Strategies for Getting the Highest Likelihood in Mixture Models
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Strategies for getting the highest likelihood in mixture models Christophe Biernacki — Gilles Celeux — Gérard Govaert. N° 4255 Septembre 2001. ISSN 0249-6399. ISRN INRIA/RR--4255--FR. THÈME 4. apport de recherche.
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(37) ¦Vitn-ckF] {(ikscelqn-lt¯ksX ]Pitheqlq{FcmksX[\x lqn rscerwkFrYlq¯ k}l!r}kF] p r2  WYX[]oxf {s{F] n>kx lqn u ckFcmlKn ithYp[{Flq iq[cehmcmksce]2r 7 M2 7 = kFX ik it{Fcers]2r ¯{slK\©kFX[] >ksXÉ\)cmydksf[{F]-x lq\)pClqn[]2nKk²¯lq{²ksX[]-xf[{F{F] n>k²®ithef[]lt¯ iq{s]-xlK\p fUks] u iqx2xlq{ u cen[²ksl)ksX ]±] ">f ikFcmlKn p. r. r. . T. c. ] T. k. _ ]. k. _. p. r. r. 2 ( ª K¬ + 2 ! >  WYX[] ,<]2rwksce\!iks]7r 2 + 2 it{F]f[p u itks] u f rscmn[gksX[]±xlKn u cmkscelqn iqhVp[{FlqVit[cehmcmksce]2r iKr xlqn u ckFcmlKn ith \)cmyUcmn[) ] ceq X>kFr2 /2]kFiqcmhe] u ¯lK{s\²f[heiKrYit{F]±n ltkqce®q]2nX ] {F]qHWYX[]2v+x iqn V]_¯lKf[n u ¯lq{YkFX[]\)l u ] hjr ]_x lqn rsc u ] {YcenkFX[cjr p itpC] { cen<§¡] he] fUy-itn u ³±l®it]2{wkª}@? ?KK¬ WYX ]²x(X iq{FiKx ks]2{scjrwkscjx r lq¯AksX[]²P iqhmKlq{FckFX[\ iq{s]g¡]2hmh u lUxf[\)]2nKkF] u ª=rw]2] ¯lK{ cen r}k(itn x ]² x ,iKx(X[heiqn iqn u {FcersX[n iq\µ@? ?
(38) Aq¬ O¾¿k-hm]7i u r+cmn q] n ] {(ithkFl rscm\)p[he]<] ">f ikFcmlKn r ¦ XViqr!kFX[]<n cex ]<p[{FlqpC] {sk}v lq¯ cen x {s]7iqrscmn[ ksX[] helqKhmceq]2hmceX[ldl u ikA]7iqx(Xgcmks]2{Fitkscelqnf[n>ksceh[r}k(ikFcmlKn it{Fck}vK¦titn u cen²\!itndvgx cm{(xf n r}k(itn x ]2r2¦ck u ] {Fcm®K]2rrs] n rsce[hm]Pp it{(it\)]kF] {]7r}kFcm\!itks]2r itn u x lqn rs] ">f[]2n>kshev cmk cjri pClqp[f heiq{ksldlKhtksl u ] {Fcm®K]A\!iyUcm\²f[\ heceq] heceX[ldl u ]2rwksce\!ikscelqn,¼ l¡]2®q]2{2¦>P©cerYdn[l n+ksl)xlqnd®K] {Fq]rshel hmv)cen-rslq\)]_rwcmksf itkscelqn r2 ] u l n ltk i u u {s]7rsr¡kFX[cercm\)pClq{skFitn>kiqrspC]2x klq¯¡©X ] {F]qA¾¿kX iKrY{s]7x] ce®q] u \²f x(XIiksks] n>kFcmlKn{F]2x ] n>kshev+itn u \!iqn>vitheqlq{FcmksX[\!r itce\)cmn )kslLrspV]2] u f[poksX ] xlqnd®K] {Fq] nVx]±lt¯,P X[cehm]p[{F]2rs] {F®dcmn cmkFr rwce\)p[hmcjxcmk}v XVi¤®q]PC] ]2ngp[{FlqpClKrs] u ª½rs] ]¡ksX[]Yx(X iqpUks]2{^lt¯9 3x , iqx(X[hjitngiqn u {FcjrwX[nVit\ ? ?
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(44)
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(98) . []\ 3. ^3_ 4K T l 4K h T h 4K T j k 4K T
(99) . Wit hmG ] A <)§¡lq\)p it{FcjrwlKn<lt¯ rwks{(iks]2qce]2r±dv<p iqcm{(r_ cmksX£ioheiq{sK]²ndf[\²V]2{ lt¯Ycmks] {(ikFcmlKn r±¯lq{ u ikFiIrs]k zn[lKcers]q. d)y n dc.
(100) @.
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(102) K gij gij. K ^ K f ^ g %^ KmYOP g ^ ^3P g %^. K f %^ g ^
(103) g g j g K j g. KmYbP g ^ j g K K g j g gij. %^3P g ^ j g g j j g. g. PSY o g jml g jQj g jml g jml. ^3_ TU h T
(104) ml TU h T UQk. []\ 3. ^3_ 4K@T U 4K@ TQ U 4K@ T U 4K@ T U l. Wit hm] <,§¡lq\)p iq{scjrslqnlt¯r}kF{Fitks]2qce]2r,dvgp iqcm{(rA cmksXLi_hjit{Fq]Yndf[\gC] {Plt¯ ckF] {(ikFcmlKn r¯lq{ u ikFirs]k¡z [ K ^ g h
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(107) g lmgh h g
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(109). KmYbP g ^ Q k g Qk h gl g j
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(114). g. PSY o ^3_ g kQU h T U@K g k@Kbk@T hQj g kQ U T lQl g kQU j T l . []\ 3. ^3_ 4K@T
(115) k 4KbU@T U
(116) 4K@ T k
(117) 4K@ T j. Wit hm5 ] ?)<)§¡lq\)p it{FcjrwlKn<lt¯ rwks{(iks]2qce]2r±dv<p iqcm{(r_ cmksX£ioheiq{sK]²ndf[\²V]2{ lt¯Ycmks] {(ikFcmlKn r±¯lq{ u ikFiIrs]k ztn[lKcers]q x lqn rscjr}k(rlq¯Pk}l\)]7iqrsf[{s]2\)] n>kFr < kFX[] u f[{(ikFcmlKn ª½cmn´\)cmndfUkF]2r(¬±lt¯PkFX[])] {Ff[pUkFcmlKn¦itn u kFX[]!iqckFcmn kFcm\)]ɪ½cmn \)cmndfUkF]2r(¬V] ¯lq{F]kFX[]In[] y>k!]2{sf[p[kscelqn [lK{)ksX[l>rw]-k}l¥] y[it\)p[he]2r)] u cersp[hji¤vq¦H¯lK{)ksX[] ¯lKf[{!\)] n>kscelqn ] u rw]2hm]7x kF] u r}kF{Fitks] Kcm]7r ¦PksX[]I]2®qlKhmfUkFcmlKn lq¯ kFX[]I\!iyUcm\²f[\ heceq] heceX[ldl u ®ithef[]2r) cmksX kFX[]ndf[\gC] { lt¯,ckF] {(ikscelqnVrª cmKf[{F]2r )iqn u ^>¬Yitn u ] u ] p cexk VlqksXIrslqhefUkscelqn rcmnIxlK\)pV] kscmkscelqn-¯lq{ ]7iqx(X u ik(i²rs]k ª ceqf[{F]2r )itn u K¬ AWYX[]_heiKr}k ] yUiq\)p[hm]_xlKn x]2{snVr9 ^V lqVrw]2{s®ikFcmlKn r¡cmn u ce\]2n rscmlKn ¨V®q]) ckFXn[lIxhe]2it{_p iq{wkFckFcmlKn[cen[Ir}kF{sf xksf[{F]¯lq{ X[cjx(X´¡]!xlKn rsc u ]2{ iks] n£x lq\)pVlKn[] n>kg³_itfVrsrsceiqn \)cmydksf[{F]q±WYX[cer u itkFiLrs]k_xlqnVx] {Fn r [cmlKhmlKqcjx iqhp[{Flt¨ he]2r lt¯Hp itksce] n>kFr²ª½rs] ])aUitn u lq{g@ ?
(118) A ¬ ceqf {s] u cersp[hji¤vdrkFX[] \!iyUce\gf[\ hecmK] hecmX[ldl u ®ithef[]2rY¯lK{kFX[]¯lqf { r}kF{Fitks]2qce]2r iqr i¯f[n x kFcmlKn lt¯ksX ] ndf[\gC] { lq¯ cmks]2{Fitkscelqn r2 r cmk²itp[pC]2iq{Fr¯{slK\©IlKnKkF]-§it{Fhel-] yUpV]2{sce\)] n>kFr2¦iohjit{Fq])ndf[\²V]2{glt¯ cmks]2{Fitkscelqn r_cjr {F] ">f[ce{s] u kFl£] n rsf[{F] irw]2n rwce[he]I\!iyUcm\²f[\ heceq] heceX[ldl u ®ithef[]KSWYXdf r2¦]Ip i¤v·itkwkF] n>kscelqn ksl£] yUpV]2{sce\)] n>kFr cmksX & [§¡lqn x hmf rscelqn r u ] {Fcm®K] u ¯{slK\n lqcjrwv-rwce\gf[hjikF] uIu itkFiLrs]k(rit{F]gx lqnU¨ {F\)] u < 7t]2.\ ¿Pitn u 2d§¡P ¿P pV]2{w¯lK{s\ kFX[] V]7r}kHcen\l>r}kHx2iqrs]2r2 WYX[]2v p[{Fl®>c u ]Yrwce\)cmhjit{A{F]2rsf[hmkFr2¦[fUk 7t]2.\ ¿P X iqr_i\lK{s])rwkFit hm])C] X i¤®dcelqf[{7 ¹ ksX ] {F cers]q¦iKr ¯lK{rwce\gf[hjikF] u¥u ik(i-rs]k(r ¦9kFX[] {F])cer±n[l rscmksf itkscelqn< X ] {F]gkFX[]rwkFiqn u iq{ u {Fiqn u lq\¢rwks{(ikF] qv7tP lqfUkFpV]2{w¯lK{s\!r ksX[])ltkFX[] {_lKn[]²rwks{(ikF] qce]2r iqn u ¦titkC]2rwk2¦qrwks{(ikF] Kv7tP qce®q]7r rscm\)ceheiq{,{s]7rwf[hmkFr kFX itn2t]2\ 4PɤWYXdf r2¦¤ksX ]YrwkFiqn u iq{ u {(itn u lq\ rwks{(ikF] KvLx itn n[ltk C]{s]7xlq\)\)] n u ] u . n-npo q l U jQj.
(119) 2^. 0Q
(120) Q. 4. −2.565. Stellar. x 10. −2.57. ML. −2.575. xEM xCEM−EM xem−EM SEMmax−EM. −2.58. −2.585. −2.59. −2.595 60. 120. 240. 480. 960 1920 Iteration number. 3840. 7680. 15360. ceqf[{F]+:<²H®qlKhmf[kscelqnlt¯YksX ]LhmlKqhecmK] hecmX l>l u ®ithef[]2r ckFXksX[]Lndf[\²V]2{glq¯Ycmks]2{Fitkscelqn r±¯lK{ kFX[]L{F]2ith u ikFi)rs]k r}k(it{(r . . . Stellar 100. 50. 50. 0. 0. −50. −50. −100. −100 x2. x2. Stellar 100. −150. −150. −200. −200. −250. −250. −300. −300. −350. −350. −400 −500. −400. −400 −500. rslqhefUkFcmlKn¥ rwlKhmf[kscelqnI ceqf[{F],)<HW ¡l!rslqhefUkscelqnVr¡cenIxlK\pC]kFckFcmlKn-¯lq{YksX ]{s]7ith u itkFi)rw] k rwkFiq{Fr ( −300. −200. −100. 0 x1. 100. 200. 300. 400. 500. −400. −300. −200. −100. 0 x1. 100. . 200. 300. 400. 500. . d)y n dc.
(121) ¤.
(122) . Geyser −1124. −1125. −1126. ML. −1127. −1128. xEM xCEM−EM xem−EM SEMmax−EM. −1129. −1130. −1131 60. 120. 240. 480. 960 1920 Iteration number. 3840. 7680. 15360. ceqf[{F]!^<²H®qlKhmf[kscelqnlt¯YksX ]LhmlKqhecmK] hecmX l>l u ®ithef[]2r ckFXksX[]Lndf[\²V]2{glq¯Ycmks]2{Fitkscelqn r±¯lK{ kFX[]L{F]2ith u ikFi)rs]k q] vUrs] { ( . . . Geyser 100. 90. 90. 80. 80. x2. x2. Geyser 100. 70. 70. 60. 60. 50. 50. 40 1.5. 40 1.5. rslqhefUkFcmlKn¥ rwlKhmf[kscelqnI ceqf[{F] :<AWl!rwlKhmf[kscelqn rYcenIxlq\)pC]kFckFcmlKn¯lK{YksX[]{F]2iqh u ik(irs]k q]2vUrw]2{ . 2. 2.5. 3. 3.5 x1. 4. 4.5. 5. 5.5. 2. 2.5. 3. 3.5 x1. . 4. . 4.5. 5. 5.5. .
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(124) ( $& !. ] X i¤®K]p[{F]2rs] n>kF] u itn u ]yUpC] {Fcm\)] n>kF] u rwce\)p[he] r}kF{Fitks] Kcm]7r¡ksl u ]2iqh ckFX-ksX[]ce\)pVlK{wk(itn>kp[{Flq[he] \ lq¯ q]kskscen[ iIrs] n rscm hm]L\!itydce\gf \ hecmK] hecmX[ldl u ®ithef[]! X[]2n£f rwcen[IkFX[]LP iqhmKlq{FckFX[\ cen£\)cydkFf[{s] n-npo q l U jQj.
(125) . 0Q
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