sxlvtxhf 4 K 4 +{, . f 5 K 5 +{, hvw lghqwltxhphqw qxo vxu xqh sduwlh gh phvxuh qrq qxo 

gx vxssruw gh K 6 1 hfl lpsoltxh txh f 5 @3/ f 4 K 4 +{, ìwdqw lghqwltxhphqw qxo vxu xqh sduwlh gh phvxuh qrq qxooh 

gx vxssruw gh K 5 1 Rq d doruv f 4 @3/ hw od idplooh +K 4 >K 5 >K 6 , hvw oleuh1 

y hvw grqf lghqwldeoh vl hw vhxohphqw vl K 4 >K 5 hw K 6 vrqw xqltxhphqw gìwhuplqìhv/ fh txl hvw oh fdv vl hw 

vhxohphqw vl y 4 > y 5 hw y 6 vrqw lghqwldeohv1[ 


99 919 Vxssruw hw frpsruwhphqw dv|pswrwltxh ghv orlv gx v|vwëph gh Shduvrq1 

919 Vxssruw hw frpsruwhphqw dv|pswrwltxh ghv orlv gx v|v0 

wëphghShduvrq1 

rqirupìphqw ã fh txh qrxv dyrqv ìwdeol fl ghvvxv/ lo frqylhqw g*dqdo|vhu ohv frpsruwhphqwv ghv glìuhqwv w|shv 

gh orl gx v|vwëph gh Shduvrq1 

W|shLhwLL=Ohv orlv ghv w|shv L hw LL +ehwdv, rqw ghv ghqvlwìv grqw oh vxssruw hvw eruqì hw fh vrqw ohv vhxohv1 

W|sh LLL = Ohv orlv gh w|sh LLL +jdppdv, rqw ghv ghqvlwìv grqw oh vxssruw hvw plqruì/ rqw xq ìtxlydohqw hq .4 

txl hvw= 

4 { t4 

+t, s t h{s { 

s 

hw xq ìtxlydohqw hq u/ od eruqh lqiìulhxuh gx vxssruw/ txl hvw 

4 +{ u, t4 

+t, s t 1 

W|sh LY hw YLL = Ohv glvwulexwlrqv gh dxfk|/ gìqlhv vxu U rqw srxu ìtxlydohqw hq .4 hw hq 4 = 

 

{ 5 

hw fhoohv gh Vwxghqw gìqlhv vxu oh píph vxssruw rqw srxu ìtxlydohqw hq .4 hw hq 4 = 

+.4, 

.4 

5 5 

s 

 

 

{ +.4, = 

5 

W|sh Y = Ohv orlv gx w|sh Y +lqyhuvh jdppdv, rqw ghv ghqvlwìv grqw oh vxssruw hvw plqruì hw rqw xq ìtxlydohqw 

hq .4 txl hvw = 

o*ìtxlydohqw hq u hvw = 

4 { t4 

+t, s t > 

4 +{ u, t4 

+t, s t h{s 

5 

s +{ u, 

W|sh YL = Ohv orlv gx w|sh YL +ehwdv gx vhfrqg w|sh, rqw ghv ghqvlwìv grqw oh vxssruw hvw plqruì/ rqw xq 

ìtxlydohqw hq .4 txl hvw = 

hw xq ìtxlydohqw hq u txl hvw = 

v t 

E+s> t, {t4 

4 

E+s> t,v s +{ u,s4 = 

 

= 


919 Vxssruw hw frpsruwhphqw dv|pswrwltxh ghv orlv gx v|vwëph gh Shduvrq1 9: 

W|sh YLLL = Ohv orlv gx w|sh YLLL +qrupdohv, rqw ghv ghqvlwìv gìqlhv vxu U hw xq ìtxlydohqw hq .4 hw hq 

4 txl hvw = 

 

4 

s h{s {5 

5 

5 5 5 = 

Oh fdv olwljlhx{ hqwuh ohv glvwulexwlrqv gh dxfk| hw gh Vwxghqw/ txl rqw ghv frpsruwhphqwv hq .4 ìtxlydohqwv/ 

vxuylhqw vxu o*lqwhuvhfwlrq hqwuh ohv ghx{ idploohv +dssduwhqdqw wrxwhv ohv ghx{ dx w|sh LY,/ lo q*| d grqf/ sdv 

gh sureoëph g*lghqwldelolwì gdqv fh fdv1 Ohv orlv ghv w|shv Y hw YL vrqw/ sdu frqvìtxhqw/ ohv vhxohv ã dyrlu ghv 

frpsruwhphqwv olplwhv hq .4 txl shxyhqw íwuh ìtxlydohqwv1 hshqgdqw/ ohv ìtxlydohqwv ghv ghqvlwìv hq od eruqh 

lqiìulhxuh vrqw elhq glìuhqwv1 Hq hhw/ vl ghx{ ghqvlwìv/ i 8 5 I 8 hw i 9 5 I 9 rqwohpíphvxssruw/û>.4`/ rqdxud 

i 9 AA i 8 1 

u 

Vl o*rq vh uìiëuh dx{ qrwdwlrqv gh od sursrvlwlrq 914;/ I 4 ^ I 5 hvw xqh idplooh gx w|sh v 4 / I 6 ^ I 8 ^ I 9 hvw gh od 

iruph v 5 hw I 7 ^ I : ^ I ; hvw gh od iruph v 6 1 Uhjdugrqv/ doruv/ o*lghqwldelolwì ghv fodvvhv ghv pìodqjhv qlv gh fhv 

wurlv idploohv1 

Qrxv dyrqv suìflvì soxv kdxw txh o*lghqwldelolwì gh od fodvvh gh wrxv ohv pìodqjhv qlv gh glvwulexwlrqv gh od 

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;1416 H{lvwhqfh hw xqlflwì orfdoh1 

dofxorqv oh mdfrelhq M 4 gx v|vwëph ghv prphqwv & 4 / frpph oh suìfrqlvh oh wkìruëph ghv irqfwlrqv lpsolflwhv/ 

dq g*ìwxglhu o*xqlflwì orfdoh gh o*hqvhpeoh ghv sdudpëwuhv gx pìodqjh1 Vljqdorqv txh gdqv oh fdv gx pìodqjh gh 

ghx{ glvwulexwlrqv qrupdohv/ od pdwulfh mdfrelhqqh dvvrflìh dx v|vwëph ghv flqt suhplhuv prphqwv fhqwuìv hvw od 

píph txh fhooh dvvrflìh dx v|vwëph ghv flqt suhplhuv prphqwv qrq fhqwuìv1 Oh gìwhuplqdqw ã fdofxohu hvw = 

 

 

% 3 % 2 

 

f f 

% 2 n j 

2% 2% 2 

 

3% 2 2 3 j 2 

 

% 

M 4 @ s 5 +4 s, 5 E% 2 n j E% 2 nj % 

2 n j 2 

3% % % 

2 3 j 2 % 2 

2 

 

% 

eE% nj % eE% e nSj % 2 nj 2 

2 nj 2 % 2 

3% e 2 3 Sj 2% 2 2 3 j 2 SE% 2 n j SE% 2 2 n j 2 

2 

% e nSj % 2 % 

e 

D 

nj 2 D 2 nSj 2 % 2 2 % D n fj % n Dj 2 % 

nj 2 2 

3% D 2 3 fj 2% 2 3 Dj 2 f% E% 2 nj f% 2 E% 2 2 nj 2 

2% 2 

> 

 

txh o*rq gìyhorssh hq = 

3 

M 4 @ s 5 +4 s, 5 E 

C 

{ < 4 .

;15 Lqyhuvlrq qxpìultxh gx v|vwëph & 4 sdu o*dojrulwkph gh Qhzwrq0Udskvrq1 ;4 

Oh vhfrqg fdv/ txl qh v*h{soltxh sdv sdu xqh gìwhulrudwlrq ìylghqwh g*xqh sduwlh gx v|vwëph/ q*dssdudñw tx*ã od 

vxlwh ghv glìuhqwhv pdqlsxodwlrqv dojìeultxhv qìfhvvdluhv ã o*lqyhuvlrq gh fhoxl0fl1 

Qrxv prqwurqv grqf tx*lo q*| d sdv xqlflwì orfdoh ghv vroxwlrqv gh & 4 gdqv oh yrlvlqdjh ghv vrxv0ydulìwìv Y 4 hw 

Y 5 gh U 5 U .5 ^3=8> 4^ = 

Y 4 @ i+{ 4 >{ 5 > 4 > 5 >s, >{ 4 @ { 5 @3j 

 

 

hw Y 5 @ +{ 4 >{ 5 > 4 > 5 >s, > m{ 4 { 5 m @ 9 s 

4@7 s 

9 < 

m4 5 m > 

hw lo idxgud dqdo|vhu suìflvìphqw oh frpsruwhphqw gx v|vwëph gdqv ohv yrlvlqdjhv gh fhv ydulìwìv dq g*dssurirqglu 

od frqqdñvvdqfh txh qrxv dyrqv gh o*hqvhpeoh ghv vroxwlrqv1 Srxu fhod/ qrxv ghyurqv lqyhuvhu oh v|vwëph srxu 

ìwxglhu od frqwlqxlwì ghv vroxwlrqv vxu ohv ydulìwìv Y 4 hw Y 5 1 

;15 Lqyhuvlrq qxpìultxh gx v|vwëph & sdu o*dojrulwkph gh 

Qhzwrq0Udskvrq1 

Dydqw g*hqwuhu gdqv gh orqjv hw idvwlglhx{ fdofxov/ vrxyhqrqv qrxv txh od pìwkrgh gh Qhzwrq0Udskvrq hvw 

xqh pìwkrgh qxpìultxh uhfrqqxh g*lqyhuvlrq gh v|vwëphv dvvxuìphqw lqyhuvleohv1 O*h{hpsoh vxlydqw looxvwuh ohv 

glìuhqwv w|shv gh sureoëphv txh o*rq shxw uhqfrqwuhu dyhf fhwwh pìwkrgh/ gdqv oh fdv gx v|vwëph ghv prphqwv1 

Srxu fhod qrxv qrxv sodfhurqv gdqv xq yrlvlqdjh gh od ydulìwì sureoìpdwltxh Y 5 1 

dofxorqv ohv prphqwv ã o*dlgh ghv ydohxuv lqlwldohv = 

{ 4 @ 4 5 4 s 

5 

9 s 9 < 4@7 

3=8 

s lql @ 3=8 / 

lql 

4 @ 5=

hw lql 

5 @ 5= 

{ 4 @ 4=3 

{ 5 @ 3=8 

s @ 3=68495 / 

4 @ 4 


;5 ;16 Lqyhuvlrq dojìeultxh gx v|vwëph1 

txl hvw oh erq hqvhpeoh gh sdudpëwuhv1 

Vl ohv sdudpëwuhv vrqw lqlwldolvìv ã = 

hw 5 @ 6> 

lo | d frqyhujhqfh yhuv = 

{ lql 

4 @ 4> 

{ lql 

5 @ 4> 

s lql @ 3=8 / 

lql 

4 @ 5=< 

hw lql 

5 @ 5= 

txl hvw xqh vroxwlrq sdudvlwh1 

Hqq/ vl ohv sdudpëwuhv vrqw lqlwldolvìv ã = 

{ 4 @ 3=;37766> 

{ 5 @ 4=78;7 

s @ 3=977849 / 

4 @ 4=7 

{ lql 

5 @ 4> 

s lql @ 3=8 / 

lql 

4 @ 6 

hw lql 

5 @ 5= 

lo q*| d sdv frqyhujhqfh 111 

Vl qrxv h{foxrqv ohv fdv rü oh v|vwëph q*d sdv gh vroxwlrq/ lo | d/ olìv ã od pìwkrgh gh Qhzwrq0Udskvrq/ wurlv 

w|shv gh fdv sureoìpdwltxhv/ txl shxyhqw vh surgxluh = 

Od surfìgxuh shxw wrxw vlpsohphqw qh sdv frqyhujhu/ ã fdxvh g*xqh pdxydlvh lqlwldolvdwlrq1 

Od Surfìgxuh shxw qh sdv frqyhujhu/ pdojuì xqh lqlwldolvdwlrq wuëv surfkh ghv yudlv sdudpëwuhv1 Gdqv fh fdv/ 

txhooh txh vrlw o*lqlwldolvdwlrq/ rq q*rewlhqgud sdv gh vroxwlrq sdu fhwwh pìwkrgh1 

Od surfìgxuh shxw frqyhujhu yhuv xq hqvhpeoh gh sdudpëwuhv dffhswdeohv/ txl q*hvw sdv fhoxl txh qrxv 

uhfkhufkrqv1 

Lo |d wurs g*doìdv gdqv o*xwlolvdwlrq g*xqh whooh pìwkrgh txl hpsífkhqw gh idluh frqdqfh dx{ uìvxowdwv rewhqxv/ 

oruvtx*rq hq rewlhqw111 Qrxv qrxv wrxuqrqv grqf yhuv xqh lqyhuvlrq dojìeultxh gh & 4 / dq g*hq frpsuhqguh ohv 

pìfdqlvphv vrxv0mdfhqwv1 

;16 Lqyhuvlrq dojìeultxh gx v|vwëph1 

Vxlwh dx fdofxo gx mdfrelhq/ qrxv vrpphv dphqìv ã vìsduhu oh fdv rü ohv pr|hqqhv vrqw ìjdohv gh fhoxl rü hoohv 

vrqw glìuhqwhv1 


;16 Lqyhuvlrq dojìeultxh gx v|vwëph1 ;6 

;1614 dv jìqìudo rü % 4 9' % 5 1 

Rq frqvlgëuh oh pìodqjh gh ghx{ glvwulexwlrqv qrupdohv Q + 4 > 4 , hw Q + 5 > 5 , grqwohvpr|hqqhvvrqwgli0 

iìuhqwhv1 Qrxv udsshorqv txh oh v|vwëph ã lqyhuvhu/ srxu xq who pìodqjh/ shxw0íwuh = 

; 

A 

& 4 @ 

A= 

rpphqêrqv sdu srvhu = 

s{ 4 . +4 s, { 5 @3 

s{ 5 4 . +4 s, { 5 5 @ P5 f +s 4 . +4 s, 5 , 

s{ 6 4 . +4 s, {6 5 @ P 6 f 6 +s 4{ 4 . +4 s, 5 { 5 , 

s{ 7 4 . +4 s, { 7 5 @ P7 f 9 

s 4 { 5 4 . +4 s, 5 { 5 5 6 s 

5 

4 . +4 s, 5 5 

 

s{ 8 4 . +4 s, { 8 5 @ P8 f 43 

s 4 { 6 4 . +4 s, 5 { 6 5 48 s 

5 

4 { 4 . +4 s, 5 5{ 5 

= ^;158` 

fh txl hvw ìtxlydohqw/ oruvtxh { 4 { 5 9@3/ã= 

@ s 4 . +4 s, 5 

hw 3 @ s 4 { 4 . +4 s, 5 { 5 > 

4 

hw 5 

@ 

@ 

3 { 5 

s +{ 4 { 5 , 

3 { 4 

+4 s, +{ 5 { 4 , = 

Oh v|vwëph dsuëv wudqvirupdwlrq v*ìfulw = 

; 

s{ 4 . +4 s, { 5 @3 

s{ A 

5 4 . +4 s, { 5 5 @ P5 f 

& 3 s{ 

4 @ 

6 4 . +4 s, { 6 5 @ P6 f 6 3 

 

s{ 

A= 

7 4 . +4 s, { 7 5 @ P7 f 9 + 3 +{ 4 . { 5 , { 4 { 5 , 6 ! # + , .# +s{ # " .+4s,{# #, 

= 

s+4s,+{ " { # , # 

R% D n E 3 R % D 2 ' D S 3 f j E% n % 2 2 3 % % 2 3 j%2 % E % n % 

2 n D 

% " % # j 2 E% n % 2 3 2jj % % 2 

^;159` 

H{sulprqv doruv/ dx pr|hq ghv wurlv suhplëuhv ìtxdwlrqv/ s> { 4 > hw { 5 hq irqfwlrq gh 3 hw 1 Ghv ghx{ suhplëuhv 

ìtxdwlrqv rq wluh = 

v 

+4 s, 

s 

{ 4 @ 

u 

hw { 5 @ vjq +{ 4 , = 

rü vjq +{, hvw od irqfwlrq txl ã { dvvrflh vrq vljqh1 

Od wurlvlëph ìtxdwlrq ghylhqw doruv = 

+P f 5 

, ^;15:` 

s 

+4 s, +P f 5 ,> 

+4 s, 5 s 

 

5 

s +4 s, " # 

@ P f 6 63 

+P f 5 , $ # 

^;15;` 

hw hq ìohydqw dx fduuì ohv ghx{ phpeuhv gh o*ìjdolwì ^;15;` = 



+4 5s, 5 

s +4 s, @ +P 6 f 6 3 , 5 

+P5 f @ D> 

,6 

fh txl qrxv udpëqh ã od uìvroxwlrq gh o*ìtxdwlrq gx vhfrqg ghjuì = 

s 5 s . 4 @3= ^;15

;16 Lqyhuvlrq dojìeultxh gx v|vwëph1 ;8 

Dlqvl/ ohv wurlv suhplëuhv ìtxdwlrqv gh prphqwv {hqw gh pdqlëuh xqltxh/ hq irqfwlrq ghv yduldqfhv/ ohv pr|hqqhv 

ghv glvwulexwlrqv dlqvl txh oh frh!flhqw gh pìodqjh = 

t 

7 +P5 f 

s @ 

,6 . +P6 f 63 , 5 . mP6 f 6 3 m 

> 

5 

t7 +P5 f ,6 . +P6 f 63 , 5 

y 

t 

4 @ P 4 vjq +P6 f 6 3 x 7 +P5 f 

, w 

,6 . +P6 f 63 , 5 mP6 f 63 m 

t 

+P 

7 +P5 f ,6 . +P6 f 63 , 5 . mP6 f 5 f , 

63 m 

y 

t 

hw 5 @ P 4 . vjq +P6 f 6 3 x 7 +P5 f 

, = w 

,6 . +P6 f 63 , 5 . mP6 f 63 m 

t 

+P 

7 +P5 f ,6 . +P6 f 63 , 5 mP6 f 5 f ,= 

63 m 

Qrxv srxyrqv gìvrupdlv qrxv dwwdtxhu dx fdofxo ghv yduldqfhv/ txl hvw soxv frpsoltxì1 Od txdwulëph ìtxdwlrq 

gh & 3 4 ghylhqw = 

 

s{ 

5 

4 . +4 s, { 5 5 

+{ 4 . { 5 , 5 5{ 4 { 5 +{ 4 { 5 , 5 @ P7 f 9 +3 +{ 4 . { 5 , { 4 { 5 , ^;164` 

 

6 + 3 , 5 . 

5 s{ 5 4 . +4 s, { 5 5 

 

s +4 s, +{ 4 { 5 , 5 = 

G*dsuìv fh txl suìfëgh/ rq ìwdeolw txh = 

{ 4 . { 5 @ P f 6 6 3 

P f 5 / 

{ 4 { 5 @ vjq +P f 6 6 3 , 

t 

7 +P f 5 ,6 . +P f 6 63 , 5 

{ 4 { 5 @ P5 

f # 

$ 

+P5 f , 6 

hw s +4 s, @ 

7 +P5 f ,6 . +P6 f 63 , 5 1 

P f 5 / ^;165` 

Qrwrqv txh gdqv od vxlwh gx gìyhorsshphqw oh vljqh gh P f 6 63 q*lqwhuylhqw soxv hw tx*lo q*hvw sdv qìfhvvdluh gh 

glvfxwhu oh fdv rü P f 6 6 3 @31 

Sdu vxevwlwxwlrq/ ^;164` ghylhqw = 

txh o*rq shxw phwwuh vrxv od iruph = 

P f5 

6 @9 35 . P f5 

5 7P f 5 .5 5 . P f 7 

+P 

f 

5 , > ^;166` 

 

P6 f5 @9 35 . 

5 +P f 5 , 5 . P f 7 6P f5 

5 

 

+P f 5 , 

hw grqf = 

 

 

P f5 

35 6 5 +P5 f ,5 . n 7 +P5 f , 

@ 

> ^;167` 

9 



rü n 7 @ P7 f 6P5 f5 hvw oh txdwulëph fxpxodqw gx pìodqjh1 

Od flqtxlëph ìtxdwlrq gh & 3 4 hvw = 

s{ 8 4 . +4 s, { 8 5 @ P8 f 43 

 

 

3 +{ 4 . { 5 , 5 { 4 { 5 { 5 { 4 + { 4 . { 5 , 

. 48 

35 +{ 4 . { 5 , 5 3 

{ 4 { 5 = 

{ 4 { 5 

Srvrqv = 

V @ P8 f 43 

 

 

3 +{ 4 . { 5 , 5 { 4 { 5 { 5 { 4 + { 4 . { 5 , . 48 

35 +{ 4 . { 5 , 5 3 

{ 4 { 5 

{ 4 { 5 

hw V 3 @ s{ 8 4 . +4 s, { 8 5= 

Vl o*rq vxevwlwxh/ gdqv V/ ohv suhplhuv uìvxowdwv/ rq rewlhqw = 

rü hqfruh = 

# # P 

V @ P8 f 43 3 f 

6 6 3 $ 

$ 

5 

P5 f . P5 f . + P6 f 6 3 , 

48 35 P f 6 63 

+P f 5 ,5 633 / 

V +P f 5 , 5 @ P f 8 +P f 5 , 5 43 3 +P f 6 6 3 , 5 43 3 +P f 5 , 6 43 + P f 6 6 3 , +P f 5 , 5 

48 35 +P6 f 6 3 , 63 3 +P5 f , 5 > 

@ P8 f +P5 f , 5 43 3 P6 5 .93 35 P 6 

@ 43 3 P6 f5 .78 35 P6 f 78 36 43 3 +P5 f , 6 . +P8 f 43 P6, f +P5 f , 5 

> 

 

 

@ 78 35 P6 f 3 43 +P5 f , 6 . P6 

f5 .78 35 

. +P f 8 43P f 6P f 5 .43P f 6 +P f 5 ,, +P f 5 , 5 > 

fh txl grqqh/ srxu qlu/ vl o*rq srvh [ @ P f 5 hw n 8 @ P f 8 43P f 6P f 5/ oh flqtxlëph fxpxodqw gx pìodqjh/ 

dofxorqv V 3 = 

V[ 5 @78 35 P6 f 3 43 [ 6 . P6 

f5 

.78 

35 . +n 8 .43P6[, f [ 5 = 

V 3 @ s{ 6 4 . +4 s, { 6 5 

{ 

5 

4 . { 5 5 

> 

@ s{ 6 4 . +4 s, { 6 

5 +{ 4 . { 5 , 5 5{ 4 { 5 > 

% P 

V 3 @ +P6 f 6 3 f 

, 6 6 3 & 

5 

P5 f .5+P5 f , > 

k 

+P6 f 6 3 , P6 f5 9 3 P6 f .< 35 .5+P5 f , 6l 

@ 

+P5 f > 

,5 

@ 

P f6 

6 9 3 P f5 

6 .< 35 P f 6 .5+P f 5 , 6 P f 6 6 3 P f5 

6 .4; 35 P f 6 5: 36 9 3 +P f 5 , 6 

+P f 5 ,5 > 


;16 Lqyhuvlrq dojìeultxh gx v|vwëph1 ;: 

@ 

rx hqfruh = 

P f6 

6 .5:P f 6 35 .5P f 6 +P f 5 , 6 6 3 

5 +P f 5 , 6 .6P f5 

6 .< 35 

+P f 5 ,5 > 

[ 5 V 3 @ P f6 

6 .5:P f 6 35 .5P f 6[ 6 6 3 5[ 6 .6P f5 

6 .< 35 > 

hw grqf = 

Od flqtxlëph ìtxdwlrq gx v|vwëph ghylhqw = 

[ 5 V 3 @5:P f 6 35 6 3 5[ 6 .6P f5 

6 .< 35 . P f6 

6 .5P f 6[ 6 = 

rx hqfruh = 

G*dsuëv ^;167` rq d = 

3 @ [ 5 +V V 3 , > 

3 @ 4; 35 P6 f 3 7[ 6 . P6 f5 .4; 35 . +n 8 .;P6[, f [ 5 P6 f6 > 

3 @ 4;35 P6 f . +n 8 .;P6[, f [ 5 P6 

f6 

7[ 6 . P6 f5 

= 

.4;35 

f*hvw ã gluh = 

3 @ 6 P6 f5 

5[ 5 . n 7 [ P 

f 

6 . +n 8 .;P6[, f [ 5 P6 

f6 

7[ 6 . P6 f5 .6+P 

f5 

6 > 

+5[5 . n 7 , [, 

3 @ 5P 6[ f 6 . n 8 [ 5 6n 7 [P6 f .5P6 

f6 

5[ 6 6n 7 [ .7P6 

f5 = 

^;167` ghylhqw doruv = 

9 5P6[ f 6 . n 8 [ 5 6n 7 [P6 f .5P6 

f6 5 

@ P 

f5 

6 5[ 5 

. n 7 [ 5[ 6 6n 7 [ .7P6 

f5 5 

> 

^;168` 

rü qrxv srvrqv dq gh uhqguh ohv fdofxov soxv fodluv = 

Rq d srxu V 5 4 = 

V4 5 @ 5P6[ f 6 . n 8 [ 5 6n 7 [P6 f .5P6 

f6 5 

hw V5 5 @ P6 f5 5[ 5 

. n 7 [ 5[ 6 6n 7 [ .7P6 

f5 

5 

= 

V4 5 @ 5P6 f6 . n 8 [ 5 . 5[ 5 

6n 7 P 

f 

6 [ 5 

> 

@ 7P6 f9 . n8[ 5 7 . 5[ 5 5 

6n 7 P 

f5 

6 [ 5 .7P6 f6 n 8 [ 5 .7P6 

f7 

5[ 5 

6n 7 [ .5 5[ 5 

6n 7 n8 P6[ f 6 > 

@ 7P6 f9 . n8[ 5 7 .7P6 f5 [ 9 .

;; ;16 Lqyhuvlrq dojìeultxh gx v|vwëph1 

@ 

45P6 f7 n 7 [ .7n 8 P6[ f 8 9n 7 n 8 P6[ f 6 > 

7P6 f5 [ 9 .7n 8 P6[ f 8 . n8 5 45n 7 P6 

f5 

[ 7 . ;P6 f6 

9n 7 n 8 P 

f 

6 [ 6 

. 

gh píph/ rq fdofxoh = 

V5 5 @ P6 f5 5[ 5 

. n 7 [ 7P 

f5 

6 . 5[ 5 5 

6n 7 [ > 

@ 

5[ 6 n 7 [ . P6 

f5 

49P 

f7 

6 . 7[ 7 . 

@ 

5[ 6 n 7 [ . P6 

f5 

7[ 9 .45n 7 [ 7 49P6 f5 [ 6 . 

@ ;[ < 57n 7 [ : .65P6 f5 [ 9 4;n7[ 5 8 .7;P6 f5 n 7 [ 7 65P6 f7 [ 6 

7n 7 [ : 45n7[ 5 8 .49P6 f5 n 7 [ 7

;16 Lqyhuvlrq dojìeultxh gx v|vwëph1 ;< 

Vljqdorqv/ dxvvl/ txh gdqv oh fdv rü P f 6 @3/ 

S +[, @;[ < .5;n 7 [ : .63n 5 7[ 8 .9n 5 8[ 7 . 

5 7 +P5 f ,6 . P6 

f5 

y 

t 

4 @ P 4 vjq +P6, 

f x 7 +P5 f 

w 

. P6 f5 6 f 

t 

m 

+P f 

7 +P5 f ,6 . P6 f5 6 fm 5 , 

y 

t 

hw 5 @ P 4 . vjq +P6, f x 7 +P5 f 

= w 

. P6 f5 6 f 

t 

m 

+P f 

7 +P5 f ,6 . P6 f5 6 fm 5 ,= 

Vl P f 6 @3/rqds @ 4 5 ohv sdudpëwuhv 4 hw 5 rqw ghv u÷ohv v|pìwultxhv hw o*rq fkrlvlud/ sdu h{hpsoh/ 4 5 1Od 

txdwulëph ìtxdwlrq ghylhqw = 

P f5 

6 @5+P f 5 , 6 . n 7 +P f 5 , = 

O*lqyhuvlrq gx v|vwëph/ gdqv fh fdv sduwlfxolhu/ vh udpëqh ã od uhfkhufkh ghv udflqhv srvlwlyhv gx sro|q÷ph = 

S yduhjdohv +[, @5[ 6 . n 7 [ P f5 

6 = 


;16 Lqyhuvlrq dojìeultxh gx v|vwëph1 

G @ { 4 { 5 

hw ] @ 4 5 = 

hw 

5 

S +[, @ G5 V 5 7[ 

4357G 9 9 

7 

P6 f +G V,+G . V,+GV .6], 

@ > 

7G 

n 7 @ +G V,+G . V,+G7 6+G 5 V 5 .7GV] .5] 5 ,, 

;G 5 

hw n 8 @ +G V,+G . V,+5G7 V 6G 5 V 6 .8G 6 ] 48GV 5 ] 48V] 5 , 

7G 5 

6 

32feH ( S f H 

3D 2 ( S ( 2 3 7 2 f . 

n2DS ( e ( 2 3 7 2 ( e 3 f ( 2 7 2 3 e2 (7~3 2 ~ 2 f S 

nSe ( e ( 2 3 7 2 2 ( e 3 . ( 2 7 2 3 2e (7~nS~ 2 f D 

3H( 2 

( 2 3 7 2 2 ( 

e b( e 3 SS( 2 7 2 n2SD7 e 3 SH( D( 2 7 3 7 

~ 

32e( 2 H( 2 3 27 2 ~ 2 ne2f(7~ nDef~ e f e 

5 

32( 2 ( e b( S n SD( e 7 2 3 ..( 2 7 e n S 7 S 6 

( 2 3 7 2 2 

9 neH( 7 2 ( e 3 H( 2 7 2 n .7 e ~ 

7 n 2( 2 D( e 3 .fe( 2 7 2 n 2fHb7 e ~ 2 : 

3D.S(7 e2( 2 3 S.7 2 ~ n fH 8 f 

= 

5 

3Sb( 2 n Sb7 2 ~ e ( e E2 ( e 3 2De( 2 7 2 n .7 e 6 

 

nE( 2 3 7 2 E(7n ~ 2 7 

n2e( 7E3eb( 2 nbD 7 2 ~ 

8 f 2 

n 2( 2 E3b ( 2 n.b7 2 ~ 2 n DHe(7~ 3 fH~ e : 

nH ( 2 3 7 2 e E(7 n~ 

e 2( e 3 D( 2 7 2 3 H(7~ 3 ~ 2 8 

f 

n2E( 2 3 7 2 D E(7 n~ S 

Rq vdlw doruv/ g*dsuëv ^;165`/ txh 

{ 4 { 5 @ G5 V 5 

7 

grlw íwuh udflqh gx sro|q÷ph S / fh txl hvw frqupì sdu od idfwrulvdwlrq gx whuph 

G 5 V 5 7[ 

gdqv o*ìfulwxuh gx sro|q÷ph1 

Srxu vdyrlu txdqg oh sro|q÷ph dgphw xqh udflqh grxeoh hq { 4 { 5 / lo idxw h{dplqhu oh uhvwh gh od glylvlrq 

sro|qrpldoh gh S +[, sdu 

+[ . { 4 { 5 , 5 @ 4 

7[ G 5 . V 5 5 

/ 

49 


;17 H{hpsohv g*lqyhuvlrq gx v|vwëph1

rü oh qxpìudwhxu hvw fdofxoì dyhf ohv sdudpëwuhv rewhqxv sdu lqyhuvlrq gx v|vwëph hw oh gìqrplqdwhxu hvw fdofxoì 

dyhf ohv sdudpëwuhv frqqxv lqlwldohphqw1 W 9 grlw/ grqf/ íwuh surfkh gh 3 +ìjdo ã 3 gdqv ohv ìwxghv wkìrultxhv,/ 

hw v*lo | d soxvlhxuv hqvhpeohv gh vroxwlrqv dffhswdeohv/ lvvxv gh o*lqyhuvlrq gx v|vwëph/ qrxv fkrlvlurqv fhoxl txl 

grqqhud oh udssruw oh soxv surfkh gh 31 Gh píph/ qrxv srxuurqv xwlolvhu/ srxu ìydoxhu ohv ìyhqwxhov fdqglgdwv 

vroxwlrq rewhqxv dyhf S yduhjdohv / o*ìtxlydohqw W 8 gh W 9 dyhf od flqtxlëph ìtxdwlrq/ 

s 

{ 8 4 .43 4 { 6 4 .48 5 

W 8 @ 

4{ 4 . +4 s, { 

8 

5 .43 5 { 6 5 .48 5 5{ 5 P 

f 

8 

 

5P8 

f 

Ohv frpphqwdluhv txh qrxv idlvrqv gdqv ohv sdudjudskhv vxlydqwv vh udssruwhqw dx{ glìuhqwv h{hpsohv sursrvìv 

hq dqqh{h/ hw grqw qrxv suìvhqwrqv ohv wdeohdx{ uìfdslwxodwliv1 

= 

;1714 dv jìqìudo +ìorljqì ghv s÷ohv gx mdfrelhq,1 

Qrxv idlvrqv uìiìuhqfh ã o*dqqh{h 41 Oh sro|q÷ph dgphw ghx{ udflqhv srvlwlyhv U 4 hw U 5 / grqw vhxoh od vhfrqgh 

grqqh xq hqvhpeoh gh sdudpëwuhv txl vrlw xqh vroxwlrq dssduwhqdqw dx grpdlqh gh gìqlwlrq ghv glvwulexwlrqv 

qrupdohv1 Oh whvw W 9 hvw qxo srxu fhw hqvhpeoh gh vroxwlrqv/ txl hvw hhfwlyhphqw fhoxl txh qrxv hvsìulrqv1 

;1715 dv rü j 4 ' j 5 +ìorljqì ghv s÷ohv gx mdfrelhq,1 

Vl R 9' fD1 

Qrxv idlvrqv uìiìuhqfh ã o*dqqh{h 51 Oh sro|q÷ph S d ghx{ udflqhv srvlwlyhv/ U 4 hw U 5 / fh txl ruh ghx{ 

srvvlelolwìv g*hqvhpeohv vroxwlrqv dx sureoëph ghv prphqwv1 Oh v|vwëph & 4 lqyhuvì sdu U 4 grqqh xq hqvhpeoh 

q*d|dqw sdv g*h{lvwhqfh gdqv o*hqvhpeoh ghv sdudpëwuhv1 O*lqyhuvlrq sdu U 5 grqqh xq hqvhpeoh txl hvw fhoxl txh 

qrxv hvsìulrqv hq lqyhuvdqw qrwuh v|vwëph sdu oh sro|q÷ph S 1 

Hq lqyhuvdqw oh v|vwëph & 4_5 sdu o*xqltxh udflqh srvlwlyh gh S yduhjdohv rq rewlhqw/ dxvvl/ o*lqyhuvlrq h{dfwh gx 

v|vwëph1 Ohv ghx{ pìwkrghv frqylhqqhqw ã o*lqyhuvlrq gx v|vwëph1 

Vl R 'fD1 

Qrxv idlvrqv uìiìuhqfh ã o*dqqh{h 61 Qrxv qrxv sodêrqv/ lfl/ gdqv oh fdv g*xqh glvwulexwlrq gh pìodqjh v|pìwultxh1 

*hvw oh fdv sduwlfxolhu rü P f 6 hw 3 vrqw vlpxowdqìphqw qxov1 Pr|hqqdqw txhotxhv suìfdxwlrqv gdqv o*xwlolvdwlrq 

gh od pìwkrgh sdu S / ohv uìvxowdwv vrqw fhx{ dx{txhov qrxv srxyrqv qrxv dwwhqguh1 Hq hhw/ oh sro|q÷ph d ghx{ 

udflqhv srvlwlyhv/ grqw xqh hvw xqh udflqh grxeoh/ txl grqqhqw wrxwhv ohv ghx{ xq hqvhpeoh gh sdudpëwuhv dffhswdeohv 

srxu xq pìodqjh gh glvwulexwlrqv qrupdohv1 Oh whvw W 9 shuphw wrxwhirlv gh wudqfkhu hq idyhxu gh o*hqvhpeoh ghv 

sdudpëwuhv wkìrultxhv1 Od glìuhqfh hqwuh ohv ghx{ hqvhpeohv gh sdudpëwuhv qh shxw vh idluh txh vxu oh prphqw 

g*ruguh 91 Lo idxw/ gh soxv/ uhpdutxhu txh oh whvw grqqh/ srxu o*hqvhpeoh lvvx gh od udflqh grxeoh gx sro|q÷ph/ xq 

uìvxowdw vx!vdphqw surfkh gh }ìur/ srxu tx*lo sxlvvh jìqìuhu ghv vrxufhv g*huuhxu gdqv ohv ìwxghv hpslultxhv1 

Od udflqh srvlwlyh gh S yduhjdohv hqjhqguh o*hqvhpeoh gh sdudpëwuhv dwwhqgx txl hvw/ ìylghpphqw/ hq dffrug dyhf 

ohv ghx{ whvwv1 


;17 H{hpsohv g*lqyhuvlrq gx v|vwëph1

Paramètres calculés : Paramètres issus de Pvarégales : 

Paramètres théoriques : Racine choisie pour l'inversion : 

Légende : 

1.0001 1.0001 0.6 1.0 1.0 5.57898E-9 oui 6.12E-10 

Annexe 7 1.0001 1.0001 0.6 1.0 1.0 2.4E-9 oui 0.0 

1.001 5.0 0.6 1.0 1.0 5.57893E-7 non 

Annexe 6 1.001 5.0 0.6 1.0 1.0 2.4E-7 oui 0.0 

2.092444442 3.0 0.5 -0.092444442 1.0 1.19343486 oui 0.0 Les deux tests aussi. 

Annexe 5 2.092444442 3.0 0.5 -0.092444442 1.0 1.19341522 oui 7.36E-7 Les deux racines sont proches 

2.092434442 3.0 0.5 -0.092434442 1.0 1.19341301 oui 0.0 

Annexe 4 2.092434442 3.0 0.5 -0.092434442 1.0 1.19341301 oui 0.0 Cette racine double caractérise V2. 

1.0 1.0 0.5 2.0 1.0 0.25 oui 0.0 T5 n'est pas calculable 

0.43301270 oui 0.066876 qui permet l'inversion. 

Annexe 3 1.0 1.0 0.5 2.0 1.0 0.43301270 oui 0.066876 Cette racine double n'est pas celle 

1.0 1.0 0.5 2.0 1.0 0.25 oui 0.0 

1.0 1.0 0.7 3.0 1.0 0.84 oui 0.0 On a aussi T5=0.0 

Annexe 2 1.0 1.0 0.7 3.0 1.0 0.84 oui 0.0 

1.0 1.0 0.7 3.0 1.0 0.26803419 non 

5.0 3.0 0.6 1.0 1.0 3.84 oui 0.0 

Annexe 1 5.0 3.0 0.6 1.0 1.0 0.54761436 non 

Moyenne F1 Variance F1 p Moyenne F2 Variance F2 racines >0 acceptable Test T6 Commentaires 

INVERSION DE S 1POUR DES MOMENTS THEORIQUES 

Ilj1 LL181 0

433

{ 5 > 4 > 5 >s, > m 4 5 m @7 

{ 5 4 hw s @3=8 

48 

> = 

Hwxgh gx v|vwëph irupì ghv txdwuh suhplëuhv hw gh od vl{lëph htxdwlrqv gh prphqwv fhqwuìv1

435 

o*ìtxdwlrq 

M 5 @6+4 s, 5 s 5 78] 8 . 468+{ 4 . { 5 ,+{ 4 { 5 ,] 7 63+{ 4 { 5 , 7 ] 6 4;+{ 4 { 5 , 8 +{ 4 . { 5 ,] 5 

6+{ 4 { 5 , ; ] +{ 4 { 5 , < +{ 4 . { 5 , 

 

ghylhqw 

M 5 @6+4 s, 5 s 5 78] 8 6[ ; ] 63[ 7 ] 6 . +4 5s, [ 468[] 7 4;[ 8 ] 5 [ < > 

txh o*rq shxw idfwrulvhu vrxv od iruph g*xq sro|q÷ph hq s/ sxlvtxh 468[ 5 ] 7 4;[ 9 ] 5 [ 43 9@3= 

M 5 @ 9[ 5 468] 7 4;[ 7 ] 5 [ ; +4 s, 5 s 5 

s 78]8 6[ ; ] 63[ 7 ] 6 .468[ 5 ] 7 4;[ 9 ] 5 [ 43 

5 +468[ 5 ] 7 4;[ 9 ] 5 [ 43 > 

, 

@ 9[ 5 # 

468] 7 4;[ 7 ] 5 [ ; +4 s, 5 s 

%s 5 4 

 

5 . 6] 48] 7 43[ 7 ] 5 [ ; $& 

5[ 5 +468] 7 4;[ 7 ] 5 [ ; 1 

, 

rpph 

48] 7 43[ 7 ] 5 [ ; @ 

3 v 8.5 s 

4 3 v 8.5 s 

4 

43 43 

Cm]m 

[ 5 D Cm]m . 

[ 5 D 

48 

48 

# 

] 5 . 49 5 s 43 8 $ 

7 

{ 4 

/ 

48 

od vrxv0ydulìwì Y 7 /ghU 5 U .5 ^3=8> 4^ = 

; 

A +{ 4 >{ 5 > 4 > 5 >s, >s@ 

Y 7 @ 

A= 

 

 

4 

5 . 6+ " 5 ,+48+ 4 5 , 7 43+{ 4 { 5 , 7 + 4 5 , 5 +{ 4 { 5 , ; , 

5+{ 4 { 5 , 5 +468+ " 5 , 7 4;+{ 4 { 5 , 7 + 4 5 , 5 +{ 4 { 5 , ; , 

hw +{ 4 { 5 , 5 

468 + 4 5 , 7 4;+{ 4 { 5 , 7 + 4 5 , 5 +{ 4 { 5 , ; 

9@ 3 

frqwlhqw od ydulìwì Y 6 lvvxh gx vrxv0fdv rü 4 @ 5 hw s @3=8/ dlqvl txh od vrxv0ydulìwì Y 3 

6/ sxlvtxh fhod fruuhvsrqg 

ã od vrxv ydulìwì vxu odtxhooh [ @5{ 4 9@3hw s @3=8 txl dssduwlhqw elhq ã Y 7 1 

< 

A@ 

A> 

Wkìruëph { 5 > 4 > 5 >s, 5 Y 7 grqw o*lqwhuvhfwlrq dyhf Y 4 ^ Y 5 

hvw qxooh/ sdu gìqlwlrq píph gh Y 7 1 

Lo qh qrxv uhvwh tx*ã lqyhuvhu oh v|vwëph ^

5 5 

/ hq irqfwlrq ghv prphqwv fhqwuìv P 

f 

5 / P f 6/ P f 7 hw P f 91 h v|vwëph v*ìfulw/ frpph qrxv o*dyrqv vljqdoì 

soxv kdxw = 

; 

3@s{ 4 . +4 s, { 5 

P5 A 

f @ s{ 5 4 . +4 s, { 5 5 . +s 4 . +4 s, 5 , 

P6 & 5 @ 

f @ s{ 6 4 . +4 s, { 6 5 .6+s 4 { 4 . +4 s, 5 { 5 , 

P7 f @ s{ 7 4 . +4 s, { 7 5 .9 

s 4 { 5 4 . +4 s, 5 { 5 5 .6 s 

5 

4 . +4 s, 5 5 

P9 A= 

f @ s{ 9 4 . +4 s, { 9 5 .48 

s 4 { 7 4 . +4 s, 5 { 5 7 .78 s 

5 

4 { 5 4 . +4 s, 5 5 { 5 5 

.48 

s 6 4 . +4 s, 6 5 

= ^

437 

sxlv = 

s{ 9 4 . +4 s, { 9 5 @ P f 9 48 

78 

48 

# 

# 

# 

 

3 

{ 7 4 { 7 5 {4 { 5 { 

6 

4 { 5 

6 

+{ 4 { 5 , 

$ 

+4 s, + 3 { 4 { 4 { 5 , 5 . s + 3 { 5 { 4 { 5 , 5 

s +4 s, +{ 4 { 5 , 5 

$ 

+4 s, 5 + 3 { 5 , 6 s 5 + 3 { 4 , 6 

s 5 +4 s, 5 +{ 4 { 5 , 6 > 

$ 

grqf = 


s{ 9 4 . +4 s, { 9 5 @ P9 f 48 3 { 6 5 . { 5 5{ 4 . { 5 { 5 4 . { 6 

4 

{4 { 5 { 

5 

4 . { 4 { 5 . { 5 

5 

. 78 

35 +4 s, { 5 4 . s{ 5 

5 5 3 { 4 { 5 +{ 4 . { 5 , . +{ 4 { 5 , 5 

{ 4 { 5 

48 +{ 4 { 5 , 

k k 

+{ 4 { 5 , 5 +4 s, 5 36 6 35 { 5 .6 3 +{ 5 , 5 +{ 5 , 6ll 

48 +{ 4 { 5 , 

k k 

+{ 4 { 5 , 5 s 5 36 6 35 { 4 .6 3 +{ 4 , 5 +{ 4 , 6ll > 

 

s{ 

9 

4 . +4 s, { 9 

5 +{4 { 5 , 5 @ 48 +{ 4 { 5 , 5 3 +{ 4 . { 5 , 

+{ 4 . { 5 , 5 5{ 5 { 4 

 

48 +{ 4 { 5 , 5 { 4 { 5 +{ 4 . { 5 , 5 { 4 { 5 

 

.78{ 4 { 5 35 +4 s, { 5 4 . s{ 5 

5 5 3 { 4 { 5 +{ 4 . { 5 , . +{ 4 { 5 , 5 

k 

ll 

ks 5 { 4 +4 s, 5 { 5 

48 +{ 4 { 5 , 

48 +{ 4 { 5 , 

. +{ 4 { 5 , 5 P f 9= 

+4 5s, 36 .6 35 

k k 6 3 5 s 5 { 5 4 +4 s, 5 { 5 5 

l k ll 

. 6 s 5 { 6 4 +4 s, 5 { 6 5 

rpph = 

s 5 { 6 4 +4 s,5 { 6 5 @ { 4 { 5 

 

s 5 { 4 +4 s, 5 { 5 

 

> 

rq d hqq = 

s 5 { 5 4 +4 s, 5 { 5 5 @ 3 

 

s{ 

9 

4 . +4 s, { 9 5 

+{4 { 5 , 5 @ .78{ 4 { 5 

k 

hw s 5 { 4 +4 s, 5 { 5 @ +4 5s, { 4{ 5 

> 

+{ 4 . { 5 , 

35 +{ 4 . { 5 , 5 { 5 { 4 

 

5 3 { 4 { 5 +{ 4 . { 5 , . +{ 4 { 5 , 5l 


5 

t7 +P5 f ,6 . +P6 f 63 , 5 

y 

t 

4 @ P 4 vjq +P6 f 6 3 x 7 +P5 f 

, w 

,6 . +P6 f 63 , 5 mP6 f 63 m 

t 

+P 

7 +P 5 , 6 . +P 6 6 3 , 5 . mP6 f 5 f , 

63 m 

y 

t 

hw 5 @ P 4 . vjq +P 6 6 3 x 7 +P5 f 

, = w 

,6 . +P6 f 63 , 5 . mP6 f 63 m 

t 

+P 

7 +P5 f ,6 . +P6 f 63 , 5 mP6 f 5 f , > 

63 m 

 

 

P f5 

35 6 5 +P5 f ,5 . n 7 +P5 f , 

@ 

= ^ 

V 6 @ 

V 7 @ 

 

78{ 4 { 5 

35 +{ 4 . { 5 , 5 { 5 { 4 5 3 { 4 { 5 +{ 4 . { 5 , . +{ 4 { 5 , 5 > 

V 8 @ 

 

48 +{ 4 { 5 , +4 5s, 36 . 6 35 6 

+4 5s, { 4 { 5 

{ 4 { 5 > 

+{ 4 . { 5 , 

V 9 @ s{ 9 4 . +4 s, { 9 

5 P 9 +{4 { 5 , 5 > 

hw frpphqêrqv ohv fdofxov= 

 

V 6 @ 48 +{ 4 { 5 , 5 3 +{ 4 . { 5 , 

+{ 4 . { 5 , 5 5{ 5 { 4 { 4 { 5 

+{ 4 . { 5 , 5 { 4 { 5 > 

 

@ 48 3 +P5 f , 5 P 

f 

6 6 3 # P 

f 

6 6 3 $ 

5 

P5 f P5 f .5+P5 f , 

# P 

48 +P5 f , 6 f 

6 6 3 $ 

5 

P5 f . +P5 f , > 

 

= 

+P5 f , V 6 @ 48 3 +P6 f 6 3 , 

+P 6 f 6 3 , 5 .5+P5 f , 6 

48 +P f 5 , 5 +P f 6 6 3 , 5 . +P f 5 , 6 > 


439 

g*rü = 

[V 6 @ 48 

 

3 +P6 f 6 3 , 

+P 6 f 6 3 , 5 .5[ 6 [ 6 +P6 f 6 3 , 5 . [ 6 

48 P f 5[ 5 +P f 6 6 3 , 5 . [ 6 > 

@ 48 P6 f 3 6 35 P6 f5 9P6 f 3 .< 35 .5[ 6 [ 6 P6 f5 9P6 f 3 .< 35 . [ 6 

48 P5[ f 5 P6 f5 9P6 f 3 .< 35 . [ 6 > 

[V 6 @ 48 P6 f6 3 9P6 f5 35 . 

 

+P5 f , V 7 @ 78 35 +P6 f 6 3 , 5 . +P5 f , 6 

5 3 +P5 f , 6 +P6 f 6 3 , 

78 

5 5 3 P5 f +P 5 f ,5 +P6 f 63 , . +P5 f ,5 +P5 f ,5 . P5 f +P 5 f , > 

grqf = 

 

[V 7 @ 78 35 +P6 f 63 , 5 . [ 6 

5 3 [ 6 +P6 f 63 , 

 

78 5 3 P5[ f 5 +P6 f 6 3 , . [ 6 P5[ f 5 5 

> 

[V 7 @ 78 P6 f5 35 9P 6 36 .< 37 . [ 6 35 5 3 P6[ f 6 .9 35 [ 6 

 

78 5 3 P6P f 5[ f 5 9 35 P5[ f 5 . [ 6 P5[ f 5 5 

> 

@ 78 :[ 6 9P5[ f 5 . P6 

f5 

35 9P6 f 36 .< 37 

 

78 5 P6[ f 6 P6P f 5[ f 5 3 . [ 6 P5[ f 5 5 

> 

@ 

78 :[ 6 9P f 5[ 5 . P f5 

6 .< 35 35 5 P f 6[ 6 P f 6P f 5[ 5 .6P f 6 35 3 


Sdvvrqv ã V 8 = 

 

V 8 @ 48 +{ 4 { 5 , +4 5s, 36 . 6 35 6 

+4 5s, { 4 { 5 

{ 4 { 5 > 

+{ 4 . { 5 , 

3 

t 

4 

7 D ' 3 D C3r} S 3 j e E2 S 3 j n E S 3 j 2 

2 S 3 j D 

53 

4 

7C 

3ƒ S 3 j ƒ 

t 

D j 3 r} E S 3 j j 2 j n j E2 S 3 j 6 

E2 S 3 j 2 

t 

8 c 

e E2 S 3 j n E S 3 j 2 e E2 S 3 j n E S 3 j 2 

 

7 D ' D r} S 3 j k3 2 S 3 j S 3 j j 

3 r} S 3 j j 2 j n j E S 2 3 j E S 2 3 j 2l c 

grqf = 

V 8 @48 

+P 6 f 6 3 , 36 

P5 f 6 35 . 6 +P5 f , 

+P5 f , > 

E S 2 3 j 7 D ' 

k 

D 3 S 3 j 

 

j 3 j 2 E3 E2 S 3 j n 2 S n E3 E2 S 3 j n 2 S E2 S 3 j E2 S 3 j 2l c 


[V 8 @ 48 

k 

+P6 f 6 3 , 36 

hw grqf = 

Hqq = 

 

 

6 35 +[ . P5, f . +[ . P5, f 6 [ [ 5l > 

@ 48 +P6 f 6 3 , 36 6 35 +[ . P5, f [ 5 [ 6 .6P5[ f 5 6P5 f5 [ . P5 

f6 

[V 8 @48 P f 6 36 6 35 [ 6 . P f 5[ 5 35 . [ 9 6P f 5[ 8 .6P f5 

5 [ 7 P f6 

5 [ 6 = 

V 9 @ s{ 9 4 . +4 s, { 9 5 P f 9 

+{4 { 5 , 5 > 

 

[ 

6 > 

@ 

@ 

@ 

@ 

k s{ 

7 

4 . +4 s, { 7 

5 { 

5 

4 . { 5 

5 . +{4 { 5 , 6l +{ 4 { 5 , 5 P9 f +{ 4 { 5 , 5 > 

kk s{ 

5 

4 . +4 s, { 5 5 

+{ 4 . { 5 , 5 5{ 4 { 5 +{ 4 { 5 , 5l 

+{ 4 . { 5 , 5 5{ 4 { 5 . +{ 4 { 5 , 6l +{ 4 { 5 , 5 

P9 f +{ 4 { 5 , 5 > 

kk+{ 4 { 5 , +{ 4 . { 5 , 5 .5+{ 4 { 5 , 5 +{ 4 { 5 , 5l 

+{ 4 . { 5 , 5 5{ 4 { 5 . +{ 4 { 5 , 6l +{ 4 { 5 , 5 

P9 f +{ 4{ 5 , 5 > 

k 

+{ 4 { 5 , +{ 4 . { 5 , 5 . +{ 4 { 5 , 5lk +{ 4 . { 5 , 5 +{ 4 { 5 , 5 5 +{ 4 { 5 , 6l . +{ 4 { 5 , 8 P9 f +{ 4 { 5 , 5 > 


43; 

3 

P6 f5 9 3 P6 f .< 35 . +P5 f , 6 P6 f5 9 3 P6 f .< 35 .5+P5 f , 6 4 

C 

P5 f D 

+P 5 f , 9 

P5 f P 9 

f +P5 f , 6 

P5 f > 

grqf = 

+P f 5 , V 9 @ 

 

87P f5 

6 .5:+P f 5 , 6 .;4 35 35 

. +P f 5 , 9 . 6P f5 

6 P f 9 

 

+P 

f 

5 , 6 . P f7 

6 > 

 

 

4;P6 f +P5 f , 6 .45P6 f6 . 43; 35 P6 

f 3 


[V 9 @ 87P6 f5 .5:[ 6 .;4 35 35 4;P6[ f 6 .45P6 f6 . 43; 35 P6 

f 

3 . [ 9 . 6P6 f5 P9 

f 

[ 6 . P6 f7 = 

Udvvhpeorqv ohv txdwuh sduwlhv gx fdofxo = 

hw grqf = 

[ +V 6 . V 7 . V 8 V 9 , @ 3 

3 @ !5 +:5[ 6

443

0 acceptable Test T5 Commentaires 

INVERSION DE S 2 POUR DES MOMENTS THEORIQUES 

Ilj1 LL191 0 


446 

0kdslwuh430 

Oh sureoëph ghv prphqwv gdqv oh fdv ghv 

prphqwv hpslultxhv1 

Dq g*looxvwuhu ohv sureoëphv txh qrxv odlvvdlhqw suìvdjhu ohv lqyhuvlrqv ghv v|vwëphv dyhf prphqwv wkìrultxhv/ 

qrxv ìwhqgrqv fhv lqyhuvlrq dx fdv gh prphqwv hpslultxhv lvvxv g*xq ìfkdqwloorq gh 433333 lqglylgxv1 Ohv udflqhv 

ghv sro|q÷phv vrqw fdofxoìhv dyhf xqh suìflvlrq gh 83333 fkluhv/ srxu dvvxuhu/ frpph qrxv hq dyrqv prqwuì od 

qìfhvvlwì soxv kdxw/ xqh suìflvlrq vx!vdqwh1 

Suìflvrqv/ hqq/ txh fh fkdslwuh q*hvw sdv xq fkdslwuh g*hvwlpdwlrq/ ã sursuhphqw sduohu/ pdlv soxvw÷w xqh 

froohfwlrq g*h{hpsohv ghvwlqìv ã phwwuh dx srlqw xqh pìwkrgh ruljlqdoh g*hvwlpdwlrq fxpxodqw ohv ghx{ v|vwëphv 

ghv prphqwv/ hw txh qrxv suìvhqwhurqv gdqv oh fkdslwuh vxlydqw1 Qrxv fkhufkrqv/ hq idlw/ ã qrxv idluh xqh lgìh 

hpslultxh vxu o*ìsdlvvhxu gx yrlvlqdjh ghv ydulìwìv Y 4 hw Y 5 srxu ohtxho o*lqyhuvlrq gx suhplhu v|vwëph ghv 

prphqwv qh grqqh sdv xqh hvwlpdwlrq deoh srxu ohv prphqwv hpslultxhv/ hw grlw íwuh vxssoììh sdu o*xqh ghv ghx{ 

dxwuhv pìwkrghv1 *hvw srxutxrl qrxv suìvhqwrqv gdqv wrxv ohv fdv/ ohv uìvxowdwv srxu ohv sro|q÷phv S hw S 45 / 

hw vl qìfhvvdluh/ fhx{ gx sro|q÷ph S yduhjdohv 1 Ohv h{hpsohv vxlydqwv looxvwuhqw gh idêrq suhvtxh h{kdxvwlyh ohv 

glìuhqwv fdv gh pìodqjhv vxvfhswleohv g*íwuh ìwxglìv/ hw qìfhvvdluh ã o*rswlplvdwlrq gx surwrfroh g*hvwlpdwlrq txh 

qrxv vrxkdlwrqv gìyhorsshu1 

4314 dv jìqìudo +ìorljqì ghv s÷ohv gx mdfrelhq,1 

h fdv gh jxuh suìvhqwì hq dqqh{h 47/ prqwuh o*h!fdflwì ghv ghx{ pìwkrghv srxu xq pìodqjh qh ghydqw 

wkìrultxhphqw sdv srvhu sureoëph1 Qrxv dyrqv/ sdu dloohxuv/ od frqupdwlrq txh o*hvwlpdwlrq sdu ohv flqt suhplëuhv 

ìtxdwlrqv hvw xq shx soxv suìflvh txh fhooh xwlolvdqw oh uhpsodhphqw gh od flqtxlëph sdu od vl{lëph1 Vljqdorqv/ hqq/ 

txhohwhvwW 9 hvw yìulì ã ghx{ ploolëphv suëv1 

Oh sureoëph ghv prphqwv gdqv oh fdv ghv prphqwv hpslultxhv1

447 4315 dv lqwhupìgldluh1 

4315 dv lqwhupìgldluh1 

Oh fdv g*xq who pìodqjh qh vhpeoh sdv sduwlfxolëuhphqw surfkh gh o*xqh ghv ydulìwìv txh qrxv dyrqv lghqwlìhv 

frpph srxydqw íwuh vrxufh g*huuhxu1 Srxuwdqw/ o*xwlolvdwlrq gh S phw hq ìylghqfh xqh lqvwdelolwì gh od pìwkrgh 

ã fhw hqgurlw1 Gdqv oh suhplhu ghv wurlv h{hpsohv suìvhqwìv hq dqqh{h 48/ 49 hw 4:/ ohv hqvhpeohv gh sdudpëwuhv 

rewhqxv sdu ohv ghx{ pìwkrghv vrqw ìtxlydohqwhv hw surfkhv gh fhoohv txh qrxv dwwhqgrqv1 Oh whvw W 9 uhpsolw vrq 

u÷oh gdqv od vhohfwlrq gx erq hqvhpeoh gh sdudpëwuhv1 Gdqv oh vhfrqg h{hpsoh/ dxfxqh ghv ghx{ vroxwlrqv lvvxhv 

gh S qh v*dssurfkh gh fhooh lvvxh gh S 45 1 Gh soxv/ od vroxwlrq gìvljqìh sdu oh whvw W 9 q*hvw sdv fhooh txl vhpeoh 

od soxv udlvrqqdeohphqw surfkh ghv sdudpëwuhv gh vlpxodwlrq1 Hqq/ oh ghuqlhu h{hpsoh/ gdqv ohtxho S q*d sdv 

gh udflqh srvlwlyh/ frquph txh o*rq hvw dssduhpphqw wurs surfkh g*xqh ydulìwì sureoìpdwltxh +suredohphqw Y 5 ,/ 

srxu srxyrlu idluh frqdqfh ã fhwwh pìwkrgh1 

4316 dv rü j ' j 2 +ìorljqì gh Y 2 rx Y e ,1 

O*remhfwli gh fh sdudjudskh hvw gh prqwuhu o*ìyroxwlrq gh od txdolwì ghv hvwlpdwlrqv/ oruvtxh/ srxu ghx{ pr|hqqhv 

wkìrultxhv {ìhv/ rq dxjphqwh ohv yduldqfhv wkìrultxhv +hq ohv pdlqwhqdqw ìjdohv,1 Od glvwulexwlrq gh pìodqjh vh 

udssurfkh dlqvl gx fdv v|pìwultxh/ txl shxw íwuh vrxufh g*huuhxu1 Gh soxv/ od txdolwì gh o*hvwlpdwlrq ghv sdudpëwuhv 

hvw dowìuìh/ gdqv ohv hvwlpdwlrqv sdu HP rx LH/ oruvtxh ohv yduldqfhv wkìrultxhv vrqw judqghv sdu udssruw ã od 

glìuhqfh ghv pr|hqqhv wkìrultxhv1 

Qrxv {hurqv/ gdqv ohv h{hpsohv suìvhqwìv gdqv ohv dqqh{hv 4; ã 5

4317 dv rü { 4 hw { 5 vrqw surfkhv +yrlvlqdjh gh Y 4 ,1 448 

Paramètres issus de Pvarégales : Paramètres issus de P12 : Racine de P12 choisie: 

Paramètres théoriques : Paramètres issus de P 

V Racine de P V choisie : 

Légende : 

1.07464487 3.09168564 0.72391464 3.01116070 0.91335100 0.74950208 oui 0.01721 

Annexe 17 

1.0 3.0 0.7 3.0 1.0 Pas de racine pou P 

V 

0.77195155 2.75498333 0.60739322 2.89014595 1.23370689 1.06993985 oui 0.00874 

Annexe 16 0.55838086 2.55239809 0.53078014 2.78588282 1.35792041 1.23574038 oui 0.00487 n'est apparemment pas la meilleure 

1.0 3.0 0.7 3.0 1.0 0.72592143 oui 0.01524 La racine de P V choisie par le test 

1.07254045 3.06072655 0.73075294 3.05270097 0.91499783 0.77147585 oui 0.00183 

Annexe 15 1.04705111 3.03885134 0.71893196 3.03461871 0.96590948 1.10372983 oui 0.01191 

1.0 3.0 0.7 3.0 1.0 0.79825763 oui 0.00140 

Moyenne F1 Variance F1 p Moyenne F2 Variance F2 racines >0 acceptable Test Commentaires 

CAS GENERAL (2) 

4.92270225 3.10778885 0.61884743 0.92739623 0.89791942 3.76515225 oui 0.01335 

Annexe 14 4.98008269 3.01868086 0.60482854 0.98130809 0.96926642 3.82183292 oui 0.00173 différence entre les deux systèmes. 

5.0 3.0 0.6 1.0 1.0 0.54601120 non Les deux ensembles illustrent la 


CAS GENERAL (1) 

Ilj1 LL1:1 0 


449 4317 dv rü { 4 hw { 5 vrqw surfkhv +yrlvlqdjh gh Y 4 ,1 

Paramètres issus de Pvarégales : Paramètres issus de P 12 : Racine de P 12 choisie: 

Paramètres théoriques : Paramètres issus de P V : Racine de P Vchoisie : 

Légende : 

1.25151315 7.35874942 0.78573391 5.65509146 5.98442463 3.26467829 oui 0.01628 

0.96523424 6.94282588 0.69124752 4.94841332 6.94282588 3.38613063 T6=0.00336 et T5=0.00985 que pour P V 

Annexe 22 1.08923689 7.11376078 0.73336313 5.23650200 6.55838727 3.36327878 oui 0.00211 Le test pour Pvarégales est moins bon 

1.0 7.0 0.7 5.0 7.0 0.98441242 oui 0.02843 

0.98888887 6.87341949 0.70423430 5.07242153 6.71195707 3.47325805 oui 0.01474 


Annexe 21 0.76793667 6.58736062 0.62650304 4.59319165 7.35733570 3.42397853 oui 0.00206 Le test pour P varégales est meilleur 

1.0 7.0 0.7 5.0 7.0 1.15562866 oui 0.02206 

0.68342115 5.72178472 0.57628467 4.24324819 7.00078 3.09434719 oui 0.02578 

1.00162274 6.10596915 0.69659531 4.92427708 6.10596915 3.25209268 T6=0.00329 et T5=0.00396 

Annexe 20 1.05018564 6.17043175 0.71368492 5.03736237 5.95786432 3.24849134 oui 0.00388 La solution de S2 se dégrade. 

1.0 6.0 0.7 5.0 6.0 1.00058277 oui 0.02412 

1.09609562 5.05903492 0.73295496 5.28323069 4.50280426 3.43159273 oui 0.01626 


Annexe 19 0.93823711 4.84155907 0.67951486 4.91973880 4.99224449 3.45223752 oui 0.00333 Le test pour P varégales est meilleur 

1.0 5.0 0.7 5.0 5.0 1.07793956 oui 0.05075 

1.02432768 4.05306643 0.70518348 5.03670172 3.96815123 3.34700817 oui 0.00045 


Annexe 18 1.02820590 4.05828756 0.70653146 5.04579476 3.95605071 3.34675593 oui 0.00012 Les trois ensembles sont équivalents 

1.0 4.0 0.7 5.0 4.0 1.05058972 non 


VARIANCES EGALES - CAS NON SYMETRIQUE. (1) 

Ilj1 LL1;1 0 


4317 dv rü { 4 hw { 5 vrqw surfkhv +yrlvlqdjh gh Y 4 ,1 44: 

Paramètres issus de Pvarégales : Paramètres issus de P12 : Racine de P12 choisie : 

Paramètres théoriques : Paramètres issus de P V Racine de P Vchoisie : 

Légende : 

3.08359495 10.8453704 0.68494833 0.23206032 6.90304353 1.75467586 oui 0.02572 

1.10020339 8.27244657 0.72383286 5.02902276 8.27244657 3.08556302 T6=0.00537 et T5=0.00397 de P V 

Annexe 25 2.66871634 11.3150539 0.82725017 -0.13014102 5.08336347 3.07609454 oui 0.00486 Le test ne choisit pas la bonne racine 

1.0 8.0 0.7 5.0 8.0 1.11947921 oui 0.00461 

3.17800505 10.5050670 0.68497068 0.09826500 6.59705860 2.04668202 oui 0.01194 

1.01816485 7.96808818 0.70316693 5.02591427 7.96808818 3.35252359 T6=0.00584 et T5=0.02534 semble donner une meilleure solution. 

Annexe 24 3.91029447 9.47813005 0.50107487 0.49796474 7.33651943 2.91098508 oui 0.00264 T6 privilégie P Valors que Pvarégales 

1.0 8.0 0.7 5.0 8.0 1.40176403 oui 0.00410 

3.62014200 9.85779265 0.57425068 0.29013280 7.14661509 2.71110500 oui 0.03298209 


Annexe 23 0.94217320 7.95842830 0.68263686 4.91308067 8.08486177 3.4160620 oui 0.00368 

1.0 8.0 0.7 5.0 8.0 1.0699469 oui 0.01236 



Ilj1 LL1

44; 4317 dv rü { 4 hw { 5 vrqw surfkhv +yrlvlqdjh gh Y 4 ,1 


Paramètres théoriques : Paramètres issus de P V Racine de P Vchoisie : 

Légende : 

3.30324882 18.503729 0.63463103 0.18356916 15.8330432 2.25669552 oui 0.05541677 

0.99999830 16.0002079 0.71639405 5.10222663 16.0002079 3.41906193 T6=0.00113 et T5=0.03503 donner une meilleure solution. 

Annexe 29 1.54482790 16.9135776 0.88432916 6.89264712 13.247605 2.92544060 oui 0.00300 T6 privilégie Pvarégales qui semble 

1.0 16.0 0.7 5.0 16.0 0.84781430 oui 0.00389 

3.42534264 18.2952048 0.61139389 0.28116721 14.7012961 2.34879036 oui 0.00564 


Annexe 28 P Vn'a pas de racine positive. 

1.0 16.0 0.7 5.0 16.0 

1.45017681 16.6007221 0.84296624 6.24345977 14.4074233 3.04137056 oui 0.01469 

1.07395556 16.01814534 0.72014096 5.10787360 16.01814534 3.27952536 T6=0.00094 et T5=0.03364 semble donner une meilleure solution. 

Annexe 27 1.60436444 16.89458054 0.88671179 6.88752314 13.3570709 2.80384831 oui 0.00083 T6 privilégie P V alors que Pvarégales 

1.0 16.0 0.7 5.0 16.0 0.82268609 oui 0.00541 

2.56049297 19.4206374 0.87030689 -0.40297715 11.6095229 0.99126648 oui 0.01221 

1.05316579 16.1689338 0.71919590 5.05234368 16.1689338 3.229922404 T6=0.00289 et T5=0.00353 de P V 

Annexe 26 2.66716938 19.3210659 0.83000347 -0.22123104 12.8540652 3.20443221 oui 0.00272 Le test ne choisit pas la bonne racine 

1.0 16.0 0.7 5.0 16.0 1.17715817 oui 0.00072 



Ilj1 LL1431 0 


4317 dv rü { 4 hw { 5 vrqw surfkhv +yrlvlqdjh gh Y 4 ,1 44< 

4317 dv rü % hw % 2 vrqw surfkhv +yrlvlqdjh gh Y ,1 

Qrxv qrxv sodêrqv gdqv xq fdguh rü ohv pr|hqqhv wkìrultxhv vrqw wuëv surfkhv1 Qrxv frpphqêrqv dyhf ghv 

yduldqfhv wkìrultxhv wuëv glìuhqwhv/ txh qrxv udssurfkrqv dx ixu hw ã phvxuh/ mxvtx*ã whqguh yhuv ghx{ glvwulexwlrqv 

lghqwltxhv/ +dqqh{hv 63 ã 6

453 4319 dv rü ohv yduldeohv dssduwlhqqhqw ã xq yrlvlqdjh gh od ydulìwì Y 7 1 


Paramètres théoriques : Paramètres issus de P V : Racine de P V choisie : 

Légende : 

0.00002336 non 

2.00344428 0.90732569 0.66031863 1.99725986 4.65849666 8.5788E-6 oui 0.4528 

3.2352E-6 non 

Annexe 33 0.00001994 non le test n'étant pas concluant. 

2.0 1.0 0.7 2.0 5.0 8.5799E-6 oui 1.69610 On ne retient pas la solution de P 

V 

0.00046085 non 

2.01262252 1.03223945 0.71515389 1.98379385 5.17189508 0.00016930 oui 0.78688 

0.00006384 non 

Annexe 32 0.00039347 non 

2.0 1.0 0.7 2.0 5.0 0.00016926 non PV n'offre pas de solution. 

0.00012675 non 

1.99929115 0.95644357 0.67654933 1.98470417 4.78025337 0.00004656 oui 0.10957 

0.00001756 non Mc,3=-0.03661 

Annexe 31 1.99778359 1.35166629 0.81875141 1.98006976 5.99496392 0.00010824 non 

2.0 1.0 0.7 2.0 5.0 0.00004656 oui 0.07170 

0.00166562 non 

1.99234068 0.98766572 0.70006352 2.04632440 9.94291514 0.00061192 oui 0.15832 

0.00023072 non Mc,3=0.30454 

Annexe 30 1.99866243 2.03686989 0.86269476 2.07054562 13.9577671 0.00142259 non 

2.0 1.0 0.7 2.0 10.0 0.00061207 oui 0.14433 


MOYENNES EGALES. (1) 

Ilj1 LL1441 0 


4319 dv rü ohv yduldeohv dssduwlhqqhqw ã xq yrlvlqdjh gh od ydulìwì Y 7 1 454 



Légende : 

0.00032765 non 

1.98881301 1.39477828 0.55820917 2.01090363 0.87556735 0.00012035 oui 0.27044 deux systèmes 

0.00004539 non proches, l'inversion est difficile pour les 

Annexe 36 1.99552815 1.23711980 0.92836941 2.03802810 0.23581681 0.00027940 non Les moyennes et variances sont très 

2.0 1.0 0.7 2.0 1.5 0.00012011 oui 0.01460 

0.00018211 non 

2.00429738 1.00869527 0.71659951 1.98624557 1.51672509 0.00006691 oui 0.11358 

0.00002522 non Mc,3=-0.00928 

Annexe 35 2.00100024 1.10105695 0.95154527 1.96288043 2.16626009 0.00015566 non 

2.0 1.0 0.7 2.0 1.5 0.00006700 oui 0.01130 

0.00048132 non 

2.00404351 1.06292532 0.87176105 1.96424523 1.75776568 0.00017707 oui 0.00500 

0.00006660 non sont équivalents. 

Annexe 34 2.00384168 1.06646236 0.88052823 1.96281222 1.78268622 0.00041134 non Les deux ensembles de paramètres 

2.0 1.0 0.7 2.0 1.5 0.00017709 oui 0.00044 



Ilj1 LL1451 0 





Légende : 

1.98058864 1.01132847 0.99483680 3.91010559 0.25659210 0.01912354 oui 0.22614 

Annexe 39 

2.0 1.0 0.7 2.0 1.1 


0.00305778 non 

0.00077491 non 

0.00058529 non de solution. 

Annexe 38 aucun des deux système ne donne 

2.0 1.0 0.7 2.0 1.1 

5.655E-6 non 

2.00089840 1.03186889 0.99984037 1.88660797 4.16760552 2.085E-6 oui 0.47034 

7.813E-7 non proches pour être différenciées 

Annexe 37 2.00088082 1.03235467 0.99999982 -1.77179499 82,0626657 5.449E-6 non Les moyennes et variances sont trop 

2.0 1.0 0.7 2.0 1.1 2.519E-6 oui 0.06385 



Ilj1 LL1461 0 


4319 dv rü ohv yduldeohv dssduwlhqqhqw ã xq yrlvlqdjh gh od ydulìwì Y 7 1 456 

Paramètres issus de Pvarégales : Paramètres issus de P 12 : Racine de P 12 choisie: 


Légende : 

0.78827180 0.96021870 0.48314645 -0.73637980 2.03091248 0.58048035 oui 0.00490 

Annexe 43 P V n ’a pas de racine positive. 

0.77246780 1.0 0.5 -0.77246780 2.0 

0.75338008 1.03561784 0.52207853 -0.82007115 1.95781281 0.61773036 oui 0.00553 

Annexe 42 0.68661584 1.09840019 0.60718480 -1.05777158 1.78554618 0.72576341 oui 0.00431 P V a deux racines proches. 

0.77246780 1.0 0.5 -0.77246780 2.0 0.47636331 oui 0.00492 

2.10050904 0.98264671 0.48999329 -0.05861019 3.01472671 1.16498215 oui 0.00067 

Annexe 41 P V n ’a pas de racine positive. 

2.09243444 1.0 0.5 -0.09243444 3.0 

2.09133500 1.00641536 0.49508118 -0.05634097 3.06998975 1.15301642 oui 0.00056 

Annexe 40 2.10536756 0.97255443 0.48036213 -0.00847831 3.11530011 1.26575051 oui 0.00414 P V a deux racines proches. 

2.09243444 1.0 0.5 -0.09243444 3.0 1.11536287 oui 0.00134 


VOISINAGE DE V 2 

Ilj1 LL1471 0 



4319 dv rü ohv yduldeohv dssduwlhqqhqw ã xq yrlvlqdjh gh 

od ydulìwì Y e 1 

Qrxv qrxv fdqwrqqhurqv/ srxu Y 7 / dx fdv v|pìwultxh rü ohv ghx{ yduldqfhv wkìrultxhv vrqw lghqwltxhv hw ohv 

sursruwlrqv ghv ghx{ glvwulexwlrqv ìjdohv/ +dqqh{hv 77 ã 79,1 hod ylhqgud frpsoìwhu o*ìwxgh idlwh dx yrlvlqdjh gh 

Y 4 1 Hq hhw/ qrxv | dyrqv hqylvdjì oh fdv rü mP f 6m 4 dyhf n 7 A 3 hw o*h{hpsoh txh qrxv wudlwrqv pdlqwhqdqw hvw 

fhoxl rü n 7 31 hwwh uhpdutxh shuphw gh phwwuh hq frqfxuuhqfh/ frpph oh prqwuhqw ohv wurlv h{hpsohv suìvhqwìv/ 

od pìwkrgh sdu S hw fhooh sdu S yduhjdohv 1 Od pìwkrgh sdu S 45 q*derxwlvvdqw sdv wrxmrxuv/ hooh qh shxw grqqhu 

vdwlvidfwlrq1 

431: rqfoxvlrq1 

O*lqyhuvlrq sdu oh sro|q÷ph S 45 gx vhfrqg v|vwëph vhpeoh wuëv frpsoìphqwdluh gh fhooh gx suhplhu v|vwëph sdu 

oh sro|q÷ph S srxu wurxyhu od vroxwlrq/ oruvtx*hooh h{lvwh/ gx sureoëph ghv prphqwv1 Lo dssdudñw/ hq hhw txh S 

grqqh gh wuëv erqqhv hvwlpdwlrqv oruvtxh ohv hqvhpeohv gh sdudpëwuhv vrqw ìorljqìv ghv ydulìwìv Y 4 hw Y 5 /doruvtxh 

o*lqwìuíw gh o*hvwlpdwlrq sdu od pìwkrgh S 45 vh suìflvh oruvtxh o*rq vh udssurfkh gh fhv ydulìwìv1 Oh vhfrqg v|vwëph 

ghv prphqwv grlw fhshqgdqw íwuh xwlolvì dyhf suìfdxwlrq hq frpsoìphqw gx suhplhu v|vwëph/ sxlvtxh qrxv vdyrqv 

txh od suìflvlrq ghv hvwlpdwhxuv ghv prphqwv hpslultxhv glplqxh dyhf o*dxjphqwdwlrq gh o*ruguh gh fhv prphqwv1 Oh 

fkdslwuh vxlydqw sursrvh xqh v|qwkëvh gh o*hqvhpeoh ghv frqwudlqwhv wkìrultxhv hw ghv sureoëphv sduwlfxolhuv vrxohyìv 

sdu oh pìodqjh gh ghx{ glvwulexwlrqv qrupdohv/ gìerxfkdqw vxu xqh pìwkrgh deoh g*hvwlpdwlrq ghv sdudpëwuhv1 


431: rqfoxvlrq1 458 

Paramètres issus de Pvarégales : Paramètres issus de P12 : Racine de P 12 choisie: 


Légende : 

1.09106412 1.15170995 0.52455407 5,09424696 0.86276401 3.99670640 oui 0.76506 

3.80174634 oui 5.00515 

1.773844448 non 

1.54463489 non 

0.00349219 non 


Annexe 46 1.00579630 1.03665526 0.50312787 5.00796237 0.97669277 4.00417662 oui 0.00022 

1.0 1.0 0.5 5.0 1.0 0.00068181 non 

3.84706114 oui 27.66 pas ces solutions. 

3.78866233 oui 34.060 Les tests sont mauvais, on ne retient 

1.68940092 non 

1.63821816 non 

0.00017001 non de P varégales. 

1.00714446 0.99873595 0.50055040 5.00720488 0.99873595 4.00011600 T6=0.001012 et T5=0.07754 T6 préconise le choix de la solution 

Annexe 45 1.00578288 0.99692069 0.50020967 5.00584061 1.00055521 4.00011477 oui 0.001014 

1.0 1.0 0.5 5.0 1.0 0.00003336 non 

1.68870807 non 

1.59655280 non 

0.00059191 non de P V 

1.00235430 0.99809997 0.50102661 5.00267799 0.99809997 4.00063055 T6=0.00227 et T5=0.25035 T6 préconise le choix de la solution 

Annexe 44 0.98848374 0.97962575 0.49755612 4.98878254 1.01665021 4.00050206 oui 0.00225 

1.0 1.0 0.5 5.0 1.0 0.00011608 non 


VARIANCES EGALES - CAS SYMETRIQUE (V 4). 

Ilj1 LL1481 0 


45: 

0kdslwuh440 

Pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv g*xqh 

glvwulexwlrq gh pìodqjh gh ghx{ glvwulexwlrqv 

qrupdohv lqfrqqxhv1 

4414 H{srvì gh od pìwkrgh sursrvìh1 

Dsuëv dyrlu uìfdslwxoì ohv glìuhqwhv pìwkrghv g*hvwlpdwlrq dyhf ohv glìuhqwv v|vwëphv/ qrxv ìqrqfhurqv od 

surfìgxuh jìqìudoh g*hvwlpdwlrq gx pìodqjh gh ghx{ glvwulexwlrqv qrupdohv/ sdu od pìwkrgh ghv prphqwv1 

441414 Hvwlpdwlrq ghv sdudpëwuhv dyhf 1 

Sdu xqh pìwkrgh dgìtxdwh/ rq fkhufkh ohv udflqhv srvlwlyhv gx sro|q÷ph = 

S +[, @ ;[ < .5; 7 [ : 45[ f5 

6 [ 9 . 63 5 7 .57 8 [ f 

6 [ 8 . 9 5 8 47; 7 [ f5 

6 [ 

7 

. 

rü 7 / 8 hw [ 6 vrqw ohv hvwlpdwhxuv uhvshfwliv gh n 7 / n 8 hw P f 61 Ohv prphqwv hw ohv fxpxodqwv vrqw hvwlpìv juåfh 

dx{ hvwlpdwhxuv hpslultxhv ghv prphqwv1 

Vl [ hvw od udflqh srvlwlyh shuphwwdqw g*rewhqlu ohv hvwlpdwhxuv ghv sdudpëwuhv +s > 4> 5> 4> 5,/ rqfdofxoh= 

3 @ 5[f 6[ 6 . 8 [ 5 6 7 [ [ f 6 .5[ f6 

6 

5[ 6 6 7 [ .7[ f5 

6 

hw @ [ f 5 [ = 

Pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv g*xqh glvwulexwlrq gh pìodqjh gh ghx{ glvwulexwlrqv qrupdohv lqfrqqxhv

45; 4414 H{srvì gh od pìwkrgh sursrvìh1 

Ohv sdudpëwuhv vrqw doruv hvwlpìv sdu = 

s @ 

t 

7[ 6 . +[ f 6 63 , 5 . m[ f 6 63 m 

5 

t7[ 6 . +[ f 6 63 , 5 

y 

t 

4 @ [ vjq +[ f 6 6 3 x 7[ 6 . +[ f 6 

, w 

, 5 m[ f 6 63 m 

t 

[ > 

7[ 6 . +[ f 6 63 , 5 . m[ f 6 63 m 

y 

t 

5 @ [ . vjq +[ f 6 6 3 x 7[ 6 . +[ f 6 

, = w 

, 5 . m[ f 6 63 m 

t 

[ > 

7[ 6 . +[ f 6 63 , 5 m[ f 6 63 m 

4 @ 3 + 5 [, 

s + 4 5 , 

hw 5 @ 3 + 4 [, 

+4 s , + 5 4 , > 

rü [ hw [ f 5/ vrqw/ uhvshfwlyhphqw/ ohv hvwlpdwhxuv gh P hw P f 51 

Orutx*lo | d soxvlhxuv vroxwlrqv vwdwlvwltxhphqw udlvrqqdeohv dx sureoëph ghv prphqwv/ qrxv fkrlvlurqv fhooh txl 

plqlplvhud oh whvw = 

W9 s 48 6 

4 .78 5 

4 { 5 

4 .48 4 { 7 

4 .{ 9 

4 . +4 s , 48 5 { 7 

5 .78 5 

5 { 5 

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5 . { 9 

5 [ 

f @ 

9 

 

> 

rü [ f 9 hvw o*hvwlpdwhxu gh P f 91 Qrwrqv/ hqq txh qrxv uhmëwhurqv wrxwh vroxwlrq srxu odtxhooh W 9 A 431 

5[ f 9 

441415 dv rü ohv yduldqfhv vrqw ìjdohv1 

Rq vh udpëqh gdqv fh fdv ã od uhfkhufkh ghv udflqhv srvlwlyhv gx sro|q÷ph gh ghjuì 6 = 

Vl [ hvw od udflqh/ rq fdofxoh / s > 4 hw 5 = 

S yduhjdohv +[, @5[ 6 . 7 [ [ f5 

6 = 

@ [ f 5 [ > 

s 

7[ 

s 

6 

. [ f5 

6 

@ 

. m[f 6m 

5 s > 

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6 

v s7[ 

4 @ [ vjq +[ f 6 , 6 

. [ f5 

6 m[f 6 

s m 

7[ 

6 

. [ f5 

6 . m[f 6 

v m[ 

s7[ 

hw 5 @ [ . vjq +[ f 6 

. [ f5 

6 

6, = 

. m[f 6 

s m = 

7[ 

6 

. [ f5 

6 m[f 6 m[ 

Pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv g*xqh glvwulexwlrq gh pìodqjh gh ghx{ glvwulexwlrqv qrupdohv lqfrqqxhv1

4414 H{srvì gh od pìwkrgh sursrvìh1 45< 

Gdqv oh fdv rü soxvlhxuv vroxwlrqv vhudlhqw vwdlvwltxhphqw dffhswdeohv/ qrxv fkrlvlulrqv fhooh plqlplvdqw o*xq ghv 

whvwv = 

W9 s 48 6 

4 .78 5 4 

@ 

{5 4 .48 4 {7 4 .{9 4 

 

 

. +4 s , 48 5 {7 

5[ f 9 

5 

.78 5 5 {5 

5 .486 

5 . {9 5 

[ 

f 9 

> 

rü 

W8 s { 8 

4 

@ 

.43 4 {6 4 .485 4 { 4 . +4 s , { 8 

5 .43 5 {6 5 .485 5 5 { [ 

f 8 

 

= 

5[ f 8 

441416 dv rü ohv pr|hqqhv vrqw ohv píphv1 

G*dsuëv rkhq/ +yrlu ^rkhq9:`, ohv sdudpëwuhv dxwuhv txh ohv pr|hqqhv +ìjdohv ã od pr|hqqh hpslultxh,/ vhurqw 

hvwlpìv sdu = 

4 

5 v 

@ [ f 5 . 4 7 9 9 

 

5 8 7 

5 

5 v 

@ [ f 5 . 4 7 9 9 

. 

5 8 7 

5 

. 7 7 

8 7 6 

5 

. 7 7 

8 7 6 

hw s @ 5 P 5 

5 . = 

4 

rx 9 hvw o*hvwlpdwhxu gx vl{lëph fxpxodqw gx pìodqjh n 9 @ P9 f 48P7P f 5 f 43P6 f5 .63P5 f6 1 

6 

8 > 

6 

8 

441417 Hvwlpdwlrq ghv sdudpëwuhv dyhf 45 1 

Sdu xqh pìwkrgh dgìtxdwh/ rq fkhufkh/ frpph srxu S / ohv udflqhv srvlwlyhv gx sro|q÷ph = 

S 45 +[, @ 4> 5> 4> 5,/ rqfdofxoh 

pdlqwhqdqw= 

3 @ 7[9 .;[ 7 7 . 

9 .5[ f5 

6 [ 6 5 7[ 5 7 7 [ f5 

6 [ .7[ f7 

6 

5[ 6 :[ 7 .43[ f5 

6 

hw @ [ f 5 [ = 

Pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv g*xqh glvwulexwlrq gh pìodqjh gh ghx{ glvwulexwlrqv qrupdohv lqfrqqxhv

463 4414 H{srvì gh od pìwkrgh sursrvìh1 

Ohv sdudpëwuhv vrqw doruv hvwlpìv/ frpph suìfìghpphqw/ sdu = 

s @ 

t 

7[ 6 . +[ f 6 63 , 5 . m[ f 6 63 m 

5 

t7[ 6 . +[ f 6 63 , 5 

y 

t 

4 @ [ vjq +[ f 6 6 3 x 7[ 6 . +[ f 6 

, w 

, 5 m[ f 6 63 m 

t 

[ > 

7[ 6 . +[ f 6 63 , 5 . m[ f 6 63 m 

y 

t 

5 @ [ . vjq +[ f 6 6 3 x 7[ 6 . +[ f 6 

, = w 

, 5 . m[ f 6 63 m 

t 

[ > 

7[ 6 . +[ f 6 63 , 5 m[ f 6 63 m 

4 @ 3 + 5 [, 

s + 4 5 , 

hw 5 @ 3 + 4 [, 

+4 s , + 5 4 , > 

rü [ hw [ f 5/ vrqw/ uhvshfwlyhphqw/ ohv hvwlpdwhxuv gh P hw P f 51 

V*lo | d soxvlhxuv vroxwlrqv vwdwlvwltxhphqw dffhswdeohv dx sureoëph gh prphqwv/ qrxv fkrlvlurqv fhooh txl 

plqlplvhud oh whvw = 

W8 s { 8 

4 .43 

@ 

4{ 6 

4 .48 5 

4 { 4 . +4 s , { 8 

5 .43 5{ 6 

5 .48 5 

5 { 5 [ 

f 8 

 

= 

5[ f 8 

rpph qrxv o*dyrqv idlw srxu S / qrxv {rqv xqh olplwh vxsìulhxuh ã od ydolgdwlrq g*xqh vroxwlrq sdu fh whvw/ hw 

qrxv uhmhwrqv xqh vroxwlrq srxu odtxhooh W 8 A 431 

441418 Pìwkrgh jìqìudoh g*hvwlpdwlrq ghv sdudpëwuhv g*xq pìodqjh gh 

ghx{ glvwulexwlrqv qrupdohv1 

Ohv sureoëphv g*xqlflwì ìwxglìv srxu ohv v|vwëphv gh prphqwv wkìrultxhv/ & 4 hw & 5 / vh frpsoltxhqw oruv gx 

sdvvdjh dx v|vwëph gh prphqwv hpslultxhv $ P q @ '+ e ,1 Hq hhw/ lo q*| d sdv frqwlqxlwì/ hq irqfwlrq ghv prphqwv gx 

pìodqjh/ ghv vroxwlrqv ghv v|vwëphv wkìrultxhv dx yrlvlqdjh gh Y 4 ^Y 5 rx Y 6 vhorq ohv fdv1 Od pìwkrgh g*hvwlpdwlrq 

txh qrxv sursrvrqv shuphw gh fkrlvlu oh v|vwëph srxu ohtxho o*hvwlpdwlrq hvw od soxv deoh/ hw shuphw gh sdoolhu 

dx{ fdv g*lqvwdelolwì suìvhqwìv gdqv ohv duwlfohv gh uìiìuhqfh Îu|hu:3` hw Îu|hu:5` grqw qrxv udsshorqv oh frqwhqx1 

Gdqv ohxu duwlfoh Îu|hu:3` frqvdfuì ã o*ìwxgh gh od vwdelolwì gh od pìwkrgh ghv prphqwv srxu oh pìodqjh gh ghx{ 

glvwulexwlrqv qrupdohv lqfrqqxhv Iu|hu hw Urehuwvrq/ hqylvdjhqw qhxi sdudpìwudjhv glìuhqwv srxu oh pìodqjh gh 

ghx{ glvwulexwlrqv qrupdohv1 Lov phwwhqw dlqvl hq ìylghqfh wurlv fdv srxu ohvtxhov od pìwkrgh suìvhqwh xq eldlv wuëv 

lpsruwdqw/ tx*lov qh sduylhqqhqw sdv ã h{soltxhu1 hv wurlv h{hpsohv vrqw vlwxìv gdqv xq yrlvlqdjh lppìgldw gh Y 5 / 

fdv grqw qrxv dyrqv glvfxwì od vwdelolvdwlrq soxv kdxw1 Qrxv vrpphv sdu frqvìtxhqw hq phvxuh gh mxvwlhu oh eldlv 

lpsruwdqw frqvwdwì sdu Iu|hu hw Urehuwvrq sdu oh idlw txh & 4 q*hvw sdv dssursulì ã o*hvwlpdwlrq ghv sdudpëwuhv gdqv 

fhv wurlv h{hpsohv1 Lo hvw ã qrwhu txh od pìwkrgh gh phvxuh gx eldlv sursrvìh sdu Iu|hu hw Urehuwvrq qh idlw sdv 

dssdudñwuh gh eldlv lpsruwdqw gdqv oh vhxo fdv gh pìodqjh fkrlvl dx yrlvlqdjh gh Y 4 1 Ohv h{hpsohv txh qrxv dyrqv 

wudlwìv gdqv oh fkdslwuh suìfìghqw prqwuhqw srxuwdqw xqh ìsdlvvhxu qrq qìjoljhdeoh gx yrlvlqdjh gh Y 4 vxu ohtxho 

o*xwlolvdwlrq gh od pìwkrgh ghv prphqwv edvìh vxu & 4 q*hvw sdv deoh1 Gdqv o*duwlfoh Îu|hu:5` gh frpsdudlvrq 

gh glìuhqwhv pìwkrghv g*hvwlpdwlrq ghv sdudpëwuhv gx pìodqjh gh ghx{ glvwulexwlrqv qrupdohv lqfrqqxhv/ Iu|hu 

hw Urehuwvrq hqylvdjhqw qhxi qrxyhdx{ h{hpsohv gh sdudpìwudjhv1 Lov frqvwdwhqw gdqv txdwuh fdv/ o*lqiìulrulwì gh 

Pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv g*xqh glvwulexwlrq gh pìodqjh gh ghx{ glvwulexwlrqv qrupdohv lqfrqqxhv1

464 

od pìwkrgh ghv prphqwv sdu udssruw ã xqh pìwkrgh gh pd{lplvdwlrq gh od yudlvhpeodqfh hw ã xqh pìwkrgh gh 

plqlplvdwlrq g*xqh glvwdqfh gx " 5 / ohv flqt dxwuhv fdv q*ìwdqw sdv gìidyrudeohv ã od pìwkrgh ghv prphqwv1 hv 

txdwuh fdv/ ã o*ruljlqh gh o*dedqgrq surjuhvvli gh od pìwkrgh ghv prphqwv srxu o*hvwlpdwlrq ghv sdudpëwuhv gh 

glvwulexwlrqv gh pìodqjhv/ vrqw/ g*dsuëv qrwuh ìwxgh/ ã wudlwhu dyhf oh v|vwëph & 5 1 Hq hhw/ ghx{ fdv vrqw lvvxv 

g*xq surfkh yrlvlqdjh gh Y 4 hw ghx{ dxwuhv g*xq surfkh yrlvlqdjh gh Y 5 1 

Od pìwkrgh g*hvwpdwlrq txh qrxv sursrvrqv idyrulvh oh v|vwëph ghv flqt suhplëuhv ìtxdwlrqv gh prphqwv/ hw 

qh suìfrqlvh oh uhfrxuv dx vhfrqg v|vwëph txh gdqv xq yrlvlqdjh gh Y 4 rx Y 5 1 hwwh pìwkrgh wluh ohv hqvhljqhphqwv 

ghv txdudqwh vl{ h{hpsohv wudlwìv lfl hw ghv gl{0kxlw fdv ìwxglìv gdqv ohv duwlfohv gh Iu|hu hw Urehuwvrq= 

D 4= Vl m[ f 6m 3=8 hw [ f 6 9@ 3= 

| Vl 7 A 3/ o*hvwlpdwlrq ylhqw gh & 5 1 

| Vl 7 3/ vroxwlrq lvvxh gh S rx S yduhjdohv plqlplvdqw oh whvw W 9 1 

D 5= Vl [ f 6 @3= 

| Vl 7 A 3/ fdv sduwlfxolhu gh & 5 rü 4 @ 5 1 

| Vl 7 3/ fkrl{ gh od vroxwlrq lvvxh gh S yduhjdohv 1 

D 6= Rq fkrlvlud od vroxwlrq gh & 5 vl = 

| Od vroxwlrq lvvxh gh & 5 yìulh = 

 

 

W Y# @43 m{ 4 { 5 m 9 s 4@7 

t 5 

9 < m 5 4 5 5 m 5/ 

| Vl & 4 qh grqqh sdv gh vroxwlrq dffhswdeoh1 

D 7= Vlqrq od vroxwlrq ylhqw gh & 4 1 

Hvwlpdwhxuv ghv prphqwv g*xq pìodqjh h{srqhqwlhooh0h{srqhqwlhooh1

466 

0kdslwuh450 

Hvwlpdwhxuv ghv prphqwv g*xq pìodqjh 

h{srqhqwlhooh0h{srqhqwlhooh1 

Oh v|vwëph ghv prphqwv/ qrq fhqwuìv/ srxu xq pìodqjh gh ghx{ glvwulexwlrqv h{srqhqwlhoohv/ gh sdudpëwuhv 

uhvshfwliv p 4 hw p 5 /hvw= 

; 

sp 4 . +4 s, p 5 @ P 4 

& 6 @ 5sp 5 4 .5+4 s, p 5 5 @ P 5 

= 

9sp 6 4 .9+4 s, p6 5 @ P 6 

= 

4514 dofxo gx mdfrelhq gh & 1 

Oh mdfrelhq M 6 gh & 6 hvw 

M 6 @ 

 

s 4 s p 4 p 5 

7sp 4 7 +4 s, p 5 5p 5 4 5p 5 5 

4;sp 5 4 4; +4 s, p 5 5 9p 6 4 9p 6 5 

> 

txh o*rq gìyhorssh hw idfwrulvh gluhfwhphqw hq = 

M 6 @45s +4 s, +p 4 p 5 , 7 = 

h mdfrelhq qh v*dqqxoh txh vl p 4 @ p 5 / f*hvw ã gluh vl o*rq q*hvw sdv hq suìvhqfh g*xq pìodqjh1 Lo q*| d/ grqf/ sdv 

srxu fh w|sh gh pìodqjh/ gh sureoëphv ìtxlydohqwv ã fhx{ vljqdoìv srxu oh pìodqjh gh ghx{ glvwulexwlrqv qrupdohv1 


467 4515 Uìvroxwlrq dojìeultxh gx v|vwëph & 6 1 

4515 Uìvroxwlrq dojìeultxh gx v|vwëph & 1 

hwwh ìwxgh d ìwì hhfwxìh sdu Ulghu +yrlu Ûlghu94` hw Ûlghu95`,1 Od uìvroxwlrq gx v|vwëph ghv prphqwv +qrq 

fhqwuìv,/ grqqh ghx{ wulsohwv vroxwlrqv fdofxodeohv olwwìudohphqw/ oruvtxh 

45 P 5 4 9 P 5 9@3hw < 6 P 5 4 5 P 5 

 

P 

5 

5 .9P 4 

 

7 P 

5 

4 6 P 5 

 

P6 . P 5 6 A 3= 

hv vroxwlrqv vrqw v|pìwultxhv +sdu od wudqvirupdwlrq p 4 :$ p 5 /hws :$ +4 s,,1 hv vroxwlrqv vrqw grqqìhv 

fl0ghvvrxv + ohv P l ìwdqw ohv prphqwv g*ruguh l ,= 

dlqvl txh 

p 4 @ 6 P 4 P 5 P 6 . s < +6 P4 5 5 P 5, P5 5 .9P 4 +7 P4 5 6 P 5, P 6 . P6 

5 

45 P4 5 9 P 5 

s @ 4 5 . 45 P4 6 

5 s < +6 P4 5 5 P 5, P5 5 .9P 4 +7 P4 5 6 P > 

5, P 6 . P6 

5 

p 5 @ 6 P 4 P 5 . P 6 . s < +6 P 5 4 5 P 5, P 5 5 .9P 4 +7 P 5 4 6 P 5, P 6 . P 5 6 

45 P 5 4 9 P 5 

> 

> 

p 4 @ 6 P 4 P 5 . P 6 . s < +6 P4 5 5 P 5, P5 5 .9P 4 +7 P4 5 6 P 5, P 6 . P6 

5 

45 P4 5 9 P 5 

s @ 4 5 . 45 P4 6 

5 s < +6 P4 5 5 P 5, P5 5 .9P 4 +7 P4 5 6 P > 

5, P 6 . P6 

5 

p 5 @ 6 P 4 P 5 P 6 . s < +6 P4 5 5 P 5, P5 5 .9P 4 +7 P4 5 6 P 5, P 6 . P6 

5 

45 P4 5 9 P = 

5 

O*lqyhuvlrq gh fh v|vwëph grqqh xqh vroxwlrq xqltxh/ lo q*hvw grqf sdv xwloh gh gìqlu xq fulwëuh gh fkrl{ srxu 

glvwlqjxhu ghv vroxwlrqv glìuhqwhv1 Od iruph ghv vroxwlrqv q*dvvxuh sdv/ srxu dxwdqw/ txh ohv vroxwlrqv rewhqxhv 

sdu lqyhuvlrq gx v|vwëph vrlhqw vwdwlvwltxhphqw dffhswdeohv1 

> 

4516 H{hpsohv g*lqyhuvlrq gx v|vwëph & 1 

O*ìwxgh gh o*lqyhuvlrq grlw vh frqfhqwuhu dxwrxu gx fdv dqqxodqw oh mdfrelhq/ f*hvw ã gluh oruvtxh ohv ghx{ 

glvwulexwlrqv frpsrvdqw oh pìodqjh vrqw surfkhv o*xqh gh o*dxwuh1 Ohv uìvxowdwv ghv h{hpsohv wudlwìv gdqv ohv 

dqqh{hv 7: ã 88/ vrqw uìfdslwxoìv gdqv oh wdeohdx Ilj1 LL1491 

Ohv uìvxowdwv ghv h{hpsohv wudlwìv srxu ohv prphqwv wkìrultxhv/ flwìv hq dqqh{h 7: hw 7;/ prqwuhqw txh 

g*ìyhqwxhoohv huuhxuv gdqv od uìvroxwlrq sdu oh v|vwëph hpslultxh/ qh ylhqgudlhqw txh g*xq pdqtxh gh suìflvlrq 

ghv prphqwv hpslultxhv1 

Oh fdv jìqìudo/ srxu ohv prphqwv hpslultxhv/ flwì hq dqqh{h 7

4516 H{hpsohv g*lqyhuvlrq gx v|vwëph & 6 1 468 

MELANGES EXPONENTIELLE-EXPONENTIELLE. 

Paramètre F1 p Paramètre F2 Commentaires 

Annexe 47 3.0 0.7 1.0 Système théorique 

3.0 0.7 1.0 

Annexe 48 1.0001 0.7 1.0 Système théorique 

1.0001 0.7 1.0 

Annexe 49 2.0 0.3 1.0 cas général 

2.046887687 0.281250057 1.010775899 

Annexe 50 1.2 0.3 1 Tous les paramètres sont bien estimés. 

1.230788933 0.2578908341 0.9972737237 

Annexe 51 1.2 0.3 1 La distribution de mélange est trop 

2.175493119 0.00951502503 1.043919526 proche d'une distribution exponentielle. 

Annexe 52 1.2 0.5 1 p est mal estimé. 

1.2720162 0.2248471473 1.045829274 

Annexe 53 1.1 0.3 1 La distribution de mélange est trop 

10.69721687 2.72E-6 1.028152786 proche d'une distribution exponentielle. 

Annexe 54 1.1 0.3 1 p est supérieur à 1 

1.033271822 1.004360216 1.959452341 

Annexe 55 1.1 0.3 1 certains paramètres sont négatifs 

-0.5128172088 -0.001361234 1.030250451 

Paramètres théoriques : 

Paramètres estimés : 

Ilj1 LL1491 0 


46: 

0kdslwuh460 

Hvwlpdwhxuv ghv prphqwv g*xq pìodqjh 

qrupdoh0h{srqhqwlhooh1 

Oh v|vwëph ghv prphqwv/ qrq fhqwuìv/ srxu oh pìodqjh g*xqh glvwulexwlrqv qrupdoh Q + 4 > 4 , hw g*xqh glvwule0 

xwlrqv h{srqhqwlhoohv/ gh sdudpëwuh p 5 /v*ìfulw= 

; 

A 

& 7 @ 

A= 

P 4 @ s 4 . +4 s, p 5 

P 5 @ s 5 4 . 4 

 

.5+4 s, p 

5 

5 

P 6 @ s 

6 4 .6 4 4 .9+4 s, p 

6 

5 

P 7 @ s = 

 

7 4 .9 5 4 4 .6 5 4 .57+4 s, p 

7 

5 

4614 dofxo gx mdfrelhq1 

Oh mdfrelhq M 7 gh & 7 hvw/ vl o*rq idfwrulvh s gdqv ohv suhplëuhv hw ghuqlëuhv frorqqhv/ hw +4 s, gdqv od ghx{lëph/ 

M 7 @ s 5 +4 s, 

 

hq dmrxwdqw ã od wurlvlëph frorqqh p 5 irlv od ghx{lëph/ rq d 

M 7 @ s 5 +4 s, 

 

> 3 6 2 f 

2> e6 2 > 2 n j 3 26 2 2 

E> 2 n j H6 2 2 > nj > 3 S6 2 > 

eE> nj > bS6 2 > e nSj > 2 nj 2 3 2e6 e 2 SE> 2 n j 

> f 

2> e6 2 > 2 n j n26 2 2 

E> 2 n j H6 2 2 > nj > n 26 2 > 

eE> nj > bS6 2 > e nSj > 2 nj 2 n.26 e 2 SE> 2 n j 

> 

 

= 

 

Hvwlpdwhxuv ghv prphqwv g*xq pìodqjh qrupdoh0h{srqhqwlhooh1

46; 4614 dofxo gx mdfrelhq1 

Vl o*rq uhwudqfkh/ doruv/ ã od wurlvlëph frorqqh 4 irlv od suhplëuh/ rq d 

M 7 @ s 5 +4 s, 

 

f 

f 

2> e6 2 3> 2 n j n26 2 2 

E> 2 n j H6 2 2 32> n 26 2 > 

eE> nj > bS6 2 3> e 3 Sj > 2 nj 2 n.26 e 2 SE> 2 n j 

Dq gh srxyrlu gìyhorsshu oh gìwhuplqdqw vxlydqw od suhplëuh oljqh/ gdqv gh erqqhv frqglwlrqv/ rq uhwluh od 

ghx{lëph frorqqh ã od suhplëuh srxu rewhqlu = 

M 7 @ s 5 +4 s, 

 

hw dsuëv gìyhorsshphqw = 

M 7 @ s 5 +4 s, 

 

fh txl shuphw g*rewhqlu = 

 

M 7 @ s 5 +4 s, 

f f f 

2> 3 e6 2 e6 2 3> 2 n j n26 2 2 

E> 2 n j 3 H6 2 2 H6 2 2 32> n 26 2 > 

eE> nj > 3 bS6 2 bS6 2 3> e 3 Sj > 2 nj 2 n.26 e 2 SE> 2 n j 

2> 3 e6 2 3> 2 n j n26 2 2 

E> 2 n j 3 H6 2 2 32> n 26 2 > 

eE> nj > 3 bS6 2 3> e 3 Sj > 2 nj 2 n.26 e 2 SE> 2 n j 

3> S n 2> D 6 2 3 SS > e 6 2 2 n b2> 6 2 3 2HH > 2 6 e 2 n 2HH > 6 D 2 3 ee 6 S 2 

n 

> e 3 H > 6 2 nS> 2 6 2 2 3 bS > 6 2 neH62 e j 3 b > 2 3 e > 6 2 3 2 6 2 2 j 2 3 b j 

^46179` 

Dq gh idfwrulvhu fh gìwhuplqdqw/ lo qrxv idxw phwwuh oh sro|q÷ph gh ghjuì wurlv hq 4 / vrxv od iruph gh dugdq1 

Srvrqv srxu fhod = 

hw ^46179` ghylhqw = 

[ @ 4 +5 4 7 4p 5 5p 5 5 , 

6 

= 

 

> 

 

 

= 

3 

M 7 @ e 3 H > 6 2 n2e> 2 6 2 2 3 ef > 6 2 n2S6 e 

2 f 

 

 

> S 3 2 > D 6 2 nSf> e 6 2 2 3 Sf > 6 2 n 222 > 2 6 e 2 3 2f > 6 D 2 n S6 S 2 

~} € 

 

4 

 

D = 

^4617:` 

Oh gìwhuplqdqw gh dugdq hvw doruv = 

@ 7D 6 .5:E 5 > 

@ 65 

> 2 3 2e > 6 2 n2Se> f 6 2 2 3 .S > b 6 2 n .ee > H 6 e 2 3 2 f2e > . 6 D 2 n SHSe > S 6 S 2 3 2.bS > D 6 . 2 

5: 

32HSH > e 6 H 2 n He > 6 b 2 3 feSe > 2 62 f n .eef> 6 2 3 .fSe 62 

2 

 

> 

grqw oh vljqh hvw wuëv gl!floh ã ìwdeolu1 Rq shxw yìulhu/ vxu ghv h{hpsohv txh fh gìwhuplqdqw q*hvw sdv gh vljqh 

frqvwdqw1 Oh qrpeuh gh udflqhv uìhoohv gx sro|q÷ph hq [ gh ^4617:` ìwdqw irqfwlrq gx vljqh gh fh gìwhuplqdqw/ 

qrxv qh srxyrqv grqqhu gh idfwrulvdwlrq soxv suìflvh gx mdfrelhq1 Od jxuh Ilj1 LL14: uhsuìvhqwh xqh sduwlh gh od 

vxuidfh 1 


4614 dofxo gx mdfrelhq1 46< 

Ilj1LL14:10Sduwlhghodvxuidfh1 


473 4615 Lqyhuvlrq dojìeultxh gx v|vwëph1 

Srxu wurxyhu xq h{hpsoh g*hqvhpeoh gh sdudpëwuhv dqqxodqw oh mdfrelhq/ lo hvw qìfhvvdluh g*hq {hu ghx{/ lfl 

p 4 @3hw p 5 @4/ hw wurxyhu ohv udflqhv frqyhqdeohv gx sro|qrph hq oh ghuqlhu sdudpëwuh/ lfl 

44 @ 5 6 5 6 

hw 45 @ 5 6 . 7 6 

# # 

s 

%4 

6 s $$& u % 

56: 

46 

46 frv dufwdq 

.5 

6 

; 

6 vlq 

# # $$& 

s 

46 frv 

%4 

6 

dufwdq 

6 s 56: 

; 

7=9;8;8= 

# # 

4 

dufwdq 

6 

6 s $$& 

56: 

3= ^4617;` 

sdu vxevwlwxwlrq ohv ghx{ ghuqlëuhv ìtxdwlrqv ghylhqqhqw = 

3@5 +4.s 6 , p 6 5 .9 +4.s, p 5 5 P 4 5 P 6 4 .6 sP 4 P 5 .6 +4.s, p 5 +5 P 5 4 . sP 5 , s 5 P 6 

^4617

4615 Lqyhuvlrq dojìeultxh gx v|vwëph1 474 

hw hq uhqrxyhodqw o*rsìudwlrq/ ^46184` ghylhqw = 

 

p 5 5 6 +4.s, 5 s 

 

5 +4.s +6.s + 

sP 7 , 

^46185` 

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ohv pìodqjhv jìqìudolvìv gh od idplooh uìxqlvvdqw ohv glvwulexwlrqv qrupdohv hw h{srqhqwlhoohv1 

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485 4616 Pìwkrgh g*lqyhuvlrq gx v|vwëph & 7 1 

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4617 H{hpsohv g*lqyhuvlrqv gx v|vwëph & 7 1 486 

D61 Vl 41 hw 51 irxuqlvvhqw ghv vroxwlrqv/ qrxv fkrlvlvvrqv fhooh gh wrxwhv ohv vroxwlrqv txl plqlplvhud oh whvw 

W 9 Qruph{ 1 

4617 H{hpsohv g*lqyhuvlrqv gx v|vwëph & e 1 

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shuwlqhqwv1 Qrxv qh grqqhurqv/ grqf/ g*h{hpsoh/ txh gdqv oh fdv jìqìudo/ hw gdqv oh fdv g*dqqxodwlrq gx mdfrelhq 

txh qrxv dyrqv flwì hq h{hpsoh soxv kdxw1 

461714 H{hpsohv g*lqyhuvlrq wkìrultxh1 

Ohv uìvxowdwv txh qrxv frpphqwhurqv lfl/ vrqw uìfdslwxoìv gdqv oh wdeohdx Ilj1 LL14;1 

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pr|hqqhv hw gh píph yduldqfhv wkìrultxhv1 Lo q*| d sdv gh gl!fxowì gdqv fhv wurlv h{hpsohv1 

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q*hvw sdv dwwhljqdeoh sdu fh v|vwëph/ hw qìfhvvlwh/ d sulrul/ oh uhfrxuv ã xqh ìtxdwlrq vxssoìphqwdluh1 

461715 H{hpsohv g*hvwlpdwlrq ghv sdudpëwuhv g*xq ìfkdqwloorq1 

Ohv uìvxowdwv rewhqxv srxu ohv prphqwv hpslultxhv hw suìvhqwìv hq dqqh{h 94 ã :3/ vrqw uìvxpìv gdqv ohv 

wdeohdx{ Ilj1 LL14< hw Ilj1 LL1531 Qrxv qrxv frqfhqwurqv/ dsuëv dyrlu gìwdlooì gdqv ohv dqqh{hv 94 ã 96/ oh shqgdqw 

ghv h{hpsohv jìqìudx{ wudlwìv gdqv oh fdguh gx v|vwëph wkìrultxh/ vxu o*ìwxgh gh o*lqyhuvlrq gx v|vwëph hpslultxh/ 

dx yrlvlqdjh gh ghx{ fdv g*dqqxodwlrq gx mdfrelhq1 Ohv h{hpsohv ghv dqqh{hv 97/ 98 hw 99 prqwuhqw txh gdqv oh 

fdv lqgxlvdqw wkìrultxhphqw xqh udflqh grxeoh/ oh gìfdodjh suryrtxì sdu ohv prphqwv hpslultxhv grqqh ghx{ 

vroxwlrqv surfkhv rx sdv gh vroxwlrq1 Gh píph/ frpph oh prqwuhqw ohv uìvxowdwv flwìv hq dqqh{hv 9: ã :3/ ohfdv 

g*lqvx!vdqfh gx v|vwëph/ hqjhqguh/ oruv gx sdvvdjh dx{ prphqw hpslultxhv/ ghv vlwxdwlrqv fulwltxhv gx w|sh gh 

fhoohv txh qrxv dyrqv gìmã uhqfrqwuìhv1 

461716 rqfoxvlrq1 

rpswh whqx gh od frpsoh{lwì g*ìfulwxuh ghv sro|qrphv txh qrxv frqvlgìurqv/ qrxv qh gìyhoorshurqv sdv gh 

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g*hvwlpdwlrq dyhf ohv prphqwv hpslultxhv1 Qrxv hq vrpphv ã xq srlqw rx qrxv frqqdlvvrqv xqh pìwkrgh g*hv0 

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dyrqv lghqwlì/ vrxv xqh iruph fhuwh frpsoh{h/ ohv grpdlqhv gh qrq dssolfdwlrq1 hwwh pìwkrgh srxuud grqf íwuh 

dssoltxìh/ frpph od pìwkrgh lqlwldoh gh Shduvrq/ dx{ sureoëphv gh frqwlqxlwì ghv vroxwlrqv gdqv oh yrlvlqdjh gh 

od ydulìwì dqqxodqw oh mdfrelhq1 


487 4617 H{hpsohv g*lqyhuvlrqv gx v|vwëph & 7 1 

Paramètres calculés : 

Paramètres théoriques : Racine retenue pour l'inversion : 

Légende : 

0.96222234 non 

Annexe 60 -2.57486722 0.87873829 0.16757485 1.11899910 0.16757485 oui 0.323474 La solution ne peut être atteinte. 

0.0 4.68584617 0.5 1.0 

0.7 oui 0.0 

0.0 0.94136684 0.7 1.0 0.7 oui 0.0 

0.52344703 oui 0.044653 Il y a une racine double 

Annexe 59 0.51675553 oui 0.332803 

0.0 0.94136684 0.7 1.0 

1.0 1.0 0.5 1.0 0.5 oui 0.0 

Annexe 58 1.0 1.0 0.5 1.0 

1.0 2.0 0.5 4.0 0.5 oui 0.0 

Annexe 57 0.10165285 oui 0.099264 

1.0 2.0 0.5 4.0 

0.98967385 oui 1.317914 

0.36081114 oui 0.322049 

0.0 2.0 0.3 1.0 0.3 oui 0.0 Il ya deux racines proches 

Annexe 56 0.10290133 oui 0.065514 

0.0 2.0 0.3 1.0 

Moyenne F1 Variance F1 p Paramètre F2 1>racines>0 acceptable Test T5normex Commentaires 

INVERSION DE S 4 POUR DES MOMENTS THEORIQUES 

Ilj1 LL14;1 0 


4617 H{hpsohv g*lqyhuvlrqv gx v|vwëph & 7 1 488 



Légende : 

0.45347870 oui 0.036774 

0.30571448 oui 0.000616 

0.02546270 0.94798025 0.30571448 1.000311 0.29549171 oui 0.223447 Il ya deux racines proches 

Annexe 66 0.27891681 oui 0.005002 

0.0 0.94136684 0.3 1.0 

0.37595574 0.93336298 0.55713404 1.10887081 0.55713404 oui 0.063909 

Annexe 65 0.29023618 oui 0.224985 l'estimation n'est pas bonne. 

0.0 0.94136684 0.3 1.0 

0.44448145 oui 0.277748 

Annexe 64 -0.51230776 0.68229771 0.29223659 0.91541446 0.29223659 oui 0.013325 l'estimation n'est pas bonne. 

0.0 0.94136684 0.5 1.0 

1.01732736 1.02979712 0.48367114 0.98803143 0.48367114 oui 0.007055 

Annexe 63 3.0795E-7 non 

1.0 1.0 0.5 1.0 

0.98817036 1.99177643 0.50000254 4.02568958 0.50000254 oui 0.003090 

Annexe 62 0.10290405 oui 0.097352 

1.0 2.0 0.5 4.0 

0.99281194 oui 1.616472 

0.36396935 oui 0.328589 

0.05477885 2.06558014 0.31039688 0.98808554 0.31039688 oui 0.007989 

Annexe 61 0.09447871 oui 0.118002 

0.0 2.0 0.3 1.0 


INVERSION DE S 4 POUR DES MOMENTS EMPIRIQUES 1 

Ilj1 LL14

489 4617 H{hpsohv g*lqyhuvlrqv gx v|vwëph & 7 1 



Légende : 

0.95811690 non 

Annexe 70 -2.59764579 0.84669134 0.16587310 1.11710361 0.16587310 oui 0.336006 l'estimation est mauvaise. 

0.0 4.68584617 0.5 1.0 

0.97928108 non 

0.52499534 oui 0.180466 paramètres proches 

-0.12174279 4.35617662 0.47942449 1.08671354 0.47942449 oui 0.048000 Il y a deux ensembles de 

Annexe 69 0.17005080 oui 0.170210 

0.0 4.68584617 0.5 1.0 

0.99711692 non 

Annexe 68 -2.59824356 0.82188741 0.09941814 1.06491402 0.09941814 oui 0.423201 l'estimation est mauvaise. 

0.0 4.68584617 0.3 1.0 

0.99841621 non 

0.31665744 oui 0.184602 paramètres proches 

-0.14891449 4.38156985 0.28572474 1.04579647 0.28572474 oui 0.030084 Il y a deux ensembles de 

Annexe 67 0.10636466 oui 0.352289 

0.0 4.68584617 0.3 1.0 


INVERSION DE S 4 POUR DES MOMENTS EMPIRIQUES 2 

Ilj1 LL1531 0 


0kdslwuh470 

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jìqìudolvì vlpsoh1 

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ghx{ glvwulexwlrqv uhqfrqwuìv srxu od idplooh uìxqlvvdqw ohv glvwulexwlrqv qrupdohv hw h{srqhqwlhoohv1 Qrxv vdyrqv 

wudlwhu lqgìshqgdpphqw fkdfxq gh fhv fdv/ vl qrxv vdyrqv o*lghqwlhu/ fh txl q*hvw/ sdu gìqlwlrq/ sdv oh fdv gdqv 

oh fdguh ghv pìodqjhv jìqìudolvìv1 

Od pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv g*xq pìodqjh jìqìudolvì/ txh qrxv sursrvrqv/ frqvlvwh ã wudlwhu/ gdqv 

fkdfxq ghv wurlv fdv/ qrupdoh0qrupdoh/ qrupdoh0h{srqhqwlhooh hw h{srqhqwlhooh0h{srqhqwlhooh/ dydqw gh fkrlvlu/ sdupl 

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pìodqjh/ vxssrvì íwuh xq pìodqjh qrupdoh0qrupdoh/ hvw idlwh sdu & 4 / hw od flqtxlëph ìtxdwlrq gh prphqwv fhqwuìv/ 

oruvtxh f*hvw & 5 txl hvw xwlolvì1 

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f 5P9 

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pìwkrgh1 

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w|shv gh pìodqjhv vrqw wrxmrxuv fkrlvlv1 Qrxv frqvwdwrqv/ dxvvl/ txh ohv pdxydlv w|shv gh pìodqjhv qh grqqhqw/ 

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Paramètres Expo-Expo : Paramètres Normale-Expo : 

Paramètres théoriques : Paramètres Normale-Normale : Mélange choisi : 

Légende : 

1.43898267 1.39337701 0.26262660 2.74228075 / 0.018751 0.041553 

2.03503477 2.60650464 0.96463754 12.25573018 2.60650464 0.259503 0.131354 seul Pvarégales donne une solution 

Annexe 73 2.0 / 0.6 3.0 / 0.0 0.0 

2.0 / 0.6 3.0 / 

0.90809461 0.61225098 0.56174646 2.94323024 / 0.008969 0.017393 les tests sont proches de zéro 


Annexe 72 1.0 / 0.6 3.0 / 0.0 0.0 

1.0 / 0.6 3.0 / 



Annexe 71 1.0 / 0.6 0.3 / 0.0 0.0 

1.0 / 0.6 0.3 / 

Moyenne F1 Variance F1 p Moyenne F2 Variance F2 Test T5 Test T6 Commentaires 

Estimation des paramètres d'un mélange généralisé 

Cas théorique du mélange Exponentielle-Exponentielle 

Ilj1 LL1541 0 


4715 H{hpsohv g*dssolfdwlrqv1 494 



Légende : 


1.0 1.0 0.6 2.0 4.0 0.0 0.0 

Annexe 76 / / pas de solution 

3.0 20.0 0.6 1.0 1.0 


1.0 1.0 0.6 2.0 4.0 0.0 0.0 


1.0 1.0 0.6 2.0 4.0 

pas de solution 

1.0 2.0 0.6 3.0 1.0 0.0 0.0 


1.0 2.0 0.6 3.0 1.0 



Cas théorique du mélange Normale-Normale 

Ilj1 LL1551 0 





Légende : 

0.0 1.0 0.6 1.0 / 0.0 0.0 

0.33674095 0.96653071 0.95363122 1.70100051 5.08933166 0.004000 c'est la solution de P12- T5 proche de 0 

Annexe 80 / / il n'y a pas de solution 

0.0 1.0 0.6 1.0 / 

1.0 1.0 0.6 1.0 / 0.0 0.0 



1.0 1.0 0.6 1.0 / 

1.0 4.0 0.6 1.0 / 0.0 0.0 

1.135446428 4.28920267 0.56570558 0.82354621 0.80514722 0.080779 on retient la solution de P12 


1.0 4.0 0.6 1.0 / 

1.0 4.0 0.6 3.0 / 0.0 0.0 



1.0 4.0 0.6 3.0 / 



Cas théorique du mélange Normale-Exponentielle 

Ilj1 LL1561 0 





Légende : 

1.40784371 1.29864702 0.24580622 2.69574622 / 0.014866 0.029218 

2.01306670 2.53131300 0.96387443 12.1473029 2.53131300 0.128986 0.253619 que l'on reconnaisse le type du mélange. 

Annexe 85 2.98032949 / 0.38213172 2.007375 / 0.009479 0.022308 l'échantillon est assez grand pour 

3.0 / 0.4 2.0 / 

0.89643206 0.59423949 0.36866601 2.96214342 / 0.004121 0.006592 

1.80979153 2.52437762 0.96513330 13.01803117 2.52437762 0.120711 0.245375 

Annexe 84 0.98722375 / 0.39334927 2.98732371 / 0.000707 0.002007 échantillon de 1000000 d'individus 

1.0 / 0.4 3.0 / 

0.95285147 0.74995525 0.39274032 2.99119584 / 0.015325 0.036278 


Annexe 83 0.99801788 / 0.40293023 2.99550292 / 0.018905 0.039580 l'échantillon est trop petit pour 

1.0 / 0.4 3.0 / 

0.28160786 0.06048700 0.58302999 0.99454964 / 0.007782 0.015534 

0.48169344 0.20862924 0.97461386 4.31015063 0.20862924 0.117735 0.247093 

Annexe 82 1.0084773 / 0.39018254 0.30401388 / 0.001544 0.003107 échantillon de 1000000 d'individus 

1.0 / 0.4 0.3 / 

0.28897659 0.06742860 0.59296923 0.99877876 / 0.014252 0.028591 


Annexe 81 1.00616939 / 0.38910866 0.30509304 / 0.015402 0.026842 l'échantillon est trop petit pour 

1.0 / 0.4 0.3 / 



Cas empirique du mélange Exponentielle-Exponentielle 

Ilj1 LL1571 0 





Légende : 

2.94830504 20.08697924 0.59645213 1.09753116 / 0.004263 les tests sont proches. 

2.95261012 19.83734829 0.61168198 1.01816235 0.69145387 / 0.001311 


3.0 20.0 0.6 1.0 1.0 

1.57950366 3.37521160 0.50766700 1.19706000 / 0.091513 

1.00030375 1.02069346 0.60995276 2.00251641 4.08666255 / 0.004711 


1.0 1.0 0.6 2.0 4.0 

pas de solution 

0.99203358 1.96799683 0.59534566 3.0025707 0.99724281 / 0.001345 


1.0 2.0 0.6 3.0 1.0 



Cas empirique du mélange Normale-Normale 

Ilj1 LL1581 0 





Légende : 

1.00837351 4.01531550 0.59904759 0.99154427 / 0.033230 

1.14451460 4.33937999 0.54985322 0.82708744 0.86343358 0.062990 

Annexe 92 / / échantillon de 1000000 d'individus 

1.0 4.0 0.6 1.0 / 

0.95365402 3.94210364 0.59933462 1.03007896 / 0.111515 on choisit le bon type de mélange 

1.10994187 4.32503767 0.55040746 0.83042874 0.86590947 0.166824 les tests sont proches, 

Annexe 91 / / 

1.0 4.0 0.6 1.0 / 

0.97934946 3.91105593 0.61015857 1.04277362 / 0.228123 on choisit le mauvais type de mélange 


Annexe 90 / / 

1.0 4.0 0.6 1.0 / 

1.00362061 4.0118766 0.60511885 2.99876241 / 0.008974 0.027084 



1.0 4.0 0.6 3.0 / 



Cas empirique du mélange Normale-Exponentielle (1). 

Ilj1 LL1591 0 





Légende : 

0.06336611 1.01099088 0.65029367 1.02821637 / 0.022567 on choisit le mauvais type de mélange 


Annexe 95 / / échantillon de 1000000 d'individus 

0.0 1.0 0.6 1.0 / 

-0.45710937 0.76668786 0.37216725 0.90448078 / 

0.030061 on choisit le mauvais type de mélange 


Annexe 94 / / 

0.0 1.0 0.6 1.0 / 

1.00900914 1.01247108 0.58893008 0.97467774 / 0.005764 0.025836 



1.0 1.0 0.6 1.0 / 



Cas empirique du mélange Normale-Exponentielle (2). 

Ilj1 LL15:1 0 


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gx whpsv gh fdofxo/ fhoohv uhodwlyhv dx{ txdwuh/ yrlu/ dx{ kxlw soxv surfkhv yrlvlqv1 

481615 Wkìruëph gh Kdpphuvoh|0olrug1 

Rq shxw grqqhu/ +yrlu Êhvdj:7`,/ xqh gìqlwlrq dqdo|wltxh ghv fkdpsv gh Pdunry/ srxu fhod/ lqwurgxlvrqv od 

qrwlrq gh foltxh1 

Gìilqlwlrq 4815= VrlwY xq yrlvlqdjh gh iruph jìrpìwultxh {ìh1 Xq vrxv0hqvhpeoh gh V hvw xqh foltxh/ srxu 

oh yrlvlqdjh Y /vl hvw xq vlqjohwrq rx vl ohv ìoìphqwv gh vrqw pxwxhoohphqw yrlvlqv1 

H{hpsoh 4814= Srxu xq fkdpsv gh Pdunry sdu udssruw dx{ txdwuh soxv surfkhv yrlvlqv/ lo | d wurlv w|shv gh foltxhv 

glìuhqwv1 Oh vlqjohwrq hvw od vhxoh foltxh g*ruguh 4 hw ohv foltxhv g*ruguh 5 vrqw krul}rqwdohv rx yhuwlfdohv1 

Vl o*rq vxssrvh txh wrxwhv ohv frqjxudwlrqv gx fkdpsv [ vrqw vxvfhswleohv gh vh surgxluh dyhf xqh suredelolwì 

qrq qxooh/ rq shxw grqqhu xqh h{suhvvlrq dqdo|wltxh gx fkdpsv hq irqfwlrq gh od iruph gx yrlvlqdjh= 

Wkìruëph 4814 +gh Kdpphuvoh|0oliirug1,= Oh fkdps [ hvw pdunrylhq/ srxu oh yrlvlqdjh Y / dyhf xqh 

suredelolwì qrq qxooh g*h{lvwhqfh gh wrxwhv vhv uìdolvdwlrqv vl hw vhxohphqw vl = 

S [ +{, @ h X+{, /+ hvw od frqvwdqwh gh qrupdolvdwlrq,/ 

^48197` 


4817 Xwlolvdwlrq ghv fkdpsv gh Pdunry fdfkìv hq lpdjhulh1 4:6 

dyhf xqh irqfwlrq ìqhujlh X grqqìh sdu 

X +{, @ [ [ 

* w +{ , > 

w5W 5G M 

^48198` 

rü W hvw o*hqvhpeoh gh wrxv ohv w|shv gh foltxhv dvvrflìv dx yrlvlqdjh Y / G w hvw o*hqvhpeoh ghv foltxhv gx fkdpsv 

gh od vruwh w 5 W hw * w hvw xqh dssolfdwlrq txl dvvrflh ã fkdtxh { @ +{ v , v5 

/dyhf5G w /xqqrpeuhuìho1 

Juåfh dx wkìruëph gh Kdpphuvoh|0olrug/ rq shxw/ srxu xq w|sh gh yrlvlqdjh {ì hw vl o*rq vh grqqh ghv 

irqfwlrqv * w / ìwdeolu xqh h{suhvvlrq gh od orl frqglwlrqqhooh gh [ d sulrul1 Hq irqfwlrq ghv * w / lqgìshqgdqwhv gh od 

srvlwlrq gh od foltxh gdqv V = 

S ^[ v @ $ l @[ Y @ { Y ` @ 

S S 

* M +{ ḷ , 

h M#9 .#/M 

nS 

h S S 

M#9 .#/M 

m@4 

* M +{ ṃ , > ^48199` 

rü { l hvw o*hqvhpeoh ghv uìdolvdwlrqv { srxu ohvtxhoohv { v @ $ l 1 

Od gl!fxowì/ srxu oh fkrl{ gh od irqfwlrq g*ìqhujlh/ frqvlvwh ã wurxyhu xq erq frpsurplv hqwuh od sulvh hq 

frpswh g*xq pd{lpxp gh w|shv gh foltxhv hw gh od srvvlelolwì g*hhfwxhu ohv fdofxov vxu oh prgëoh1 Vl od iruph ghv 

yrlvlqdjhv/ hw grqf fhooh ghv foltxhv/ hvw vlpsoh/ rq frqqdñw od orl gh [ ãxqhfrqvwdqwh +txh o*rq qh shxw sdv 

fdofxohu, suëv1 Sdu frqwuh/ lo hvw srvvleoh gh fdofxohu h{solflwhphqw ohv orlv frqglwlrqqhoohv gh [ v vdfkdqw [ YL /hw 

vlpxohu/ juåfh ã o*ìfkdqwloorqhxu gh Jleev/ +yrlu ^Jhpdq;7`,/ ghv uìdolvdwlrq gx fkdpsv [1 G*dsuëv ^Vdo}hqvwhlq \ , hvw shuplv sdu ghx{ frqfhvvlrqv frqfhuqdqw od prgìolvdwlrq gx euxlw1 Qrxv 

vxssrvhurqv/ dlqvl/ txh = 

+\ v , v5V 

hvw xqh idplooh gh yduldeohv lqgìshqgdqwhv frqglwlrqqhoohphqw ã wrxwh uìdolvdwlrq gh [1 

O +\ v @[, hvw dvvlploodeoh ã O +\ v @[ v , / srxu wrxw v dssduwhqdqw ã V1 

Vrxv fhv k|srwkëvhv/ od glvwulexwlrq prgìolvdqw oh euxlw = 

v*ìfulw = 

S { +|, @ \ v5V 

S [ +\ , @ S +\ m [, > 

S +\ v @ | v m [ v @ { v , @ \ v5V 

i {L +| v , > 

rü i {L +| v , hvwodghqvlwìprgìolvdqwoheuxlwsrxuohsl{hov1 


4:7 4817 Xwlolvdwlrq ghv fkdpsv gh Pdunry fdfkìv hq lpdjhulh1 

Od orl gx frxsoh +[> \ , hvw doruv = 

S ^+[> \ , @ +{> |,` @ fh X+{,. S 

L#V 

oq+i {v +| v ,, 

> 

^4819:` 

rü X/ od irqfwlrq g*ìqhujlh/ d ìwì gìfulwh soxv kdxw hw f hvw xqh frqvwdqwh gh qrupdolvdwlrq1 

O*h{suhvvlrq ^4819:` shuphw gh gìgxluh txh S ^[ @ { m \ @ |` hvw xqh glvwulexwlrq gh Pdunry/ fh txl dxwrulvh od 

vlpxodwlrq ghv uìdolvdwlrqv gh [ vhorq od orl d srvwhulrul/ sdu ghv dojrulwkphv dssursulìv1 

481715 Prgh ghv pdujlqdohv d srvwhulrul +, 

Oh PSP/ txh qrxv dyrqv gìql soxv kdxw/ v*ìfulw = 

;v> e{ PSP 

v 

@dujpd{ 

e{ L 

S +[ v @ { v @\ @ |, = ^4819;` 

O*dydqwdjh gh fhwwh pìwkrgh hvw txh od irqfwlrq gh shuwh sìqdolvh o*huuhxu frpplvh sursruwlrqqhoohphqw dx qrp0 

euh gh vlwhv huurqìv1 rpph/ oh fdofxo gluhfw ghv S +[ v @ { v @\ @ |, hvw whfkqltxhphqw lpsrvvleoh/ o*dojrulwkph gh 

Pdurtxlq hw do/ +yrlu ^Pduurtxlq;:`,/ shuphw g*dssurfkhu oh PSP/ sdu xqh dssolfdwlrq gluhfwh gh o*ìfkdqwloorqqhxu 

gh Jleev1 

O*ìfkdqwloorqqhxu gh Jleev1 

*hvw xqh surfìgxuh lwìudwlyh hw vwrfkdvwltxh/ gìyhoorsìh sdu Jhpdq hw Jhpdq gdqv ^Jhpdq;7`/ shuphwwdqw gh 

vlpxohu xqh vxlwh gh uìdolvdwlrqv gh [ frqyhujhqw hq orl yhuv xqh glvwulexwlrq gh Jleev1 

Udsshorqv txh o*rq frqvlgëuh ghx{ fkdpsv doìdwrluhv [ @ +[ v , v5V 

hw \ @ +\ v , v5V 

rü V hvw o*hqvhpeoh ghv 

sl{hov1 kdtxh [ v hvw ã ydohxuv gdqv o*hqvhpeoh ghv fodvvhv @i$ 4 >===$ n j hw \ v hvw ã ydohxuv gdqv U= 

Dsuëv od q lhph lwìudwlrq gh o*ìfkdqwloorqqhxu gh Jleev/ od frqjxudwlrq gx fkdpsv vlpxoì vhud qrwìh = 

{ +q, @ i{ v +q, >v5 Vj > 

hw o*rq fkhufkh ã sdvvhu ã od frqjxudwlrq { +q .4,1 Od frqjxudwlrq { +q .4, vhud lqlwldolvìh ã { +q,1 

Rq qrwhud/ gh soxv/ X w +{,/ od uhvwulfwlrq gh od irqfwlrq ìqhujlh X +{, dx{ whuphv gh od vrpph frqwhqdqw oh sl{ho w1 

O*dojrulwkph/ suìvhqwì pdlqwhqdqw/ shuphw gh vlpxohu xq fkdps [ vxlydqw od orl frqglwlrqqhooh ã \ @ | = 

hfkdqwloorqqhxu gh jleev = 

Ohv ìwdshv gh ^J 5` ã ^J 8` vrqw suìvhqwìhv srxu oh sl{ho v 3 ã o*lwìudwlrq q. 

4 hw vrqw ã uìsìwhu/ ã fkdtxh lwìudwlrq srxu wrxv ohv sl{hov gh o*lpdjh hw 

mxvtx*ã fh tx*xq fulwëuh g*duuíw suìgìiql vrlw yìulilì1 

^J 4` uìdwlrq g*xqh frqiljxudwlrq lqlwldoh { +3,1 

^J 5` Srxu wrxw $ l 5 / rq fdofxoh/ frqglwlrqqhoohphqw ã \ @ | = 

s l @ h XL ! +{+q.4,,.oq+i $ l 

+| v !,,= 

^J 6` Wludjh gh t 5 ^3> 4` vxlydqw od orl xqliruph vxu fhw hqvhpeoh1 

^J 7` Gìwhuplqdwlrq gh o*xqltxh m who txh m4 S S 

s l t n S 

s l m s l 1 

^J 8` Doruv { v! +q .4, @ $ m 

l@4 

l@4 

hw rq sdvvh dx sl{ho vxlydqw1 

l@4 


4817 Xwlolvdwlrq ghv fkdpsv gh Pdunry fdfkìv hq lpdjhulh1 4:8 

O*ìfkdqwloorqqhxu gh Jleev hvw/ dyhf o*dojrulwkph gh Phwursrolv/ +yrlu ^Phwursrolv86`,/ txh qrxv q*xwlolvhurqv 

sdv lfl/ oh srlqw fhqwudo gh qrpeuhxvhv surfìgxuhv g*hvwlpdwlrq gdqv oh fdguh gh sureoëphv idlvdqw lqwhuyhqlu xqh 

prgìolvdwlrq sdu ohv fkdpsv gh Pdunry 

O*dojrulwkphghPdurtxlqhwdo1 

Juåfh dx{ k|srwkëvhv frqvhqwlhv oruv gh od prgìolvdwlrq/ txl shuphwwhqw o*xwlolvdwlrq gh o*ìfkdqwloorqqhxu gh 

Jleev/ lo ghylhqw srvvleoh gh phwwuh hq sodfh xq dojrulwkph vwrfkdvwltxh gh fodvvlfdwlrq1 hw dojrulwkph/ gìyhorssì 

gdqv oh fdguh gh od uhfrqvwuxfwlrq g*lpdjhv/ vh gìurxoh hq wurlv ìwdshv/ od suhplëuh ìwdqw od soxv orqjxh1 

dojrulwkph gh pdurtxlq hw do1 

Oh qrpeuh Q gh uìdolvdwlrqv gh [/ vlpxoìhv vhorq od orl frqglwlrqqhooh ã 

\ @ |/ oruv gh o*ìwdsh ^P 4` hvw gìwhuplqì sdu frpsurplv hqwuh od qìfhvvlwì 

gh frqyhujhqfh gh o*dojrulwkph gh Pdurtxlq hw do1 hw oh frþw hq whpsv fdo0 

fxo gh fkdtxh xwlolvdwlrq gh o*ìfkdqwloorqqhxu gh Jleev1 

^P 4` Vlpxodwlrq/ sdu o*ìfkdqwloorqqhxu gh Jleev/ gh Q uìdolvdwlrqv/ { 4 > 111>{ Q 

gx fkdps [ vxlydqw od orl frqglwlrqqhooh ã \ @ |1 

^P 5` Hvwlpdwlrq/ srxu wrxw v 5 V/ gh od orl gh [ v / sdu ohv iuìtxhqfhv g*ds0 

sdulwlrq ghv fodvvhv gdqv o*ìfkdqwloorq = { 4 v> 111 >{ Q v 1 

^P 6` Hvwlpdwlrq/ srxu wrxw v 5 V/ gh [ v sdu od fodvvh pd{lplvdqw od orl rewhqxh 

hq ^P 5`1 

481716 Pd{lpxp d srvwhulrul +, 

Lo v*ìfulw = 

e{ PDS @dujpd{ S +[ @ {@\ @ |, = ^4819

4:9 4818 Xwlolvdwlrq ghv fkdñqhv gh Pdunry fdfkìhv 

Dojrulwkph gx uhfxlw vlpxoì1 

Vl o*rq lqwurgxlw gdqv o*ìqhujlh gx frxsoh +[> \ ,/ xwlolvìh hq ^4819:`/ oh sdudpëwuh W dsshoì whpsìudwxuh/ od 

qrxyhooh ìqhujlh vhud = 

% 

X W @ 4 X +{, [ & 

oq +i {L +| v ,, > 

W 

v5V 

hw od orl gh od glvwulexwlrq pdunrylhqqh g*ìqhujlh X W / 

S W ^+[> \ , @ +{> |,` @ fh " 9 

% 

X+{, S 

& 

L#8 oq+i +| {v v,, 

/ 

oruvtxh W whqg yhuv 3/ S W frqyhujh yhuv od glvwulexwlrq xqliruph vxu o*hqvhpeoh G gh wrxwhv ohv uìdolvdwlrqv gh [ 

pd{lplvdqw od orl d srvwhulrul1 Vl od frqjxudwlrq rswlpdoh gx fkdps hvw xqltxh/ od glvwulexwlrq olplwh fruuhvsrqg 

ã xqh glvwulexwlrq gh Gludf yhuv odtxhooh rq hvsëuh idluh frqyhujhu S W 1 hwwh frqyhujhqfh hvw dvvxuìh/ g*dsuëv ohv 

ìwxghv wkìrultxhv/ vl od gìfurlvvdqfh gh W v*hhfwxh vx!vdpphqw ohqwhphqw/ f*hvw ã gluh w|sltxhphqw hq 4 

oq+q, /+yrlu 

^Kdmhn;;`,1 

Dsuëv od q lhph lwìudwlrq gh o*dojrulwkph gx uhfxlw vlpxoì/ od frqjxudwlrq gx fkdpsv vlpxoì vhud qrwìh= 

{ +q, @ i{ v +q, >v5 Vj > 

hw o*rq fkhufkh ã sdvvhu ã od frqjxudwlrq { +q .4,1 Od frqjxudwlrq { +q .4, vhud lqlwldolvìh ã { +q,1Rq qrwhud/ 

gh soxv/ X w +{,/ od uhvwulfwlrq gh od irqfwlrq ìqhujlh X +{, dx{ whuphv gh od vrpph frqwhqdqwv w1 

dojrulwkph gx uhfxlw vlpxoh 

hw dojrulwkph hvw xq ìfkdqwloorqqhxu gh Jleev rü/ ã o*lwìudwlrq q/ od irqf0 

wlrq g*ìqhujlh hvw X WG 1Ohv ìwdshv gh ÛV 5` ã ÛV 8` vrqw suìvhqwìhv srxu oh 

sl{ho v 3 ã o*lwìudwlrq q.4 hw vrqw ã uìsìwhu/ ã fkdtxh lwìudwlrq srxu wrxv 

ohv sl{hov gh o*lpdjh hw mxvtx*ã fh tx*xq fulwëuh g*duuíw suìgìiql vrlw yìul0 

ilì1 

ÛV 4` krl{ g*xqh vxlwh +W q , q 

gìfurlvvdqwh yhuv 3 hw fuìdwlrq g*xqh frqilj0 

xudwlrq lqlwldoh { +3,1 

ÛV 5` Srxu wrxw $ l 5 / rq srvh { v! +q .4, @ $ l hw rq fdofxoh = 

s l @ h " 

9 G." ^XL ! +{+q.4,,.oq+i$ l+| v !,,`= 

ÛV 6` Wludjh gh t 5 ^3> 4` vxlydqw od orl xqliruph vxu fhw hqvhpeoh1 

ÛV 7` Gìwhuplqdwlrq gh o*xqltxh m who txh m4 S S 

s l t n S 

s l m s l 1 

ÛV 8` Doruv { v! +q .4, @ $ m 

l@4 

l@4 

hw rq sdvvh dx sl{ho vxlydqw1 

Od vroxwlrq gx PDS hvw gluhfwhphqw lvvx gh fhw dojrulwkph txl v*dyìuh wuëv orqj ã frqyhujhu/ pdlv txl shxw 

wrxwhirlv grqqhu g*h{fhoohqwv uìvxowdwv1 

l@4 


4818 Xwlolvdwlrq ghv fkdñqhv gh Pdunry fdfkìhv 4:: 

4818 Xwlolvdwlrq ghv fkdñqhv gh Pdunry fdfkìhv 

481814 Lqwurgxfwlrq 

O*lqfrqyìqlhqw pdmhxu gx prgëoh ghv fkdpsv gh Pdunry hvw txh ohv dojrulwkphv g*dssur{lpdwlrq tx*lov lqgxlvhqw/ 

vrqw lwìudwliv hw xwlolvhqw ghv whfkqltxhv gh vlpxodwlrq gh Prqwh0duor1 Hq hhw/ o*dojrulwkph gh Pduurtxlq hw do1 

hvw edvì vxu xqh xwlolvdwlrq lwìudwlyh gh o*ìfkdqwloorqqhxu gh Jleev/ txl hvw wuëv jrxupdqg hq whpsv fdofxo1 

Dq gh uìgxluh od txdqwlwì gh fdofxov hw srxu rewhqlu ghv dojrulwkphv qrq lwìudwliv/ rq shxw xwlolvhu ohv fkdñqhv gh 

Pdunry fdfkìhv/ hq vh edvdqw vxu ohv sulqflshv jìqìudx{ gìyhorssìv gdqv ^Slfndug;3`/ ^Ghylmyhu;8`/ ^Ghylmyhu;:` hw 

^Ghylmyhu;;`1 Od vwuxfwxuh/ hq oljqh/ ghv fkdñqhv q*hvw sdv lqwxlwlyhphqw dgdswìh ã od frqvwuxfwlrq elglphqvlrqqhooh 

ghv lpdjhv1 Lo hvw/ fhshqgdqw/ srvvleoh/ juåfh dx fkrl{ g*xq erq sdufrxuv gh o*lpdjh/ g*rewhqlu ghv uìvxowdwv 

dssuìfldeohv/ vxuwrxw frpswh whqx gx jdlq gh whpsv1 *hvw o*xwlolvdwlrq g*xq sdufrxuv gh Shdqr txl d ìwì fkrlvl srxu 

sdufrxulu o*lpdjh +yrlu Êhqplorxgmn rü = 

d lm @ S +[ q.4 @ $ m @[ q @ $ l , = 

^481:5` 

Rqrewlhqwhqhhw= 

S +[ 4 @ $ l" >[ 5 @ $ l# > ===> [ q @ $ lG , @ l" d l"l # 

===d lG"lG= 

^481:6` 


4:; 4818 Xwlolvdwlrq ghv fkdñqhv gh Pdunry fdfkìhv 

Ilj1 LLL15;1 0 Sulqflsh gx sdufrxuv gh Shdqr1 


4818 Xwlolvdwlrq ghv fkdñqhv gh Pdunry fdfkìhv 4:< 

481816 Orl gh +[/\,1 

rpph fh ixw oh fdv srxu ohv fkdpsv/ qrxv vxssrvrqv txh ohv +\ q , vrqw lqgìshqgdqwv frqglwlrqqhoohphqw ã [ 

hw txh od orl gh fkdtxh \ q frqglwlrqqhooh ã [ hvw od orl frqglwlrqqhooh ã [ q 1 

+\ l , 4lq 

hvw xqh idplooh gh yduldeohv lqgìshqgdqwhv frqglwlrqqhoohphqw ã wrxwh uìdolvdwlrq gh [1 

O +\ l @[, hvw dvvlploodeoh ã O +\ l @[ l , / srxu wrxw l lqiìulhxu ã q1 

^481:7` 

Rq qrwh i 4 >i 5 > ===> i n ohvghqvlwìvgh\ q uhvshfwlyhphqw frqglwlrqqhoohphqw ã [ q @ $ 4 >===>[ q @ $ n = 

$ohv i l gìqlvvhqw wrxwhv ohv orlv gh \ frqglwlrqqhoohv ã [1 

Od ghqvlwì k/ gh od orl gx frxsoh +[> \ , hq { @ +$ l" > ===> $ lG , hw | @ +| 4 > ===> | q , v*ìfulw = 

k +{> |, @ l" i l" +| 4 , d l"l # 

i l# +| 5 , ===d lG"lGi lG +| q , = 

^481:8` 

481817 Pìwkrgh gx 1 

rqvlgìurqv od fkdñqh gh Pdunry fdfkìh +[ v >\ v , v5V 

/rüV hvw xq hqvhpeoh gh sl{hov rugrqqìv sdu oh fkrl{ 

gx sdufrxuv gh o*lpdjh1 Txlwwh ã rsìuhu xqh uìlqgh{dwlrq/ rq vxssrvhud txh od fkdñqh hvw lqgh{ìh sdu ohv hqwlhuv 

v 5i4> ===> mVmj1 

dofxo gx 

Vrxv ohv k|srwkëvhv g*lqgìshqgdqfh ìqrqfìhv fl0ghvvxv/ rq vrxkdlwh fdofxohu 

v +l, @ S ^[ v @ $ l @\ @ |` > 

^481:9` 

juåfh ã od irupxoh = 

v +l, @ 

v +l, v +l, 

> ^481::` 

nS 

v +m, v +m, 

m@4 

rü 

v +l, @ S ^[ v @ $ l >\ 4 @ | 4 > ===> \ v @ | v` hvw od suredelolwì iruzdug 

hw v +l, @ S 

\ v.4 @ | v.4 > ===> \ mVm @ | mVm > @[ v @ $ l hvw od suredelolwì edfnzdug1 

txh o*rq fdofxoh juåfh ã o*dojrulwkph gh Edxp hw Zhofk gìyhorssì gdqv Êdxp:3`1 

Od vroxwlrq gx PSP vhud grqqìh/ doruv/ sdu = 

k l 

e[v @ $ l +, 

 

 

v +l, @ pd{ v +m, = ^481:;` 

4mn 

Suredelolwìv iruzdug hw edfnzdug 0 Dojrulwkph gh Edxp0Zhofk1 

Ohv suredelolwìv iruzdug hw edfnzdug rqw ìwì lqwurgxlwhv sdu Edxp hw do1 +yrlu Êdxp:3`,1 Ohxu lqwìuíw uìvlgh 

gdqv ohv uhodwlrqv gh uìfxuuhqfhv txl shuphwwhqw ohxu fdofxo1 


4;3 4818 Xwlolvdwlrq ghv fkdñqhv gh Pdunry fdfkìhv 

Od suredelolwì iruzdug v*rewlhqw ã od vxlwh g*xqh uìfxuuhqfh prqwdqwh lqgxlwh sdu od uhodwlrq = 

v.4 +m, @ 

# [ 

l 

v +l, d lm 

$ 

i m +| v.4 , > 4 m n> 4 v mVm = ^481: 4 l n> mVm v 4= 

^481;3` 

dojrulwkph gh edxp0zhofk 

hw dojrulwkph hvw hqwlëuhphqw gìwhuplqlvwh1 

ÊZ 4` lqlwldolvdwlrq gh od sdvvh prqwdqwh = 

4 +m, @ m i m +| 4 , srxu 4 m n= 

ÊZ 5` sdvvh prqwdqwh = fdofxo ghv v.4 +m, srxu 4 m n/ 4 v mVm = 

ÊZ 6` lqlwldolvdwlrq gh od sdvvh ghvfhqgdqwh = 

mVm +l, @4 srxu 4 l n= 

ÊZ 7` sdvvh ghvfhqgdqwh = fdofxo ghv v +l, srxu 4 l n/mVm v 4= 

ÊZ 8` fdofxo ghv v +l,/ rx soxw÷w gh 

ÊZ 9` fdofxo ghv 

3 v +l, @ v +l, v +l,, srxu 4 l n> 4 v mVm = 

e{ v @duj pd{ 

4ln 

v +l, @duj pd{ 

4ln 

3 v +l, /srxu 4 v mVm = 

O*hvwlpdwhxu gx PSP hvw doruv grqqì/ srxu wrxw v/ sdu e [ v @ e{ v 1 

PSP xwlolvh grqf oh fdofxo gluhfw ghv suredelolwìv pdujlqdohv d srvwhulrul/ g*rü oh jdlq gh whpsv/ sdu udssruw 

dx{ pìwkrghv srxu ohv fkdpsv1 Dq g*ìylwhu ohv sureoëphv olìv ã od gìfurlvvdqfh/ yhuv 3/ wuëv udslgh ghv txdqwlwìv 

pdqlsxoìhv/ rq uhpsodfh ohv suredelolwìv mrlqwhv sdu ohv suredelolwìv d srvwhulrul +yrlu ^Ghylmyhu;8` ,1 

481818 Pìwkrgh gx 1 

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hw oh PDS hvw grqqì qdohphqw sdu 

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DojrulwkphghYlwhuel1 

hw dojrulwkph +yrlu Îruqh|:6`hw ^Ylwhuel9:`, hvw wuëv xwlolvì hq uhfrqqdlvvdqfh gh od sduroh rx hq uhfrqqdlvvdqfh 

gh fdudfwëuhv1 O*dojrulwkph gh Ylwhuel hvw frqiurqwì ã xqh gìfurlvvdqfh h{srqhqwlhooh/ +yrlu Ôhylqvrq;6`,/ yhuv 3/ 

ghv suredelolwìv pdqlsxoìhv1 Ohv sureoëphv gìfrxodqwv gh fhwwh gìfurlvvdqfh rqw ìwì/ hq sduwlh/ uìvroxv sdu Ghylmyhu 

hw do gdqv ^Ghylmyhu;8`/ hq xwlolvdqw/ frpph suìfìghpphqw/ ohv suredelolwìv d srvwhulrul/ soxw÷w txh ohv suredelolwìv 

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dojrulwkph gh ylwhuel 

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Prgìolvdwlrqv vxu dueuhv1 

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sdvvhu sdu 1 

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4;7 4914 Judskhv hw suredelolwìv1 

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fodvvltxhv1 

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prqwuh Ilj1 LLL15

4915 H{hpsohv gh judskhv gh gìshqgdqfhv1 4;8 

4915 H{hpsohv gh judskhv gh gìshqgdqfhv1 

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ohv judskhv gh gìshqgdqfhv1 

H{hpsoh 4914 +fkdñqh gh Pdunry fdfkìh1,= h judskh ìwdeolw ylvxhoohphqw ohv k|srwkëvhv idlwhv hq ^481:7`1 

Y 

X 

1 2 3 ........ N-1 N 

Ilj1 LLL1631 0 Judskh gh gìshqgdqfh g*xqh fkdñqh gh Pdunry fdfkìh1 

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gìshqgdqfh g*xqh ghv yduldeohv/ frqglwlrqqhoohphqw dx uhvwh ghv yduldeohv gx fkdpsv/ ã xqh gìshqgdqfh sdu udsruw 

ã vhv txdwuh soxv surfkhv yrlvlqv1 

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gìshqgdqfh g*xqh ghv yduldeohv/ frqglwlrqqhoohphqw dx uhvwh ghv yduldeohv gx fkdpsv/ ã xqh gìshqgdqfh sdu udsruw 

ã vhv kxlw soxv surfkhv yrlvlqv1 

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ghv uhodwlrqv gh gìshqgdqfh hqwuh ohv yduldeohv/ fh txl shxw íwuh frqvlgìuì frpph xqh idleohvvh gx prgëoh1 Qrxv 

dssruwhurqv txhotxhv suìflvlrqv vxu fh srlqw/ gdqv oh fkdslwuh vxlydqw1 


4;9 4916 Od vwuxfwxuh g*dueuh1 

Ilj1 LLL1651 0 kdpsv gh Pdunry uhodwli dx{ ; soxv surfkhv yrlvlqv1 

4916 Od vwuxfwxuh g*dueuh1 

Ohv dueuhv iruphqw xqh vrxv idplooh sduwlfxolëuh gh od idplooh ghv judskhv1 

491614 Gìqlwlrqv1 

Gìilqlwlrq 4918= Xqjudskh klìudufkltxh hvw xq judskh frqqh{h < sduwlwlrqqì hq ìfkhoohv/ qxpìurwìhv gh 3 srxu 

od soxv jurvvlëuh/ ã Q srxu od soxv qh1 

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ã xq qrpeuh frqvwdqw gh vrpphwv1 

Ohv txhotxhv h{hpsohv vxlydqwv grqqhqw xqh lgìh gh od ydulìwì ghv judskhv dssduwhqdqw ã od idplooh ghv judskhv 

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491615 H{hpsohv1 

Oh judskh gh Ilj1 LLL166/ dsshoì judskh gh Ndwr +yrlu ^Ndwr f*hvw xq dueuh txhofrqtxh1 O*dueuh 

krprjëqh hvw suìvhqwì vrxv od iruph gx txdgdueuh/ Ilj1 LLL1681 

491616 Lqwhusuìwdwlrq gh o*dueuh 

O*hqvhpeoh V ghv qrhxgv gh o*dueuh hvw sduwlwlrqqì hq ìfkhoohv V q /ghodsoxvqh3/ ã od soxv jurvvlëuh Q1 Rq 

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gh vhv ghvfhqgdqwv1 

b Xq judskh hvw glw frqqh{h/ vl srxu wrxw frxsoh gh vlwhv Erc | lo h{lvwh xq fkhplq shuphwwdqw g*doohu gh v ã w1 

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Ohv ihxloohv vrqw ohv vrpphwv gh o*ìfkhooh 3 

' r 


4916 Od vwuxfwxuh g*dueuh1 4;: 

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Ilj1LLL16610JudskhghNdwr1 

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Ilj1 LLL1671 0 Dueuh txhofrqtxh1 


4;; 4916 Od vwuxfwxuh g*dueuh1 

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hvw gìwhuplqdqwh srxu ohv dojrulwkphv vxu ohv vwuxfwxuhv gh w|sh dueuh1 

491617 Sdvvdjh g*xq fkdpsv doìdwrluh ã xq dueuh krprjëqh 

Oh olhq hqwuh od vwuxfwxuh wulglphqvlrqqhooh ghv dueuhv hw fhooh gh o*lpdjh q*hvw sdv dssduhqw hw lo hvw qìfhvvdluh 

gh suìvhqwhu fodluhphqw o*dgdswdwlrq gx vtxhohwwh g*dueuh ã o*ìwxgh g*xq sureoëph hvvhqwlhoohphqw elglphqvlrqqho 

frpph fhoxl gx wudlwhphqw g*lpdjh1 

Sulqflsh gh od prgìolvdwlrq 

Lo v*djlw gh frqvwuxluh xq dueuh grqw ohv ihxloohv lqglfhqw ohv yduldeohv gx fkdpsv1 Rq uhjurxsh txdwuh ã txdwuh 

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doruv dssoltxìh dx qrxyhdx fkdpsv irupì sdu ohv sëuhv1 Rq uìsëwh dlqvl gh vxlwh o*rsìudwlrq wdqw tx*hooh hvw srvvleoh 

mxvtx*ã o*dqfíwuh xqltxh/ txh o*rq dsshooh od udflqh u1 Srxu ohv lpdjhv 589 589 rq frqvwuxlw grqf xq txdgdueuh ã 

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Ghv dueuhv sduwlfxolhuv1 

Oh txdgdeuh1 

Oh txdgdueuh hvw xq dueuh krprjëqh srxu ohtxho fkdtxh sëuh d txdwuh ghvfhqgdqwv1 Vd vwuxfwxuh g*ruguh txdwuh 

hvw elhq dgdswìh dx{ lpdjhv txl irqw frxudpphqw 5 q 5 q sl{hov1 

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Ilj1 LLL1681 0 Txdgdueuh ã wurlv qlyhdx{1 


4916 Od vwuxfwxuh g*dueuh1 4;< 

Oh gldueuh1 

Oh gldueuh/ txl q*d sdv ìwì ìwxglì mxvtxh oã/ suìvhqwh od sduwlfxodulwì gh jìqìuhu/ xqh ìfkhooh vxu ghx{/ xq ihxloodjh 

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gx uhjurxshphqw ghv ov1 Od v|pìwulh ã ghx{ iuëuhv gh fh prgëoh shuphw g*hqylvdjhu ehdxfrxs soxv gh frqvwuxfwlrqv 

glìuhqwhv gx eudqfkdjh/ txh od v|pëwulh ã txdwuh iuëuhv txl lqgxlw qdwxuhoohphqw xq uhjurxshphqw hq fduuì1 Qh 

vhudlw0fh tx*hq jdugdqw fhwwh edvh fduuìh/ lo | d wurlv idêrqv gh uhjurxshu ghx{ ã ghx{ txdwuh shwlwv0ov1 

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Ohv ov vrqw hqjhqguìv dowhuqdwlyhphqw gdqv oh vhqv odwìudo hw gdqv od surirqghxu Ilj1LLL1691 

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Ilj1LLL1691 0 Wudqvplvvlrq gurlwh1 

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4

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prgëoh ã ghx{ fodvvhv/ srxu ohtxho ohv suredelolwìv gh frqvhuydwlrq gh od fodvvh vrqw ohv píphv srxu ohv ghx{ fodvvhv/ 

rq rewlhqw ghv lpdjhv gx w|sh gh fhoohv gh Ilj1LLL1751 

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oh prgëoh hq idlvdqw gìshqguh ohv suredelolwìv gh wudqvlwlrq gh o*ìfkhooh1 

Oh txdgdueuh ìyroxwli1 

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uìvroxwlrq1 Dq gh qh sdv pxowlsolhu ohv sdudpëwuhv/ ohv suredelolwìv gh wudqvlwlrqv lqwhuìfkhoohv gìshqghqw g*xq 

sdudpëwuh frqvwdqw vxu o*dueuh/ g*xq sdudpëwuh {ì %/ hw gh od uìvroxwlrq1 Sdupl ohv glìuhqwhv iruphv gh irqfwlrqv 

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ã qrv dwwhqwhv/ sduwlfxolëuhphqw hq fh txl frqfhuqh od frqfhqwudwlrq ghv }rqhv gh fodvvhv xqliruphv1 Qrxv dyrqv/ 

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vrqw grqqìhv ã o*ìfkhooh qA3, = 

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4 

rü V hvw o*hqvhpeoh ghv qrhxgv gh o*dueuh/ V q hvw o*ìfkhooh q1 Qrxv suìvhqwrqv oh prgëoh jìqìudo ghv dueuhv gh 

Pdunry fdfkìv/ hq frqvlgìudqw o*h{lvwhqfh g*xqh revhuydwlrq ã fkdtxh qlyhdx gh o*ìfkhooh1 

Qrxv qrwhurqv { hw |/ ohv uìdolvdwlrqv uhvshfwlyhv gh [ hw \ / { q hw | q ìwdqw ohv uìdolvdwlrqv uhvshfwlyhv gh [ q hw 

\ q 1 Sdu dloohxuv/ v hvwo*xqltxhsëuhghv hw v . o*hqvhpeoh gh vhv ov/ v hvw wrxmrxuv o*hqvhpeoh gh vhv dqfíwuhv 

hw v o*hqvhpeoh gh vhv ghvfhqgdqwv1 Hqq od udflqh vhud qrwìh u1 

Srxu dvvxuhu txh +[> \ , hvw pdunrylhq vxu o*dueuh/ rq idlw ohv txdwuh k|srwkëvhv vxlydqwhv = 

D41 [ hvw pdunrylhq hq ìfkhooh= 

S [ q @ { q @[ n @ { n >Q n q .4 @ S [ q @ { q @[ q.4 @ { q.4 = 

^4:1;7` 

D51 Srxu frpsoìwhu od suhplëuh sursulìwì/ qrxv vxssrvhurqv txh = 

Q\ 

S ^\ @ |@[ @ {` @ S ^\ q @ | q @[ q @ { q` = 

q@3 

^4:1;8` 

Od vwuxfwxuh gx surfhvvxv/ hq ìfkhooh/ vh udssurfkh/ iruphoohphqw/ gh fhooh g*xqh fkdñqh gh Pdunry fdfkìh1 

D61 D fkdtxh qlyhdx g*ìfkhooh/ q/ 3 q Q 4/ ohv suredelolwìv gh wudqvlwlrq lqwhu0ìfkhooh shxyhqw vh idfwrulvhu 

hq 

S [ q @ { q @[ q.4 @ { q.4 @ \ 

j q +{ v >{ v , > 

^4:1;9` 

v5V G 

rü j q hvw xqh irqfwlrq gìqlvvdqw od suredelolwì gh wudqvlwlrq ã o*ìfkhooh q/ { v hw { v ìwdqw uhvshfwlyhphqw ohv 

uìdolvdwlrqv gh [ v hw [ v 1 Qrxv xwlolvhurqv/ gdqv od vxlwh/ fhoohv gìqlhv sdu ^4:1;6` hw ^4:1;5`1 

D71 Hqq/ ã fkdtxh qlyhdx g*ìfkhooh/ q/ 3 q Q/ = 

S ^\ q @ | q @[ q @ { q` @ \ 

v5V G i q +| v @{ v , > 

^4:1;:` 

rü i q hvw od ghqvlwì gx euxlw ã o*ìfkhooh q/ { v hw | v ìwdqw uhvshfwlyhphqw ohv uìdolvdwlrqv gh [ v hw \ v 1 

Gh pdqlëuh dqdorjxh ã fhooh ghv fkdñqhv gh Pdunry fdfkìhv fodvvltxhv/ fh prgëoh vh suëwh doruv dx gìyhorsshphqw 

ghv pìwkrghv ed|ìvlhqqhv gh uhvwdxudwlrq fodvvltxhv PSP hw PDS1 

Dssolfdwlrq ã o*lpdjhulh1

4:16 Dojrulwkphv gh fodvvlfdwlrq vxu ohv dueuhv gh Pdunry fdfkìv1 4\ u @ | u , pd{ 

{)K S +[ Au @ { Au >\ Au @ | Au m [ u @ { u , > 

% 

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% 

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w5u. 

pd{ 

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@ { v , 

pd{ S +[ w @ { w >\ w @ | w m [ v @ { v , 

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fh txl shuphw g*hhfwxhu xqh uìfxuuhqfh vxu ohv w +{ v ,1 

Srvrqv/ sdu dqdorjlh dyhf ohv fkdñqhv gh Pdunry/ 

e[ v +{ v , @duj pd{ S +[ v @ { v >\ v @ | v m [ v @ { v , \ 

w +{ v , = 

{ L 

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O*dojrulwkph gh Ylwhuel0Odihuwì txh qrxv gìfulyrqv pdlqwhqdqw/ frpsruwh/ oxl dxvvl/ ghx{ skdvhv sulqflsdohv/ 

o*xqh prqwdqwh o*dxwuh ghvfhqgdqwh1 


4v5 V 3 = 

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v +{ v ,1 

^YO 6` Lqlwldolvdwlrq gh od sdvvh ghvfhqgdqwh = 

e{ u @dujpd{ i Q +| u m { u , \ 

v +{ u , = 

{ K v5u. 

^YO 7` Sdvvh ghvfhqgdqwh = srxu Q 4 q 3 rq wluh ghv fdofxov gh ^YO 5`/ 

O*hvwlpdwhxu gx PDS hvw doruv = 

;v 5 V q > e{ v @ e [ v +e{ v , = 

e[ @ +e{ v , v5V 

== 

hw dojrulwkph grqqh od vroxwlrq h{dfwh gx PDS1 Lo hvw wuëv udslgh pdlv wuëv fkhu hq hvsdfh pìprluh1 Gh 

soxv od idêrq grqw ohv huuhxuv vrqw sìqdolvìhv q*hvw sdv wuëv dgdswìh ã od vwuxfwxuh ghv judskhv klìudufkltxhv1 Xqh 

huuhxu vxuyhqdqw ã xqh ìfkhooh jurvvlëuh hvw/ lqwxlwlyhphqw/ soxv sìqdolvdqwh tx*xqh huuhxu vxuyhqdqw ã xqh ìfkhooh 

qh1 *hvw srxutxrl Erxpdq hw do/sursrvhqw gdqv Êrxpdq\ v @ | v , S +[ v @ { v m \ @ |, > ;v 5 V> 


4:16 Dojrulwkphv gh fodvvlfdwlrq vxu ohv dueuhv gh Pdunry fdfkìv1 4[ v @ { v m \ v @ | v , 

S +[ 

S +[ v @ { v m \ v @ | v , 

v @ { v m \ @ |, > ;v 5 V= 

~} € 

! L +{ L >{ L , 

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S +[ v @ { v m \ v @ | v ,/ ; +{ v >{ v , 5 5 1Hqhhw/ 

hw grqf = 

S +[ v @ { v >[ v @ { v m \ v @ | v , @ S +[ v @ { v m [ v @ { v , S +[ v @ { v m \ v @ | v , > 

! v +{ v >{ v , @ +[ v @ { v m [ v @ { v , S +[ v @ { v m \ v @ | v , 

S 

S +[ v @ { v m [ v @ { v , S +[ v @ { v m \ v @ | v , = 

{ L 5 

^4:1;;` 

Od suhplëuh txdqwlwì hvw rewhqxh sdu = 

S +[ v @ { v m [ v @ { v , @ S +[ v @ { v m [ v @ { v , S +[ v @ { v , 

> 

S +[ v @ { v , 

rü ohv S +[ v @ { v ,/ ;v 5 V/ vrqw fdofxoìv sdu od uìfxuuhqfh = 

S +[ v @ { v , @ [ 

S +[ v @ { v m [ v @ { v , S +[ v @ { v , / 

{ L 5 

fhwwh uìfxuuhqfh ìwdqw lqlwldolvìh/ ã od udflqh/ sdu ohv sdudpëwuhv gx prgëoh = 

S +[ u @ $ l , / ;$ l 5 1 

Od vhfrqgh txdqwlwì hvw lvvxh g*xqh dxwuh uhodwlrq gh uìfxuuhqfh prqwdqwh1 Gìwdloorqv ohv fdofxov = 

S +[ v @ { v m \ v @ | v , @ S +[ v @ { v , S +\ v @ | v m [ v @ { v , 

> 

S +\ v @ | v , 

S +[ v @ { v , S +\ v @ | v m [ v @ { v , T 

@ 

S +\ w @ | w m [ v @ { v , 

w5v. 

> 

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hw grqf = 

S +[ v @ { v m \ v @ | v , 

S +[ v @ { v , 6 S +\ v @ | v m [ v @ { v , T S +\ w @ | w >[ v @ { v , 

w5v. 

@ 

> 

S +\ v @ | v , 

@ S +[ v @ { v , 6 S +\ v @ | v m [ v @ { v , 

S +\ v @ | v , 

\ 

^S +\ w @ | w , S +[ v @ { v m \ w @ | w ,` > 

w5v. 


533 4:16 Dojrulwkphv gh fodvvlfdwlrq vxu ohv dueuhv gh Pdunry fdfkìv1 


S +[ v @ { v m \ v @ | v , 

@ 

T 

S +\ w @ | w , 

w5v. 

S +[ v @ { v , 6 

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~} € 

4@]+|L, 

S +\ v @ | v m [ v @ { v , \ 

S +[ v @ { v m \ w @ | w , > 

w5v. 

txh o*rq ìfulud = 

S +[ v @ { v m \ v @ | v , @ S +[ v @ { v , 6 S +\ v @ | v m [ v @ { v , 

] +| v , 

\ [ 

S +[ v @ { v m [ w @ { w , S +[ w @ { w m \ w @ | w , = 

w5v. { M 5 

dq gh idluh dssdudñwuh od uìfxuuhqfh prqwdqwh1 

G*dsuëv od iruph gx udssruw ^4:1;;`/ qrxv srxyrqv rphwwuh oh fdofxo ghv ] +| v , = 

dojrulwkph gh edxp0zhofk hw odihuwh 

O*dojrulwkph gh fdofxo gx PSP vh gìurxoh frpph vxlw = 

ÊZO 4` Lqlwldolvdwlrq gh od uìfxuuhqfh prqwdqwh = 

S +[ v @ $ l m \ v @ | v , @ +[ v @ $ l , S +\ v @ | v m [ v @ $ l , 

S 

S +[ v @ $ m , S +\ v @ | v m [ v @ $ m , >v5 V3 = 

$ m5 

ÊZO 5` Sdvvh prqwdqwh = srxu 4 q Q 4/ ;v 5 V q / rq gìgxlw gh od uìfxu0 

uhqfh prqwdqwh ohv fdofxov gh = ! v +{ v >{ v ,1 

ÊZO 6` Lqlwldolvdwlrq gh od sdvvh ghvfhqgdqwh = 

e{ u @dujpd{ 

{ K 

S +[ u @ { u m \ u @ \ @ |, = 

ÊZO 7` Sdvvh ghvfhqgdqwh = srxu Q4 q 3/ rq wluh ghv fdofxov gh ÊZO 5`/ juåfh 

ã od uhodwlrq gh uìfxuuhqfh ^4:1;;` = 

;v 5 V q > e{ v @dujpd{ S +[ v @ { v m \ @ |, = 

{ L 

O*hvwlpdwhxu gx PSP hvw doruv = 

;v 5 V> f [ v @ e{ v = 

O*dojrulwkph fl0ghvvxv pdqlsxoh ghv txdqwlwìv vx!vdphqw judqghv srxu qh sdv íwuh ylfwlph ghv sureoëphv 

g*xqghu rz vljqdoìv gdqv oh fdv ghv fkdñqhv gh Pdunry fdfkìhv1 

Gdqv ^kduglq33`/ D1 kduglq d gìyhorssì xqh pìwkrgh dqdorjxh dx VPDS/ dsshoìh VPSP/ dq gh uhqguh 

soxv ìtxlwdeoh od sìqdolvdwlrq ghv vlwhv ìuurqìv1 Vhv wudydx{ frqfhuqhqw dxvvl ohv dueuhv wurqtxìv1 Lo v*djlw gh 


4:16 Dojrulwkphv gh fodvvlfdwlrq vxu ohv dueuhv gh Pdunry fdfkìv1 534 

txdgdueuhv dpsxwìv gh ohxuv uìvroxwlrqv ohv soxv jurvvlëuhv1 Oh fkdpsv rewhqx ã od uìvroxwlrq gìqlh/ doruv/ frpph 

ìwdqw od soxv jurvvlëuh/ hvw frqvlgìuì frpph xq fkdpsv gh Pdunry1 h prgëoh vh mxvwlh sdu o*revhuydwlrq txh 

ohv qlyhdx{ ohv soxv qv gh o*dueuh vx!vhqw ã fdudfwìulvhu od vwuxfwxuh g*xqh lpdjh1 Ohv phloohxuv uìvxowdwv gh 

vhjphqwdwlrq vrqw dwwhlqwv/ gdqv oh fdguh g*xq txdgdueuh frpswdqw qrupdohphqw kxlw qlyhdx{/ srxu xq qrpeuh gh 

qlyhdx{ lqiìulhxu rx ìjdo ã txdwuh1 hv wudydx{ vrqw frqupìv sdu fhx{ gh Zlooldpv hw Ihqj/ +yrlu ^Zlooldpv ^4:1 { Q u @ $ l 

$ = ^4:1


S \ q 

 

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q 

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@ $ D q { 

q 

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S 

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q 

v @ $ m @ v @ $ l >{ q.4 

v @ $ m 

S 

D q { 

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$>$ ! 5 

fh txl shuphwwud gh frqvhuyhu ohv fdofxov gìfulwv gdqv o*dojrulwkph suìfìghqw/ hq uhpsodêdqw/ ã fkdtxh ìwdsh/ D q 

sdu D 3 q1 Rq yìulh dlvìphqw txh ohv suredelolwìv gh wudqvlwlrq d srvwhulrul v*h{sulphqw/ hq hhw/ gh od píph idêrq hq 

irqfwlrq gh D 3 q hw gh D q / dyhf o*lqwìuíw vxssoìphqwdluh txh ohv D 3 q/ uhqrupdolvìv ã fkdtxh ìwdsh/ uhvwhqw vx!vdphqw 

judqgv srxu íwuh jìuìv sdu od pdfklqh1 

Uhpdutxrqv/ hqq/ txh oh fdofxo ghv suredelolwìv d srvwhulrul/ gìfulw gdqv od sdvvh ghvfhqgdqwh gh o*dojrulwkph/ 

shxw íwuh phqì vlpxowdqìphqw ã fhoxl ghv D 3 q1 hwwh uhpdutxh shuphw/ oruv gh o*h{ìfxwlrq gx fdofxo/ g*ìylwhu xqh 

grxeoh pdqlsxodwlrq lqxwloh hw frþwhxvh hq whpsv fdofxo/ gh fhuwdlqhv txdqwlwìv1 

4:1617 Vhjphqwdwlrq vxshuylvìh vxu ohv dueuhv1 

Dq gh whvwhu o*h!fdflwì gh o*dojrulwkph gh fodvvlfdwlrq/ rq shxw oh whvwhu vxu ohv lpdjhv vlpxoìhv sdu ohv 

glìuhqwv w|shv g*dueuhv1 

Vhjphqwdwlrqv sdu txdgdueuhv1 

Vl o*rq euxlwh/ dyhf xq euxlw jdxvvlhq gh uìiìuhqfh/ ghv uìdolvdwlrqv vlpxoìhv sdu ohv ghx{ w|shv gh txdgdueuhv/ 

ohv uìvxowdwv rewhqxv grqqhqw xqh erqqh lgìh gh od glìuhqfh hqwuh ghv ghx{ prgëohv1 Ilj1 LLL177 suìvhqwh ohv 

vhjphqwdwlrqv gh ghx{ lpdjhv surfkhv/ od suhplëuh/ srxu oh txdgdueuh fodvvltxh/ od vhfrqgh/ srxu oh txdgdueuh 

ìyroxwli 

Vhjphqwdwlrqv sdu gldueuhv1 

Ohv glìuhqwv euxlwdjhv/ ghv lpdjhv/ Ilj1LLL176/ txl vrqw sursrvìv ã od vhjphqwdwlrq/ Ilj1 LLL178/ shuphwwhqw 

gh whvwhu od vhjphqwdwlrq sdu ohv gldueuhv gdqv oh fdv gh glvwulexwlrqv qrupdohv euxlwdqwhv glìuhqfldeohv sdu ohxuv 

pr|hqqhv rx sdu ohxuv yduldqfhv1 Lo dssdudñw fodluhphqw txh od pìwkrgh hvw deoh/ lqgìshqgdpphqw ghv sdudpëwuhv 

gx euxlw rx gh fhx{ gx prgëoh1 


4:16 Dojrulwkphv gh fodvvlfdwlrq vxu ohv dueuhv gh Pdunry fdfkìv1 536 

Quadarbre 

Quadarbre 

classique 

classique 

bruit 

bruit 

gaussien 

gaussien 

N(0,1)-N0,1) 

N(0,1)-N(0,1) 

Image 

Image 

segmentée 

segmentée 

Erreur : 8,3 % 

Quadarbre évolutif 

Quadarbre évolutif 

bruit gaussien 

N(0,1)-N0,1) 

bruit gaussien 

N(0,1)-N(0,1) 

Image segmentée 


Erreur : 2,3 % 

Ilj1 LLL1771 0 Vhjphqwdwlrqv g*lpdjhv vlpxoìhv sdu txdgdueuhv1 

Od vhjphqwdwlrq sdu oh gldueuh gurlw looxvwuh oh fdv rü ohv ghx{ fodvvhv/ gh o*lpdjh/ vrqw wuëv h!fdfhphqw 

glìuhqflìhv/ pdlv rü ohv frxohxuv ohv uhsuìvhqwdqw vrqw lqyhuvìhv1 h skìqrpëqh hvw irqfwlrq gx euxlwdjh/ 

sxlvtx*rq shxw uhpdutxhu txh od fodvvh eodqfkh gh o*lpdjh vhjphqwìh fruuhvsrqg ã od fodvvh od soxv fodluh gh 

o*lpdjh euxlwìh1 Od vhjphqwdwlrq g*lpdjhv gh v|qwkëvh lvvxhv gx prgëoh ã 6/ 8 rx : fodvvhv/ prqwuh txh od txdolwì 

gh od vhjphqwdwlrq hvw wrxw dxvvl suredqwh/ oruvtxh oh qrpeuh gh fodvvhv dxjphqwh1 Gdqv ohv h{hpsohv gh od jxuh 

Ilj1LLL179/ ohv fodvvhv vrqw euxlwìhv dyhf ghv glvwulexwlrqv qrupdohv gh pr|hqqhv ìfkhorqqìhv gh 4 hq 4/ hwgh 

píph yduldqfh 41 Ohv vhjphqwdwlrqv ghv lpdjhv ã 6 hw 8 fodvvhv vrqw hhfwxìhv srxu ghv gldueuhv gurlwv/ doruv txh 

ohv vhjphqwdwlrqv ã : fodvvhv vrqw suìvhqwìhv/ gdqv o*ruguh/ srxu ohv gldueuhv gurlwv/ dowhuqìv hw gldjrqdx{1 

Hq frqfoxvlrq/ lo dssdudñw txh ohv glìuhqwv h{hpsohv wudlwìv sdu od pìwkrgh PSP gh fodvvlfdwlrq srxu ohv 

glìuhqwv gldueuhv gh Pdunry fdfkìv/ dvvxuhqw g*xqh fhuwdlqh delolwì gh od pìwkrgh1 Srxu íwuh ìwhqgxh ã g*dxwuh 

w|shv g*lpdjhv/ od pìwkrgh gh fodvvlfdwlrq qìfhvvlwh xqh skdvh suìdodeoh g*hvwlpdwlrq ghv sdudpëwuhv gx euxlw/ hw 

ghv sdudpëwuhv gx prgëoh1 hw dvshfw hvw wudlwì gdqv oh fkdslwuh vxlydqw1 



Diarbre droit 

Bruit N(1,1)-N(1,2) 


Erreur 1,7% 

Diarbre alterné 

Bruit N(0,1)-N(2,1) 


Erreur 1,8% 

Diarbre diagonal 

Bruit N(0,1)-N(1,1) 


Erreur 2,9% 

Diarbre diagonal 

Bruit N(0,1)-N(1,1) 

N(2,1) 


Erreur 4,2% 

Ilj1 LLL1781 0 Vhjphqwdwlrqv g*lpdjhv vlpxoìhv sdu gldueuhv1 


Segmentation à 3 classes : bruitage gaussien N(0,1) - N(1,1) - N(2,1) : 

Diarbre droit : 

Erreur : 2.6 % 

Segmentation à 5 classes : bruitage gaussien N(0,1) - N(1,1) - N(2,1) - 

N(3,1) - N(4,1) : 



Segmentation à 7 classes : bruitage gaussien N(0,1) - N(1,1) - N(2,1) - 

N(3,1) - N(4,1) - N(5,1) - N(6,1) : 



Diarbre alterné : 


Diarbre diagonal : 


Ilj1 LLL1791 0 Vhjphqwdwlrqv g*lpdjhv vlpxoìhv sdu ohv gldueuhv1

539 

Hvwlpdwlrq ghv sdudpëwuhv gdqv ohv prgëohv fdfkìv1

53: 

0kdslwuh4;0 

Hvwlpdwlrq ghv sdudpëwuhv gdqv ohv prgëohv 

fdfkìv1 

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gh [ hw 5 o*hqvhpeoh gh fhx{ frqfhuqdqw ohv orlv gh \ frqglwlrqqhoohphqw ã [/ txh o*rq vxssrvh jìqìudohphqw 

jdxvvlhqqhv1 hwwh hvwlpdwlrq vh idlw ã sduwlu gh od vhxoh revhuydwlrq ghv uìdolvdwlrqv gh \ 1 

4;14 dv ghv fkdpsv gh Pdunry1 

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suredelolwìv frqglwlrqqhoohv/ s l / tx*lo idxw hvwlphu srxu gìfuluh 4 / 

s l @ S ^[ v @ $ l @Y v gh w|sh ` / ;v 5 V= 

^4;1

53; 4;14 dv ghv fkdpsv gh Pdunry1 

gh yrlvlqdjh/ od svhxgr0yudlvhpeodqfh o hvw gìqlh sdu = 

o +{> |, @ \ v5V 

S +[ v @ { v @[ YL @ { Y8 >, S +\ v @ | v @[ v @ { v , > 

hw qrxv uhfkhufkrqv = 

e HP @dujpd{ H ô +{> |, @\ @ |>` = 

 

^4;1 |, @\ @ |> +t,l = 

^4;1 

w5V 

S 

 

S 

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l 

w5V 

S 

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rü Q t l hvw oh qrpeuh g*rffxuhqfhv gh od frqjxudwlrq [ v @ $ l srxu Y v gh w|sh / gdqv od uìdolvdwlrq gh [ ã 

o*lwìudwlrq t1 Srxu vlpsolhu o*ìqrqfì gh o*dojrulwkph/ srvrqv = 

l@4 

s t l @ S 

+J, +[ v @ $ l @[ YL /dyhfY v gh w|sh , 

hw t w +l, @ S 

+J, +[ w @ $ l @\ @ |, = 

O*dojrulwkph HP Jleevlhq vh gìurxoh frpph vxlw = 


4;14 dv ghv fkdpsv gh Pdunry1 53< 

dojrulwkph hp jleevlhq 

Ohv ìwdshv ^HPJ 5` ã ^HPJ 7` vrqw suìvhqwìhv ã o*lwìudwlrq t hw vrqw ã uìsìwhu mxvtx*ã frqyhujhqfh +mxvtx*ã fh tx*xq 

fulwëuh g*duuíw vrlw yìulì,1 

^HPJ 4` Lqlwldolvdwlrq gh +3, 1 

^HPJ 5` Wludjh g*xq judqg qrpeuh gh fkdpsv sdu ìfkdqwloorqqhxu gh Jleev vxly0 

dqw od orl = 

S 

 

[ @ {@\ @ |> +t, = 

^HPJ 6` Dssur{lpdwlrq/ sdu od orl ghv judqgv qrpeuhv/ gh t w +l, hw Q t l 

$ l 5 / w 5 V hw wrxw w|sh gh yrlvlqdjh 1 

^HPJ 7` dofxo gh s t.4 

l / t.4 l / 5 t.4 

sdu ohv irupxohv suìfìghqwhv1 

l 

srxu wrxw 

Oh fkrl{ gh o*lqlwldolvdwlrq ghv sdudpëwuhv gx prgëoh lq xhqfh od udslglwì gh frqyhujhqfh gh o*dojrulwkph1 Lo | 

d dxvvl ghv ulvtxhv gh frqyhujhqfh yhuv xq pd{lpxp orfdo ìorljqì gx pd{lpxp devrox1 

4;1415 O*dojrulwkph gx judglhqw vwrfkdvwltxh C71 

JV uhfkhufkh gluhfwhphqw oh pd{lpxp gh od orj yudlvhpeodqfh orj Ô +[> \ ,`1 Rq fkhufkh oh sdudpëwuh 

pd{lplvdqw orjÔ `/ srxu o*ìqhujlh +> {> |, hw 3 +> {> |,/ vrq judglhqw sdu udssruw ã 1 

\rxqhv prqwuh gdqv ^\rxqhv;;` hw ^\rxqhv; |,` @ H ^ 3` H ^ 3 @\ @ |` > 

^4;1 \ ,/ hw t hvw xqh yduldeoh vxlydqw od orl frqglwlrqqhoh gh [ 

sdu udssruw ã \ @ |/ fhv yduldeohv shxyhqw íwuh vlpxoìhv/ dyhf o*ìfkdqwloorqqhxu gh Jleev/ hq irqfwlrq gh od ydohxu 

frxudqwh gh t 1 Od uìhvwlpdwlrq vh idlw sdu= 

t.4 @ t . 

g 

t .4^3 +> D t >E t , 3 +> t >|,` > 

^4;1433` 

rü g hvw xqh frqvwdqwh/ txl shxw/ sdu h{psoh/ íwuh fkrlvlh lqyhuvhphqw sursruwlrqqhooh dx qrpeuh gh sl{hov gh 

o*lpdjh1 

4;1416 O*dojrulwkph Lwhudwlyh rqglwlrqqdo Hvwlpdwlrq +U., 

*hvw xqh surfìgxuh jìqìudoh g*hvwlpdwlrq ghv sdudpëwuhv gìyhorssìh sdu Slhf}|qvnl +yrlu ^Slhf}|qvnl 


543 4;14 dv ghv fkdpsv gh Pdunry1 

txl uìdolvh od phloohxuh dssur{lpdwlrq1 Oh sulqflsh g*LH frqvlvwh grqf/ sdu xq surfìgì lwìudwli/ ã fdofxohu/ vl fhod 

hvw srvvleoh/ fhwwh hvsìudqfh frqglwlrqqhooh rx ã o*dssurfkhu vl fhod v*dyëuh qìfhvvdluh/ sdu xqh gìpdufkh vwrfkdvwltxh 

gh wludjh g*ìfkdqwloorq vxlydqw od orl d srvwhulrul srxu oh sdudpëwuh frxudqw1 D fkdtxh lwìudwlrq rq uìhvwlph/ grqf/ 

oh sdudpëwuh frxudqw sdu = 

+t.4, @ H 

+J, 

k 

e +[> \ , @\ @ | 

l 

= 

Lo hvw srvvleoh g*hvwlphu vìsduìphqw rx hq sdudooëoh ohv ghx{ frpsrvdqwhv e 4 hw e 5 gh e = 

k 

+t.4, 

4 @ H 

+J, e4 +[> \ , @\ @ |l 

hw +t.4, 

5 @ H 

+J, 

k 

e5 +[> \ , @\ @ |l 

= 

G*xqh idêrq jìqìudoh/ ohv hvwlpdwhxuv fkrlvlv srxu ohv sdudpëwuhv gx euxlw vhurqw ohv prphqwv hpslultxhv1 

Qrwrqv txh oruvtxh o*hvwlpdwhxu e txh o*rq fkrlvlw hvw o*hvwlpdwhxu gx pd{lpxp gh yudlvhpeodqfh/ LH hw HP 

vh uhmrljqhqw vl o*rq shxw idluh hqwuhu o*rsìudwhxu duj pd{ gdqv o*hvsìudqfh frqglwlrqqhooh +yrlu ^Ghopdv

4;15 dv ghv fkdñqhv gh Pdunry1 544 

hw/ srxu ohv sdudpëwuhv gx prgëoh = 

f 5 

l 

+t.4, 

@ 

qS 

\ m e +t.4, 

l 

m@4 

qS 

4 +[m@$ l, 

m@4 

5 

4+[m @$ l , 

srxu wrxw v 5 V/ $ l 5 / S +J.4, +[ v @ $ l @[ YL /dyhfY v gh w|sh , @ 

H 

kQ t l @\ @ |> +t,l 

^4;1435` 

nS 

H 

kQ t l @\ @ |> +t,l> 

rü Q t l hvw oh qrpeuh g*rffxuhqfhv gh od frqjxudwlrq [ v @ $ l srxu Y v gh w|sh / gdqv od uìdolvdwlrq gh [ ã 

o*lwìudwlrq t1 

l@4 

4;15 dvghvfkdñqhvghPdunry1 

Qrxv h{srvrqv gdqv fh txl vxlw ohv sulqflsdohv pìwkrghv g*hvwlpdwlrq ghv sdudpëwuhv gdqv oh fdguh ghv fkdñqhv 

gh Pdunry/ fdguh txl hvw surfkh gh od prgìolvdwlrq vxu dueuh txl idlw o*remhw gh qrwuh ìwxgh1 h sdudjudskh hvw 

iruwhphqw lqvsluì gh o*duwlfoh gh Ehqplorxg hw Slhf}|qvnl Êhqplorxg 

w5V ! 

rü V 3 hvw o*hqvhpeoh V sulyì gx ghuqlhu ìoìphqw gh od fkdñqh gh Pdunry hw rü 

w +l> m, @ S +[ w @ $ l >[ w.4 @ $ m @\ @ |, 


545 4;15 dv ghv fkdñqhv gh Pdunry1 

vh fdofxoh sdu/ 

w +l> m, @ w +l, d lm i m +| w.4 , w.4 +m, 

> 

nS S 

i w +| w.4 , n w +m, d mo 

o@4 

m@4 

^4;1437` 

w d|dqw ìwì gìql hq ^481::`/ 

Srxu ohv sdudpëwuhv gx euxlw rq dxud = 

+t.4, 

l 

hw 5 +t.4, 

l 

@ 

@ 

S 

w5V 

S 

w5V 

S 

w5V 

+t, 

w +l, | w 

+t, 

w +l, 

+t, 

w +l, 

S 

w5V 

 

| w +t.4, 

l 

+t, 

w +l, 

5 

= 

^4;1438` 

rpswh whqx gh fh txh qrxv dyrqv gìmã yx vxu ohv fkdñqhv gh Pdunry fdfkìhv/ o*dojrulwkph HP gh Edxp0Zhofk 

vh gìurxoh gh od idêrq vxlydqwh = 

dojrulwkph hp gh edxp0zhofk 

Ohv ìwdshv ^HPEZ 5` ã ^HPEZ 7` vrqw suìvhqwìhv ã o*lwìudwlrq t hw vrqw ã uìsìwhu 

mxvtx*ã frqyhujhqfh +mxvtx*ã fh tx*xq fulwëuh g*duuíw vrlw yìulilì,1 

^HPEZ 4` Lqlwldolvdwlrq ghv sdudpëwuhv +3, 1 

^HPEZ 5` dofxo ghv suredelolwìv iruzdug hw edfnzdug srxu ohv sdudpëwuhv frxudqw1 

^HPEZ 6` dofxo gh 

+t, 

w +l, srxu wrxw w 5 V 3 hw $ l 5 

hw +t, 

w +l> m, srxu wrxw w 5 V 3 hw +$ l >$ m , 5 5 = 

^HPEZ 7` Uìhvwlpdwlrq ghv sdudpëwuhv = 

+t.4, 

l >d +t.4, 

lm > +t.4, 

l > 5 +t.4, 

srxu wrxw +$ 

l l >$ m , 5 5 > 

juåfh dx{ irupxohv h{srvìhv suìfìghpphqw1 

Oh fkrl{ gx fulwëuh g*duuíw gh fhw dojrulwkph hvw dxvvl glvfxwì gdqv Êhqplorxg[ w.4 @ $ m , = 


4;15 dv ghv fkdñqhv gh Pdunry1 546 

Hq hhw/ vl o*rq fkrlvlw ohv iuìtxhqfhv hpslultxhv frpph hvwlpdwhxuv gh fhv sdudpëwuhv/ 

f lm +[, @ 4 [ 

mV 3 4^[M @$ 

m 

l >[ w."@$ m`> 

w5V ! 

od irupxoh gh uìhvwlpdwlrq/ ã od +t .4, lhph lwìudwlrq vhud = 

f +t.4, 

lm 

@ 4 [ k 

mV 3 S [ w @ $ l >[ w.4 @ $ m @\ @ |> +t,l > 

m 

w5V ! 

fh txl qrxv udpëqh/ vl o*rq uhsuhqg od sdudpìwulvdwlrq xwlolvìh srxu HP/ ã xqh uìhvwlpdwlrq srxu ohv frpsrvdqwhv 

gh od pdwulfh gh wudqvlwlrq txl hvw = 

d +t.4, 

lm 

@ 

S 

w +l> m, 

S 

+t, 

w +l, = 

w5V ! 

w5V ! +t, 

Od uìhvwlpdwlrq ghv d +t.4, 

lm / txh o*rq vdlw fdofxohu dqdo|wltxhphqw/ vh idlw grqf/ srxu LH hw PLH/ghodpíph 

idêrq txh srxu HP1 Ohv sdudpëwuhv gx euxlw ìwdqw hvwlpìv sdu ohv prphqwv hpslultxhv/ ã sduwlu ghv ìfkdqwloorqv 

vlpxoìv vhorq od orl d srvwhulurul hq irqfwlrq gx sdudpëwuh frxudqw/ fhwwh pìwkrgh hvw ã od irlv gìwhuplqlvwh hw 

vwrfkdvwltxh1 hfl od glìuhqflh gh HP hw VHP hw frqvwlwxh xq mxvwh plolhx{ hqwuh ohv dydqwdjhv hw ohv lqfrqyìqlhqwv 

gh fkdfxqh gh fhv ghx{ dxwuhv pìwkrghv1 

*hvw suìflvìphqw dx qlyhdx gh od vlpxodwlrq gh od uìdolvdwlrq gh od fkdñqh vhorq od orl d srvwhulrul hq irqfwlrq gx 

sdudpëwuhv frxudqw/ txh vh vlwxh od glìuhqfh hqwuh LH hw PLH1 O*dojrulwkph PLH glëuh gh od surfìgxuh 

jìqìudoh hq fh tx*lo sursrvh/ soxw÷w tx*xqh vlpxodwlrq vxlydqw od orl d srvwhulrul/ xq wludjh/ ã fkdtxh lqvwdqw w gh 

o*lwìudwlrq t/ vxlydqw od orl pdujlqdoh d srvwhulrul/ +t, 

w +l,1 

dojrulwkphv lfh hw plfh 

Ohv ìwdshv ^LHP 5` ã ^LHP 7` vrqw suìvhqwìhv ã o*lwìudwlrq t hw vrqw ã uìsìwhu mxvtx*ã frqyhujhqfh +mxvtx*ã fh 

tx*xq fulwëuh g*duuíw vrlw yìulì,1 

^LHP 4` Lqlwldolvdwlrq ghv sdudpëwuhv +3, 1 

^LHP 5` dofxo ghv suredelolwìv iruzdug hw edfnzdug srxu ohv sdudpëwuhv frxudqwv1 

^LHP 6` dofxo gh 

hw 

+t, 

w +l, srxu wrxw w 5 V 3 hw $ l 5 

+t, 

w +l> m, srxu wrxw w 5 V 3 hw +$ l >$ m , 5 5 = 

^LHP 7` Uìhvwlpdwlrq ghv sdudpëwuhv = 

+t.4, 

l 

>d +t.4, 

lm 

srxu wrxw +$ l >$ m , 5 5 = 

^LHP 8` Vlpxodwlrq g*xqh uìdolvdwlrq gh [/ hq irqfwlrq gh o*dojrulwkph LH 

rx PLH fkrlvl/ vhorq oh sdudpëwuh frxudqw +t, 1 


547 4;16 dv ghv dueuhv gh Pdunry fdfkìv1 

^LHP 9` Hvwlpdwlrq sdu ohv prphqwv hpslultxhv ghv sdudpëwuhv gx euxlw= 

+t.4, 

l 

hw 5 +t.4, 

l 

@ 

@ 

S 

| w 4^[M @$ l` 

w5V 

S 

4^[M@$ l` 

w5V 

S 

w5V 

| w +t.4, 

l 

S 

4^[M@$ l` 

w5V 

srxu wrxw $ l 5 

5 

4^[M @$ l` 

srxu wrxw $ l 5 = 

4;1516 O*dojrulwkph 7. 

Od yhuvlrq vwrfkdvwltxh gh o*dojrulwkph HP +yrlu Êurqldwrzvnl;6`/ ^hohx{;9`/ ^hohx{

4;16 dv ghv dueuhv gh Pdunry fdfkìv1 548 

dlqvl txh ohv suredelolwìv gh wudqvlwlrq sëuh0ov ã o*ìfkhooh q/ 

j q l>m +, @ S { q v @ $ l >{ q.4 

v4 @ $ m 

/+uhvs1k 

q 

l>m +, vxlydqw oh prgëoh,/ 

hqwlëuhphqw gìqlhv/ gdqv qrwuh prgëoh/ sdu od frqqdlvvdqfh gx sdudpëwuh lqwurgxlw dx fkdslwuh suìfìghqw1 

$ Ohv sdudpëwuhv gx euxlw/ w|sltxhphqw/ ohv pr|hqqhv hw yduldqfhv ghv glvwulexwlrqv qrupdohv = 

srxu wrxw $ l 5 > l hw l gìqlvvdqw i l 1 

4;1614 Dojrulwkphv . vxu oh gldueuh1 

O*dojrulwkph HP txh qrxv gìyhorssrqv lfl/ hvw lqvsluì gh fhoxl txh qrxv yhqrqv gh gìfuluh vxu ohv fkdñqhv gh 

Pdunry1 Rq srxuud frqvxowhu ã fh vxmhw Êrxpdq +t, 

mV q > ^4;143:` 

m 

+t.4, 

rx +t.4, 

@ 

@ 

4 

mm S 

f5 

4 

mm S 

f5 

mV f m 

mV f m 

% [ 

f5 

% [ 

f5 

k j 

mV f 

f 4 

m l>l j 

f 

l>l +t.4,l& / Q ^ ^4> Q` > 

k k 

mV f 

f 4 

m l>l k 

f 

l>l +t.4,l& / Q ^ ^4> Q` = 

^4;143;` 

^4;143 +t, 

v5V ! 

S 

^4;1443` 

[ v 3 @ $ l m \ @ |> 

+t, 

v5V ! S 

S 

| v +t.4, 

l 

v5V ! 

S 

v5V ! S 

5 

S 

[ 3 v @ $ l m \ @ |> +t, 

= ^4;1444` 

[ v 3 @ $ l m \ @ |> 

+t, 

hw dojrulwkph/ hqwlëuhphqw gìwhuplqlvwh/ vh gìurxoh frpph vxlw = 


549 4;16 dv ghv dueuhv gh Pdunry fdfkìv1 

dojrulwkph hp vxu oh gldueuh 

Ohv ìwdshv ^HPG 5` ã ^HPG 7` vrqw suìvhqwìhv ã o*lwìudwlrq t hw vrqw ã uìsìwhu 


^HPG 4` Lqlwldolvdwlrq ghv sdudpëwuhv +3, 1 

^HPG 5` dofxo g*xqh uìdolvdwlrq [ vhorq oh sdudpëwuh frxudqw t1 

^HPG 6` Hvwlpdwlrq ghv sdudpëwuhv gx prgëoh = 

 

+t.4, 

l 

@ S [ u @ $ l m \ @ |> +t, srxu wrxw $ l 5 

$ 

hw +t.4, 

@ 

S 

f5 

mV f m j 4 

f 

# S 

L#8 B S+[ B L @$ lm[ B." 

mm S mV f m 

f5 

L @$ m>\ @|> +J, , 

mV G m 

^HPG 7` Hvwlpdwlrq ghv sdudpëwuhv gx euxlw = 

/ ohv j f gìshqgdqw gx prgëoh1 

+t.4, 

l 

@ 

hw 5 l 

+t.4, 

@ 

S 

| v S 

[ v 3 @ $ l m \ @ |> +t, 

v5V ! 

S [ v 3 @ $ l m \ @ |> +t, 

v5V ! S 

S 5 

| v +t.4, 

l S 

[ v 3 @ $ l m \ @ |> +t, 

v5V ! S 

S 

[ v 3 @ $ l m \ @ |> +t, = 

v5V ! 

4;1615 Dojrulwkphv U. hw U.1 

Qrxv phwwrqv hq sodfh/ vxu fh píph prgëoh ghx{ dxwuhv surfìgxuhv g*hvwlpdwlrq ghv sdudpëwuhv/ LH hw 

PLH1 Ohv sdudpëwuhv gx prgëoh vrqw fdofxoìv gh idêrq gìwhuplqlvwh/ frpph srxu HP/ juåfh ã ^4;1439`0^4;143 ^4;1446` 

fdofxoìhv srxu xqh uìdolvdwlrq gx fkdps rewhqxh/ sdu vlpxodwlrq vhorq od orl gh [ d srvwhulrul/ dyhf ohv sdudpëwuhv 

hvwlpìv ã o*lwìudwlrq t1 SrxuPLH/ qrxv vlpxorqv xqltxhphqw xqh uìdolvdwlrq gh [ 3 / vhorq ohv orlv pdujlqdohv d 

srvwhulrul1 Hq fhod/ od yhuvlrq vxu dueuh/ gh fhw dorulwkph/ glëuh oìjëuhphqw gh fhooh vxu ohv fkdñqhv1 

hv dojrulwkphv vh uìvxphqw ã = 

dojrulwkphv lfh hw plfh vxu oh gldueuh 


4;16 dv ghv dueuhv gh Pdunry fdfkìv1 54: 

Ohv ìwdshv ^GLH 45` ã ^GLH 47` vrqw suìvhqwìhv ã o*lwìudwlrq t hw vrqw ã uìsìwhu 


^GLH 44` Lqlwldolvdwlrq ghv sdudpëwuhv +3, 1 

^GLH 45` dofxo vhorq oh sdudpëwuh frxudqw +t, /g*xqh uìdolvdwlrq gh [ sdu 

od orl d srvwhulrul/ srxu LH hw g*xqh uìdolvdwlrq gh [ 3 / vhorq ohv orlv 

pdujlqdohv d srvwhulrul/ srxu PLH1 

^GLH 46` Hvwlpdwlrq ghv sdudpëwuhv gx prgëoh = 

 

+t.4, 

l @ S [ u @ $ l m \ @ |> +t, srxu wrxw $ l 5 

$ 

hw +t.4, 

@ 

S 

f5 

mV f m j 4 

f 

# S 

L#8 B S+[ B L @$lm[B." 

mm S mV f m 

f5 

L @$m>\ @|>+J, , 

mV G m 

^GLH 47` Hvwlpdwlrq ghv sdudpëwuhv gx euxlw= 

+t.4, 

l 

@ 

hw 5 l 

+t.4, 

@ 

S 

| v 4^+{ ! L , t @$ l` 

v5V 

S 

! 4^+{ ! L , J @$ l` 

v5V ! 

S 

| v +t.4, 

l 

v5V ! S 

v5V ! 4^+{ ! L , J @$ l` 


5 

4^+{ ! L , t @$ l` 

= 

4;1616 Dojrulwkph 7. vxu oh gldueuh1 

Rq qh fdofxoh sdv ohv suredelolwìv d srvwhulrul hq fkdfxq ghv qrhxgv/ pdlv rq vlpxoh xqh uìdolvdwlrq gh o*dueuh ã 

sduwlu gh od suredelolwì gh od udflqh hw ohv suredelolwìv lqwhuìfkhoohv fdofxoìhv oruv gh od sdvvh prqwdqwh1 Rq frqvlgëuh/ 

sdu frqvìtxhqw/ ohv hvwlpdwhxuv hpslultxhv/ edvìv vxu ohv iuìtxhqfhv g*dssdulwlrq ghv glìuhqwv frxsohv sëuh0ov/ 

gdqv od uìdolvdwlrq vlpxoìh gh o*dueuh/ srxu ohv sdudpëwuhv gx prgëoh/ hw ohv pr|hqqhv hw yduldqfhv hpslultxhv/ 

^4;1445` hw ^4;1446`/ srxu ohv sdudpëwuhv gx euxlw1 

dojrulwkph vhp vxu oh gldueuh 

Ohv ìwdshv ^GVHP 5` ã ^GVHP 7` vrqw suìvhqwìhv ã o*lwìudwlrq t hw vrqw ã uìsìwhu 


^GVHP 4` Lqlwldolvdwlrq ghv sdudpëwuhv +3, 1 

^GVHP 5` Vlpxodwlrq g*xqh uìdolvdwlrq [ d srvwhulrul/ vxlydqw oh sdudpëwuh 

frxudqw +t, 1 

^GVHP 6` Hvwlpdwlrq ghv sdudpëwuhv gx prgëoh = 

 

+t.4, 

l @ S [ u @ $ l m \ @ |> +t, srxu wrxw $ l 5 


54; 4;16 dv ghv dueuhv gh Pdunry fdfkìv1 

hw +t.4, 

@ 

S 

f5 

mV f m j 4 

f 

# S 

$ 

L#V f 4^[ v($ f l`4^[ v f." m` 

S 

f." 

L#8B 4^[ 

v ($ m` 

mm S mV f m 


f5 

^GVHP 7` Hvwlpdwlrq ghv sdudpëwuhv gx euxlw= 

+t.4, 

l 

@ 

S 

v5V ! | v 4^+{ ! L , t @$ l` 

S 

v5V ! 4^+{ ! L , J @$ l` 

S 

hw | v +t.4, 

l 

5 +t.4, v5V 

l @ 

! S 

v5V ! 4^+{ ! L , J @$ l` 

5 

4^+{ ! L , t @$ l` 

= 


0kdslwuh4

553 4 4, hw 

Q +4> 4,1 Qrwuh remhfwli hvw gh gìwhuplqhu o*dojrulwkph oh plhx{ dgdswì ã fkdfxq ghv prgëohv g*dueuhv/ hw gh suìflvhu 

ohv grpdlqhv g*dssolfdwlrq ghv ghx{ prgëohv1 Srxu fhod/ qrxv idlvrqv wrxuqhu fhqw irlv/ ohv kxlw dojrulwkphv vxu 

fkdfxqh ghv flqt lpdjhv/ euxlwìhv lqgìshqgdpphqw/ srxu fkdfxqh ghv fhqw vhjphqwdwlrqv/ dyhf ohv glvwulexwlrqv 

qrupdohv fl0ghvvxv1 

Oh wdeohdx Ilj1LLL17; uìfdslwxoh/ srxu fkdtxh lpdjh hw fkdtxh dojrulwkph/ o*huuhxu pr|hqqh vxu ohv fhqw vhj0 

phqwdwlrqv/ dlqvl txh o*huuhxu plqlpdoh hw o*huuhxu pd{lpdoh frpplvhv dx frxuv ghv fhqw h{sìulhqfhv1 Qrxv | 

grqqrqv ìjdohphqw/ o*huuhxu joredoh frpplvh sdu fkdtxh dojrulwkph/ dx frxuv ghv flqt fhqw vhjphqwdwlrqv1 

Image SEM type 1 ICE type1 MICE type1 EM type 1 SEM type 2 ICE type2 MICE type2 EM type 2 

3.41% 3.51% 3. 43% 3.51% 3.42% 3.40% 3.42% 3.43% Erreur mini. 

4.29% 4.63% 4.66% 4.48% 4.72% 4.50% 4.46% 4.64% Erreur maxi. 

3.91% 4.01% 3.98% 3.97% 4.01% 3.97% 3.95% 3.97% Erreur moy. 

4.12% 4.23% 4.28% 4.03% 4.09% 4.02% 4.02% 4.12% Erreur mini. 

6.41% 7.50% 6.45% 6.61% 6.52% 6.68% 7.84% 6.73% Erreur maxi. 

5.04% 5.48% 5.32% 5.23% 5.16% 5.32% 5.22% 5.34% Erreur moy. 

7.66% 8.14% 8.23% 7.84% 7.71% 7.60% 7.87% 7.67% Erreur mini. 

9.88% 11.16% 11.08% 10.08% 9.81% 10.16% 10.35% 10.17% Erreur maxi. 

8.86% 9.44% 9.41% 8.95% 8.75% 8.88% 8.88% 8.94% Erreur moy. 

13.50% 14.87% 14.41% 13.83% 13.26% 13.78% 13.79% 13.89% Erreur mini. 

15.39% 16.91% 16.82% 15.64% 15.05% 15.77% 15.80% 15.71% Erreur maxi. 

14.51% 15.79% 15.62% 14.88% 14.15% 14.79% 14.68% 14.96% Erreur moy. 

29.75% 32.23% 31.99% 30.37% 27.03% 29.87% 29.31% 31.15% Erreur mini. 

31.91% 51.46% 47.42% 33.10% 32.24% 35.05% 34.01% 48.82% Erreur maxi. 

30.66% 34.06% 33.88% 31.47% 28.46% 31.61% 31.45% 32.98% Erreur moy. 

12.60% 13.76% 13.64% 12.90% 12.11% 12.91% 12.83% 13.24% Erreur globale 

Ilj1 LLL17;1 0 rpsdudlvrq ghv pìwkrghv g*hvwlpdwlrq1 

Ohv dojrulwkphv edvìv vxu oh VHP hw ohv gldueuhv vwrfkdvwltxhv grqqhqw gh phloohxuv uìvxowdwv txh ohxuv frq0 

fxuhqwv/ dyhf xqh glìuhqfh vljqlfdwlyh srxu oh VHP gh w|sh 5/ vxu od ghuqlëuh lpdjh1 

Oh wdeohdx Ilj1LLL17

4

555 4

4

557 4

4

559 4

4

55; 4

4

563 4

4

565 4

4

567 4

4

569 4

4

56< 

0kdslwuh530 

Oh prgëoh ghv pìodqjhv jìqìudolvìv1 

5314 Suìvhqwdwlrq gh od pìwkrgh1 

Mxvwlìh sdu ohv hvwlpdwlrqv ghv sdudpëwuhv vxu ghv lpdjhv uìhoohv lvvxhv g*LUP/ gh vdwwholwhv rx gh udgduv/ 

^Pdhw

573 5315 Hwdorqqdjh gh od pìwkrgh sursrvìh1 

5315 Hwdorqqdjh gh od pìwkrgh sursrvìh1 

Vl qrxv uhsuhqrqv ohv h{hpsohv txh qrxv dyrqv wudlwìv soxv kdxw/ hq qh suìmxjhdqw sdv/ fhwwh irlv/ ghv glvwulexwlrqv 

euxlwdqwhv xwlolvìhv/ qrxv srxyrqv ìsurxyhu fhwwh qrxyhooh pìwkrgh g*hvwlpdwlrq srxu ohv pìodqjhv jìqìudolvìv1 

rpswh whqx gh od pìwkrgh g*hvwlpdwlrq ghv pìodqjhv jìqìudolvìv txh qrxv sursrvrqv/ hhfwxdqw ghv hvwlpdwlrqv 

gh sdudpëwuhv frpsoëwhv srxu ohv glìuhqwv w|shv gh pìodqjhv/ oh sureoëph gx fkrl{ qdo gx w|sh hvwlpì hvw fuxfldo1 

Lo hvw lqwìuhvvdqw gh frqvwdwhu txh od pìwkrgh gh whvw hhfwxh oh erq fkrl{ gx w|sh gh pìodqjh gdqv ohv txdwuh fdv 

suìvhqwìv 

Od glìuhqfh hqwuh ohv ghx{ dojrulwkphv VHP4 hw VHP5 q*hvw sdv vljqlfdwlyh/ hw od vroxwlrq od soxv udlvrqqdeoh/ 

srxu idluh xq fkrl{ hqwuh ohv lpdjhv fodvvlìhv ã sduwlu ghv sdudpëwuhv lvvxv gh fhv ghx{ dojrulwkphv g*hvwlpdwlrq/ 

hvw gh suhqguh fhooh srxu odtxhooh o*hqvhpeoh ghv sdudpëwuhv hvwlpìv plqlplvh oh whvw g*dgìtxdwlrq dyhf ohv prphqwv 

hpslultxhv1 Ohv glìuhqwv uìvxowdwv gh vhjphqwdwlrq hhfwxìv gdqv xq fdguh gh pìodqjhv jìqìudolvìv vrqw uìiìuhqfìv 

jxuhv Ilj1LLL19; ã Ilj1LLL1:41 Ohv vhjphqwdwlrqv suìvhqwìhv vxu od suhplëuh oljqh vrqw rewhqxhv sdu ohv dojr0 

ulwkphv gìyhorssìv vxu oh gldueuh 4/ fhoohv gh od vhfrqgh oljqh ylhqqhqw gx gldueuh 51 Oh judskltxh gx pìodqjh 

hvwlpì txl dffrpsdjqh ohv lpdjhv vhjphqwìhv hvw fhoxl gx pìodqjh/ hvwlpì sdu o*dojrulwkph VHP4 rx 

/ plqlplvdqw oh whvw g*dgìtxdwlrq dyhf ohv prphqwv hpslultxhv1 

Gdqv oh wdeohdx gh od jxuh Ilj1 LLL19:/ qrxv suìvhqwrqv ohv uìvxowdwv ghv hvwlpdwlrqv ghv sdudpëwuhv/ srxu ohv 

txdwuh lpdjhv whvwv/ sdu ohv pìwkrghv jìqìudolvìh lvvxhv gh VHP vxu ohv dueuhv gh w|sh 4 hw 5/ dlqvl txh fhoohv 

rewhqxhv sdu od pìwkrgh ghv prphqwv jìqìudolvìh ìqrqfìh gdqv od ghx{lëph sduwlh1 Oh idlw txh qrxv wudydloolrqv 

ã xqh uìvroxwlrq g*lpdjh gh 45; 45; q*hvw sdv idlw srxu idyrulvhu od pìwkrgh ghv prphqwv/ txl ìfkrxh ã fkrlvlu 

oh erq w|sh gh pìodqjh gdqv oh fdv gx pìodqjh h{srqhqwlhooh0h{srqhqwlhooh1 h uìvxowdw ìwdlw suìylvleoh/ frpswh 

whqx ghv uhpdutxhv txh qrxv dyrqv idlwhv/ gdqv oh fdguh gh fh w|sh gh pìodqjh/ gdqv od ghx{lëph sduwlh1 Ohv 

dojrulwkphv lwìudwliv g*hvwlpdwlrq ghv sdudpëwuhv grqqhqw g*h{fhoohqwhv hvwlpdwlrqv gdqv ohv txdwuh fdv suìvhqwìv 

lfl/ hw frpph qrxv o*dyrqv gìmã vljqdoì/ vhpeohqw deohv dx qlyhdx gx fkrl{ gx w|sh gx pìodqjh1 D prlqv txh qrxv 

qh glvsrvlrqv g*xqh lpdjh gh judqgh uìvroxwlrq/ w|sltxhphqw vxsìulhxuh ã 845 845/ od pìwkrgh ghv prphqwv qh 

shxw íwuh udlvrqqdeohphqw xwlolvìh1 

Oh prgëoh ghv pìodqjhv jìqìudolvìv1

5315 Hwdorqqdjh gh od pìwkrgh sursrvìh1 574 

Type et 

Méthode 

Image paramètres SEM type 1 SEM type 2 des 

théoriques 

moments 

N-N N-N N-N N-N Type 

0.0 0.05 0.01 -0.04 Moyenne 1 

1.0 1.13 1.05 0.97 Variance 1 

0.60 0.61 0.61 0.60 p 

2.0 1.22 1.97 2.00 Moyenne 2 

1.0 1.04 1.14 1.04 Variance 2 

N-E N-E N-E N-E Type 

1.0 0.99 1.00 1.01 Moyenne 1 

2.0 1.99 2.02 1.93 Variance 1 

0.28 0.27 0.28 0.30 p 

1.0 0.99 0.98 0.98 Moyenne 2 

/ / / / Variance 2 

E-E E-E E-E N-E Type 

1.0 1.01 1.03 0.91 Moyenne 1 

/ / / 0.58 Variance 1 

0.68 0.64 0.67 0.62 p 

3.0 2.83 2.84 2.86 Moyenne 2 


N-E N-E N-E N-E Type 

1.0 1.00 1.00 0.99 Moyenne 1 

1.0 1.03 1.03 1.00 Variance 1 

0.70 0.70 0.70 0.74 p 

1.0 1.01 1.02 1.06 Moyenne 2 


Ilj1 LLL19:1 0 Hvwlpdwlrq ghv sdudpëwuhv gdqv oh fdguh ghv pìodqjhv jìqìudolvìv1 



Bruit N(0,1)-N(2,1) 

N-N : Erreur 7.81 % 

Test : 0.54 

N-E : Erreur 16.34 % 

Test : 20.94 

E-E : pas de solution 

N-N : Erreur 7.10 % 

Test : 0.44 

N-E : Erreur 11.19 % 

Test : 27.25 


Graphique du mélange estimé par SEM2 type N-N. 

: histogramme de fréquences. 

: courbe du mélange estimé. 

Ilj1 LLL19;1 0 Vhjphqwdwlrq g*lpdjh vxu ohprgëohghvpìodqjhvjìqìudolvìv1 



Bruit E(1)-N(1.2) 

N-N : Erreur 47.47 % 

Test : 6.72 

N-E : Erreur 4.04 % 

Test : 0.66 


N-N : Erreur 33.99 % 

Test : 4.40 


Test : 0.49 


Graphique du mélange estimé par SEM2 type N-E. 



Ilj1 LLL19


Bruit E(1)-E(3) 

N-N : Erreur 16.03 % 

Test : 329 

N-E : Erreur 30.06 % 

Test : 276 

E-E : Erreur 8.36 % 

Test : 30 

N-N : Erreur 17.30 % 

Test : 329 

N-E : Erreur 32.37 % 

Test : 176 

E-E : Erreur 8.40 % 

Test : 29 

Graphique du mélange estimé par SEM2 type E-E. 



Ilj1 LLL1:31 0 Vhjphqwdwlrq g*lpdjh vxu ohprgëohghvpìodqjhvjìqìudolvìv1 



Bruit N(1,1)-E(1) 

N-N : Erreur 23.43 % 

Test : 4.06 


Test : 0.001 


N-N : Erreur 41.06 % 

Test : 2.50 


Test : 0.01 


Graphique du mélange estimé par SEM1 type N-E. 



Ilj1 LLL1:41 0 Vhjphqwdwlrq g*lpdjh vxu oh prgëoh ghv pìodqjhv jìqìudolvìv1 


579 5316 Ohfwxuhghfduwrqvg*rujxhgheduedulh1 

5316 Ohfwxuhghfduwrqvg*rujxhgheduedulh1 

531614 Ghvfulswlrq g*xq fduwrq1 

Vxu xqh ghpdqgh gh o*dvvrfldwlrq Dujhqwltxh 45 / hw dq g*looxvwuhu od udslglwì hw od suìflvlrq ghv dojrulwkphv gh 

vhjphqwdwlrq g*lpdjhv gìyhoorsìv vxu ohv dueuhv gh Pdunry fdfkìv/ qrxv qrxv vrpphv lqwìuhvvìv ã od plvh dx srlqw 

g*xq surwrfroh gh ohfwxuh g*xq fduwrq g*rujxh gh eduedulh ã 5: wrxfkhv1 h w|sh gh fduwrq phvxuh 487pp gh odujh1 

Od qrwh od soxv kdxwh/ txl hvw xq Uh: hvw vlwxìh ã 8pp gx erug jdxfkh gx fduwrq1 kdtxh qrwh hvw frqvwlwxìh sdu 

xq wurx uhfwdqjxodluh/ gh 6> 8pp gh odujhxu/ hw grqw od orqjxhxu hvw irqfwlrq gh od gxuìh gh od qrwh gdqv od sduwlwlrq 

ruljlqdoh/ hw gh od ylwhvvh fkrlvlh srxu oh gìohphqw vwdqgdug gx fduwrq1 Ghx{ qrwhv yrlvlqhv/ gdqv od vxffhvvlrq 

qrq fkurpdwltxh ghv qrwhv g*xq who rujxh/ vrqw vìsduìhv/ vxu oh fduwrq/ sdu xq ìfduw gh 5pp/ fhtxlsrvlwlrqqhod 

qrwh od soxv edvvh/ xq Gr7/ ã

5316 Ohfwxuh gh fduwrqv g*rujxh gh eduedulh1 57: 

Ilj1LLL1:510Lpdjhghfduwrq41 

Ilj1 LLL1:61 0 Lpdjh gh fduwrq 5+wdfkìh,1 


57; 5316 Ohfwxuhghfduwrqvg*rujxhgheduedulh1 

531616 Vhjphqwdwlrq gh o*lpdjh gx fduwrq 41 

Ohv wurlv lpdjhv Ilj1LLL1:7/ Ilj1LLL1:8 hw Ilj1LLL1:9/ gìgxlwhv gh o*lpdjh fduwrq 4 hw rewhqxhv vdqv dyrlu 

uhfrxuv ã xq dojrulwkph gh vhjphqwdwlrq/ prqwuhqw o*lpsrvvlelolwì g*h{sorlwhu gluhfwhphqw o*lqirupdwlrq frqwhqxh 

gdqv oh fklhu gh fdswxuh1 O*lpdjh fodvvdqw ohv sl{hov hq ghx{ fdwkìjrulhv hvw fodluhphqw lqxwlolvdeoh/ ã fdxvh gh 

od sduwlh qrluh gh od jdxfkh gh o*lpdjh1 Od uìsduwlwlrq ghv fodvvhv gdqv ohv lpdjhv Ilj1LLL1:8 hw Ilj1 LLL1:9/ qh 

shuphw sdv / qrq soxv/ xqh ohfwxuh udlvrqqdeoh1 D prlqv gh shuihfwlrqqhu oh glvsrvlwli g*dftxlvlwlrq ghv lpdjhv/ oh 

uhfrxuv ã xqh pìwkrgh vwdwlvwltxh gh vhjphqwdwlrq g*lpdjhv dssdudñw frpph xqh vroxwlrq vhfrxudeoh1 

Ilj1 LLL1:71 0 H{sorlwdwlrq gluhfwh gh o*lqirupdwlrq hq ghx{ fodvvhv1 

Srxu ghv udlvrqv gh udslglwì gh od vhjphqwdwlrq/ hw gdqv xq vrxfl gh txdolwì ghv uìvxowdwv/ lo hvw qìfhvvdluh gh 

prufhohu o*lpdjh ruljlqdoh ã od uìvroxwlrq 653653/ hq58 vrxv0lpdjhv ã od uìvroxwlrq 97971 hwwh vroxwlrq shuphw 

gh uìdolvhu od vhjphqwdwlrq gh o*lpdjh gh fduwrq hq xq shx prlqv gh ; vhfrqghv/ fh txl ìtxlydxw ã xqh lqwhusuìwdwlrq 

hq whpsv uìho gh o*lpdjh dftxlvh sdu od fdpìud1 Od udslglwì ghv dojrulwkphv gh vhjphqwdwlrq gìyhorssìv vxu ohv 

dueuhv gh Pdunry/ shuphw/ vdqv shuwh gh whpsv/ g*dmrxwhu xqh glphqvlrq vwdwlvwltxh ã o*lqwhusuìwdwlrq ghv grqqìhv 

lvvxhv gh od fdswxuh1 


5316 Ohfwxuh gh fduwrqv g*rujxh gh eduedulh1 57< 

Ilj1 LLL1:81 0 H{sorlwdwlrq gluhfwh gh o*lqirupdwlrq hq wurlv fodvvhv1 

Ilj1 LLL1:91 0 H{sorlwdwlrq gluhfwh gh o*lqirupdwlrq hq txdwuh fodvvhv1 



Ohv whqwdwlyhv gh vhjphqwdwlrq gh o*lpdjh hq soxv gh ghx{ fodvvhv gdqv oh fdv jdxvvlhq/ dq g*rswlplvhu od txdolwì 

gh o*lqwhusuìwdwlrq txh o*rq yhxw hq idluh/ v*dyëuhqw shx iuxfwxhxvhv1 Hq hhw/ xqh vhjphqwdwlrq hq wurlv fodvvhv/ 

Ilj1LLL1::/ srvh/ frpph qrxv o*dyrqv yx vxu o*lpdjh Ilj1 LLL1:8/ oh sureoëph gh o*dhfwdwlrq ghv sl{hov gh od fodvvh 

julvh lqwhupìgldluh1 Od vhjphqwdwlrq hq txdwuh fodvvhv/ Ilj1 LLL1:;/ soxv lqwìuhvvdqwh/ suryrtxh ghv ìfkhfv/ ã fdxvh 

gx wurs judqg qrpeuh gh fodvvhv/ oruv gh o*hvwlpdwlrq ghv sdudpëwuhv gh fhuwdlqhv }rqhv gh o*lpdjh1 

Ilj1 LLL1::1 0 Vhjphqwdwlrq gh o*lpdjh fduwrq 4 hq 6 fodvvhv1 

Ilj1 LLL1:;1 0 Vhjphqwdwlrq gh o*lpdjh fduwrq 4 hq 7 fodvvhv1 


5316 Ohfwxuh gh fduwrqv g*rujxh gh eduedulh1 584 

Od vhjphqwdwlrq vhud grqf phqìh vhorq ghx{ fodvvhv/ fh txl shuphwwud xqh frqvwuxfwlrq qdwxuhooh gx fklhu 

ox/ frpph qrxv o*dyrqv gìfulwh soxv kdxw1 Od suhplëuh vhjphqwdwlrq vhorq ghx{ fodvvhv/ Ilj1 LLL1:


Vljqdorqv dxvvl/ tx*xqh vhjphqwdwlrq/ gh fkdfxqh ghv 58 vrxv0lpdjhv/ sdu od pìwkrgh gh pìodqjhv jìqìudolvìv/ 

vìohfwlrqqh/ ohv 58 irlv/ oh w|sh gh pìodqjh qrupdoh0qrupdoh1 Ohv vhjphqwdwlrqv gdqv ohv fdv qrupdoh0h{srqhqwlhooh/ 

hw h{srqhqwlhooh0h{srqhqwlhooh vrqw sursrvìhv srxu lqirupdwlrq1 Od suhplëuh/ Ilj1 LLL1;4/ hvwwuëvsurfkhghfhooh 

rewhqxh gdqv oh fdv qrupdoh0qrupdoh/ ã fhod suëv txh ohv fodvvhv wurx vrqw/ hq jìqìudo/ soxv odujhv1 Od frqqdlv0 

vdqfh wkìrultxh ghv }rqhv gh o*lpdjh vxvfhswleohv gh sruwhu od fodvvh wurx/ shuphw fhshqgdqw xqh wuëv erqqh 

lqwhusuìwdwlrq gh fhwwh lpdjh1 Od vhfrqgh/ Ilj1 LLL1;5/ dx frqwudluh/ uìgxlw od odujhxu gh fhv fodvvhv/ txl dssdudlv0 

vhqw wuëv qhwwhphqw1 Ohv wudyhuv gh fhwwh vhjphqwdwlrq dssdudlvvhqw/ qrwdpphqw gdqv od sduwlh jdxfkh gh o*lpdjh/ 

rx od uhvwulfwlrq h{wuëph ghv }rqhv gh fodvvhv wurx/ idxvvh o*lqwhusuìwdwlrq txh qrxv srxuulrqv hq idluh1 

Ilj1LLL1;410Lpdjhfduwrq4vhjphqwìhvxuohprgëohghw|shQ0H1 

Ilj1LLL1;510Lpdjhfduwrq4vhjphqwìhvxuohprgëohghw|shH0H1 



531617 Vhjphqwdwlrq gh o*lpdjh gx fduwrq vdoh1 

Oh fduwrq txh qrxv hvvd|rqv gh oluh lfl/ elhq tx*lo vrlw wuëv vlpsoh g*xq srlqw gh yx pxvlfdo/ vrxoëyh ghx{ 

w|shv gh gl!fxowìv/ txl qh wrxfkdlhqw sdv oh suhplhu fduwrq1 Lo d g*xqh sduw/ ìwì wdfkì sdu oh yhuqlv xwlolvì srxu 

gxuflu hw ìylwhu o*xvxuh ghv erugxuhv ghv wurxv/ hw lo d/ g*dxwuh sduw/ ìwì solì ã soxvlhxuv hqgurlwv1 hv sureoëphv 

vh pdqlihvwhqw/ dx qlyhdx gh o*lpdjh ã vhjphqwhu/ sdu ghv wdfkhv dqrupdohphqw fodluhv srxu oh yhuqlv/ hw sdu ghv 

wudlwv vrpeuhv txh o*rq shxw frqirqguh dyhf ohv wurxv/ srxu ohv solxuhv1 

Od frpsoh{lfdwlrq gx wudydlo ghylhqw fodluh oruvtxh o*rq whqwh gh vhjphqwhu o*lpdjh fduwrq vdoh sdu od pìwkrgh 

ghv pìodqjhv jìqìudolvìv/ txl/ ã qrxyhdx qh uhwlhqw txh ohv hvwlpdwlrqv gh w|sh qrupdoh0qrupdoh1 O*lpdjh vhjphqwìh/ 

Ilj1LLL1;6/ q*hvw sdv h{sorlwdeoh srxu xqh lqwhusuìwdwlrq pxvlfdoh fruuhfwh gx fduwrq1 

Ilj1 LLL1;61 0 Lpdjh fduwrq vdoh vhjphqwìh sdu od pìwkrgh ghv pìodqjhv jìqìudolvìv1 

Lo dssdudlw/ hq idlw/ txh vl od vhjphqwdwlrq vxu oh prgëoh h{srqhqwlhooh0h{srqhqwlhooh/ Ilj1LLL1;7/ q*hvw sdv soxv 

shuwlqhqwh/ lo q*hq hvw sdv gh píph gh fhooh hhfwxìh vxu oh prgëoh qrupdoh0h{srqhqwlhooh suìvhqwìh Ilj1 LLL1;81 

hwwh ghuqlëuh vhjphqwdwlrq lghqwlh ohv wurlv qrwhv/ frpph oh prqwuh vrq lqwhusuìwdwlrq/ Ilj1LLL1;9/ hw q*hqwudñqh 

sdv o*dssdulwlrq duwlflhooh gh idxvvhv qrwhv1 



Ilj1 LLL1;71 0 Lpdjh vhjphqwìh vxu oh prgëoh gh w|sh H0H1 

Ilj1 LLL1;81 0 Lpdjh vhjphqwìh vxu oh prgëoh gh w|sh Q0H1 



Ilj1 LLL1;91 0 Lpdjh fduwrq vdoh cclqwhusuìwìh**1 



531618 Xqh pìwkrgh gh ohfwxuh1 

Oh sureoëph txl vh srvh gdqv fh fdv wuëv sduwlfxolhu gh vhjphqwdwlrq g*lpdjhv uìhoohv/ w|sltxhphqw dgdswì dx 

fdv ghv pìodqjhv jìqìudolvìv/ f*hvw txh oh fulwëuh gh fkrl{ jìqìudo txh qrxv dyrqv gìfulw gdqv oh fkdslwuh suìfìghqw/ 

lqgxlvh xq fkrl{ gh vhjphqwdwlrq txh qrxv vdyrqv lqdssursulì1 Lo frqylhqw/ grqf/ gh gìqlu xq fulwëuh gh fkrl{ 

srxu od pìwkrgh ghv pìodqjhv jìqìudolvìv/ vsìfltxhphqw dgdswì dx sureoëph txl qrxv lqwìuhvvh1 O*lqwhusuìwdwlrq 

gx fduwrq ghpdqgh/ hq hhw/ xqh judqgh suìflvlrq gdqv od vhjphqwdwlrq/ txh o*rq shxw dwwhlqguh juåfh ã xqh erqqh 

frqqdlvvdqfh ghv frqwudlqwhv d sulrul1 

Od qrxyhooh pìwkrgh g*hvwlpdwlrq ghv sdudpëwuhv gh pìodqjhv jìqìudolvìv txh qrxv sursrvrqv/ vsìfldohphqw 

gìqlh srxu od vhjphqwdwlrq g*lpdjhv gh fduwrqv g*rujxh gh eduedulh/ pxowlsolh sdu ghx{ od gxuìh wrwdoh gh od 

vhjphqwdwlrq/ phqìh lqlwldohphqw vxu xqh vhxoh idplooh gh glvwulexwlrq1 Oh qrxyhdx fulwëuh frqvlvwh ã fkrlvlu/ srxu 

fkdfxqh ghv 58 vrxv0lpdjhv/ fhooh ghv ghx{ vhjphqwdwlrqv lvvxhv ghv hvwlpdwlrqv srxu ghv pìodqjhv gh w|sh Q0Q hw 

Q0H/ txl dxud oh prlqv vrxyhqw dwwulexì od fodvvh wurx/ ã ghv yduldeohv txh qrxv vdyrqv qìfhvvdluhphqw fodvvìhv 

fduwrq1 

hwwh surfìgxuh/ dssoltxìh ã o*lpdjh fduwrq vdoh/ shuphw hhfwlyhphqw gh fruuljhu/ sdu oh fkrl{ gh od vhjphq0 

wdwlrq lvvxh gx w|sh Q0H/ ohv txdwuh vrxv0lpdjhv gìjudgdqw suìfìghpphqw oh uìvxowdw gh od vhjphqwdwlrq Q0Q1 

O*lpdjh dlqvl rewhqxh/ Ilj1LLL1;:/ rswlplvh od vhjphqwdwlrq hw sdu frqvìtxhqw/ od txdolwì gh od ohfwxuh gx fduwrq1 

Qrwrqv/ srxu frqfoxuh/ txh fhwwh pìwkrgh/ dssoltxìh ã o*lpdjh fduwrq 4/ shuphw g*derxwlu/ frqirupìphqw ã fh 

tx*rq srxydlw suìyrlu/ ã od píph vhjphqwdwlrq txh fhooh rewhqxh gdqv oh sdujudskh suìfìghqw1 

Ilj1 LLL1;:1 0 Lpdjh fduwrq vdoh srxu xqh vhjphqwdwlrq rswlplvìh1 

hwwh pìwkrgh gh ohfwxuh gh fduwrqv g*rujxh gh eduedulh/ whvwìh vxu soxvlhxuv fduwrqv frpsohwv/ grqqh gh wuëv 

erqv uìvxowdwv1 Qrxv qh srxyrqv/ elhq vþu/ ìylwhu ohv huuhxuv vxu od gxuìh gh fhuwdlqhv qrwhv/ rx o*dssdulwlrq gh 

txhotxhv idxvvhv qrwhv1 Ohv uhwrxfkhv pxvlfdohv ã dssruwhu vxu oh fklhu lqwhusuìwì vrqw fhshqgdqw udlvrqqdeohphqw 

shx iuìtxhqwhv1 Ohv huuhxuv vrqw/ jìqìudohphqw/ frqfhqwuìhv vxu ghx{ rx wurlv phvxuhv/ srxu ohvtxhoohv ohv frqwudlqwhv 

gh yluwxrvlwì gx prufhdx/ rx od txdolwì orfdoh gh o*dftxlvwlrq gh o*lpdjh vhjphqwìh/ shuwxuehqw od fodvvlfdwlrq1 

Ohv sulqflsdohv olplwhv gh fhwwh whfkqltxh gh ohfwxuh gh fduwrqv g*rujxh gh eduedulh/ vrqw fhoohv {ìhv sdu oh 

surwrfroh g*dftxlvlwlrq gh o*lpdjh1 Xq uìvroxwlrq gh fdswxuh vxsìulhxuh/ shuphwwudlw ìylghpphqw gh jdjqhu hq 


5316 Ohfwxuh gh fduwrqv g*rujxh gh eduedulh1 58: 

delolwì1 Xqh h{sorlwdwlrq/ hq sdudooëoh/ ghv glìuhqwv fdqdx{ UYE g*xqh fdswxuh frxohxu/ rswlplvìh sdu od srvvlelolwì 

gh froruhu oh vxssruw/ txl fdudfwìulvh od frxohxu gh od fodvvh wurx/ hvw dfwxhoohphqw hq frxuv gh gìyhorsshphqw1 


Dsshqglfh D= Wkìruëph irqgdphqwdo 

gh Ureelqv1 

Vrlw oh yhfwhxu gh p / | @ +| 4 >===| p , hw J +|, xqh irqfwlrq gh uìsduwlwlrq grqw od suredelolwì vxu E p dvvrflìh 

ã J/ hvw1 RqvhgrqqhxqhirqfwlrqI +{> |, whooh txh/ srxu suhvtxh wrxw |/ I +{> |, hvw xqh irqfwlrq gh 

uìsduwlwlrq/ gh {/ hw srxu wrxw { xqh irqfwlrq eruìolhqqh gh |1 Od suredelolwì vxu E q dvvrflìh ã od irqfwlrq gh 

uìsduwlwlrq I +{> |,/ gh{/ vhud qrwìh | 1 

Wkìruëph = Od irqfwlrq 

] 

K +{, @ I +{> |, gJ +|, 

F ^531447` 

hvw xqh irqfwlrq gh uìsduwlwlrq gh U q 1 Vl hvw od suredelolwì vxu E q dvvrflìh ã K/ srxu wrxw V 5E q / | +V, hvw 

xqh irqfwlrq eruìolhqqh gh | hw 

] 

+V, @ 

F | +V, gJ +|, = 

58< 

^531448` 

Gìprqvwudwlrq = Rq qrwhud / od fodvvh gh wrxv ohv eruìolhqv V gh U q whov txh | +V, hvw xqh irqfwlrq eruìolhqqh 

gh |1 Prqwurqv txh hvw qrupdoh = 

0VrlwV 4 >V 5 > 111 xqh vxlwh g*hqvhpeohv glvmrlqwv gh 1 VlV @ .4 V 

V q /rqd= 

# $ .4^ 

| +V, @ | V q @ 

q@4 

q@4 

.4[ 

q@4 

| +V q , > 

txl hvw xqh vìulh frqyhujhqwh gh irqfwlrqv eruìolhqqhv hw grqf xqh irqfwlrq eruìolhqqh hooh píph1 

0VrlwV 4 V 5 111 xqh vxlwh gìfurlvvdqwh g*hqvhpeohv gh 1 VlV @ .4 W 

V q /rqd= 

| +V, @ | 

# .4 

_ 

q@4 

q@4 

V q 

$ 

@ olp 

q$4 | +V q , > 

txl hvw od olplwh g*xqh vxlwh gh irqfwlrqv eruìolhqqhv hw grqf xqh irqfwlrq eruìolhqqh hooh píph1 

hvw grqf xqh fodvvh qrupdoh1 

rpph | +V { , @ I +{> |, hvw xqh irqfwlrq eruìolhqqh gh | srxu wrxw {/ frqwlhqw wrxv ohv V { /rq d grqf 

@ E q 1 

Ru rq prqwuh idflohphqw +yrlu o*duwlfoh gh Ureelqv Ûreelqv7;`, txh vl hvw xqh phvxuh vxu E q /txh | hvw xqh 

suredelolwì vxu E q hw txh +U p , @4/doruv +U q , @41 Rq hq gìgxlw grqf txh od irqfwlrq gìqlh hq ^417` hvw 

xqh suredelolwì vxu E q dvvrflìh ã od irqfwlrq gh uìsduwlwlrq K +{, gìqlh hq ^416`1 [ 

= Lghqwldelolwì gh od fodvvh gh phvxuhv surgxlwv1

Dsshqglfh E= Lghqwldelolwì gh od 

fodvvh gh phvxuhv surgxlwv1 

Vl o*rq qrwh/ hq vh uìiìudqw dx{ qrwdwlrqv gh od gìqlwlrq ghv fodvvhv gh pìodqjhv/ = 

x q>pq 

@ 

+ 

I {> h 

 

= I {> h 

@ 

q\ 

I {> l > rü I , 

{> l 5 x 4>p > 4 l q > 

l@4 

srxu h @ 4 >===> q 5 qp @ 

p p ~} 

p 

€ 

> 

q irlv 

rq gìqlw xqh fodvvhx q>pq sduwlfxolëuh/ vxu qp / grqw od glvwulexwlrq jìqìultxh hvw od phvxuh surgxlw I 

{> h 

1 

Hq idlw/ x q>pq hvw od idplooh ghv glvwulexwlrqv mrlqwhv ghv ìfkdqwloorqv gh q yduldeohv lqgìshqgdqwhv d|dqw fkdfxqhv 

xqh glvwulexwlrq gdqv x 4>p 1 Qrxv dyrqv doruv oh uìvxowdw vxlydqw = 

Wkìruëph = Vl od fodvvh gh wrxv ohv pìodqjhv gh x 4>p hvw lghqwldeoh/ doruv srxu wrxw qA4/ odfodvvhghv 

pìodqjhv gh x q>pq hvw dxvvl lghqwldeoh1 Uìflsurtxhphqw/ vl srxu dx prlqv xq qA4/ od fodvvh gh wrxv ohv pìodqjhv 

gh x q>pq hvw lghqwldeoh/ lo hq hvw gh píph srxu x 4>p 1 

Gìprqvwudwlrq = Od uìflsurtxh hvw ìylghqwh sxlvtx*lo vx!w/ srxu I 5 x 4>p / gh pxowlsolhu sdu q4 T 

I +{ l > 3 ,/ 

l@4 

3 5 p / ohv ghx{ phpeuhv gh 

] 

] 

I +{> , gJ +, @ I +{> , gJ e +, / ;{/ 

594 

srxu dyrlu L ! L ! J @ L ! L ! J/ e L! ìwdqw od irqfwlrq gh uìsduwlwlrq gh od suredelolwì qh fkdujhdqw txh oh srlqw 

3 /hwgrqfJ + 3 , @ J e + 3 ,/ srxu wrxw 3 5 p 

Prqwurqv txh vl x q>pq hvw lghqwldeoh/ doruv lo hq hvw gh píph gh x q.4>p+q.4, 

/ fh txl ìwdeolud od suhplëuh 

sduwlh gx wkìruëph 5181 Vxssrvrqv/ srxu I 5 x q>pq hw I 5 x 4>p /txho*rqd/srxuwrxw+{> |, 5 U q U q = 

] +G.4,F I {> h ] 

I +|>, gJ h> @ 

+G.4,F I {> h 

I +|> , gJ 

e 

h> 1 ^531449` 

Rq qrwh J 5 +, hw J e 5 +, ohv glwulexwlrqv pdujlqdohv/ vxlydqw / ghJ hw J e 

uhvshfwlyhphqw1 J h m hw J e 

h m 

 

vrqw ohv suredelolwìv whoohv txh srxu wrxw / J h m hw J e 

h m vrqw ghv irqfwlrqv gh uìsduwlwlrqv gh od yduldeoh 

 

h / hw srxu wrxw h / J h m hw J e 

h m vrqw suhvtxh sduwrxw ghv irqfwlrqv phvxudeohv gh 1 Doruv ^531449` ghylhqw 

rü 

] 

] 

F I +|>, K +{> , gJ 5 +, @ 

F I +|> , K e +{> , gJ e 5 +, / 

K +{> , 

@ 

] GF I {> h 

g e J h m 

^53144:` 

/ ^53144;` 

= Lghqwldelolwì gh od fodvvh gh phvxuhv surgxlwv1

595 

Rq shxw uììfuluh ^53144:` 

rü 

eK +{> , @ 

] GF I {> h 

g e J e h m 1 

] 

] 

F I +|> , gM { +, @ 

F I +|> , g M e { +, / 

^53144 }, gJ 5 +}, J 5 +, / ^531453` 

eK +{> }, g e J 5 +}, e J 5 +, 1 

Sdu od frqyhujhqfh grplqìh gh ^53144pq ìwdqw lghqwldeoh/ x 4>p 

hqjhqguh xqh fodvvh gh pìodqjhv lghqwldeohv1 rq d grqf / g*dsuìv ^53144 }, gJ 5 +}, @ 

] 

4 

4 

eK +{> }, g e J 5 +}, 1 

^531454` 

G*xq dxwuh f÷wì/ sdu sdvvdjh ã od olplwh lqqlh/ hq {/ gdqv ^53144;` sxlv gdqv ^53144:`/ rq d/ srxu wrxw |/ 

] 

] 

F I +|> , gJ 5 +, @ 

F I +|> , g J e 5 +, / 

fh txl lpsoltxh frpph fl0ghvvxv txh 

J 5 @ e J 5 1 

^531455` 

G*dsuëv ^531455` hw ^531454`/ rq d/ srxu suhvtxh wrxw / K +{> , @ e K +{> ,1 Juåfh ã ^53144;` hw ã o*lghqwldelolwì gh 

x q>pq/ rqd/doruv/srxusuhvtxhwrxw/ 

J + m, @ e J + m, / 

fh txl/ frpelqì ã ^531455`/ grqqh qdohphqw J +> , @ J e +> ,/ x q.4>p+q.4, 

hvw elhq lghqwldeoh1[ 

= Wkìruëph g*lghqwldelolwì gh Whlfkhu1

Dsshqglfh = Wkìruëph 

g*lghqwldelolwì gh Whlfkhu1 

Wkìruëph = Vrlw x @ I l +{, @ I {> l >{5 U q > l 5 p xqh idplooh gh irqfwlrq gh uìsduwlwlrq srxu 

ohvtxhoohv lo h{lvwh xqh wudqvirupdwlrq / gh grpdlqh gh gìqlwlrq V / whooh txh P = I :$ / vrlw olqìdluh hw 

elmhfwlyh1 Vxssrvrqv gh soxv tx*lo h{lvwh/ vxu x/ xq ruguh wrwdo/ qrwì Q/ who txh I 4 R I 5 lpsoltxh = 

V " V # >= 

l > 4 l nj hw x 5 @ iI 5>m > 4 m n 3 j / 

dlqvl txh ghx{ yhfwhxuv f 4 @ +f 4>l , 4ln 

hw f 5 @ +f 5>m , 4mn 

! whov txh = 

n[ 

n[ 

[ 

n ! 

;{ 5 U q > f 4>l I 4>l +{, @ f 5>m I 5>m +{, / dyhf 

[ 

n ! 

f 4>l @ 

l@4 

m@4 

l@4 

m@4 

f 5>m @4hw ;l> 4 l n> f 4>l A 3 hw ;m> 4 m n 3 >f 5>m A 3= 

^531456` 

Rq shxw/ txlwwh ã uhjurxshu/ gdqv fkdtxh phpeuh/ ohv ìoìphqwv lghqwltxhv gh x hw ã uìlqgh{hu od vrpph/ 

vxssrvhu txh ohv I 4>l hw I 5>m vrqw rugrqqìv gdqv xq ruguh vwulfwhphqw furlvvdqw/ f*hvw ã gluh txh I 4>l R I 4>m hw 

I 5>l R I 5>m vl lm1 Rq shxw dxvvl/ wrxmrxuv vdqv shuwh gh jìqìudolwì/ vxssrvhu txh I 4>4 Q I 5>4 1Gdqv fh fdv/ vl 

I 4>4 R I 5>4 /rqdV V ">" #>" 

hw grqf sdu wudqvirupdwlrq sdu P gh ^531456` rq d = 

f 4>4 . 

n[ 

l@5 

 

4>l +w, 

f 4>l @ 

4>4 +w, 

[ 

n ! 

m@4 

 

5>m +w, 

f 5>m / ;w 5 W 4>4 @ w 5 V ">"> 4>4 +w, 9@ 3 > ^531457` 

4>4 +w, 

hw sdu sdvvdjh ã od olplwh/ oruvtxh w $ w 4 / gdqv o*ìjdolwì ^531457`/ rq d f 4>4 @3fh txl frqwuhglw ohv k|srwkëvhv gh 

^531456`1 Rq d/ sdu frqvìtxhqw/ I 4>4 @ I 5>4 + dx vhqv gh od uhodwlrq g*ruguh Q ,/ fh txl hqwudñqh/ sdu elmhfwlylwì gh 

P/ 4>4 @ 5>4 hw sdu od olqìdulwì gh od wudqvirupdwlrq sdu P gh ^531456` rq d = 

+f 4>4 f 5>4 , . 

n[ 

l@5 

 

4>l +w, 

f 4>l @ 

4>4 +w, 

[ 

n ! 

m@5 

 

5>m +w, 

f 5>m / ;w 5 W 4>4 @ w 5 V ">" 

4>4 +w, 

> 4>4 +w, 9@ 3 = ^531458` 

Sdu sdvvdjh ã od olplwh/ oruvtxh w $ w 4 / gdqv o*ìjdolwì ^531458`/ rq d f 4>4 @ f 5>4 hw grqf ^531456` ghylhqw 

;{ 5 U q > 

n[ 

[ 

n ! 

f 4>l I 4>l +{, @ f 5>m I 5>m +{, 1 

l@5 

m@5

597 

Hq uìsìwdqw fh udlvrqqhphqw plq +n> n 3 , irlv/ rq vh udpëqh/ hq vxssrvdqw sdu h{hpsoh txh nAn 3 /ã 

;{ 5 U q > 

n[ 

l@n ! .4 

f 4>l I 4>l +{, @3= 

Rq prqwuh doruv/ frpph suìfìghpphqw/ txh f 4>l @3/ n 3 l n/ fh txl hvw lpsrvvleoh g*dsuìv ^531456`1 Rq d grqf 

n @ n 3 / fh txl dfkëyh gh prqwuhu o*lghqwldelolwì gh K 3 1[ 

= QV g*lghqwldelolwì gh \dnrzlw} hw Vsudjlqv1

Dsshqglfh G= QV g*lghqwldelolwì gh 

\dnrzlw} hw Vsudjlqv1 

Vrlw od idplooh x @ I l @ I {> l > l 5 p >{5 U q / g*xqh glvwulexwlrq sdudpìwultxh grqqìh1 

Wkìruëph = Od fodvvh gh wrxv ohv pìodqjhv qlv gh x hvw lghqwldeoh vl hw vhxohphqw vl x hvw xqh idplooh 

olqìdluhphqw lqgìshqgdqwh vxu U 47 1 

nS 

Gìprqvwudwlrq = Vl d l I l @3/ d l 5 U hw ohv I l glìuhqwv ghx{ ã ghx{/ rq shxw vxssrvhu/ txlwwh ã uìlqgh{hu 

l@4 

ohv d l / tx*lo h{lvwh P/ 3 P n/ whotxhd l 3 +, l P1 doruv= 

P[ 

n[ 

md l m I l @ md l m I l = 

l@4 

l@P.4 

^531459` 

Ohv I l ìwdqw ghv irqfwlrqv gh uìsduwlwlrq/ I l +{, $ 

PS 

md l m @ 

l@4 

n S 

l@P.4 

{$+.4>>.4, 

md l m @ fA31 Lo vx!w doruv gh srvhu d 3 l @ mdlm 

f 

hw o*rq d = 

P[ 

d 3 lI l @ 

l@4 

n[ 

l@P.4 

41 Hq sdvvdqw ã od olplwh gdqv ^531459`/ rq d grqf 

f*hvw ã gluh ghx{ uhsuìvhqwdwlrqv glìuhqwhv gx píph pìodqjh + ohv I l ìwdqw glìuhqwhv ghx{ ã ghx{ ,1 Lo hvw grqf 

qìfhvvdluh txh x vrlw olqìdluhphqw lqgìshqgdqwh1 Vl f*hvw oh fdv/ x hvw xqh edvh gh o*hvsdfh yhfwrulho kxl/ vxu 

U/ hqjhqguì sdu x1 rpph Kkxl/ o*xqlflwì gh od gìfrpsrvlwlrq g*xq ìoìphqw gh kxl vxu vd edvh x/ dvvxuh 

o*lghqwldelolwì gh K1 Od frqglwlrq hvw grqf vx!vdqwh1[ 

d 3 lI l / 

&S 

e x hvw xqh idplooh olqìdluhphqw lqgìshqgdqwh vxu -/ vl& M ( % M - (E@ c uuuc@ & M - & ( ' c 8 ' 8 c @ 8 E% 'fi 

E@ c uuuc@ & 'Efc uuuc f1 

'

59: 

Dsshqglfh H= Ohv prphqwv1 

H141 

Prphqwv hw prphqwv fhqwuìv1 

O*hvsìudqfh pdwkìpdwltxh g*xqh yduldeoh doìdwrluh/ [/ gìqlh vxu o*hvsdfh +> I>S,/ hvw gìqlh/ vl hooh h{lvwh/ 

sdu = 

] 

P @ H ^[` @ 

 

[ +$, S +g$, 1 

Vl/ o*rq qrwh od phvxuh gh suredelolwì dvvrflìh ã od yduldeoh doìdwrluh/ hw txh o*rq frqvlgëuh od irqfwlrq eruìolhqqh 

j gìqlh vxu U/ rq dxud = 

] 

H ^j +[,` @ 

U 

j +{, +g{, 1 

Vl/ gh soxv/ od yduldeoh d xqh ghqvlwì/ i/ o*hvsìudqfh gh j +{, hvw grqqìh sdu = 

] 

H ^j +[,` @ 

Vl/ hqq/ I hvw od irqfwlrq gh uìsduwlwlrq gh [/rq d = 

U 

] 

H ^j +[,` @ 

U 

j +{, i +{, g{1 

j +{, gI +{, 1 

Oh prphqw g*ruguh n 5 Q / vl lo h{lvwh/ g*xqh yduldeoh doìdwrluh [ hvw gìql sdu = 

P n @ H [ n ] 

@ 

hw oh prphqw fhqwuì g*ruguh nA4/ hvw/doruv/gìqlsdu= 

U 

] 

{ n +g{, @ 

U 

{ n gI +{, 1 

k 

Pn f @ H +[ H ^[`, nl ] 

] 

@ +{ P 4 , n +g{, @ +{ P 4 , n gI +{, 1 

U 

U 

Vljqdorqv txh/ sdu gìqlwlrq/ oh suhplhu prphqw fhqwuì hvw qxo/ hw txh ohv prphqwv fhqwuìv g*ruguh lpsdlu ghv 

glvwulexwlrqv v|pìwultxhv vrqw wrxv qxov1 

H151 rqglwlrqv vx!vdqwhv ã od fdudfwìulvdwlrq g*xqh 

glvwulexwlrq sdu vhv prphqwv1 

Vrxv fhuwdlqhv frqglwlrqv/ xqh glvwulexwlrq hvw gìqlh gh pdqlëuh xqlyrtxh sdu vhv prphqwv1 Oh wkìruëph 

vxlydqw/ grqw rq wurxyh od suhxyh gdqv Êloolqjvoh|9;`/ hvw od vroxwlrq od soxv frqqxh dx Sureoìph ghv prphqwv gh 

Vwlhomhv1 

Wkìruëph = Vrlwxqhphvxuhghsuredelolwì// gìqlh vxu oh píph hvsdfh txh suìfìghpphqw hw d|dqw ghv 

prphqwv/ P n /qlvãwrxvohvruguhv1 

=Ohvprphqwv1

59; 

Vl od vìulh hqwlëuh V = 

V +{, @ [ n4 

P n { n 

d xq ud|rq gh frqyhujhqfh srvlwli/ doruv hvw od vhxoh phvxuh gh suredelolwì d|dqw ohv prphqwv P n / n 41 

Vl xqh phvxuh gh suredelolwì/ / dvvrflìh ã xqh glvwulexwlrq I / yìulh ohv frqfoxvlrqv gh fh wkìruëph/ od glvwul0 

exwlrq hvw gìwhuplqìh sdu vhv prphqwv1 

Glìuhqwhv ìwxghv shuphwwhqw gh frpsoìwhu fh wkìruëph1 Wrxw g*derug/ o*duwlfoh gh Gkdupdgklndul ^Gkdupdgklndul98` 

prqwuh txh fhwwh frqglwlrq q*hvw sdv qìfhvvdluh1 O*duwlfoh gh udphu hw Zrog/ ûdphu69`/ grqqh xqh frqglwlrq 

vx!vdqwh ã od uìvroxwlrq gx sureoëph ghv prphqwv gh Vwlhomhv/ gdqv oh fdv ghv yduldeohv doìdwrluhv gh glphqvlrq 

51 Hqq/ gdqv vrq olyuh/ Udr/ Ûdr:6`/ grqqh xqh vìulh gh frqglwlrqv vx!vdqwhv hq irqfwlrq gx udqj gh od yduldeoh 

doìdwrluh ìwxglìh1 

H161 rqglwlrqv qìfhvvdluhv hw vx!vdqwhv ã o*h{lvwhqfh 

g*xqh glvwulexwlrq fruuhvsrqgdqw ã xqh vxlwh gh prphqwv1 

Gdqv ohv ghx{ duwlfohv ^Kdpexujhu53` hw ^Kdpexujhu54`/ Kdpexujhu gìyhorssh xqh QV dx sureoëph ghv 

prphqwv gh Vwlhomhv/ gdqv oh fdv sduwlfxolhu ghv phvxuhv srvlwlyhv furlvvdqwhv d|dqw xq qrpeuh lqql gh srlqwv gh 

furlvvdqfh1 hfl v*dssoltxh grqf/ dx{ irqfwlrqv gh uìsduwlwlrqv ghv yduldeohv doìdwrluhv fkdujhdqw xq qrpeuh qrq 

ql gh srlqwv1 Dydqw gh gìyhorsshu fh wkìruëph/ frpphqêrqv sdu gìqlu/ srxu xqh vxlwh gh uìhov +v n , n5Q 

/odvxlwh 

ghv gìwhuplqdqwv/ n 5 Q = 

G n +vn , n5Q @ Gn @ 

 

n$ 

 

4 v 4 ====== v n 

v 4 v 5 ====== v n.4 

= 

=== ====== ====== ====== 

v n v n.4 ====== v 5n 

Gìilqlwlrq = xqh vxlwh gh uìhov +v n , n5Q 

hvw gìqlh srvlwlyh vl/ srxu wrxw n 5 Q/ oh gìwhuplqdqw G n hvw srvlwli1 

Iruwv gh fhv suìflvlrqv/ qrxv srxyrqv ìqrqfhu oh wkìruëph vxlydqw = 

Wkìruëph = Xqh frqglwlrq qìfhvvdluh hw vx!vdqwh srxu tx*lo h{lvwh xqh irqfwlrq gh uìsduwlwlrq/ I / gìqlh vxu 

U/ d|dqw xq qrpeuh lqql gh srlqwv gh furlvvdqfh hw whooh txh = 

hw txh od vxlwh +P n , n5Q 

vrlw gìqlh srvlwlyh1 

] 

P n @ 

U 

{ n gI +{, / n 5 Q> 

Oh wkìruëph gh Ulhv}/ txl vxlw/ jìqìudolvh oh wkìruëph gh Kdpexujhu1 Lo xwlolvh od qrwlrq gh vxlwh qrq qìjdwlyh 

grqw od gìqlwlrq hvw grqqìh fl0ghvvrxv1 Lo shuphw gh vwdwxhu vxu wrxv ohv w|shv gh glvwulexwlrqv hw gh udlvrqqhu vxu 

xqh vxlwh qlh gh prphqwv1 

= Ohv prphqwv1

59< 

Gìilqlwlrq = xqh vxlwh +v n , 4nq 

/rüq .4/ hvw qrq qìjdwlyh uhodwlyhphqw ã o*lqwhuydooh `d> e^/ rüd 4 

hw e .4 vl srxu wrxw sro|q÷ph U/ ghghjuì q/ 

who txh = 

rq d = 

U +{, @ d 3 . d 4 { . ======d q { q >{5 `d> e^ > 

;{ 5 `d> e^ / U +{, 3> 

d 3 . d 4 v 4 . ======d q v q 3= 

Oh wkìruëph gh Ulhv}/ srxu ohv glvwulexwlrqv v*ìqrqfh doruv frpph vxlw = 

Wkìruëph = Xqh QV srxu tx*lo h{lvwh xqh irqfwlrq gh uìsduwlwlrq/ I/ srxu odtxhooh = 

] 

P n @ { n gI +{, / 4 n q .4 

hw txh od vxlwh +P n , 4nq 

vrlw qrq qìjdwlyh/ q ìwdqw/ elhq vþu/ vxssrvì íwuh sdlu1 

Yrlu dxvvl ^Dnklh}hu98`/ ^Dnklh}hu;;` hw Ôdqgdx;:`1 

U 

H171 

xpxodqwv1 

Ohv fxpxodqwv g*xqh glvwulexwlrq vrqw gìulyìv ghv prphqwv hw qrxv shuphwwhqw gh uhqguh fhuwdlqhv pdqlsxodwlrqv 

dojìeultxhv soxv fodluhv1 

Gìilqlwlrq = Vl xqh glvwulexwlrq I srvvëghghvprphqwvqlvãwrxvohvruguhv/rqgìqlodvxlwh+n u , u5Q 

ghv 

fxpxodqwv sdu = 

h{s 

# 4 

[ 

n u hvw oh fxpxodqw g*ruguh u gh od glvwulexwlrq I 1 

u@4 

$ 

n u w u 

@ 

u$ 

4[ P u w u 

> 

u$ 

Shx xwlolvdeohv vrxv fhwwh iruph/ ohv fxpxodqwv yìulhqw od sursrvlwlrq vxlydqwh = 

Sursrvlwlrq = srxu wrxw +u> m, 5 Q Q /rqd= 

u@4 

CP u 

Cn u 

@ u 

m 

 

Pum = 

=Ohvprphqwv1

5:3 

Juåfh ã fhwwh sursulìwì/ rq d dx udqj m @4= 

CP u 

Cn 4 

@ uP u4 > 

fh txl shuphw gh wluhu/ sdu uìfxuhqfh = 

P 4 @ n 4 > 

P 5 @ n 5 . n4> 

5 

P 6 @ n 6 .6n 4 n 5 . n4> 

6 

P 7 @ n 7 .7n 4 n 6 .6n5 5 .9n4n 5 5 . n4> 

7 

hw dlqvl gh vxlwh 11111 

Srxu ohv prphqwv fhqwuìv rq rewlhqw ghv uhodwlrqv vlplodluhv = 

3 @ n 4 > 

P5 f @ n 5 > 

P6 f @ n 6 > 

P7 f @ n 7 .6n5> 

5 

P8 f @ n 8 .43n 6 n 5 > 

P9 f @ n 9 .48n 7 n 5 .43n6 5 .48n 6 n5> 

5 


fh txl shuphw g*h{sulphu ohv fxpxodqwv hq irqfwlrq ghv prphqwv fhqwuìv = 

n 4 @ 3> 

n 5 @ P5> 

f 

n 6 @ P6 f > 

n 7 @ P7 f 6 +P5, f 5 > 

n 8 @ P8 f 43P6P f 5> 

f 

n 9 @ P9 f 48P7P f 5 f 43 +P6, f 5 .63+P5, f 6 > 


Rq srxuud wurxyhu ghv gìyhorsshphqwv vxssoìphqwdluhv vxu ohv fxpxodqwv gdqv oh olyuh gh Nhqgdoo hw Vwxduw 

^Nhqgdoo;:`/ rx gdqv ^Kdehupdq

Dsshqglfh I= Ohv prphqwv 

hpslultxhv1 

I141 

Gìqlwlrq ghv prphqwv hpslultxhv1 

Vrlw o*ìfkdqwloorq [ 4 > ===> [ q gh od glvwulexwlrq I 1 Vl o*rq vxssrvh txh srxu wrxw l/ 4 l q/ [ l 5 U/ frqvlgìurqv 

od gurlwh uìhooh pxqlh gh od wulex/ E/ gh vhv eruëolhqv hw gìqlvvrqv vxu +U> E, od phvxuh gh suredelolwì S q = 

;E 5E/ S q +E, @ 4 q 

q[ 

L +[ l >E, dyhf L +[> E, @ 

l@4 

4/vl[ 5 E 

3/ vl[ @5 E = 

Hq idlw/ S q hvw xqh suredelolwì frqfhqwuìh hq ohv [ l hw whooh txh od suredelolwì gh fkdfxq g*hx{ hvw ìjdoh ã 4 q 1 S q hvw 

dsshoìh phvxuh hpslultxh dvvrflìh ã o*ìfkdqwloorq [ 4 > ===> [ q 1 Ohv prphqwv hpslultxhv +uhvs1 ohv prphqwv fhqwuìv 

hpslultxhv, g*xq ìfkdqwloorq lvvx gh od glvwulexwlrq I / vrqw ohv prphqwv sdu udssruw ã od phvxuh gh suredelolwì 

hpslultxh dvvrflìh ã fhw ìfkdqwloorq1 Oh suhplhu prphqw hpslultxh/ [/ hvw od pr|hqqh gh o*ìfkdqwloorq1 Oh prphqw 

hpslultxh g*ruguh n hw oh prphqw fhqwuì hpslultxh g*ruguh n vrqw grqqìv/ uhshfwlyhphqw/ sdu = 

[ n @ 4 q 

hw [ f n @ 4 q 

q[ 

[ l n / 

l@4 

q[ 

+[ l [ 4 , n 1 

l@4 

I151 rqyhujhqfh ghv prphqwv hpslultxhv yhuv ohv 

prphqwv wkìrultxhv1 

Rq vh grqqh xq ìfkdqwloorq/ [ 4 / 111/[ q / lvvx g*xqh glvwulexwlrq I 3 dssduwhqdqw ã xqh idplooh I gh glvwulexwlrqv1 

Vxssrvrqv txh J hvw xqh irqfwlrqqhooh gh od iruph = 

] 

J +I , @ k 

 

j +{, gI +{, > 

rü j hvw xqh irqfwlrq eruìolhqqh hw k xqh irqfwlrq frqwlqxh dx srlqw d @ U j +{, gI 3 +{,1 Qrxv srxyrqv/ doruv/ 

gìqlu od irqfwlrq j q = 

j q @ 4 q 

q[ 

j +[ l , = 

Rq qrpphud/ sdu dloohxuv/ vwdwlvwltxh gh w|sh L / xqh vwdwlvwltxh Vq gh w|sh = 

rü I q hvw od irqfwlrq gh uìsduwlwlrq dvvrflìh ã S q 1 

l@4 

V q @ J +I q , @ k +j q , >

5:5 

Wkìruëph = Vrxv ohv k|rwkëvhv suìfìghqwhv = 

rx hqfruh = 

J +I q , $ 

s=v= 

J +I 3 , / txdqg q $4> 

V q @ k +j q , $ 

s=v= 

k +d, /txdqgq $4= 

Hq sduwlfxolhu/ ;n 5 Q/ 

hw 

[ n @ 4 q 

q[ 

l@4 

[ l n $ H [ n 

s=v= 

4 / txdqg q $4 

[ f n @ 4 q 

q[ 

l@4 

k 

+[ l [ 4 , n $ H +[ 4 P 4 , nl /txdqgq $4> 

s=v= 

f*hvw ã gluh txh ohv prphqwv hpslultxhv fhqwuìv hw qrq fhqwuìv frqyhujhqw suhvtxh vþuhphqw yhuv ohv prphqwv 

wkìrultxhv/ uhvshfwlyhphqw fhqwuìv hw qrq fhqwuìv1 

Vl o*rq vxssrvh/ gh soxv/ txh k hvw gìulydeoh dx srlqw d/ hwtxh 

] 

j 5 +{, gI 3 +{, 4/ 

rq shxw ìwdeolu oh wkìruëph vxlydqw = 

Wkìruëph = Vrxv ohv k|srwkëvhv suìfìghpphqw ìqrqfìhv/ 

s q +V 

 

q k +d,, @, k 3 +d, 3> #> 

rü 3> # hvw xqh glvwulexwlrq qrupdoh fhqwuìh hw gh yduldqfh = 

] 

5 @ 

+j +{, d, 5 gI 3 +{, = 

Hq sduwlfxolhu/ srxu wrxw n 5 Q/ 

s q +[n P n , @, 3> #/rü 5 @ P 5n P 5 n 

hw 

s q +[ 

f 

n P f n, @, 3> # B 

/rü 5 f @ P f 5n +P f n, 5 = 

Wrxv fhv gìyhorsshphqwv shxyhqw íwuh uhwurxyìv gdqv Êrurynry;:`1 

= Od pìwkrgh ghv prphqwv1

Dsshqglfh J= Od pìwkrgh ghv 

prphqwv1 

J141 Hvwlpdwlrq sdu pìwkrgh gh vxevwlwxwlrq1 

rqyhujhqfh1 

Rq vh grqqh xq ìfkdqwloorq/ [ 4 / 111/[ q / lvvx g*xqh glvwulexwlrq S dssduwhqdqw ã xqh idplooh S gh glvwulexwlrqv1 

Rq vxssrvh txh oh sdudpëwuh / lqfrqqx hw fdudfwìulvdqw S vh uhsuìvhqwh sdu xqh irqfwlrqqhooh J gh od glvwulexwlrq 

S / f*hvw ã gluh = 

@ J +S , = 

Rq qrpphud/ doruv/ hvwlpdwhxu gh vxvevwlwxwlrq gh / o*hvwlpdwhxu q who txh = 

q @ J +S q , = 

Udsshorqv/ sdu dloohxuv/ tx*xq hvwlpdwhxu q hvw glw frqyhujhqw yhuv v*lo frqyhujh hq suredelolwì/ hw iruwhphqw 

frqyhujhqw v*lo frqyhujh suhvtxh vþuhphqw1 

Srxu ohv hvwlpdwhxuv gh vxevwlwxwlrq/ rq shxw ìwdeolu oh wkìruëph vxlydqw = 

Wkìruëph = Vrlw o*ìfkdqwloorq +[ q , q4 

lvvx gh od glvwulexwlrq I 31 Vxssrvrqv txh @ J +S ,/ rüJ hvw xqh 

irqfwlrqqhooh gh od iruph = 

] 

J +S , @ k 

 

j +{, gI +{, > 

rü j hvw xqh irqfwlrq eruìolhqqh hw k xqh irqfwlrq frqwlqxh dx srlqw d @ U j +{, gI 3 +{,1 

Doruv/ q @ J +S q, hvw xq hvwlpdwhxu iruwhphqw frqyhujhqw1 

J151 Qrupdolwì dv|pswrwltxh ghv hvwlpdwhxuv gh 

vxevwlwxwlrq1 

Udsshorqv tx*xq hvwlpdwhxu q @ q>4> ===> q>n 

gh @ +4 > ===> n , hvw glw dv|pswrwltxhphqw qrupdo gh pdwulfh 

5 3/ vl od glvwulexwlrq olplwh gh s q + q , hvw xqh glvwulexwlrq qrupdoh +pxowlydulìh, fhqwuìh gh pdwulfh 5 1 

Srxu ohv hvwlpdwhxuv gh vxevwlwxwlrq/ rq shxw ìwdeolu oh wkìruëph vxlydqw = 


lvvx gh od glvwulexwlrq I 31 ìwdqw gìql sdu xqh irqfwlrqqhooh gx w|sh 

gx wkìruëph suìfìghqw/ rü j @ +j 4 > ===> j v , hw k +w, @ +k 4 +w, > ===> k n +w,,/ +rüw @ +w 4 > ===> w v ,, dyhf ohv k|srwkëvhv 

vxssoìphqwdluhv txh k dgphw hq d @ +d 4 > ===> d v , +rü d m @ U j m +{, gI 3 +{,, ohv gìulyìv sduwlhoohv = 

k l>m @ Ck l 

Cw m 

+d, / 4 l n/ 4 m v/

5:7 

hw txh 

] 

j l +{, j m +{, gI 3 +{, 4/ 4 l v/ l m v1 

Doruv/ q hvw xq hvwlpdwhxu dv|pswrwltxhphqw qrupdo gh pdwulfh = 

5 @ Kg 5 K W > 

rü g 5 @ ^g l>m` hvw xqh pdwulfh fduuìh v v/ v|pìwultxh/ dyhf 

hw K @ ^k l>m`4ln 1 

4mv 

g l>m @ H ^+j l +[ 4 , d l , +j m +[ 4 , d m ,` > 

Uhpdutxrqv txh 5 hvw xqh pdwulfh nn/ doruvtxhg 5 hvw xqh pdwulfh vv1 Yrlu Êrurynry;:` rx ^Kdehupdq ===> j n ,/ hw qrwrqv 

] 

p m +, @ j m +{, S +g{, @ H ^j m +[ 4 ,` > srxu 4 m n= 

krlvlvvrqv/ gh soxv/ od irqfwlrq j gh whooh vruwh txh o*ìtxdwlrq = 

p +, @ w/ dyhfw @ +w 4 > ===> w n , > 

dgphwwh xqh xqltxh vroxwlrq frqwlqxh @ p 4 +w,/ gdqv o*hqvhpeoh ghv ydohxuv gh 1 Qrxv srxyrqv/ doruv/ gìqlu 

od irqfwlrq j q = 

j q @ 

# 

4 

q 

q[ 

j 4 +[ l , > ===> 4 q 

l@4 

$ 

q[ 

j n +[ l , = 

G*xqh idêrq jìqìudoh o*hvwlpdwhxu q @ p 4 +j q , hvw dsshoì hvwlpdwhxu gh sdu od pìwkrgh ghv prphqwv1 Oh fdv 

oh soxv vrxyhqw xwlolvì hvw fhoxl rü j n +{, @ { n hw ohv ìtxdwlrqv vrqw ohv ìtxdwlrqv ruglqdluhv ghv prphqwv1 Ohv 

ìtxdwlrqv ghv prphqwv fhqwuìv/ fruuhvsrqgdqw dx fdv rü j n +{, @ +{ P, n / hqwuhqw dxvvl gdqv oh fdguh gh fhwwh 

pìwkrgh1 Qrxv qrwhurqv uhvshfwlyhphqw q hw f q fhv ghx{ ghuqlhuv w|shv g*hvwlpdwhxuv1 

Lo dssdudñw fodluhphqw txh ohv hvwlpdwhxuv ghv prphqwv vrqw ghv hvwlpdwhxuv gh vxevwlwxwlrq hw tx*lo vrqw/ 

ãfhwlwuh/iruwhphqw frqyhujhqwv1 Od qrupdolwì dv|pswrwltxh grlw íwuh ìwxglìh dx fdv sdu fdv hq yìuldqw ohv 

k|srwkëvhv gx wkìruëph suìfìghqw rx/ gdqv oh fdv oh soxv vlpsoh/ fhoohv gx wkìruëph vxlydqw = 

l@4 

= Od pìwkrgh ghv prphqwv1

5:8 


lvvx gh od glvwulexwlrq I whooh txh 

P l +, @ H 

 

[ 

l 

4 

 

4> 4 l 5n= 

Vl o*rq vxssrvh gh soxv txh p dgphw hq d @ ^P 4 +, > ===> P v +,` ohv gìulyìv sduwlhoohv = 

k l>m @ Cp l 

C m 

+d, / 4 l> m n1 

hw txh K @ ^k l>m`4l>mn 

hvw gh udqj n1 

Doruv/ q hvw xq hvwlpdwhxu dv|pswrwltxhphqw qrupdo gh pdwulfh = 

rü @^ l>m`4l>mn 

/dyhf 

5 @ K 4 

K 

W 4 

> 

l>m @ P l.m P l P m = 

= Ohv prphqwv gh pìodqjhv1

5:: 

Dsshqglfh K= Ohv prphqwv gh 

pìodqjhv1 

K141 Htxdwlrqv ghv prphqwv ghv pìodqjhv gh ghx{ 

glvwulexwlrqv1 

Vl o*rq frqvlgìuh/ srxu od foduwì gx sursrv/ ghx{ glvwulexwlrqv I 4 hw I 5 / pìodqjìhv hq sursruwlrqv uhvshfwlyhv/ 

s hw4 s1 Vl[ 4 vxlw xqh glvwulexwlrq I 4 hw [ 5 xqh glvwulexwlrq I 5 / rq dxud/ srxu wrxw q 4 = 

sH ^[ q 4 ` . +4 s, H ^[ q 5 ` @ P l{ 

q / 

rü Pq 

l{ hvw oh q lhph prphqw gh od glvwulexwlrq gh pìodqjh1 hwwh ìtxdwlrq hvw od q lhph ìtxdwlrq gh prphqwv g*xq 

pìodqjh gh ghx{ glvwulexwlrqv1 Od q lhph ìtxdwlrq ghv prphqwv fhqwuìv g*xq pìodqjh gh ghx{ glvwulexwlrqv hvw/ 

vrxv xqh iruph euxwh 

sH 

k [4 

P4 

l{ q 

l 

k [5 

. +4 s, H P4 

l{ q 

l 

@ Pq l{ / 

rü Pq 

l{ hvw oh q lhph prphqw fhqwuì gh od glvwulexwlrq gh pìodqjh1 

Vl o*rq srvh = { 4 @ H ^[ 4` P4 l{ hw { 5 @ H ^[ 5` P4 l{ fhv ìtxdwlrqv ghylhqqhqw = 

Od suhplëuh ìtxdwlrq hvw/ doruv = 

hw srxu qA4 = 

sH ^+{ 4 . +[ 4 H +[ 4 ,,, q` . +4 s, H ^+{ 5 . +[ 5 H +[ 5 ,,, q` @ P l{ 

q = ^53145:` 

s{ 4 . +4 s, { 5 @3> ^53145;` 

s{ q 4 . +4 s, { q 5 @ P l{ 

q 

 

q[ lq 

 

sH [ 

l 

4 { 

ql 

4 . +4 s, H [ 5 

l 

{ 

ql 

5 = 

l@5 

^53145 5 4 

hw Q 

 

5 > 5 5 

/ hq sursruwlrqv uhvshfwlyhv s hw 

+4 s,/ oh qrpeuh ghv sdudpëwuhv ã hvwlphu ìwdqw 8/ qrxv whqwhurqv g*dssoltxhu od pìwkrgh ghv prphqwv sdu oh 

= Ohv prphqwv gh pìodqjhv1

5:; 

v|vwëph = 

; 

A 

A= 

3@s{ 4 . +4 s, { 5 

P l{ 

P l{ 

P l{ 

5 @ s{ 5 4 . +4 s, { 5 5 . +s 4 . +4 s, 5 , 

6 @ s{ 6 4 . +4 s, { 6 5 .6+s 4 { 4 . +4 s, 5 { 5 , 

7 @ s{ 7 4 . +4 s, { 7 5 .9 

s 4 { 5 4 . +4 s, 5 { 5 5 .6 s 

5 

4 . +4 s, 5 5 

P8 l{ @ s{ 8 4 . +4 s, { 8 5 .43 

s 4 { 6 4 . +4 s, 5 { 6 5 .48 s 

5 

4 { 4 . +4 s, 5 5{ 5 

= ^531463` 

Qrxv yhuurqv fhshqgdqw tx*lo qrxv idxgud dyrlu uhfrxuv ã xqh vl{lëph ìtxdwlrq gh prphqwv/ txl hvw = 

P l{ 

9 @ s 48 6 4.78 5 4 { 5 4.48 4 { 7 4.{ 9 4 

. +4 s, 

 

485 { 7 5.78 5 5 { 5 5.48 6 5. { 9 5 

= 

K161 rqyhujhqfh ghv hvwlpdwhxuv ghv prphqwv ghv 

pìodqjhv1 

G*dsuëv fh txh qrxv dyrqv yx gdqv o*Dsshqglfh J/ ohv hvwlpdwhxuv ghv prphqwv ghv pìodqjhv vrqw iruwhphqw 

frqyhujhqwv/ od qrupdolwì dv|pswrwltxh ìwdqw ã yìulhu juåfh dx wkìruëph fl0ghvvxv1 

= Oh Wkìruëph ghv irqfwlrqv lpsolflwhv1

Dsshqglfh L= Oh Wkìruëph ghv 

irqfwlrqv lpsolflwhv1 

L141 

Pdwulfh mdfrelhqqh1 

Vrlw xqh irqfwlrq i @ +i 4 > ===> i n ,/rüi = 

H U t $ I U n 

w @ +w 4 >==>w t , $ +i 4 +w, > ===> i n +w,, = 

Vl i hvw xqh irqfwlrq glìuhqwldeoh/ f*hvw ã gluh vl fkdfxqh ghv frpsrvdqwhv gh i hvw gìulydeoh sdu udssruw ã fkdfxqh 

gh vhv yduldeohv/ od pdwulfh mdfrelhqqh gh i/ hvw od pdwulfh = 

Cil 

Mdf+i, @ 

4ln 

= 

Cw m 4mt 

VlohvglphqvlrqvghH hw I vrqw ohv píphv/ oh mdfrelhq gh i hvw oh gìwhuplqdqw gh Mdf+i,1 

L151 

Wkìruëph g*lqyhuvlrq orfdoh ghv irqfwlrqv1 

Oh wkìruëph ghv irqfwlrqv lpsolflwhv grqqh xqh frqglwlrq ã o*lqyhuvlrq g*xq v|vwëph/ tx*lo vrlw olqìdluh rx qrq1 

Wkìruëph = Vrlhqw H>I>J wurlv hvsdfhv gh Edqdfk1 Vrlw/ hqfruh/ xqh irqfwlrq i/ frqwlqxhphqw glìuhqwldeoh 

g*xq vrxv hqvhpeoh rxyhuw D gh H I gdqv J1 Vrlw/ hqq/ xq ìoìphqw +{ 3 >| 3 , 5 D who txh i +{ 3 >| 3 , @31 Vl oh 

mdfrelhq gh i +{ 3 >=, qh v*dqqxoh sdv hq | 3 / doruv lo h{lvwh xq yrlvlqdjh rxyhuw X 3 gh { 3 lqfoxv gdqv H/ whotxhsrxu 

wrxw vrxv hqvhpeoh X/ rxyhuw hw frqyh{h/ gh X 3 lo h{lvwh xqh xqltxh irqfwlrq frqwlqxh x gh X gdqv J/ whoohtxh= 

x +{ 3 , @ | 3 hw i +{ 3 >x+{ 3 ,, @3= 

Rq srxuud frqvxowhu ^Glhxgrqqì:3` srxu gdydqwdjh gh suìflvlrqv1

Dsshqglfh M= Od pìwkrgh gh Qhzwrq 

gh uìvroxwlrq ghv v|vwëphv dojìeultxhv 

qrq olqìdluhv1 

Doruv txh o*rq glvsrvh gh pìwkrghv jìqìudohv gh uìvroxwlrq ghv v|vwëphv dojìeultxhv olqìdluhv/ o*lqyhuvlrq ghv 

v|vwëphv dojìeultxhv qrq0olqìdluhv qh shxw/ oh soxv vrxyhqw/ sdv íwuh uìdolvìh olwwìudohphqw1 Od vhxoh srvvlelolwì hvw/ 

doruv/ gh vh wrxuqhu yhuv xqh pìwkrgh qxpìultxh hw lwìudwlyh gh uìvroxwlrq1 Sdupl fhv pìwkrghv/ ohv soxv h!fdfhv 

vrqw fhoohv gìulydqw gh od pìwkrgh gh Qhzwrq/ grqw qrxv suìvhqwrqv lfl oh sulqflsh +yrlu sdu h{hpsoh Êdwhv;;` hw 

^Nuhvv dyhf +{ { 3 , % +{, $ 

{${ ! 

31 

Vl xq srlqw { 4 hvw vx!vdphqw surfkh gh { 3 rq shxw qìjoljhu oh ghuqlhu whuph gh o*ìjdolwì fl0ghvvxv hw frqvlgìuhu 

txh = 

5;4 

i +{ 4 , @3vh udpëqh ã { 4 @ { 3 i 3W +{ 3 , 4 

i +{3 , = 

Od pìwkrgh lwìudwlyh gh uìvroxwlrq gh Qhzwrq frqvlvwh ã frqvwuxluh od vxlwh { q gìqlh sdu = 

{ q.4 @ { q î 3 +{ q ,`4 i +{ q , ãsduwlug*xq{ 3 5 G> 

txl/ vrxv fhuwdlqhv frqglwlrqv frqyhujhud yhuv xq }ìur gh qrwuh v|vwëph1 Oh wkìruëph vxlydqw/ suìflvh ghv frqglwlrqv 

vx!vdqwhv gh frqyhujhqfh gh od pìwkrgh gh Qhzwrq1 

Wkìruëph = Vrxv ohv k|srwkëvhv ìqrqfìhv fl0ghvvxv/ G ìwdqw xq rxyhuw/ vl i hvw ghx{ irlv frqwlqxhphqw 

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