Choosing Models in Model-based Clustering and Discriminant Analysis

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(1)Choosing Models in Model-based Clustering and Discriminant Analysis Christophe Biernacki, Gérard Govaert. To cite this version: Christophe Biernacki, Gérard Govaert. Choosing Models in Model-based Clustering and Discriminant Analysis. RR-3509, INRIA. 1998. �inria-00073175�. HAL Id: inria-00073175 https://hal.inria.fr/inria-00073175 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Choosing Models in Model-based Clustering and Discriminant Analysis Christophe Biernacki, Gérard Govaert. No 3509 Octobre 1998 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(38) . & . ". a Ed. ". ". ". ". & k. G. ". & . Wfoec[{U±Guwviml?imvMhMacimkIn)v[+egfKh[ p hMi4hMX[nracZ%degfKh[ p Z1i p [+egn+« L. AIC. nb. of situations (among 73). 20. AIC3. 20. BIC. 20. 0.654. ICOMP. 20. 0.645. 20. 0.635. 0.621. 0.647. 15 (0.0853). 15 (0.0873). 15 (0.0949). 15 (0.102). 15 (0.0808). 10. 10. 10. 10. 10. 5. 5. 5. 5. 5. 0 0 0 0.5 1 0 Relative performance NEC. 0.5. 1. 0. 0. 0.5. EC. 20. 1. 0. E. 20. 0.5. 1. 0. 0.5. 20. 0.578. 0.645. 0.634. 15 (0.116). 15 (0.12). 15 (0.121). 15 (0.0893). 15 (0.0921). 10. 10. 10. 10. 10. 5. 5. 5. 5. 5. 0. 0. 0.5. 1. 0. 0. 0.5. 1. 0. 0. 0.5. 1. 0. 1. C. 20. 0.583. 0. CLM. 20. 0.577. 0. 0. 0.5. 1. 0. 0. 0.5. 1. 9acmdv[]m± 9imvG[7fmsXs vajhM[+vacimk!KXagnrhMimDvfmZoZ1[;fok4Kfoecd[mVfok p mack4lIfov[+kVhMX[7nr[7n+Dnyhfok p fmv p4p [;VagfKhajiDk imhX[#vfohMaciªimhMX[Z4ackacZ%dZ½Z1acns egfmnnrajµIs;fKhMacimk%vfohM[#iKm[;vhX[#Z4agns egfmnnrajµIs+fohMacimk,vfohM[iohX[Z1i p [+e sXiDnM[+kbqhMX[s+vMajhM[;vMacimk ¤a¨hX hX[nfoZ1lec[]nM\a =+[ G «. ë ë Ý3qrîþï. &.

(39) 7. %&-&#$S %& )Y.L %b%B

(40) %&#. L. AIC. 20. BIC. 20. 0.828 Nb situations (parmi 73). AIC3. 20. ICOMP. 20. 0.843. 20. 0.845. 0.841. 0.832. 15 (0.074). 15 (0.0648). 15 (0.07). 15 (0.0865). 15 (0.0779). 10. 10. 10. 10. 10. 5. 5. 5. 5. 5. 0 0 0 0.5 1 0 Relative performance NEC. 0.5. 1. 0. 0. 0.5. EC. 20. 1. 0. E. 20. 0.5. 1. 0. 0.5. 20. 0.621. 0.772. 0.738. 15 (0.163). 15 (0.145). 15 (0.149). 15 (0.107). 15 (0.122). 10. 10. 10. 10. 10. 5. 5. 5. 5. 5. 0. 0. 0.5. 1. 0. 0. 0.5. 1. 0. 0. 0.5. 1. 0. 1. C. 20. 0.629. 0. CLM. 20. 0.605. 0. 0. 0.5. 1. 0. 0. 0.5. 1. 9acmdv[}{U± 9imvG[7fmsXs vajhM[+vacimk!KXagnrhMimDvfmZoZ1[;fok4Kfoecd[mVfok p mack4lIfov[+kVhMX[7nr[7n+Dnyhfok p fmv p4p [;VagfKhajiDk imhX[#vfohMaciªimhMX[Z4ackacZ%dZ½Z1acns egfmnnrajµIs;fKhMacimk%vfohM[#iKm[;vhX[#Z4agns egfmnnrajµIs+fohMacimk,vfohM[iohX[Z1i p [+e sXiDnM[+kbqhMX[s+vMajhM[;vMacimk ¤a¨hX hX[nfoZ1lec[]nM\a =+[ G «. b &>&. Z4agn;«Gv;« ³}zM ³}zM ´zM zM¬Qu ¶}· ·# · : ¬ <b« m ;« m «j; <b«ÖL; <b«c; « ;V « o\ « m « ; 7« m « m Wfoej[]± 9imv)[;fDsXs va¨h[+vajiDk!ohX[Z4[7fok®kbdZ,I[;vOim!lfovfoZ1[ h[+vnG¥iDvwhMX[nM[+ec[;s hM[ p Z4i p [;eI¤ajhMXhMX[ nfoZ1lec[]nr\a =+[ G «GW#X[]Z4[7fokkbdZ%?[+v#iolfovfoZ1[+hM[+vn)¥imvhMX[ªhvMd[dk p [+vecqVack4Z1i p [+e¿agn«c+\« 5. &. Z1acn;«Gv;« ³}zM ³ zM ´zM zM¬ u ¶}· · · : ¬ « ;m ;«<m{ « \V « o\ <b« ;'< « D <b« D « \'; «Öo « ; < « \ < Wfoej[ª\]± 9imv)[;fDsXs va¨h[+vajiDk!ohX[Z4[7fok®kbdZ,I[;vOim!lfovfoZ1[ h[+vnG¥iDvwhMX[nM[+ec[;s hM[ p Z4i p [;eI¤ajhMXhMX[ nfoZ1lec[}nM>a =;[ G « W#X[}Z1[;fokkbdZ,I[;vwio¿lfmvfmZ1[ hM[;vn¥iDvGhMX[}hMvd[dk p [+vejqbackZ1i p [+eIacnO«c+\«. 5. b &>&. å¥ìë!åçæ.

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(47) %&#. L. AIC. Nb situations (among 37). 20. AIC3. 20 0.7499 (0.1099). 15. BIC. 20 0.7658 (0.0979). 15. 20 0.7841 (0.08705). 15. 15. 10. 10. 10. 10. 5. 5. 5. 5. 0. 0. 0 0.5 1 Performance relative. 0. 0.5. 1. 0. 0. 0.5. VC−. 0. 1. 0. 0.5. 1. VC+. 20. 20 0.8004 (0.08548). 15. 10. 5. 5. 0. 0.7871 (0.07569). 15. 10. 0. 0.7706 (0.09479). 0.5. 1. 0. 0. 0.5. 1. 9acmdv[± 9imvG[7fmsXs vajhM[+vacimk!KXagnrhMimDvfmZoZ1[;fok4Kfoecd[mVfok p mack4lIfov[+kVhMX[7nr[7n+Dnyhfok p fmv p4p [;VagfKhajiDk imhX[#vfohMaciªimhMX[Z4ackacZ%dZ½Z1acns egfmnnrajµIs;fKhMacimk%vfohM[#iKm[;vhX[#Z4agns egfmnnrajµIs+fohMacimk,vfohM[iohX[Z1i p [+e sXiDnM[+kbqhMX[s+vMajhM[;vMacimk ¤a¨hX hX[nfoZ1lec[]nM\a =+[ G Z «. &. Z1acn;«Gv;« ³}zM ³ zM ´zM <U« D 7« m <U« \V ;« <b« D{ ;«c; «c+\ Wfoej[ ±M9imv,[;fDsX¯s va¨h[+vajiDk!hX[Z1[7fok kbdZ%?[+v4iolfmvfmZ1[ hM[;vn~¥imvhX[ nM[+ec[;s hM[ p Z1i p [+eO¤ajhMX nfoZ1lec[~nra\=+[ G[Z «GW#X[]Z1[;fmkkbdZ,I[;violfovfoZ1[ h[+vn)¥imv#hMX[]hMvd[~Z1i p [+e¿agn4<b«ÖK\I« 5. D. `. D3d. &. Z1acn;«Gv;« ³}zM ³ zM ´zM «jD 7« m <U« m\ <b«Öo <b« \V <b« ; « o\ Wfoej[;±M9imv,[;fDsX¯s va¨h[+vajiDk!hX[Z1[7fok kbdZ%?[+v4iolfmvfmZ1[ hM[;vn~¥imvhX[ nM[+ec[;s hM[ p Z1i p [+eO¤ajhMX nfoZ1lec[~nra\=+[ G «GW#X[]Z1[;fmkkbdZ,I[;violfovfoZ1[ h[+vn)¥imv#hMX[]hMvd[~Z1i p [+e¿agn4<b«ÖK\I«. 5. D. `. D3d. &. å¥ìë!åçæ.

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(50) ( )+*,-/.%& 0+ ) 1#324 56&-. L Nb situations (among 37). 20. 15. AIC. AIC3. 20 0.9485 (0.03224). BIC. 20 0.9555 (0.03361). 15. 20 0.9579 (0.03455). 15. 10. 10. 10. 10. 5. 5. 5. 5. 0. 0 0.5 1 Performance relative. 0. 0. 0.5. 0. 1. 0. VC−. 0.5. 1. 0. 0. 0.5. 1. VC+. 20. 20 0.9035 (0.05964). 15. 10. 5. 5. 0. 0.9303 (0.03531). 15. 10. 0. 0.9577 (0.03475). 15. 0.5. 0. 1. 0. 0.5. 1. 9acmdv[#\± 9imvG[7fmsXs vajhM[+vacimk!KXagnrhMimDvfmZoZ1[;fok4Kfoecd[mVfok p mack4lIfov[+kVhMX[7nr[7n+Dnyhfok p fmv p4p [;VagfKhajiDk imhX[#vfohMaciªimhMX[Z4ackacZ%dZ½Z1acns egfmnnrajµIs;fKhMacimk%vfohM[#iKm[;vhX[#Z4agns egfmnnrajµIs+fohMacimk,vfohM[iohX[Z1i p [+e sXiDnM[+kbqhMX[s+vMajhM[;vMacimk ¤a¨hX hX[nfoZ1lec[]nM\a =+[ G « Z4agn;«Gv;« ³}zM ³}zM ´zM lI[;vry« m« mD « <o{ « mV « D o\ « D ; « D' < « ' <m{ kb!«$lfmv;« ' <U« D ;D« m m« <o <L;« ;m <mU« \D <mU« m 7{m « m Wfoej[ <b±| fKhMaciiohX[)Z1ackacZ%dZ Z1agnMs+ecfDnMnMajµIs+fohMacimkvfKh[OiKm[;vhXagn Z1agnMs+ecfDnMnMajµIs+fohMacimkvfKh[Oimhfoack[ p ¤ajhMXBhMX[4nr[;ej[7sh[ p Z1i p [+e ¢ lI[;vry«Ö£Bfmk p Z1[;fmkQkbdZ%?[+viowlIfovfoZ1[ h[+vn#¥imvªhMXagnZ1i p [+e ¢ kb!«lfov7« £ ¤ajhMX nfoZ1lec[;n#ack p ajZ1[;knracimPk X G Z « Z1acn;«Gv7« ³}zM ³}zM ´zM l?[+vMy« m« mD « ;DD « <o{m « <mo « <mU « DD{ « <K D kb!«GlIfov7« <U«j7 7« m <U« D ;« m ;« <K « Y ; <b« <K Wfoej[#±| fKhMaciiohX[)Z1ackacZ%dZ Z1agnMs+ecfDnMnMajµIs+fohMacimkvfKh[OiKm[;vhXagn Z1agnMs+ecfDnMnMajµIs+fohMacimkvfKh[Oimhfoack[ p ¤ajhMXBhMX[4nr[;ej[7sh[ p Z1i p [+e ¢ lI[;vry«Ö£Bfmk p Z1[;fmkQkbdZ%?[+viowlIfovfoZ1[ h[+vn#¥imvªhMXagnZ1i p [+e ¢ kb!«lfov7« £ ¤ajhMXkiDkU§ ¡ fodnnragfoks egfmnn#nMfmZ4lej[7n#ioGnr\a =+[ G[Z « &. 5. D. `. D d. D. `. D d. &. 5. ë ë Ý3qrîþï. &.

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