1229579

Sur quelques extensions des chaînes de Markov cachées
et couples. Applications à la segmentation
non-supervisée de signaux radar.
Nicolas Brunel
To cite this version:
Nicolas Brunel. Sur quelques extensions des chaînes de Markov cachées et couples. Applications à la
segmentation non-supervisée de signaux radar.. Mathématiques [math]. Université Pierre et Marie
Curie - Paris VI, 2005. Français. �tel-00011302�
HAL Id: tel-00011302
https://tel.archives-ouvertes.fr/tel-00011302
Submitted on 5 Jan 2006
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THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6
Spécialité : MATHÉMATIQUES APPLIQUÉES
présentée par
Nicolas Brunel
en vue de l'obtention du titre de docteur de l'université Paris 6
Sujet :
Sur quelques extensions des chaînes de Markov cachées et couples.
Applications à la segmentation non-supervisée de signaux radar.
soutenue le 5 décembre 2005
devant le jury composé de :
Rapporteurs
Directeurs de thèse
Examinateurs
Invité, co-encadrant
Eric MOULINES
ENST
Bernard PRUM
Université d'Evry
Paul DEHEUVELS
Université Paris 6
Wojciech PIECZYNSKI
INT
Alain HILLION
ENST Bretagne
Daniel PIERRE-LOTI-VIAUD
Université Paris 6
Frédéric BARBARESCO
THALES Air Defence
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Q ]qŠZuWmSX\^]hkNe l¶¬ Œ1b1o>\^UXŒ5yvSVe k“¬ WP]akNd$o>e$]ak5Zu]>’e ]5\<aS“Zuoao>e$lGlPe†qn\^]aS+’e$S“Y”\NkNy$lGevS Xj\NkNy†lPe$S+kNeTjDZ^qXz<\[{ƒ€‘Z^’ŠŒ5dvS
µ‰¨‘qX5WGUž]akNd$o>e$]ak5Z^]<U @jDj€‘Z[§ž¨‘ ŸM5¦AZ”k5e†]aSXW UndkN`o5qn\1’†e$SnSV>S‘’†\^5oalGe
e$SVU
X
1
p(x) =
XS
(Ys )s∈S
∀s ∈ S, p(ys |x ) = p(ys |xs )
Q
c∈C
ψc (xc )
XS
ZS = (Xs , Ys )s∈S
p(z) = p(x)
=
Y
Y
s∈S
p(ys |x )
ψec (zc )
“¦ evSh‡ˆ\^]>’ŸUXWP\^]>S SX\^]-Ud†t-ZulPe$Sfµ SXW ]“¬ e$SVU£oaZ<Sh5]/SXWG]5t<lGe†UX\<] SXWG]5\<] ]5\^aS£Zc{^\^]>S
M&³Z^q’†\^]aSXd$Á-5e$]-U lGeho5qn\N’evSXSXaS#)r\^WP]<U e$SVUse†]a’†\^qne¤5]`’ŠŒaZuY‰o`kNehjDZuqnz^\[{‰qne†lmZ[UXWP{^e†§
Y‰e†]-UZu!Y"$Y‰egSXbNSrUny†Y‰egk5eƒ{<\^WmSVWP]aZut<eƒÁ-5eh’e$lG5W•kNe MN¦“ehSVa’$’yvS‘kaZu]aS‘lPe$SZuo5oalGWm’†ZuUXWP\^]aS‘kNe$S‘Y‰\NkNy†lPe$S
jDj€‘Zu§©¨‘hqne†o&\<SXeSV5q#lGeŽ‡8ZuWGU”Á-5el¶¬ Zo5qnWG\<qXWFY‰Z^qXz<\[{1WGe$]evSrU V’†\^] )r5t<5dZu oaqX\N’evSXSXaSk“¬ \<_aSVe$qX{[ZuUXWP\^]
•lPZpkNe†]>SVWGUXd‰ZŽo&\<SVUXe$qXWP\^qnW SVe#‡8Z^’UX\^qnWmSVe#qXe$lPZuUXWP{^e†Y‰e$]<UfZuY "†Y‰e”SXbNSr§
UXy$Y‰e‰k5e#{<\^WmSVWP]aZut<e$S Á-5e MA€Fe$’†W:o>e$qXY‰eUkNe‰’\<]5]aZ[PUnqXe#lGevSgSXo&d$’WG„>’$Z[UXWP\^]>SƒlP\N’†Z^lGevSƒkNe
eU
kN\<]a’k•¬ Zuo5oalGWmÁ-5e†q”lGevSfUne$’ŠŒ5]5WmÁ-5e$S#kNe!SXWPY£5lmZ[UnWG\<]/kNeUrb1o&e!j€‘j€ ]5\^UnZuY‰Y‰e†]-U£l¶¬ d$’ŠŒaZ^]<UnWGlPlP\^]5]5e$5q#k5e
gWP_5_aS$M^žl5evSrU o&\<SnSVWP_5lPekNeSXWGY#5lGe$q lPe$S ’ŠŒaZ^Y‰oaS³Zho>\-SrUne†qnWG\<qXW eUFoaZuqXUnZ^]<U k•¬ e$SVUXWPY”e$q lPe$S³o5qn\^_>Zu_5WPlGWGUXdvS:Y%ZuqX§
t^WP]aZ^lGevSFZ#o>\-SrUne†qnWG\<qXW o&\^aq‘lPeh’†Zulm’al3k5pj &Fj \^Žk“¬ e†¢1oalG\<qXe$qFlmZ£oaqX\<_aZu_5WPlPW UndgZ£o>\-SrUne†qnWG\<qXW ]a\uUnZ^Y‰Y”e$]-U
o&\^5qflmZ`kNdUne†qnY”WP]aZuUXWP\^] kNe%SV\<]Y%Z[¢NWPY£5Y PO†­Ni Ov¡^ĶM“‚ƒe%o5lPaS lGevShY”d†UXŒ5\NkNevSfj€‘j€ o>e$5{^e$]<U "†UXqne
NUnWGlPWPSXd†evS”o>\<5q‰[email protected]¬ e$SVUXWPY%Z[UXWP\^] kNej\NkNy†lPe$S‰jDZ^qXz<\[{1WGe$]aS”€‘Z<’ŠŒ5d$S”SV\<W U‰o&\^5q‰5]5e`evSrUnWGY%Z[UnWG\<]Ho5lPe†WP]5e$Y”e$]-U
_aZcb<d$SXWGe$]5]5e SX\^WGUso>\<5qlPegY%Zu¢1WPY£aYkNeh{1qnZ^WPSXe†Y#_5lmZu]a’†e xiuÄ ©M
¦“\<qnSnÁ-5e evSrU5]ae’ŠŒaZc ]5ekNejDZ^qXz<\[{ ]a\^aSh\<_NUXe$]5\^]aSha]5e’ŠŒaZc ]5ekNej Zuqnz^\[{’†Z<’ŠŒ5d†e%µ`_5qn5WGU
WP]akNd†o&e†]>k5Zu]-U ¶€‘jw€‘Z[§ž¨F eUl¶¬ ZulPt^\^qnWGUXŒ5Y‰eHVZc{[Zu]-UX§©Z^qXqnWGy$qXeL ˆ\^ \¹ T¼%7 ,n¼<º T¼% 7 {^\^WPqSXe$’ŸUnWG\<]`L5M L5M L o&e†qnY”e†U‰k5ep’$Zulm’5lPe†q”qnZ^o5WmkNe†Y‰e†]-U‰eU‰e¢5Z^’UXe$Y”e$]-U#lmZ o5qn\^_>Zu_5WPlGWGUXdpZ o&\<SVUXe$qXWP\^qnW ’†\^]-UXqŠZuWPqXe$Y‰e†]-U
Zu5¢p’ŠŒaZ^Y”o>SY%Zuqnz^\[{1WPe†]aS‘t^d$]5d†qŠZu5¢3M5‚ƒe£oalGaS WPl+evSrUƒo>\-SXSXWG_alGekNe#’†Zulm’alGe$qe¢5Z^’UXe$Y”e$]-UlGevSo5qn\^_aZ^_5WPlGWGUXdvS
Y%Zuqnt^WP]aZulPe$S
ZuWP]aSXW>Á-5e
MN€Fe$lPZh‡ˆ\<5qn]5W U‘kN\^]a’¤kNe$S‘ZulPt^\<qXWGUXŒ5Y‰evS³kNeƒ’$Zulm’5l>qnZ^o5WPk5e
kNevS:e$SVUXWPY‰ZuUXe$5qnSB_aZcb<d$SXWGe$]aSB’lmZ^SnSVWmÁ-5e$S 8k5Z^]aS:lPe’$Z^SB\ e$SVUB„>]5W M \<aS:qXevkNd†Y‰\<]<UnqX\<]aS k5Z^]aSBlPZhSXe$’UXWP\^]
SX5WG{[Z^]<Une ’†e$Sqnd$’aqnSXWG\<]aSk5Zu]>SlGe’$Z^kNqneho5lG>St^d†]ad†qŠZul•kNe$S¤’ŠŒaZc ]5e$SkNej Zuqnz^\[{%’\<5o5lPe ¶€‘j€F\ ŸM
c∈C
ψec (z)
ψc (x)
ψ{s} (xs )p(ys |xs )
p(y |x ) =
Q
s∈S
ψe{s} (z) =
c
ZS
XS
p(ys |xs )
V
p(x |y )
p(x)
p(x |y )
XS
p(x |y )
p(xn |y )
LE4K:361C86
*
arg maxx p(x |y )
'
X
!
MK6 2/?A#MK? @MK6:68A#?F&36 GH67L& (BE @: ? 6 :'&)A#4K786 2!2'&): 8?F& 8:36 [email protected] LE4K2)4K7A 6 2
L^L
"
H
\<aS qŠZ^SnSXe†Y£_alG\<]aS³k5Zu]>ST’e†UVUneƒSXe$’ŸUnWG\<]%lGevS o5qn\^oaqXWPdUnd$S³‡ˆ\<]ak5Z^Y”e$]-UnZulPe$STkNe$SFY‰\1k5y†lPe$ST’†\^5oalGevSTÁ-5el¶¬ \<]
o&e†NUUnqX\<5{^e$q SX\^>SkNe$S‡ˆ\^qnY‰e$SsoaZ^qV‡ˆ\<WPSsk5W 3d†qne†]-UXevSk5Z^]aSlPe$SZ^qVUnWP’†lGevS PO$5O OvLu­ OvL^« ¡^ OcL^¡ e†U¤]5\^>S
e†]k5\^]5]5\<]aS³lPe$S k5d†Y‰\^]aSVUXqŠZ[UnWG\<]aS$M \<aS³Y‰eUXUX\^]>S e†]%o>ZuqXUXWm’5lPWGe$q³e†]%d${-WmkNe$]a’elPeslPWGe$]%e†]-UXqnesY%Zuqnz^\[{1WPZ^]5WGUXd
kNo5qX\N’†e$SnSVaSƒ’†Z<’ŠŒ5deUhlP\^W:’\^]>kNW UnWG\<]5]5e$lGlPe#kNevS¤\<_aSXe†qn{cZuUXWP\^]aS¤kNe#Y%Z^]5WGy$qXe£o5lG>SƒoaqXdv’WmSVe£Á-5e”k5Zu]>S POvLu­!¶M
\<aShkN\^]a]5\^]aShZ^aSXSXWBlGevSfkNd†Y‰\^]>SrUnqnZuUXWP\^]aS ˆ]5\<] e†¢1WmSVUnZu]-Une$Shk5Zu]>SƒlPe$ShZ^qVUnWP’†lGevS¤o5qnd$’†d$kNe$Y‰Y”e$]-Uh’†W Und$S ƒk5e
l¶ ¬ Z^lGt<\^qnW UnŒ5Y‰egkNef²ƒW Une†qn_5W“o&\^5qslPe$Ss’ŠŒ>Zc ]5e$Ss’†\^5o5lPe$SsZ^WG]>SVW“Á-5ekNehlmZ‰o5qn\^o5qnWGd†UXdfLNMxLNMPO^M
" # " #%$
F"
J “¦ \<qnSnÁ-5elPe$Sh’ŠŒaZ^Y”o>SƒZ^lGdvZ[UX\<WGqne$Sƒ\^_>SVe$qX{<d eU’$Z^’ŠŒ5d SV\<]-UƒUne†lmSgÁ-5e”lPe#o5qn\N’evSXSXaS)r\^WP]<U
SV\<W U”5] ’ŠŒ>ZuY‰oHkNepjDZuqnz^\[{ ]5\<aS”kNWGqn\^]>S£Á-5e
‡ˆ\^qnY‰e†]-U#5] ’ŠŒaZuY‰oHk5e!jDZ^qXz<\[{
’\<5o5lPe^M5‚gZu]aSlPef’†Z<kNqXehkNehlmZ£Y‰\NkNd†lPWmSXZuUXWP\^] SXW•]5\^>ST‡8Z^WPSX\^]>SFa]5WPÁ-5e$Y‰e†]-U[email protected]¬ Œ1b1o>\^UXŒ5yvSVehkNehY%Zuqnz^\[{1WPZ^]5WGUXd
kNe ]5\^aSsk5WGqn\^]aSsÁ-5eh]5\<aSsZc{^\<]aS‘5]`Y‰\1k5y†lPehk5ejDZuqnz^\[{%’\<5o5lPe^M
hÌ[7 "F7>?v6ˆ2v687&0 ' m' 8 /RS'HTWU!HV'H&Q_ '?O !H WT !HTW W L'HNOQ`!H#O" ‰ºŠ»5¼ 7 %¼ <%¹ ;G¼[¸¶3¾ >¸J[ %¼ G¹Š½ Ÿ½Ÿ¸ + ºŠ»5¼ 7 %¼ <%¹ ·3¹u¾ ¸
+
;G¼[¸¶3¾ >¸ [ ºŠ¹ 7[¾ ¸¶¾8¹ ;3 a¸ [ 0
B %¹ a½ŠZ¸ X¼u¸@¾8¹ 0
YS
ZS =
XS
XS , YS
(Xs , Ys )s∈S
ZS
ZS = (Xs , Ys )s∈S
V
YS
V
XS
p(xs xS\s , y )
=
p(xs , ys xS\s , yS\s )
p(ys xS\s , yS\s )
∝ p(xs , ys |xVs , yVs )
∝ p(xs |xVs , ys , yVs )
Q ] qne†{[Z^]a’ŠŒ5e ]“¬ evSrUpo5lPaSŽ]ad$’evSXSnZuWPqne†Y‰e†]-UlP5W §©Y "†Y‰e Y%Z^qXz<\[{-WPe†] ˆ]a\^]N§ž’\<]akNWGUXWP\^]5]5e$lGlPe†Y‰e$]<Upµ
oaZuqŠ’eÁ-“¬ WGlƒ]“¬ e$SVUŽÁ<aelGeDoaqX\N’evSXSXaS‰Y%Zuqnt^WP]aZ^lk“¬ 5] oaqX\N’evSXSXaS‰Y%Zuqnz^\[{1WPe†]“M³‚ƒeDo5lPaS e†] Un\^NUne
t^d$]5d†qŠZulPWGUXdg]5\^aSsZc{<\^]aS
M
\<aSh\<kNadS”„>]5o>WP\<Sn5SX\^{^]a\<]aSgS”lPe$’Sh\^j]>Sr\NUnqXkN5y†WPlPqne$e!SfkNj evS”Zuqn’Šz^Œ>\[Zu{1Y‰WGe$oa]aS‰Sg€‘’\<Z<5’Šo5Œ5lPd$e$S S#8e†j ] jZ •€‘ZuZ WP _5lGWmSnSXZ^]<U”lGevS”’Œ-\^b1Y‰o&Y‰\uUne‰Œ5y$dSXUŠe$ZuS‰]-UhkNevk5S”e$Sgj o5jqn\1€‘’†e$Z[Sn§ž¨‘SV>ŸS M
o& \^5q‘lGevSXÁ-5e$lGlPe$S evSrU‘5]!’ŠŒ>ZuY‰o!kNegjDZuqnz^\[{”qne†lmZ[UXWP{^e$Y‰e†]-UFµ#5]!SXbNSrUny†Y‰eƒkNeg{^\<WPSXWP]aZut<e M1¦“evS‘\^_aSXe†qX§
{[Z[UnWG\<]aS:]5esSV\<]-U:o>Z^SA‡ˆ\^qŠ’d$Y”e$]-U:WP]akNd$o>e$]ak5Z^]<Une$S³’†\^]akNWGUXWP\^]a]5e†lPlGe$Y‰e†]-U µ eU³5]aest^d†]ad†qŠZulPWPSnZ[UnWG\<]£SVWPY‰o5lPe
kNeglPZ#lP\^W•’\<]akNWGUXWP\^]5]5e$lGlPehkNe e$SVUkNegSX5o5o&\<SXe†qsÁ-5eh’u¬ e$SVUZ^aSnSVW35]p’ŠŒaZuY‰oŽY%Z^qXz<\[{-WPe†]%qne†lmZ[UnWG{<e†Y‰e†]-Uµ
’†\^]akNWGUXWP\^]a]5e†lPlGe$Y‰e†]-Uƒµ 8kN\<]-UlPZ%kNe$]aSXW Und£S†¬ d$’†qXWGU¤SX\^aS‡ˆ\<qXY‰eh‡8Z<’ŸUX\<qXWmSXd†e
ŸM&‚ƒZ^]aS’†e
’†Z<Sr§©lmµ ]5\^>SsZc{^\^]>S XS
YS
p(ys |x ) 6= p(ys |xs )
(XS , YS )
V
XS
XS
V
YS
XS
Q
p(x, y)
1W’ŠŒaZ<Á-5eŽ‡ˆ\<]a’ŸUnWG\<]
ψc,x (yc )
= p(y|x)p(x)
Y
Y
=
ψc,x (yc )
ψc (xc )
]5ekNd$o>e$]ak kNe Á-“¬ µDUnqnZc{<e†qŠSlPe$S%’†\-\<qnk5\^]5]5d$e$S”Zuo5o>ZuqXUXe†]>Zu]-U”µ ]5\^aS
c∈C
ψc,x (y)
c∈C
x
c∈C
c
" " ZuaqX\<]aSZulP\^qŠS
L^
H ∝
p(z)
Y
ψc (zc )
cZ {^ev’
M>€F\^Y‰Y‰eglmZ#lG\<W3k5e SXe¤‡8Z<’ŸUX\<qXWmSXegSXe†lP\^]!lGefSXb1SVUXy$Y‰egkNef’†lGWmÁ-5e$S ’^¬ evSrU
kN\<]a’g5]’ŠŒaZuY‰o`kNejDZuqnz^\[{‰qne†lmZ[UnWG{<e†Y‰e†]-Uµ M
]ae‘Y‰d†UXŒ5\NkNet^d$]5d†qŠZulPeFk5e’\^]>SrUnqXa’UXWP\^]”kNesjDj€‘Zƒo5qX\<o>\-SVd$e‘kaZu]aS POv¡ 8t^d†]ad†qŠZulPWPSnZu]-U:5]5eY‰dUXŒa\1k5e
o5qn\^o&\<SXd†e¤kaZu]aS «< F’\<]aSXWPSVUXe¤µ#’\<]aSVWmkNd$qXe$q 5]Ž’ŠŒaZ^Y‰oY%Zuqnz^\[{1WGe$] qne†lmZ[UXWP{^e$Y‰e†]-UFµ a]e†]aSXe†Y#_5lPe
kNe‘’lPWPÁ-5evS
‡ˆ\<qXY%Zu]-UB5]5eTo>ZuqXUXWGUXWP\^]£kNe eUB5]5eF‡ˆ\^]a’UXWP\^] M 1W1lPZs‡ˆ\^]a’UXWP\^] SXeFkNdv’\<Y”o&\<SXeFSXe§
lP\^]”lGevSB’†lGWmÁ-5e$S³SX\^>SAlmZƒ‡ˆ\^qnY‰e
ZulP\^qŠSBlGes’ŠŒaZ^Y”o
e$SVUFa]ŽjDj€‘ZaM ]ŽY”\[b<e†]%o5lPaS‘kNWPqXev’ŸUTevSrUZulP\^qŠS kNeƒSX5o5o&\<SXe†q‘Á-5e e$SVUF5]5eƒY‰e$SX5qXeƒkNe gWP_5_aS‘SVe$lG\<]
5] SXbNSrUny†Y‰ekNe#{^\^WmSXWG]aZ^t^evSsZ^k5Z^oNUXd<M>‚gZu]aSƒ’e£’$Z^SV§¶lmµ ]a\^aS¤]“¬ Zc{<\^]aSoalGaSƒlmZ‰Y‰Z^qXz<\[{1WPZ^]5W UndfkNe Y%ZuWmS
]5\<aSo&\^5{<\^]aS¤’ŠŒ5\^WmSVWPq UXe$lBÁ-“¬ WPl+qne†o5qnd$SXe†]-UXef_aWGe$]lPZ%qnd$Z^lGWGUXd<M>‚gZu]aS GOv5O O†­! lmZ%kNe$]aSVWGUXd kN
’\<5o5lPeZ‰dUnd’ŠŒ5\^WmSVWPekNehUne†lPlGe£SX\^qXUXefÁ-5eflmZ‰lG\<W+’\<]akNWGUXWP\^]5]ae†lPlGeSX\^WGU5]D’ŠŒaZ^Y”opt<Z^aSnSVWPe†] \<qlGevS’ŠŒaZ^Y”o>S
t<Z^aSnSVWPe†]aS ˆe$]#o>ZuqXUXWm’5lPWGe$q:lGevSAY‰\NkNy$lGevSB’†\^]ak5W UnWG\<]5]5e†lmS:ZuNUn\u§©qXd$t^qne$SnSVWG‡8S 3qXevSrUnW Un5e†]-U³_5WPe†]‰’e$qVUŠZuWP]5e$S+UXe†¢-Un5qne
kNevSWGY%Zut<e$S {^\<WGq # WP]5z1lGe$q GOc¡uÄ©M
ψc (zc ) = ψc,x (yc )ψc (xc )
c∈C
C
ZS
V
V
WS
(ci )1≤i≤Q
S
f (x, w) = (fi (xci , wci ))1≤i≤Q
f
f
(XS , YS = f (XS , WS ))
p(z)
XS
p(y|x)
" #" #"
p(z)
# \<aSFUXqŠZuWGUX\<]aSk5Zu]aSs’†eUXUXehoaZuqXUXWPefkNe$S¤€‘jw€F\”SVUnZuUXWP\^]5]aZ^WGqne$S‘e†Usk5eglPe†aqnSo5qn\^oaqXWPdUnd$SFe$SnSVe$]<UnWGe$lGlPe$S ]5\^§
UnZ^Y‰Y”e$]-UBlPe$SBo5qn\N’d$k55qXevS+kNes’$Zulm’5l1kNe$SBo5qn\^_>Zu_5WPlGWGUXdvSAZ¤o>\-SrUne†qnWG\<qXW¶Mc¦+Z¤SVUnZuUXWP\^]5]>ZuqnW UndT]“¬ e$SVUBo>Z^SB]5d$’†e$SnSXZ^WGqne
o&\^5q d†UnZ^_5lGWPqT’†e$S kNe$qX]aWGy$qXevS Y%ZuWmS³e$lGlPeo&e†qnY”e†UTkNe¤SXWGY‰o5lPWG„ae†q lmZho5qnd$SXe†]-UnZuUXWP\^]‰kN%Y‰\NkNy†lPeseUFk5esY”e†UVUnqXee$]
d†{1WmkNe†]>’elPe$SFkNW 3d†qne†]>’e$STY%Z )re†aqXevS Zc{^ev’lGevS‘€‘j€‘ZaM \<aSFNUXWPlGWmSX\^]aSF’e$STo5qn\^o5qnWGd†UXdvS³o>\<5qTY‰e†UVUXqnee$]d†{1WG§
kNe$]a’elPe$STlGWPe†]>STe$]<UnqXesY%Z^qXz<\[{-WmZu]aW Undk5o5qn\1’†e$SnSV>S WP]akNd$o>e$]ak5Zu]>’eƒ’†\^]ak5W UnWG\<]5]5e†lPlPe¤kNe$ST\^_aSXe†qn{[Z[UnWG\<]aS
eUsY%Zuqnz^\[{1WmZu]5WGUXdƒkN!oaqX\N’evSXSXaS‘’\<5o5lPe^Ma€Fe$S‘o5qn\^o5qnWPdUXdvS‘\^]-U‘d†UXdho5qnd$SXe†]-UXd$e$S‘k5Zu]>SFlPef’†Z<ST\ !l¶¬ evSVo>Z^’egkNe$S
dUŠZ[UŠS e$SVUT„a]5W3k5Z^]aSTlPe$S‘ZuqXUXWm’lPe$S GOcL[­ ¡u! e†lPlGevSTqne$SVUXe$]<U‘d†t-ZulPe†Y‰e†]-UT{[ZulmZu_alGevSTlP\^qŠSXÁ-5e e$SVUF5]!e$SXoaZ^’†e
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T
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∀φ, G(U, φ) = Eφ [T(V) |U ] − φ
G
T̃φ
T̃φ
g
G
G(u, φ) = 0
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(Zs )s∈L
L
(Zs )s∈L
/
X
pL (φ) =
s∈L
log p(zs |(zt )t6=s , φ)
L
"
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X
s∈L
log p(Zs |(Zt )t6=s , φ)
#
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X
s∈L
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##
E [∇φ log p(Zs |(Zt )t∈Vs ∩Lc , φ) |(Zt )t∈Vs ∩Lc ]
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∇φ L(φ)
sL (φ) = ∇φ pL (φ)
L
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L(φ)
=
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∇φ L(φ)
X
s∈S
log p(zs |(zt )t6=s , φ)
L
h
i
e |φ |y
h(φ1 , φ2 ) = Eφ1 ∇φ L(φ)
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Hv
Hu
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H∈Hu
Hu
g
Hv
Hu
Hu
u
u
G(u, φ)
H (φ |φ0 , u )
φ0
i
h
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H (φn+1 |φn , u ) = 0
(φn )n≥0
G
φ̃
H φ̃ φ̃, u = G(u, φ̃)
g
Hu
G(u, φ)
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G(φ) = H(φ, φ, u)
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θ = ( 1b , a − 1)
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ϕ(θ) = −(θ2 + 1) log(θ1 ) + log(Γ(θ2 + 1))
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θ = κξ
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k
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k,n
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d†t-ZulPe!µ e†U%k5e$S”lG\<WPS#t<Z^Y‰Y‰Z WG]1{<e†qŠSVepkNe!Y‰\[b<e†]5]5e M 1W
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(
E [V ]
1
(a−1)b
=
E [log V ] = −ψ(a) − log(b)
b
b=
1
a
a
a−1
1
ν
U ∼ γ( ν2 , ν2 )
Tm,Σ,ν
f (y, (m, Σ, ν)) =
−1/2
Γ( ν+d
2 )|Σ|
d/2
(πν) Γ( ν2 )
1+
2q(y)
ν
Y
− ν+d
2
ν>2
VT,Σ,ν (Z) =
lmZ#Y‰\[b<e†]5]5ehd†UnZu]-U M
1W
Z^lG\<qnS Z”5]5eflP\^W“kNehUrb1o&e ]5\^UXd$e
m
U ∼ iγ(a,
1
a)
ν
×Σ
ν−2
Z
Km,Σ,a
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o&\^5qUn\^NU Z^WG]aSXW“Á-5efSXZ”{[ZuqnWPZ^]a’ehÁ-5W“{[Zu5U 2aa |Σ|−1/2
f (y, (m, Σ, a)) =
(2π)d/2 Γ(a)
r
q(y)
a
!a− d2
p
Ka− d 2 aq(y)
2
Kα
a
VK,Σ,a (Y ) = Σ
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Z^SXb-Y‰oNUn\uUnWPÁ-5eh’lmZ^Sn SVWmÁ-5efkNeflmZ£‡ˆ\<]a’ŸUnWG\<] ˆ{^\<WGq x¡ ˆ­aMPO$« o>e$qXY‰eU£k5e%Y”\<]-UXqne†qfÁ-5e%lGevSÁ-5e†ae$SkNdv’qn\^WmSXSXe†]-UgUX\< )r\^5qŠSfµ`5]5e%{1WGUXevSXSXe‰e¢No>\<]5e†]-UnWGe$lGlPe ˆl¶¬ dvÁ-5W §
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ν
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m
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Kα
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a− d+1
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√
akyk
r
π −x
e
2x
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" " J F +
f (r) =
I
u
σ2
0
γ(1, σ 2 )
I
R +∞
r2
exp − 2σ
2
r
σ2
−u σy2
exp
Ka,σ2
0
0
I = U −1/2 I
I
fI 0 (y) =
g
g(u)du
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M
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º > + ¾"v¾ >¸ + ½Ÿ¸¶¾ . ; G ¼ X¹ ^¾ %¼[¸¶¾8¹ 7 /G¼YG¹u¾ 7 ½f¾ a¸ a½Š¾ ¸$B½ a%¼ + /G¹uJ¾ :<¼ %W¼ 0
7 B1. a¾ g½ a¹ G¼ %¹ 7 B- ¾m½¼u¸@¾8¹ 87 ½
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¸ž¼[¸¶¾8¹ 7 \¹ + ¾33 ¾m½ G¼ ŠºŠ¹ &¼u¾m½n½†¼ &º 7 Žº†¾1,- Ÿ½ †¸ + ½ :B B-n¼% a¸MG¼= nºŠ¹ &¼u¾m½Š½¼ &º Y7 \¹ ;½ 0
@A¼ 7 5½Ÿ¾ ¸6B
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B;X¼ Ÿ½ºŠ¹
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√ ay
2aa y (a−1)/2
Ka−1 2
1
fI (y, (a, σ )) = 2
2
σ Γ(a) aσ
σ
2
2
Cy
a−1
e
√
ay
−2 σ
#" #
K0,σ2 ,a
"
fI 0
K0,σ2 ,a
(y)
K0,σ2 I2 ,a
+
fI (y) ∼
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kNevS!o>ZuqŠZuY‰yUnqXevSo&\<SXW UnWG\<]N§¶dv’ŠŒ5e†lPlPe
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M \^>S!]a\^aS
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e$SVU’\<]5]15e lmZ kNW 3d†qne†]-UnWPZuUXWP\^] kNe`lPZwlP\^tu§©{1qnZ^WPSXe†Y#_5lmZu]a’†e 8k5Z^]aS‰lPe`’$Z^S%k•¬ 5] d$’ŠŒ>Zu]-UXWPlGlP\^] [email protected] Wmk•M
skN\<]5]5ehlPeSVbNSVUXy†Y‰efSX5WG{[Z^]<U E(m, Σ, ϑ)
(m, Σ)
(m, Σ, ϑ)
h
y =
(y1 , . . . , yN )



 m=


 Σ=
ˆ­aMxLu¯ PN
i=1 w(ri )yi
P
N
i=1 w(ri )
1
N
PN
i=1
w(ri )(yi − m)(yi − m)
0
cZ {^ev’
eU
M
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qn\^_5aSVUXevSXSXe 8e†]DUne†qnY‰e$SgkNe”SXe†]>SVWP_5WPlGWGUXd”µŽkNevSƒ{[Z^lGe$5qnSƒZu_&e†qnqnZ^]-UXe$S }5O SXW:lPZ‡ˆ\^]a’UXWP\^] kNd$’†qX\uPUhZ^SnSVe {1W Une^M3Q ]e 3eU lmZ%‡ˆ\<]a’ŸUnWG\<]
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ri = Σ−1/2 (yi − m)
2
0
(r)
w(r) = − hh(r)
w
r 7→ w(r)
w(r) =
ψ(r)
r
zi
[i Ä
#
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kNd†„a]5WPefoaZuq w
m
Σ
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s,
|s| ≤ a
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e$SVUWGY‰o&\<SnSVWP_5lPe Œ5\<qXY‰WmSo>\<5q¤Á-5e†lmÁ-5e$S
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d†t-ZulPeµ M‚gZu]>S!lPe ’\^]-Une¢1UXeqnZ<k5Zuq a]5eY‰d†UXŒ5\NkNewW Und†qŠZ[UnWG{<eoaZ^qVUnWP’†5lPWGy$qXe$Y”e$]-U!WP]<Und†qne$SnSXZ^]-UXe Z dUnd
kNd${^e†lP\^oao>d$e”o&\^aqflGevS 1 “² k5e”Y‰\[b<e†]5]5e‰]15lPlGe Zuoao>e$lGd$e Ve$SVUXWPY%Z[Une†5qkN o&\^WP]-Uf„5¢Ne 8{^\<WGq i^i O<O$« ŸM
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lmZ^Á-5e†lPlPeg WPl“evSrUo&\<SnSVWP_5lPekNe’\<]aSVUXqn5WG qneg5]5eSX5W UnekNefY%Z[UnqXWm’ef’\<]1{^e†qnt^evZu]-UF{<e†qŠSs’efo&\^WP]-Us„5¢Ne^M ƒN UnqXeSnZ
‡8Z^’†WGlPWGUXd£k•¬ WPY”oalGd$Y”e$]-UnZ[UnWG\<] ’†eUgWG]ak5d†o&e†]ak5Z^]a’e#kNelmZ‰‡ˆ\^qnY‰efoaZuqŠZuY‰dUnqXWmÁ-5ekNe
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g
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lPe$Ss’†Z<SskNef lmZ#lP\^W eU M
N
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Eφn [∇ϑ log(g(Ui , ϑ))|y] = 0
i=1
a(ϑ) =
m Σ
(k)
(k)
a
ϑ
ϑ
Ui
Yi
U
∇ϑ log (g(U, ϑ))
U
p(u |y ) ∝ ud/2 e−up(y) g(u)
Q
U
p(u |y ) ∝ u
d
2 +a−1
1
exp −u(q(y) + )
b
i[«
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ν
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wi
ν
2
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2 , 2q(yi )+ν )
Ui
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
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=
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Eφ [log(Ui ) |y ] = ψ( d+ν
)
2 ) − log(
2
8­aMxLu« N
(n)
(n)
ν +d
ν
ν
1 X
ν +d
(n)
(n)
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2
2
N i=1
2
2
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ϕ(x) = ψ(x)−log(x)
ν
ϕ̃(n) = 1+ N1
PN
i=1
(n)
(n)
log(wi )−wi
(n)
+ψ(
νi
+d
2
ν
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2
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N
PN
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=
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1
N
PN
i=1
yi − m(0)
0
ν (0) = 100
q (n) (yi ) = 12 (yi − m(n) )0 Σ(n)−1 (yi − m(n) )
(n)
(n)
wi = 2q(n)d+ν
(yi )+ν (n)
n
(yi )1≤i≤N
Σ(n+1)
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ϕ(n)
yi − m(0)
ν
m(n+1)
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wi
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d
u( 2 −a)−1 exp −(uq(y) + ua )
p(u|y) = q
p
a− d
2K
2( q(y)
a− d (2 aq(y))
a )
2
U
Q
M
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
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√
K d −a+1 (2 aq(y))
q 2
√
q(y)
aq(y))
a K d −a (2
=
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2
Eφ [ U1 |y]
q
=

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q(y)
a K d −a−1 (2
2
K d −a (2
2
q
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√
aq(y))
aq(y))
q(y)
a , a)
M
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= H(
H(x, a) = − log(x) + ∂α log Kα (2ax)|α= d −a
2
a
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ϕ(a) = ϕ(n)
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N
PN
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i=1
(n)
wi
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r
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yi − m(0)
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2
(n)
q
(yi )
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(n)
a
2
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√
n
1
N
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0
0
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#
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#
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"
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N
ν1 = ν2 = ν3 = 10
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K2
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T1
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p(y1 |x1 )
ρ
"
ρ1
r
r
0
ρ2
#
Skl
Skl
2d × 2d
(ρi )i=1,2
0
ρ 2 − r ρ1 r > 0
E(mk , Σk , ϑ)
(Y1 , Y2 )
(X1 = k, X2 = l)
p(y1 |x1 )
E(mkl , Σkl , ϑ)
fx11 (y11 ), . . . , fxd1 (y1d )
p(y1 |x1 )
Fx11 (y11 ),. . . , Fxd1 (y1d )
(Y1 , Y2 )
(X1 , X2 )
F (y11 , . . . , y1d , y21 , . . . , y2d |x1 , x2 ) = Cx1 ,x2 (Fx11 (y11 ), . . . , Fxd1 (y1d ), Fx12 (y21 ), . . . , Fxd2 (y2d ))
Cx1 ,x2
C[1..d],x1 ,x2
= C x1
C[d+1..2d],x1 ,x2
= C x2
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Cx1
Cx2
p(y1 |x1 )
p(y2 |x2 )
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p(y1 |x1 )
P
2d
∀u ∈ [0, 1]2d , C(u) = ϕ−1
i=1 ϕθ (ui )
θ
P
P
d
2d
C[1..d] (u1 , . . . , ud ) = ϕ−1
C[d+1..2d] (ud+1 , . . . , u2d ) = ϕ−1
i=1 ϕθ (ui )
i=d+1 ϕθ (ui )
θ
θ
(Y1 , Y2 )
A ⊂ [1..M ]
ρ1
ρ2
ρ
C
C CA
ρA = (ρij )i,j∈A
"
#
ρ1 r
M = 2d
ρ=
0
r ρ2
C[1..d]
C[d+1..2d]
M
C
ρ
(d × d)
(U1 , . . . , UM )
g = (g1 , . . . , gM )
(V1 , . . . , VM ) = (g1 (U1 ), . . . , gM (UM ))
(Vi )i∈A
g = (Φ−1 , . . . , Φ−1 )
Φ
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k
d
ρk
ρkl =
(ρk )1≤k≤K
"
ρk
rkl
0
rkl
ρl
(rkl )1≤k,l≤K
2d
#
rkl
ρkl
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Y
s
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n≥1
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Yn
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(n )n≥2
Yn+1
= f (Y n , Xn , n+1 )
X
n+1
(Y1 , . . . , Yn )
f
X
X


 Xn+1
∀k, Ak ∈
d
Bk ∈
d


Yn+1
∼ q(xn , ·)
= AXn Yn + BXn + n+1
Z = (X, Y)
Yn
Y
Xn
X
Xn+1
Xn
Xn+1
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k•¬ e†qnqXe$5q£e$]HY‰\NkNeŽ]a\^]N§žSVao>e$qX{1WmSVd$e^MA‚gZu]>S#’†ep’†\^]-UXe†¢-Une WGlZuoaoaZuqŠZcmU#’lmZuWPqne†Y‰e†]-U”Á-[email protected]¬ e$SVUXWPY%Z[UnWG\<]HkNe$S
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A=
γ(1; 0, 5)
(X1 = k, X2 = l)
ρ
k 6= l
N = 1000
ρ > 0, 5
"
0, 8 0, 2
0, 2 0, 8
γ(2; 1)
#
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(τn )1≤n≤9
H (ρn )1≤n≤9
(ρ̃n )1≤n≤9
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− :
∗−∗ :
−− :
ρ > 0, 5
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ρ = 0, 4
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m̂
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∀k, l,
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kl
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i=1
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pi (k, l)∇θkl f (yi , θkl ) = 0
(n)
i=1
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(n+1)
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pd (y |1, 1)
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P =
pd (y |2, 1)
0, 4
0, 15
0, 15
0, 4
pd (y |2, 2)
γ(3; 3) pd (y |1, 2 )
γ(2; 2, 5)
W (1, 2)
#
N = 500
ρ11 = ρ22 = 0, 1
ρ12 = ρ21 = 0, 5
ρ11 = ρ22 = 0, 5
ρ12 = ρ21 = 0, 8
Yn
γ
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|Θ|
p(y1 |x1 , x2 )
ηkl
|Υ|
2
K × (K − 1) + K × |Θ| + K × |Υ|
p(y1 |x1 )
|Φ
2
|Φ
|=
| = K × (K − 1) + K × |Θ|
|Φ
| = (K − 1) + K 2 × |Θ| + K 2 × |Υ|
5
|Υ| = 1
θkl = θk
|Φ
ρkl = 0
| = 30 |Φ
| = 55
|Φ
| = 99
k 6= l
K × (K − 1) + K × (|Θ| + |Υ|)
|Θ| = 2
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sr (t) = As(t − ∆T )e2iπ∆f t + b(t)
y(t) =
R
sr (u)s?e (u − t)du
f0
Kt2
τ τ
, ], se (t) = S0 e2iπ(f0 t+ 2 )
2 2
∀t ∈] −
k
Damb =
vk
cTr
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vamb =
λ
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λ
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2π
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+
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λ
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λ
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
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 H0 : sr (t) = b(t)

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Pd
H1 : sr (t) = As(t − ∆T )e2iπ∆f t + b(t)
H1
Pf a
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∀n, Yn =
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(ak )1≤k≤p
Pp
P (z) = 1 − k=1 ak z −k
(n )n∈
(R(n) = E [X0 Xn? ])n∈
S(f ) =
P
n∈
R(n)e2iπnf
I = − 21 , 12
S(f ) =
(a1 . . . ap )
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Y
ρn =
σb2
k=1
ak e−2iπkf |
2
0
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(µk )1≤k≤p
σb2
(µk )1≤k≤p
(y1 , . . . , yni )
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k
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R(0)
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1 σb2
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µn
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| |z| ≤ 1}
(ρn )n≥1
(µn )n≥1 ∈ D∞
p
p
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µ
µ
qP
µ
kµk =
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p
i=1
|µi |
2
µ=
µ
µ
kµk
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σ02
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Q
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Y
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c
Zn = (Xn , µn )
(
(kµn k , µn )
#" #
p(zn+1 |zn ) = p(xn+1 |xn )p(µn+1 |xn+1 )
p(zn+1 |zn ) = p(xn+1 |xn )p(kµn+1 k |kµn k , xn , xn+1 )p(µn+1 |xn+1 )
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„at<[email protected]Ä5M « ŸM




0, 1 − 0, 9i
0, 3 + 0, 8i




µ1 = 
0, 4i
0, 5
 µ2 = 

0
0, 1 + 0, 2i

0, 9
0, 05 0, 05

A =  0, 05 0, 9
0, 1 0, 1
kµ2 k = 1, 03 kµ3 k = 0, 87



0, 1 − 0, 9i


µ3 = 
0

0

0, 05 
0, 8
kµ1 k = 0, 98
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kµk
(kµk , µ̄)
c = c
⊥
µ̄
µ̄
µ̄
µ̄
kµk
µ̄
kµk
kµk
kµ1 k kµ2 k
kµ3 k
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lPe$Ss’lmZ^SnSXe$S$M
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(kµk , µ̄)
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M
3
−
E(→
r)
→
−
r
→
−
k
→
−
r
−
→
− →
−
E(→
r ) = E 0 ei r · k
→
−
r
t
E0 e
→
−
−
i(ωt+→
r·k)
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(eh , ev , ez )
→
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(eh , ev ) eh
→
−
k
ev
z
E(z) = Eh (z)eh + Ev (z)ev
Eh (z)
Ev (z)
(
Eh (z) = Eh0 eikz eiδh
Ev (z) = Ev0 eikz eiδv
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→
−
k
→
−
k
(eh , ev )
ψ
χ
tan(2ψ)
\
tan(χ)
2Eh0 Ev0
2 − E 2 cos(δ0 )
Eh0
v0
aξ
= ±
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=
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2aξ , 2aη
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α
#
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2π
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g0
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 
 g1   |Ev |2 − |Eh |2
 
g=
 g  =  2<(E E ∗ )
v h
 2  
g3
2=(Ev Eh∗ )

4


|Ev0 |2 + |Eh0 |2
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
  |Ev0 |2 − |Eh0 |2 
=

  2E E cos(δ) 
v0 h0
 

2Ev0 Eh0 sin(δ)
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
g = g0 
 sin(2ψ) cos(2χ) 

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sin(2χ)
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χ
g0
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χ=
π
4
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d=

√
g12 +g22 +g32
g0


1−d
d



 0 
 cos(2ψ) cos(2χ)


g = g0 
 0  + g0  sin(2ψ) cos(2χ)



0
sin(2χ)






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g3
arcsin( )
2
g0
g3
1
arctan( )
2
g2
χ =
# #"
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=
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