Download THËSE - Le serveur des thèses en ligne de l`INSA de Toulouse
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!" # # " &" " !" # ! ! * 0 1 3*4 ' % # "# # & !" +, - " -) ! ./* ' *' # #( # # )! *' $ ' ! # ! " # " # +- ) - $ &- + !- ) , - , ! 2 !! ! Remerciements Les travaux présentés dans ce mémoire ont été réalisés au Laboratoire d'Analyse et d'Architecture des Systèmes du Centre National de la Recherche Scientifique (LAAS-CNRS) au sein du groupe Diagnostic, Supervision et Conduite qualitatifs (DISCO). Je tiens tout d'abord à remercier les directeurs successifs du LAAS, Messieurs Jean-Claude Laprie et Malik Ghallab pour leur accueil. Je tiens à remercier Monsieur Joseph Aguilar-Martin et Madame Louise Travé-Massuyès, successivement responsables du groupe DISCO, de m'y avoir accueilli pendant ces années de thèse. J'adresse toute ma gratitude à mes directeurs de thèse Madame Audine Subias et Monsieur Joseph Aguilar-Martin pour leur collaboration inestimable, leur disponibilité et les nombreuses conversations et conseils qui m'ont aidée et orientée non seulement dans mon travail mais aussi dans la vie quotidienne. Je voudrais les remercier aussi pour toute la confiance qu'ils ont eu en moi, pour leur patience et leur soutien. Je suis très reconnaissante de l'honneur que m'ont fait Madame Sylviane Gentil, Professeur à l'INP de Grenoble et Monsieur Jean-Philippe Cassar, Professeur à l'Université de Lille 1, en acceptant la tâche d'évaluer en qualité de rapporteur les travaux présentés. J'exprime également ma reconnaissance à Messieurs André Titli, Professeur à l'INSA de Toulouse, Fernando Jimenez, Professeur à l'Université des Andes, Colombie et Madame Nuria Agell, Professeur à l'Universitat Ramon Llull, Barcelona, pour avoir accepté de prendre part au jury. Je remercie également à Madame Marie-Véronique Le Lann, Professeur à l'INSA de Toulouse, pour ses remarques, ses conseils et son soutien et en fin pour avoir accepté de prendre part au jury. Je leur remercie tous pour l'intérêt qu'ils ont porté à ces travaux. Une grande partie de la thèse a été financée par le projet européen CHEM. Je voudrais remercier les partenaires du projet et toutes les personnes qui m'ont apporté leur collaboration. Je tiens à remercier tout particulièrement toutes les personnes qui ont contribué au développement de l'outil SALSA. Je tiens à exprimer ma gratitude à Bernard Franc pour sa disponibilité et son aide chaque fois que j'ai rencontré un problème informatique. Durant ces années passées au LAAS, et à Toulouse, j'ai pu apprécier l'amitié de certaines personnes qui, au-delà de l'ambiance très amicale qu'elles ont créée, m'ont toujours apporté leur soutien dans les moments critiques. Je voudrais remercier très sincèrement Emmanuelle Despontin pour ses encouragements et son amitié (je n'ai pas assez de mots pour lui exprimer ma gratitude). Je souhaiterais aussi remercier mes collègues du groupe DISCO en particulier Miguel Gonzalez et Victor-Hugo Grisales qui ont toujours été prêts à répondre à mes interrogations de nature très diverse. A mes colocataires de bureau, Antonio et Hector je dis merci pour tout ce que nous avons pu partager. Mes remerciements à tout le personnel technique et administratif du LAAS, avec une attention particulière envers Eliane Dufour et Christian Berty pour leur amabilité, leur efficacité et leur temps. Enfin, il y a des choses que ne peuvent pas être remerciées par écrit, néanmoins je voudrais remercier Juan-Carlos d'avoir été toujours à mes cotés dans cette aventure, de croire en moi plus que personne d'autre, de ne pas m'avoir laissé baisser les bras dans les moments les plus difficiles et bien sûr d'avoir partagé aussi les moments les plus heureux. A ma famille, pour leur soutien et leur compréhension : là où je serai je sais qu'elle sera toujours près de moi. ! "! #! % &! '' ! % # ( # ) % % % # * * + ! #,# -( # #1+ # ( -* # 2 0 -* # 3 *! ! '# % -* # ! ## % ,# -( # 6 + #1 %% *,# %% *,# # ! % # % # # # )! # ! 3 # ! 3 *! ! " 5 ( ! # ' ! ** # % % % 5* ##! '! ! ! . / # (! 43 *! ! '# #1+ # 0& 5 * #! # ! # % ! " !. ! ! '# ,# -( # '* 0 ) "! #! / 0 %%0 ) 5 $ # !* 0 ) % # # ( $ # # $ # 7 ! ! * # 8 579 ) #( # ) ! ## # ! *# ' ( # : ! ! +* ( ! - # ! " ! ; # .! 5 8 * ( ( %& ) * # # % 540 , # ?* % # % 0& ## ( *! 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" #1 "! #! # !. # !9 8 ! * # ( * 6 # ' ! ** * " # *! # * # (! # . * ( * Surveillance DETECTION DE DEFAILLANCES DIAGNOSTIC (Origin ou causes) RECONFIGURATION (Décision et proposition d’actions ) Acquisition mesures 4 )( . PROCEDE * # "! #! 1 )P ( ! & # =( * ! # HL 0/:I commande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ystème 4 # R # # 414 ! ** ! #! * # * ! " . -* ?! . ! ** # * #! # !J Résidus Génération de résidus θˆ 7 ! ! #J -.* # ( Yˆ θˆ ' ! ** # Y Modèle estimé #! )( ! 3 # 6 )! 6 * &! " # # Sorties θ +G ! ' #,# -( ( ) . ( ## # U " * ( * 6 # #4# J Entrées * +# # #,# -( # * # # 1 # * * #. ! #! #* ! #' # # #! # ( ! ! 3 ## ' ! ** ! #* # # ! ! * )( ! ) # ! !# 1 ! #( # 5 # ( # * ! 1* " -* # # !# "! #! . ! #! +* 6 ' # #! # ' ! # !#! J ( # r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Σ 3!# * ! # 1 !J* ! '' # * * ( * # ! # #J ( * 6 # *# # ! # *! ! " ! # * ( # G * # 1 ( -* * #J #! , # J * -* # # , . !# 3 +! *! # +# )( ! 3 # ! ( 5 # 4*! + ( *! . " * # ' ! ** # ## E # -# # * # #! # # ! ## # *! (! # ! '! 3 # " # 6 ; . # ( *! # # ( +* # ) . ( ! # # -* # !# ' 6 * " ##! +* J ! # 3! )# Σ( #! ! -* # ( )! (! 3 # # ( *! # ! !* # * ( * ' ! * J " ! ## # ! * *,! 3 # # ( # ' # #! . *! # # ( Σ9 ; # ! # ) " ! #! # 5 * !'* * # ( # !# # * ! '! 3 J ! * ?! * ( # # # 8 ( ( ##! 3 # (+* ( ! * ! *! # * ! ( * 6 # * # ' # " 3! * Σ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i = 1,2,..., K # 3! '!! ## ./ H >: I ( # 8 9 * !! ( # * #3 *# * # # . ! *! - ( # #,# -( . ( * 6 ! # 3 ! ((! ## # * #! # ## # ' ! '* * # -.* # # ! 3! 1 ! * # # # ( * 6 # # # #4#,# -( # ! +* ! " ( ! '! ! 1 # #,# -( # * #,# -( # 8 9 -* . / # # ! 6 * # ! ! # *!. ! # ! 3 J ( ! # i = 1,2,..., K 6! ( " ' # * ! *! # , *, <( # *! ! ! ## . (! ## # * #! # # #J ( ! J ! #9 * " * # # ( ! " !** ( ! #9 3 ! ! ## ! - * ) (( -.* # '* * ##! 3 Ri : SI x est Ai ALORS yi = f i ( x ), 5 *! # ! ! ' 6 !+* # *!. ! #! 3 # ( ! # * # * ( )! 3 # . 1 ! .% ! 3 J . # ( ( * #! # *!. ! # ! 3 # (+* # '* # !# '* ( (( ! # # # -.* # # ( # #,# -( # * # # Y * # -.* # # -.* # # * .! 3 ? I 3! # (+* # '* #,# -( J # " ! # #,# -( # 1 + # # (+* Ci # +# * 6! ( Ri : SI x est Ai ALORS y est Bi # 5 ( (! ## # # * AY Ai ( * 6 # ! # ! 3 # ( ! # # # !+* # # YJ * # ! ## ' M # #,# -( # # (+* # '* ! # " H0 0 * #,# -( 8*! ( * 6 # # A * # #$%&! *!. ! # ! 3 # 3! • # #,# -( # # * *! # ! # # " -.* # ! ! " ! # !. +# ) ( # ! -* # # • * '* # 3! Y* # ( * # -* ( ! ! ( , ! " # ( -.* # '* * + ! #! " HV C/ I ; * !' ! # ! 1* ! * #! # ; * # * * "! #! ! ## ( * 6 # * ( * # #! ( * # 8 # )# ! " (! * # R ' ! # *! 6 # ' ! ( # ! # * #! #J * # 5) ! "! #! ' ! # fi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•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• ( ! * 6 # # * # # * ' J ! ' ! ** ( * # !( ( ' ! ** . ** ! '! P J 1 # # 6 *! 3 ) '!J ! * # , "-* # 5) ! "! #! ! 1* ! #! # #,# -( # ( * 6 # # 7 ! * # #,# -( # # !* # # # !)(! 3 ! 3 6 6 # J " * # * ( # ! ## # # # ( * # # # # +# ! *! # # # 6 # *! # * ( # # # )3 ( ! #J ( * 6 # H5 * * ! J " ! ! ## . ' ! ** -* # ' H C; //I *# ! " /%I * # '!!! 3 HLT / I -" " " #" " " " # !# ! ( ! ! # • * # & " # * ' * # # ( # # !*! # # ** 6 ! ## ( " ( * # # " # ** 4(R( * # !' # 6 •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ase de Règles Moteur d’Inférence UTILISATEUR Base de Faits Base de Connaissances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α, β, γ, λ, µ9 5 ! J * 3 ** 9 H5A + ! ! # # ! " 3 *! '! # ( * # ! # 4*! (! I ! ! '# 5 " ** 6! -( 8\+ +! *! 3 H0 ! # % , # # 5 * ( ( *! #( J ! # (+* E # 3 *! ! '# ! # ' # (+* * # ! *! # # + 0 * 6 (! - ! ( # # ! ! !#! J ! #! . ! '' + ! !# # S ( * , * . ' ! ** # +!! # * ( *! '! ! #! . * # # # * " # " ## # * J ) !+* # ! # 9 ; * J * ( *! #( # ) 6 * # . I 1 * '! * # #! . ! '! ! " # # * + # # '! * 8" ! ?! . " ! ** # 3 # #! 9 # ! # 6 *!! ( ) ! *! # J 5) ! "! #! (+! ! # H5A ! 1* " ! #! # ( # #,# -( # ) ( * 6 # #J * !. # ! !# ! # !** # I Α Β Χ ∆ Ε φ γ η ι ϕ κ λ µ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étection de défaillances Analyse du Modèle ! *! # (! 3 J # ** + "! # 6 ! J !. " * ( J 1 "! #! '! . 5 ! . * # '! ! #! # )-# 0 Actions suggérées TB3.2a TB5.4 TB5.4 TB7.1 TB3.4 TB7.3 TB3.1 TB3.7 TB7.4 Mesures F F F F F F 2 " ! 3 *! ! " # # ## # 8 ! # #9 C ! # 3 # ( ** # C ! * ( # C # ! 3 *! ' ( + # # * (( # ! # ! 3 ( * ! 4" !+* 8 *,# #5 ( # # 7 ! ! * #9 5 # ! ( -* ' ! * * *,# 3 *! ! " # * &!. # ! ' ! ** # 1 ! # #! ( ! # # 8+ # ( ** #9 &!. # ! ' ! ** # . # ! * ( # . )# # 6 A ! * * # # #!! # # *,# # ! '!+! *! ! 1* * # #! ! # ( * # # J+ % 4 ; ! . ! * G 5 ! 1 * 6 Explication à l'opérateur ! J ! #J # * TB5.2 F% F% F% F% # # # # ! # "! # J Composant défectueux # . )! 3 # )! 3 # * # G 5 J ! # ) ! # # # !!! ! " # # * G # # * ( TB3.2b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mesures TB3.1 Génération d'épisodes fenêtre temporelle TB3.2a enveloppes 4 ! # ) # #* TB 7.1,3,4 TB3.6 Etat du Classification qualitative Procédé des états Tendances ' ; TB3.7 Commande statistique multi-variable ACP Bandes temporelles TB3.3 Décomposition des tendances fenêtre temporelle # Episodes # ! . ! *# 1 + # ! # ( # )! # ) ! 3 # # 1 + # # )! # ! 3 # ?! . % +G ! ' . * ' ! ## * # (! ! " ' !* # ' ! ** # * # ( * # ( ) # 1 ## #! ! # )! # ! " J ! " ! *! # ! " # ! 3 # ! # ( * " !. ( ) " # # * # '! # # # ( J # # # # ! # ! # ) # +*! * # 3 # **-* #J # "- !. ! # ! ! ## * * # ! 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J * # * # "! #! # 8= " ! ( # # ( ( , # ( 1* ! *# * 1 )3 3 ( # !R ! # 1 ( . * ( * # ) . ( ) J 3 !# # # !R ( ) % ( * 6 # "! #! * # * ( ! #! # #,# -( # # " * ( ! # # ! # ! #! ! * !J * P # * 3 ** !' ! 1* # # ## # 5 ) # * ##! '! ! * # 7 )! #! ** & ( # !J 3 !# (+* # * # # * + (! *! ! '' * -* # ! # ! * .! 1 ( #J ! '! ! # )! 3 # ! * 5) ! "! #! ! 1* ! #! # #,# -( # ( * 6 # 5) ! ; C 0 A& 5 (( & 5 # * " +* # # ! #J # 3! ( ( ! # ! 3 # ( * # ! ## ( -* # # ! #! # 7* # ## # # # . !# ( * " * " ! ** J ,#' ! # ( ! (! J * * # 63 *# # 6 .! *! . * ( ( ; #J + # (+ ! ! #,# -( # . !# O ! * # * #. ! " #[ ## # * ! '! )( ! * # <J * # ( # ! # # # J 1 ! # # # #* # * ! '! J ' !R ! # * 6 * ! ! ## # ( * ( # ( ! " ! ** (!! (! # ' # # ( # + # !*# !! * # ! #3 # # ! & ! # ! J * # "! F !*# # )! #J +G ! ' # # #,# -( 1 . !# #[ #( J * #J # 5 A (( * # . !# =) ( !#=J )! # ! # ! (! ; B ? 5 ( " ! ** '! 7A; J # #! ! # # # ' ! ** ! # # $ 5) ! +G ! ' ( * # .! )3 #! * " * ! ( ; #! ! # ! #! ! # ( ! ! # # # ! 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( * ( -* ( ` # J * * ! ## "! ! +# #,( " ; ( ! #! ! ! " ( * ## " !. * # +# ( 7 ! #J ( ! - . ( # * # (( * '' # * " ' ! ** J !. ## ! ''! ! # ( ! # 414 ! # ! * *,# # # ! ## ! " * 6 * # * ?! . * # ! # # # # #! . ! '! ! ! #! J * # ( ! # * ! # (( # !' * # ! " " ! ' ! ** ## ! !* * "! ! # ! * #! # !* . ## # # + 3 ! * "! #! ' ! ** # !J # ! J ! * # " ** ! ## # #! J ! .! 7 # + 7 "! ! ( 1 + <( # ) # *! . # )( # ' ! ** <( ! * = 5 #J * " #,( ! #! ! !# 1 * 6 <( ! #' '' ( ! # ' #,( ' ! # ! # 1 * R #,( # # ( * # # # ''! # < ! # ! ( # ! # ! #2 #! <( #,( <( !. # ! 8" ! _ * # 9J #! ! ! ## # # 5) ! ; # ## # .! " Z ( 2 R # ! *! # # !* ( * " ! ** " ** # ) * # "! ! !R # / * ##! '! ! LA7 HB ::IJ * # . +# G " J # * ( * ! 4 !& !! # %! *# 3 , J ( )# ' ! *# I (! # 1 G ## # # 1+ # #J # H AC * 3 '' * # ' ! ** #! -* ' M J ' ( ( -* ( ! ! - 1 ( ! " ( & ! " & ! # !#!3 ' #! * ! # * # (! * .! ( # ! ) ' M #J # ' ! %9 ( # # !! # * # ! # +*!# ' ! '' # J * ! ## . # # # # * # +# " " J ## 1 ! # 1 * ' M 1 )! 3 #! ## # ( 8_ ! # ?! * ( J * " ** :9 3 # *! # " ** "! ! J ( * !! # #! ! # ! 3 # #! (! * ! # 8: #" ! # ( ! ! '! ! ) ! . ! "! # 8 !! . 9J # ! )# ## )# !#! J* # ! # .! # 1 * ! ## . # ! "! R " # ( # * ! ** J * * # ! * # .! 1 * ## # " -* J ! *! # 1 # ( # ! <( # # (! * !! ( 1 ' ! " * * ( * # -* + * 6! # * #! # -* ( ( . * ( # 0 # & (R( # # . ! ! ## . + # # # #! (! *! ' !J ! * 6! # * ! (! # ! '!! # # (+* # )! 3 # 3 !(!! (! # * # # 3 !( 6! (! # J ! # * + -" # (! # * ! # " ! ! " x∈ℜ J # * ! * 9> )! 3 # ## , * ## #9J * ##! '! ! #9 1 * ## * * +!! # , #J ( ! #! # # ( # * * # * ## # # ) +G # +G # 1 3! ## # J * 3 ** # 8 ! # ( *J * # ( '( 3 * ) "! # # #,( (( ! # ! +*! ! *! # 3! " * ( " ( * . * # 1 *! # ! - ! # ! . *! ' M 4# # H? V/$IJ * 6 ' M #J & # ! , ! ## . % ## )# ( ## #J * # # ) D * # * # 8 . ( * # # (+* X = {x1 ,..., x N } # ! ! * # .! )# 3 # >: !! ## (+* # # . +! * .* ! # = #" * # * ( #J (+ * . 6! (! H> ;/ I :9 # 8 H] //I % 5) ! ,! 3 ( J * ! #' ! 6 # * * ## * ## • 5) 3 ! " # ! # # * ##! '! ! * 8 * ## 9 5 # . # ! " * # 8 . " ( * # ! # ! !)( * ! ##! +* + # # * # # * # ) # ! '' # # #. # ##! +* 9 * # +G # 3 ! # +G # #9 # ) +! ** ( # # (R( # R !( !+! *! * # ! # ! ! " * . +G # 6 8* , #J 3 ! !! # * # ! R # )! #!# # * (! # # 1 G # # * ## #9J # 1 (R( * ! (! # ! ( + ! ( # ! " ! # * ## # ##! +* # * / =9J 5) 3 * ( ! # ! # * ( 6 * ( # # !J !#! J )3 ## # # ! . . # # (( * ## !)( 2 DknA ( # 8= * # # (( + * # ( # '!! k =1 n =1 ! # * ( ! * ## #=9 '! 6 #1* " ** . ( * # *! # )3 . [ ) * ( # 3 !* ! # * # # 3! # 3 !' * # 3 * # # * # * # ' M ##! +* J , # ! # ! # (+* # ' ! vk * ! * # .! * ## ! * # * (!! (! # ! K N # * * # # (+* +* # #* # * ( ! # ! 3 # # # * # !! 3 1 !! # ! # '!! * ! 3 . . ! # J= AY / # . ! (+ *. * * # ( *! # . # . # !)( '!! * # / # . *! 1 J ! * ' # ##! . x ∈ℜD # * ( *. ) # (+ " !# (!! (! # * 5 ! " # + )3 # ( +G ! * ( # )3 #! ( * # 1 *! * # H052$ I # 414 ! 3 X = {x1 ,..., x N } # ) ?70 8= # . * # * ! . R ) '( *. * # # 8* ! # ! 3 # ! " # ' ! * ' ( ! #! (! * !# * ## #. ) ( .- # 1+ # '!!1 # ) ( .- (R( . 1 # ( " ! ** 9 R . ! 3 * # 8 . . J .! !- # • 5) 3 * " # # " # ; ! # * . ! J ! '( , 2 DknA 8%9 * +G xn * , vk 5) ! ; # .! * # " ! ** 1+ # ' * 3 ! ( # . 8 9 *!! ; # )! 6 ) # % * ##! '! ! = ( x n − v k ) ⋅ A⋅( x n − v k ) 2 2 Dkn = xn − vk A ( T A # (! -# (( * )! 6 A=I J # 8 9 * ( ! 3! # A * ! # HF F/:I 2 Dkn = ( x n − v k ) ⋅( x n − v k ) A T 5 ! # # * !! ( ! ), !. * # ), 4 **!# K * #J ?! * ( * ! *! # ! A=M * # " 5 * ( # * 3 # & " *S ! ! # #! ; ! # ! !* ( R '!! (( * ! ! # !. . # * # - # * ## # ! ( ! " 1 N ! T N n =1 ( x n − x )⋅( x n − x ) 8$9J x * # ( # ! * ## # +! ! J '!! A ! # 1 * " * !* ( , ( # * # " 8$9 * 0 )* # # # MJ # 0 # # ! # * . ( # (+* +! # #* # ), 4 **!# # (( ?! . 4 ! # AJ # * # * !" ** ! )! 6J # * # * # #J * ( J ! ( #/ −1 ! # !)( 6 6 # M= & # * *. **-* # J A # (+* ! *! # 4# )- # 3! * # * # 8 9 HF F/:I * ( # * % 5) ! S*. ; # .! !)( #! ( *! '! # # ( ! 4 ,/ < 7 " 1+ # ( ) # # #" * ##! '! ! # <=/=> # # ? -" @ 8 ## " ! ** , * " ; * # # "# A " xn " # # " # vk -" " @ (" -" -" , '# # ? / " ' t 0R( #! * ( * # ' R ( '! . 3 ! # * 3 6 '!!! #! # !1 ! *! # J ( * ## # . * ## # )3 94%0: ! * *. 5) 3 * ## ! - . # ! 4 ." HF L: IJ ## " 8%9 G #J ! * " (!! ( ( .* + * 5 ! # * 1 +*-( * %70 ; !)( ! ( * J '( ! (! # ! ( * ! ( * *. Y* ! # '* * *. # . ! # * 8 " *!! 9 #J * *. (!! (! # ! # ! !)( # ! * # /4( , J FCM ! # # * ( # # (+* (( !- ( 6! (! # ! * . ' ! ! " J # / +G & ! !)( # !)( ! # !7 # ! 4 * ## #J ( ! # ( 4 # # J FCM ( X , U ,V ) = K N k =1 n=1 2 (µ kn )m DknA 8 9 AY • • X U # * # * ( ( ! #> * ( ! !! ''!! • V • m ∈ (1, ∞ ) # * x1d x1D x 1n x nd x nD x 1N x dN x ND '* 8 ! ( )3 ( ! # !! # * # ! # # x11 #! , # * . *! ! #9 ='* = '* # * !! /6>9 xn 1 * * ( vk !# * #! " µ kn ∈ [0,1] 6 '!! * * ## 7%J # * ## #J !! 8# " * .( 3! # #! " # . - * ( (( ''!! 7 '! 6 1 (# 3 J 5) ! ; # * #' ! +* # .! ! " * # " ! ** 1+ # 6 ( ) # # %% * ##! '! ! J ! * , " ( * # * ## # S*. !)( #! ( *! '! # # ! 4 ## # ." , ( ; ,/ " # B1 < k < N J # # 8 l = 1,2, -" < # * # N = v k( l ) (µ ) ( l −1) m kn n=1 N n =1 # "# -" A U (0 ) # " ? # "m J # ( xn ) ( l −1) m µ kn vk (l ) , 1≤ k ≤ K 2 D kn "& ## 2 DknA = ( x n − v k( l ) ) T A( x n − v k( l ) ), 1 ≤ k ≤ K , 1 ≤ n ≤ N @ U (l ) ( " # Si DknA > 0 pour 1 ≤ k ≤ K , 1 ≤ n ≤ N , 1 (l ) µ kn = K ( DknA / DhnA )2 /( m −1) k =1 (l ) Sinon µ nk = 1 (distance est 0, appartenance est 1) C " -" ' # + * # C # '# # * ( #. ( # " . ! " + # " #' ; # #! +* # 1 * * !( . # * J # # 0 )* ! 3 # ! '' # *. ! )! 3 # 4# * ' ! #J # 5 (R( J 1 # /IJ 3 ! +! #8 3 * 3 # * *. !)( ( # / ! # ! 8$99J . !!!* * # ! 3 #! +! *! # * ## # * ## # # # 3 ! *# # J * !!!*! # ! # !! # "! #! # # ' ! (- 1 ! . a3 * (! * # ( ) ! *! # ( !#J ) " : ! # ** # #!* # 8= #" 1 * !!!*! # ! " ( #J ( !! . # ( !7 ) # # ! ( !)( !)( # . "! # R J * *. > ## * HC; #( * ## # 3 ! ! " J #! # ( +* * ##! '! ! )! #!1 * " 6 * #! # ( ! # ! + ! " & 6! # * U (l ) , U (l ) − U (l −1) < ε J B ε '# ! ## (+ # # # - ! '! 3 * ## # #! . ! '! ! " # # "! # # !! 5 !R J ! # R 3! % 5) ! ! " ! 0 # # # #* # (+* ! * 6 # # !)( # ( : ) '!! , ! "! # # (+* ), # ' # . # + +! *! # # # * 1* # ! ! '!# , # * ##! '! ! # ! # ! 3 # # H;5 /:I )3 # ! #" ! *! # * # 3 * # ! + # " * -.* # ! " * ( * * ( # # # * ! " %IH5 , " ! 3 * " !! # # ! # * * . H0 5/ I * " # # ' !J # # #!! -# " # # J ! + # # # J B #! * 6 ! # & %I ; # ' " *! 3 # 1 * ! # . " ! # ! + ! ?! #) # " * ## ! + # xJ )-# H0 " # + +! *! J * ), * # B #! * * # +G J ' (R( ( ! #" * * # #! " f i ( x ) # ( 6! ( * ' ! # ! # ! + ! -" * , # #!! # !- 4 ! ( '!!# *! . Y! *# # #! * # " * ! # * # * ## # * ## # ! * # * # # # (! ! *! # * ##! '! ! 1 * ! ##! ! # ! '' * # *! # * ##! '! ! ! # ! + ! # # * * # ! 1 +! ## 3! ! ##! +* + !1 % ! ** # # * ! # * ), ) * # ' M ! ! ! ## . (+! ! # # > # (+* # # * # #( ! # " ?! . ! +* # * ## # ( # '!! * ##! '! ! * # ! '' ) # # +G # ! ## . , xD ) 5 5 * 3 ** * * 3 )3 9 f i H0 5/ I ; # # # # * ' ! * )! 3 # ! # ! 3 # !# G! +G * # * ## 5 # # # # )! 3 # # # ! " + +! *! 8' ' ! * # #9 * # ! J # !! . $ '!!# 8 *! # * ), '# ( # * ##! '! ! (+ (! ! # ! + ! ! (! 6 ##! +! *! ! * ), # # # # # (+* #J # ( 6! (! # * ## # # * ! # * ( # • ; ( , ) ( ) ?! # ) J * * ## # 6' M # )! #! ) * *, ' M 3 # .* + * 6 # ; ( , ! + # x = ( x1 , x 2 , ! "! # * # ' * * ##! '! ! ! ! ' ), * G! . '' ! '' ! " # ! 3 " & * q * ## # ! • ; 6 , # # ! + # ( ( P ! *# # # ! ( # ! + # ##! +* 6! -( ! * # # 1+ # * ## 7 * ( " ! ** # -" : %$I) ' # )3 * # # ! # ! 3 # !# : # $ H? : G # H0 5/ IJ # *. .! 6 0!)! (( # ! * 8 * ## #9 # ; B!.!! #3 * ! + # )3 * # B!.!! J * # ), * , ' ! '* * # !! # !#! J # 5) ! ; # .! * # " ! ** 1+ # ( ) # % * ##! '! ! SI Largeur de pétale < 3.272 – 0.325*Longueur de pétale, ALORS Versicolor SI Largeur de pétale > 3.272 – 0.325*Longueur de pétale, ALORS Virginica 4 # 5* ##! '! ! # )! 3 # # ! # ! 3 # =( * ##! '! ! ( * 6! # * ! # # (! * * " #% # # " # ! O )3 " ** # ), )-# # # ( H&;& I # * * # * ## " " * +G # * ( *! # ! * # " # "- ( # # # * .! # * # J # * # R # ( (+ ! #3 ! " ! '! # %D<9 ! ## . !+* # # ## ! ! ( ( # 8 * ##! '! ! # )# ! ! "! # # # # # # * # .! " * +G x ′ J ! x′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• * # * ## # 3 * ## ! " # ''! # # * ##! '! ! # + + $ : 8 ) # U # ! + )# , +! ! # (! ! ## # + #9 5) 3 # " R (( * ## # 1 ! ## # U # "## J # 6 # ! ! "! 1 6! # ! # * # ! # 63 ** # * # ! ! "! # # ! # ! " * #9 + '' ! ) # # # ! # # '' 6 8 1 * 6 ! #! J # # ) : & ( )! # J 1 ! #! E" 1+ # * * ##! '! ** # ! ! * #J • # ) !!!* #S !! # # !! # ! # # " ! ** !)( # ! #! J ! ! "! # ( * # *. *! # # ! .! ! ! #! 9 # ! 3 *! # 8" ! ?! . U " * + ; ##! . # ! " R 6 ( * # 8! ! "! #9 +*!# 1 * " # ! '# # J * ## # # ## ! J • )3 U ! #! J * ! + nœud racine ! A = bleu nœud décision B B < 4.5 4.5 B K=x K=y B 8.1 B < 8.1 C K=x C = vrai C = faux nœud feuille K = x 4 *. * *. * # !)( 5 * # )! 6 * !)( &% # # + # ! #! ! + 1 6 (! 1 )3 # (+* ! ## . ! # ! + ! * J ! ## J # *)( . ## !J * ! + * # ) ** ! #! 9 ] " * ! 3 !# ! ! + * !' ! ! "! # + # * ( I # H2; :$I * # # * *. * 4) ( .! 2 !* 9 H&;& ! #! )! #! 3 * * ## * 5 . !- ! J * 7 K=y + 8 # * 8 6 ( * 87# * # * !" $ )! #! A A = rouge B !R !)( # !)( # &% # * )! #!J # (! 6 * # ! ! "! # * # !J H0 / IJ * *. ! *. ! !)( # 414 ! ( * ! ! # * 5) ! ; # .! * # " ! ** 1+ # Entropie ( A) = ( ) # % * ##! '! ! nbk n log 2 bk nb nb − k 8:9 nb × Entropie nN Entropie Moyenne = b 8/9 AY • # * nb (+ # ! ! "! # nbk # * (+ • n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• * # ! ! "! # *!# 1 ( * # ! + #* #+ # ! " ( ! 3 # ! + ! #! * " * ! *! # ' ! # * ! + ) ' # # * # # ! # ( !# ) ' + !( , ! J G 1 • * * 3 * * )3 # * # ! ! "! # # !J ! * ' * ! #* ! ! "! # 1 # ! " ! + + ! J • * # * # 8 + # #9 * # #,(+ *! 3 # ; 1 . # + -.* # * * # # # ! #! ! + 9 1 * # * * ##! '! ! '# # 3 ' ! # & ! ** ! J * *# * * * #J ! *# " * ! ## ! #$8 " # " ! ( * 6 + # S ! # #! ( * #J ' !J * # "! # # / %J 3 ! J # * 1( (! # # + # * # \ *! # " # ( -* 981(: S )( ! # ^J ' * # 6 # !# * # #' # ! ( S * #S.! S ! # ! ## . " * ( )! 3 # +!* .! 3 #J " # # (+* ** # )( ! 0 5 ** # 4 ) ( 7! # -* +! ! %: 5) ! Y )3 ' ( ! #( #, ! 3 # 5 ( S ! ## . J # ! #( (! " 6 # # )# ! ( * ' # # * ' ( " * # -.* * ##! '! ! ! '! ! # ( * ! # 6! # # P / :J . Q 6 ) S ' . ! *! # #9 * wi1 xn win xN wiN ' ! # ( S ) U !# #, " $ # ! " ! ! yi # * * * ! 3 # 8b!JNJb!9 5 ui f 8" ! ?! . * * # * # # ! # # N w in x n 8 9 y i = f (u i ) 8 9 4 n =1 f ui )( . ! " ! * # 8 9' * # " # x1 # * # 5) 3 * ! " ! * # * #! # ! & S * # # ! # J +* # # )( " * ! # # ' !)( # J 1 # 86 JNJ6 9J -* . ) # ( # *. S ! 3 # * ! " ! * # ; ? S ( ++ ) ui = 4 / /J + # ,! 3 ( ( #(! #1 1+ # /:I !# #, #[ ( ! * " ! ** # # # ! # # U ?! . # ! * # !! # # # (( .! * J* H7 ( )3 # ! " # # )! +* ; J ; * # ! *! # yi * # ! #! .( K J 8 9 ' 8 9' ! . ! 1# ##! ! *J 8+9 ' ! *! !J 5) ! ; # .! * # ! ! ! * ! ## . # ! R # * + # ! 3 # # * # * * + ' * . ! (! # 1 G # !# 3 ! * * * J ! # ( " ! J 8?! . 9 "! # HF * ! '' ! # # 8 d i 9J ! ! '' # ( # ( yi ! '! # # 3 − di ) * !# G#3 1 1 # * * # * ! *. !)( ## , * * # !#J # / I G# * 1 2 E= # ' # * ; # # ( * # # * ! ## . 6! # * ( # ' ##! +* # * + ' M J * ) ' !J ! ## . J !)( * 5 * ! ## . )! 3 # ! & U %/ * ##! '! ! ! 6 ( * # )! 3 # ( *! # [ !! * * # *. (! * # (( +* # * # !* ) * # * ( # * # #! # 3 # ! '' ! ! # 1+ # #! 6! # )! 3 # ( " ! ** 8 yi 9 # * !# )! 3 ! 3 2 8 9 3 réglage poids erreur (attendue-réelle) 1 entrée sortie réelle 4 ' # 6 " ! ## . w in # * !!!*! # # ! " ! ) ! " 8 3 E # 3 J * ) . ( * # * # # 6 * !( + # # ! * # # * . 5) 3 U ! 8 * * * 8 F?9 HA # * * # * # F? * # # J !# ! J * * J # 414 ! * * ! ' ! " ! ! f 7 " 3 ! ## . # ) 87059 H ( !# # ! " ( * ! 4 # * ! # # ! " * # ' ! # * ! " /$I ' M # J * !* * ) . ( !# . 6 ( * ( * # * 5 ( # # " ! ** -# 3 * 99 # 4 ' ! # 3 ) #" * ! *! # # # 1 U )3 # ! ## . 4 )3 δ ip * 2 sortie attendue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• 5 !# ! ( 6 7 # ! # #7 0& # ( . ( ) # * # 4B! * " I #, ) * .! HC ( !. HT IJ * # # !*# H> 0 %I ( ) J # * # ! 1 + # . * # # # 414 ! # " * ## # * ## * # * # #! ! # # . * ! . # ! J ! # * ## # (( ! * # G 1 ' ! + # # 9 . ( # 5 ) # * * * ! ! "! # % # " * # * * .! 3 . 6! # # # ! # (+* * ##! '! ! 1 3 + +* *! ! "! # !! * .! 3 # * # & 0# # # # +! ! #J " # * #! * # 1 " * ##! '! ! ! + ! ! * 8" ! _ J ( +! # " ! + ! # 1 ! * # ! ! "! # # # ! ! "! 9 1 ' ! # 3 * ' ! ) * ( # 5 ! # ! * # * ## # 5 H7 ( *! ) . * #! # !! * ## #J ! # #J * ( ! S. ##! ' # ' *1 ! ! "! *# 3 # # * ( ! ** *,# * ! :/I # T ! ## ( * # 6! ( ! " #J +# * ! + # ! + # 3 *! ! " # 8 ) " * ! '' ! # # * ( ! " # ! .! # !) * ( " '* -# //I !#!3 ** + : IH7 # 1 * H C; /:I ! ''! * !J ! # ( ( ! # ( ) '' • 7* # #* ) . ! * 40 ! ** ! " ! #J 1 * ) )! # # # #J 7! J N9H C; : IH& ( *! . ) * .! 3 # HT #* 9 ]# # # # !! ! *! # '# 0d (+ )-# # +! 3 ! + # * # *! ! "! + 1 6 #" 1 (! ! ! # ! 3 # 0 * # * # ! 2 ( , #3 # # # * .! 3 # J ! !# !# / I 0& Z ( ! * '' * ##! +! *! 4# "! # * # 3 8 ! + ! ! ! "! * ! ## . 4 ! ## . * ## :! ** # S'' ! ## . 9 !** [ * # )# !!!*! # ?! . ) # # 8 c E J 0& * ( "# & ! ## . J * #! # ! 1+ # )(02( $#" # 8 ; " ! ** 1 3 ! * 5 ! # -# * ## ! + # !* 1 * ! + ! "! # 8 " ! ## . # # 1 # ! (R( ( ! '! ! ! ! "! 1 # ' ! #J ' ## ' ! 3! ! # 2 !3 ! * J 3 ! * # !* * ## * ## 6! # 9 ' M ( # !J (! 1 5) ! ; # .! * # " ! ** 1+ # ( ) # * ##! '! ! DAM1 ( x1 , C1) DAG 1 DAM2 ( x2 , C1) ! "! C1 DAMd ( xd , C1) x1 DAM1 ( x1 , C2) DAG 2 DAM2 ( x2 , C2) x2 MAX C2 Classe Cl DAMd ( xd , C2) DAM1 ( x1 , CK) xd DAG K DAM2 ( x2 , CK) CK DAMd ( xd , CK) 4 #) @ 0 # & . * J # " ! + # " ! "# " ) #" * # ( *! # ! # ! *! )3 {x1 , ! ! "! ( # ! # ! " # 1* " * ( # ! , # 3 ! - 1 ##! # " X ) " ) # ! # x jMAX 3 * # " * # +# " * " * * # # # ! " x jMIN 3 (!! ( * # # J *# 3 3 *! ! ' 8#,(+ *! 3 9 ! ! '#J * # " * ( 6! ( * ( *! # ! * # # # # xj = AYJ * ## ! ! " #J ( ! #! * 6! # " !J " ** H J I 5 # , xD} 5 # * # " * * ! ! "! 1 #" #3 ) 7 * ! " ' M # ! '' " G # # # xˆ j − x jMIN 8 %9 x jMAX − x jMIN ( * ( J x̂ j # * " * + # * xj " * ( *! # ! ! "! )3 ( () !#!3 ! + * ## J 3 7% ) * # (R( # 0& !R ! # ( ) 7% * ! # ! 3 # 3 ( * ## 3 *! ! '#J * 1 # / .! * # * ! " # +! # & 0# # 6! # # 7 R 3 * # ! ! "! # # ! ! "! # * * ## # * * # (- " ! # # *! # ! '' * # (( * * '' # 8 # <) < # 7% ) ) * ## # * # & 0# 7 ! #3 # ! # * ! J # 3 * # ! ! '# 5) ! • ; # .! 2(0 5 "< & * #! # ' ! #' " ! ** # * # .! # # # # * , * # % * ##! '! ! # ! # ! *! # ) * 3 ! ! 'J ! * 6! # # & 0# # ! # !! " ! * 4 6 # ! ( ! #! '* * F , #! ! # ! + ! ( Y* ! # 3! + 6 1+ # # * ! # 3! 4 ** * # + +! *! # ' J ! ( 4 ( # @ * . ! ( ! !#-3 ( ! ' # ! 3 J + # H C; : I 5 # 414 ! J ! * # * ! * ' 3 ! ! , # + +! *! +!<(!* J *! 3 ! 1− x j DAM( x j , ρ kj ) = ρ kj x j (1 − ρ kj ) & # * 3 ! (- * ( *! # # ! 7% 3 ! " 4 T ! ##( ! # ! = ##! 4 + % HT # # /:I # * ## # & 0 " * # ! ' ' ( * ' ! # # ! # " * !! * # /9 !! (R( # * (- xj " * * * ## % d kj 8 d kj = x j − ckj # 9 8 $9 4 ?! * ( J # * # (( , # Y* ! # ! + ! . ! # ! + ! , Y µ kj σ kj2 # ! # /9 * ! * ## % **-* ! # / ( ! # J * # .! ?! . * J ( 4 **!# K * ?! . R ##! +* ! *! # #! ' ! ##! DAM ( x j , C kj ) = e ), # ! ( *J ! * # − ( 1 * 6! (! DAM ( x j , ρ kj ) = ρ kj 1−d kj (1 − ρ kj ) Y* # ( * c kj * " * 8" ! ?! . 3! * * 9 # 414 ! 3 ! * , ! '! # 1 * # xj ! ! ρ kj ! ! ' ( # 6 R( #J ) =J ! # 8 # ! 5 6" * +! * 3 !* # *! ! "! ! #* +# 9J * ## +# # " 8 # * !'* * ( ) 1* 1 2 2σ kj ! " ( ( x j − µ kj ) 2 J 1 * " " * ' 6 6 # ( , ! J * # 1 * !. , ' " * ## # * # , ! # ! + ! 8 ! ' # 9 ! ( 8" ! * ! * # * * # & 0 5 ! *! # # ! 5) ! * * ##! '! ! 6 # ! !# ' !#! J * #J # * ( # ! # * ( . * ( * 7 # ") * # 6 # **-* ( * * ! ## . # ! F! (!* ?* 9? !! * # ! C # !! # * # * # ! # ( * ## # # (! #[ J # 1+ # ) # * ##! '! ! Y* # ! ! "! # # ['! * ( 9? 4 " ! ** / # * # * ## # # # (+* ?! . ' # **!# # 9 +9 ? * * # ' ! " ' ! # ( .! # ' # ) * # ( ! # . #* " . ; ! # # # # ( ' * # ' ! 6 ! 6 6 # # # , # !# . ##! J * # ! # "! # 8% * ## #9 ! F! (!* 5 ##! * #' ! #& 0 * #3 ! ! ' 5) ! • ; # .! 2(0 5 3 ' 3 " ! ** " < # # ' * # ( ! ! "! *! # 5) 3 ** 4 ! # +# " ( #3 3! ( # * * ## * " *! * # # * ##! '! ! ! + # ## 7%J ( ( ) * ## *! # # # (+* *! J * R 3 !! ! 3 7% ( # # ( πi " * )3 & 0J * ! # 3 *! ! ' ! 1 1 * * # * # # ) " * *! +G # # ( J * # * # Ψ kj = (π 1 ,..., π i ,..., π m ) 7 ( 1+ # ( # * ' # * " *! * ( #! . 3 * DAM x j ,Ψ kj = π i 4) 0 # ; ' ! # 3 .* + * 8 # 3 2 9 + * * . ! '# # . *S * ) ! S ! αJ (- 5 (( * + # * " # & 0# * * .! 3 * +!! # * ## ! ! ## # 8 :9 #3 * # & 0# !R # # ! ' ( * # ( ! *! # # # (+* # '* # 8 4 #J * . ! 3 (( ! '# (! 6 # H7 *# ( 9 '' * ! α ( 8! # * ## * # ' M J * ## * * #! #J # * "! ! ! "! & C # > 7 # !! 7 * 8 4 5 # # # ! * ! * * ## " * ## * # # ! * & C 5J '!! 9 5 # # # # * 3 G # ! S ! * , 9 8 # * & * # * * # ! ! "! * ## ! ## . * #3 ! ! '# # * 3 * # # ! ! #3 # # 6! # # ! 9J * * ( # 414 ! > ! #* # 4 # " * '' ! * (!! ( ( 7# ψ NIC , j = ( m1 ,..., m1 ,..., m1 ) # "! # * ## J 1 1 2 ! # 8 " ' ! ! "! # & C# ! + ρ NIC , j = ! # & 0 # ' (! ** # ! ' 1 *)( . # #! * * ( 6! ( * ! ! "! # ! ! "! 1 # * # ! ! "! # " * E # ( 6! ( ( + # 4 )! #! ! ** J #!* * # * # * ## #J & C 1 * # & C # 1 * (- 3!# # &! '' # ! ! "! ! * !R * .* + * + # ' * 8 /9 !#! J #!* " * α =1 ! ( 9J . ! + ! # * # * ## # # ! + ! (R( # * * S # ! 9 + & C # 3 3 *! ! '# * ## * # # * 9 5 * ## " 8 8 ( * , ! # ' ! +* 5 * ## * ( 8 6! . )! #!# (! - " # !! !' . 4 # R ! ! "! J * 7 1* * # ' " α ∈ [0,1] J 1' # * ##! '! ! 6! . ! '# '* 1 * * ! 9 αJ * )! #! # & C / I 5 # ! *S ! ! J * DAGα (DAM 1 ,..., DAM d ) = αT (DAM 1 ,..., DAM d ) + (1 − α )S (DAM 1 ,..., DAM d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k ρ ki = xi 8 x∈C k 9 INDIVIDUS Xn = (x1, x2,…, xd) C0: Classe Non-Informative ENSEMBLE de CLASSES Degré d'Adéquation Marginal DAM (xj / Ck) Mise à jour Création classe Cl Cl= Ck+1 j = 0,…,d Degré d'Adéquation Global DAG (X / Ck) NON OUI C0 = Cl Affectation à la classe l Cl / DAG (X / Cl) DAG (X / Ck) k = 0,…,K RECONNAISSANCE AUTO-APPRENTISSAGE 4 # )( * ! ## . # ) 3 3 ! * # '' * ( ( * !# +! * # ! A *. ?! . ! ** # #* # ' M # 3 ! + ! # ! 3 # !)( . * *. * 0& & !** J * * ( 0& !)( ! ## (! # 1 G " * * #* # * !#!3 # * # ! * ## * # * * 4 ! ## . J * ## " ! -# # '' # !! * * ## 1 5) ! ; # .) " 5- .! * # " ! ** 1+ # ( ) # * ##! '! ! 0 3! * * # ' !#! J ! * ' # ! )! #! # 3 * * ! ! ! '#J * # * " (- # 1 ( 1 G # & 0# J # " # # # # * ρˆ kj = ρ kj + & # * # # ! ( # * ' ' M +!! # * *. * 8 N +1 ! #! " * x j − ρ kj " ** J * " d̂ kj ! # !)( ĉ kj R ! ##! 6 (- * # * ! " # 6 (- x j − c kj 8 N +1 dˆ kj = x j − c kj' ρˆ kj = ρ kj + & # * # ' µ̂ kj ( , * ! " ! * # #* # #J + ( * J # " # * * * 9 " ** σˆ kj2 2 # (x + j x j − µ kj 8 N +1 − µ kj ) − σ kj2 2 8 $9 N +1 * 9 (+ * ( # ! + # ! # 3 *! ! '#J * # (( 1 * ## 3! πi * ( * *! (! # 1G # # * ( , δi = * ' M J* # (( ' ' 3 #* #' 8 δi −πi " 9 8 :9 N +1 3 +# δi ! 1 si modalité présente 0 autrement πˆ i = π i + & 8 N +1 ! # ! + ! σˆ kj = σ kj & 9 8 %9 1 − d kj' − ρ kj µˆ kj = µ kj + 2 #J ! *! # cˆ kj = c kj + % R + +! *! ρ̂ kj " ** * 9 # # ! # . * 1 : 5) ! & # * & C# # * 4 ##! +* 5 ! ! "! " ** * ## " * (+ * ! (! J * ( " ! ** # 1+ # * ** # ! J # # # G & C 1 * ## # ! '! ! 7 ( * # # # (+* # # ( 5 ! '! . # ! * ( 1 * + ! * 3 * ## # !J ** #J * " ** * ##! '! ! 5J * ! #!- * J # * * ( !N0 = 0J * * ## * # ) ( 6! ( * ! * (!! ( ( * ## ( " * * ## J * # # # ! " * # " ** ! * '' " ** * # # * " ** * ( * ( ' ! # * .! ! ## . J 3 5 1 * >G '! ! ' # ## # ! # ! 3 # ; ; # ! # + +* * ## # ! . # 6! # # * ) # ) ( 6! ( ( #;8 " '! ( (( # "! #! J # # +# `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ε ( ! 3 ( e #,(+ *! 3 * ##! '! ! # ! # ! '* ! 3 ! 3 e #,(+ *! 3 ! ! ! Z ! ! ! Z ! " * 444 ' ! * , 1+ # ! # 5* ## # * # ! ( F # ! S .! # ! R ? ! '# ! " ! ) ) # ! ! & 0 α # ! e '* # * ## # ! f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•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• ! * # ##! +* +# " 3 ! # •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• # * ! # # ! 3 # • *! #J # * 6 ' ! # * # # !! # " ; " * # ! ( )! 6 # . ! ! ( ' ! ! *! # + # 3 ( $ 5) ! ( * & # !J * # " * +# #J * # . ) # ) % ! '! ! # ! ** ! # # : # . * ! 3 ( # #' !# * # # ! ##! +* # ! + ( *! # 1 * # *# * # " ! ** ! # ( *! '! # ! !# + * ! # * . # # " +* # a) b) c) 4 # 5) ! 6 ' ! " ** " ! G# '!! G#3 1 3 * .* . *! *! ! * " ** * *! # " ( *! # ! # ! " ** # # ! # 3 !! # " R #! (! *S !'* # # ! 3 ! ! ' ! ! '# # ## ( & (R( J * * ! # ! # * ## # # 0 & ( '! ! # ! '' # ' * # # ( * * ! (! ! '' !! # ( # ! ## *# * # * # ' ! ## . # ## #J 6 ! ' # # * #! # #! ! # !! * ( J * # # ! *! # ! # * # # +* ' ! ! ## *! " " ! '! ! .! # * '' # ! '' 6 J 3 # # 5) ! % ! '! ! • * 6 # • ! * # # ! ! *! # •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• * #" * # , ( * *S3 # 99 R # # # ! # ! 8 • #!* # " * # 6 99 # * # # # * * # # ! ! ## ! " ' ! " ** H # * ' ! # ! + ! , " 8" ! J # # " # *! ! 3 J # # ! " * # ! , 9[* (! # " * * #! # 8 # 8 3 # ! 8 •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α = 0 .3 9 3 + 1 3! # )! #! # ! # #! ( !# ! + # # ! * # .! 6! . ##* # # " !α =1 1 * " ! # * ! '' ! *! # * * # " 45 ! ( !! # ! #' ! # ( * 8?! . % !! # ! ** # (+ & ! ' # * ?! . ! ! '! 3 * ## # # * # #! . ! '! ! " * ## # ( * # ' ! # ! '! # J 9[ * + # # " % +9J J .( # (( 6! . * ## # 6 ( * J* * 6 ! ( ( * # ! ! ( 9 & J * # ' * ##! '! ( !! # * 3 ** # # # ( 0!! ( ( 8 4 #! !#! J* ! 3 ( * ##! '! # 9J ! * # α * ## ( !# * ' ! ## . ! !! J ! - J * # 6 ' ! # # # # !! " ! J * " 8_ % I # * ## [ # * 6! . :J ( !*J ) ! * * ## [ 6 ( * 4 . # * 1 * ! #! * # # * # ! 1 (R( # 3 # J * 6 # # * * " ' ! ** ' ! ** * ( # ! G 1 ( ! % # ! # * ## .! * ( !# # ! # 1 & # J 1 ?! . 0& . ! ! "! ! 99 ( * # max ( x, y ) 6! . ! ! "! 8α = 0 9 & min ( x, y ) & 0 * " # ! 9 x+ y − x⋅ y * ##! '! # ( 8 x⋅ y !! # # * 6! . ! + # 8 4 # ! D5 ! 9 ! ## . ! '' . S ( x, y ) + +! *! # 0!! ( ( Z0 6! ( ( # . T ( x, y ) !Z (( # ' 8?! . % J * ##! '! ! 3 ! + " $ 5) ! ' ! * 6 ( " ! " # ' ! % <* ( ( ! '! ! * 6! . (! # !! # #' * *. !)( !! # ''! # (( +* ! ' ! *# * # * ##! '! ! * # . h1 h2 Qp Up 4 # 6 ( * # # !! +9 α = 0 .3 9 α = 0 .6 9 α = 0 .8 * ##! '! ! " ! '' ! + # #" * # ( ! " *! # ! * # + # !# 9 # " ! ** α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b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• ( • " ! '! * # (- * # ! "! J # * # ! '! #!* # ! ! "! # ( * ! # ! 3 # # ! + • " * ##! '! #* ! '! # . * ( ! # ! 3 # '' # ! " ( * ! ## * # J + J * # ! ( ! ** J ( * R )# * * ## . ! 3 *! # ! # 1 * 3 ** J ! # ! 3 # # ! ! "! # # 6 # * ## # h2 Qp Up 9 # # !! +9 * ## # ! ( " # # ! + # * 6 éléments ayant changé de classe éléments rejetés 9 #$ ! ## # ! *# h1 4 * ! ## -# " = G= * ##! '! ! " ! ## . !! . 5) ! % ! '! ! & +# +# # " " * # ! # ! *! # # !** '' * ## 3 ) # *! . 1 G#3 1 3 (! # 1 G J +# # !! ## 3! '!! ' ( # # # ) ! +* 1 * ## ! '' ! * ## # #!* # * )# # *! # # ( * # . * # # +* J ! 1 ) ! +* * #! # ' ! # * (( # (+* * ( & * ##! '! #J # * (R( ) ! +* ! # ! ) . ' M # (!* # * J ! * " 6 g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gB * 83 *! ! '9J ) gB * 83 )3 # # ! ! '9J ! ! '9J 2 g * # " * * ## ! # ! ## ' + # ! ! '# ! # # * # ! " 83 : # 9 & 2 gB * 83 -# # 8_ % % ) gB * * ## ( * , ! # * # # 3 )3 # # 3 83 *! ! '9J 83 *! ! '9 * !)( # * ## # # , * ## # )g *. # ! # 3 *! ! '# 1 * ! # ! 3 # , ! # ! 3 # * ## #J # '! * 83 * # -# * * # # ! ( , * # 3 *! ! '#J # 3 *! ! '# *! * !( * ! # ! ! '# ρ kj # ! * * , 3 Y )3 83 *! ! '9J % # ! ! '# # !! J ! ## . # ! * ## )3 *S# * )3 )3 #* ! ! '9J ! ! '9 ?! . ; g )g $ 5) ! % f ! '! ! a 0.4 h g g 0.3 g g # #' a h f l ! *# * # h l a a l f l " ! ** 0.3 g g g g g g g g Episodes : g = γ , f = φ, a = α, h = η, l = λ 4 ` * # # 7 ! * # * # # # ! # ! ! * * # * ( !# #* h1 ! ## . ! J 3 *# # * # * ## J * ! " ! # ! # ! 6 3! ' !J ! * # 1 )3 ! # ! (! !! . # 414 ! # * ## # 7 # ! !# * !3 ! ! (! # * # )3 * #1 * ## # * 6 # # * * # * ## # + '! * ! # ! (! !'* '! * ##! +* !#!* * # * * # * # ! * " * # . ! # * ## # h2 Qp Up 4 7 ( !# #; 7 * 6 ( * * ! # ! (! ! '' # '! * # * ## # " ?! . % = ?= J * # ! # ! ! #3 #3 =K = # # " * ! ! '# # * # * # )3 ! # ! (! * ## * # # 5 # !' #! . ! '! ! " # ( ! # ( ** # 3 ! 1 *S6 ! S ! # ! '! ! # (R( ' ! * # * ## # * ##$E ; # !' 0 ! # ! * # # ! ! "! # 1 # . ( 6 ! ! "! # S 3 ! .* + * * ! " 1 * )3 * # *S ! ! "! * * ##! '! ! * ## # 1* & # 6 * ## # * ## % *! # ( 6! # J # * ! J 8 /9 * # *! . # #3 6 # * # * # 5) ! % ! '! ! # #' ! *# * # DAG11 $ " ! ** DAG1K DAG nk 8 /9 DAG N 1 ##$ DAG NK . & @ 3 ( # # " * ?! . % J "! # 3 6 # ! .* + * 8& C9 * # # ''! # (( !' #3 ! # #! . ! '! ! " # * ( * ?! . % # * * ! . ! )! 3 6 * 0 *J ! ** # * ##! '! ! 6 * ## # # ! ! "! # * * 5 1 !#!3 * ## ( ( ( ( # & C * # 6 +! ! ! '! 8" ! & C # ! # !. #! * ! * ## 7# ? 9J " * #! # * ## # ##! +! *! . ! ( * * ## # 7< & C# )J 3 * # & C# ! # #! (! * !# * 6 !# ! ** # * * # " * * #! # * ## # 5 # *,# 6 !# ! ! "! 3 • ' . (( & C # ( # # ! ( )3 # * ## #J * 5 # ! # *! . *! # • * ## * ( # * ( . # ## #J * 90(:J * 92( : !! !+* # & ( 6 * * #* * ## 7A ; * * ! # ! ! ! # ?! . % * # # 6! (! # & C# 6! (! ! "! * ## # * ## # 8_% % + # * (R( ( * ! ! * # #! # ! " # 9 * " * ; ! # # C1 C4 NIC C5 C2 C3 4 ##$ . # '* 3 # *! # ! # & C# )3 * ## 0 = *S6 # ' B! # = 6 # * ## # * ## # -* 3! # '!! # # α " H C; /:I # * * ! (+ 1 * * # . S ! ! "! # 3 !# " * # ) ! # ! * '! 6 # (+* # $$ 5) ! % ! '! ! # #' ! *# * # " ! ** 9 α " L< G +9 α " L G MN 9 α " LG G 4 #@ 5 S * ! # ! 3 # J # 414 ! 3 * ## # . # & C# # # α # α " L<) 4 # * ## # 1 # ' ! ' ! # # # * ## # *# 3 * ## 3S 1 * # 1 *S α " ; * ( ! ** *,# ! % : 99 J # * # ! . # # * ## # ! ! # # +* ! # ! 3 # *! # # # #! (! *! ! # ! ## ! # 7 ! "! 8_ % % ** ( #!- ! #G! # 8?! . # * ! ! 1 ! # * # * ## # # * #! . ! '! ! 'J * '! * # (! ! ## . J * ! # '' " )# *! # % : 99J ** # # ( # ! * ##!+! -# #' " # + * 4 * ## # "! (" " ! ( * 6 ( * * #! # . # * 3 J α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• * # •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`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déviation h1 h2 Qp Up 9& # *! . éléments non reconnus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h1 Qp Up h2 +9 7 4 # ! *,# '! * A 5 # # * * ! # ! # " ** + # * ## J '! * # * ## # #,( <( 4 * # 414 ! ! ## . # ! ## . # # # "! # "! # ! ' ! 'J ! # ! 3 #J * # # nouvelle situation 95 * " 5) ! % ! '! ! ! * # * * J " * . * ! #! +*-( # ! 3 ( ; +G ! ' # S! *S6 ' * !. " ** #! ( 1 G ! J '! * * #,# -( * ## * ! * " R ' ! ** # # !. % 9J * 6 * # " * # =) = 3 ! # ! ( ! # ' ! ** * ! .! 9 ! 8?! . # ! ( * # ! # ! 3 J # * ! " ! (( '! * * #! #! +! # #8 ' ! ** * (( ! # # * ! ' ! #3 ( " ** " " ! ** ! '! * # # * 6 ( * # .! * # ! ## . J # * # !' # 3 ( & *# " ** ( # ! )# ' ! ** # #' "! ! # # .! ! *! # # ! # # ! ( * ( ! '! J ** # #J * ! G ( * ! '! 3 ! * ( ! * -* # ! "! # # # # )& () ( 7 ( ! *! # * ) (( 4( + " ( ) )! # ! ' # 5 * ! # F * ! ! ! * 3! ! '! " ! ** ! # " # •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onnées : entraînement Espace de Description Profil & détails des classes # # *,# Mode d'Apprentissage Données : test Matrice Appartenances - DAG paramètres LAMDA Affectation états fonctionnement OUI NON VALIDATION Modèle de Référence 4 # • ; ) # *! . *! . # +# " # * 3 ** J 1 ! # J *! ! # # *! # 8" ! ?! . * * # % -# # 45 *! . ! ( 9 5 -* "! ! 1 -* * + # ! "! J 1* ) # *! . * * # 8 ##! . ' ' ! '! ! '' 3 ( # "! # +# " ! ** " ! #J ( * # 59 Données en ligne Reconnaissance état courant Modèle de Référence 4 & ANALYSE # # *! . *S ! ## . 6 F # ! " 4 ! '! ! # ) # *! . # ' M ! ## #' * # ! ! ** *! . * *# * # ! * * ! *! # ! #;% 5 * # # ) ! " ! ** # # 1 * # # ( )# 1 *! # * ! ## . # 7 ! '! ! # # !J # * * + ! " (! # # ! '' ( #J # #' (! -* # ( #J * # ' " ! # J *# ! '' # # 414 ! 5) ! % • * '!!! 3 •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•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f ! & # # ! 1* # ! * 4 h ! ! ! # ! # '!! ##! (! * ! ( ' # ! ' ' ! # *[ =Y 9 ' ! J * " #J * X = {X 1 ,..., X i ,... X K } * ! # " " & ## ) #!! # #3 #!! ( -* # ! # #!! # ! ( -* -* # ! J # # " # # # R ** 3 * * #( # * # #J 3 ! #! # . * ! * # 1 6 !# (! # 1 . * + # " # 3 * ! ( # !'!! + # ( # # # 414 ! 3 t i +1 − t i = ∆ = constante #" ! # ( (R( Xi #!! #,# -( 3 # # !! J # t i +1 > t i 6 # # * # # U #* # # * #! # (+* uij = 1 # #! 1 #! 3! #!! 3 3! # 3 *# #!! # & ( J " ! uij = 0 [ * #3 ** ##! +* #! # * # !# # ' uij * " ' ! # {t1 , t 2 ,..., t i ,.t i +1 ,...} #J " 3 # ( # . *! - J + # #J 8% 9 # (+* " ** # # ! * # -* # 1 ( # #* ( ( " # !# ( ##! +* *# # " # # 3! * # ( ( ( ** 4(R( Xj 1 # 3 AY 1 3! S ( # ! ( *- S = {U , X , f } # ! * # *! ! "! #! ! # , (! 3 # ! 8X ! ( # * + ! 3 ( #! ! # * ! S = {U , X , X , f , id } J Y id * ## # ! # ! 3 # ! 3!# h ! * # 1 #! '! . # ( # ' ' ! * #! # ! ## . h ! R " #! ##! +! *! # !#! J +# 8% 9 * " ! ** ##! " h: X →Y !* ' " ! ** 8% 9 # ! 1* * # f : X ×U → X !! . # ( * " ! ## . ! '! ! J # # #J 3 3 #3 !# # *! ! # 8_ ) ! !! " # 8%%9 ( ) . * ! -* #, ) ! ** . J # 9J 1 # J* # ##! +! *! ! ! * # ! 1 ## ! ( uii = 1 ∀i # 5) ! ! '! ! $ 0 ( ! " #,# -( G ! ' ! ## # ' ! ! +*! "! ## !# ! ( #! # # $ 6 ( * * ( )! )! # ! * # # 3 * ) . #J *# # +* ( (! 3 ** # # " & ## ! *J ## ! * J ! * 3 ! * ! ## )! ! * # ! '' # 6 ! * * ## (( # 6 , ! # ' M J * ! * 5 1 ** ( * "! ! # # # # * # * # * # Q) * # ( ! * ! ( * '' ) # # 8h # ! - * ##! +* # ! # ! *J * !* " *iJ h #!! # * ( *iJ h # i ##! +* # # ! # ( !# # ## ! # *# ' #! " , (! 3 J ! * # ! * ?! . # ! 6 ( #! 1 # * 3 ** ' ! * # ! - ! # ## # " ! ** Q) ( #! *# ! ! - # )( # ' #' J * * + ! '! ! J * #,# -( * # 3 * ! ## # # 5 *! # ! '' )3 '!!! !* # h ## i9J * ! (! '!!J . ! # 3 ! * * 6 * # ''! # # 1 # * 6 ( * ! #! # ' ( " * ! # ! 3 4 " ! ## J * # ( #1* 7 ( ! ! # # !* . ! ## ' ' " ** # #! * * 3 *! # * !' ## ! # # ## ( , (! 3 * # .! ' #!! # " * : ! # " ! ** 9 #,# -( J ! # ! " * # # )( # #! ! 3 ( * # ' !3 J 3 ! # #3 * #!! # ' " * -# ! * " ! # # " ! ** # # > ## ( # * 6 ' ! +* # 1 # * # ; * # ' # #,# -( ( # 3! # " * ! # ! - #J ! * # ## # J *! # : 5) ! ; #,# -( *!! #J " # 414 ! 3 (- #1 # #! ! # !. # ! '' ! * * # # "! # ! * # J ## ! ! (! # ( 1 * # # '!! # # ) * # ?! . J ! * h ## i <* +* # * #* ( ) . ( )! J # * 6 ( * J * '' 6 ##! ' 3! # * # " ! ** J # 3 *! " ' # ! # # # ( ! ( * # )!# # ! # 3! 3! J -* * #J , (! 3 "! #! # ## # ! '# $ 0 # # * #!! ## Ø # ! 6 ( * . # ! ) # # ?! . ! * +* ( # * ' ! # g # (+* U = {e, f , g} f,g ? e,f,g 4 $ 6 ( * . 0 ' ! " ( # F ( * " ! #!! {A, B, C , D}× {e, f , g} → f (.,.) → {A, B, C , D} $ )J ( X = {A, B, C , D} e # ( # (+* e,f,g f # ! " # ' ! condition initiale e . . *! 6 ( * * !# #! !- # #* " ! ** * +* * '*!J #!! # 3 ! #* 6 ( * J# ! 4 * # ( R) " * , ! - # 3 !# * # #!! P# '# * * # (( # , # • # ( # 6! # +* #J P# '# * ! '! 3 6! # • # * ! '! ! ) # ! #!! # #!! # #!! ' . & & F & 5 5 5 5 & & & & ?! . 5 . J 6 ( * J * # # 3 H F F F5IJ H F F&IJ H F F F &IJ H F &IJ H F F F F FNI & # " # G '! ! ' j ( 3 # #! # # * (( 5) ! ! '! ! !! * !!!* !" ! 7 * " # . # 5 ! +* # +# + # # ! [ ] # # $ J * # * 0 ** 3 # # #8 "& 9 . H F F F &IJ ! * ! ## (+ F" # 3 ! " * * # # 3 6 ( * ! ! ! 3 9 3 * # # TB = TBij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e% 5* ## ##! ! " 5* ## " 6 # -* ! * ' # # ! '! ! # ! " * # ! ## # # ( * • + & +# " !J # # *! ! '! ! !# # ( ) #J ( ! # # # !! * #! # # ' " ( # #!! # # # # # ! ! ( ) # * .! * +# ( # #! # " 8 * ## 9 # # # # " 1 * # # * ( #!! # J " ! * *! # # " ** # !J Y TBij # !J ** ( ( ! * J ! * # + TBij ≠ 0 ( * ##! +! *! ! # + 3 [ # # # # 3 # [ ] # ! ( TB = TBij * , # #!! + # # ( ! " *! " J* # * " ( #!! # 3 !! ( *! 3 * * # #!! # " *! ! • #J ! + * 6 5 #J 3 ! #J 1 J * # #!! # " ! # ! 3 # ! '! ! 6 (! - (+ !* . # #!! # ( # * ! # 1 # # # )! # ! ! )! ) ! # # " 6 # ( #!! # #J # * ! '! ! # # *! 3 J # * # (! ! * ##! '! ! & ( " 3 * ! " # ! & *! ( ! 3 ( !# # . * ( # # #!! J ! + # !R ! *! * # ( * ! ! 3 * J ** (+ : 5) ! * ( * ( # * ! ! '! ! 8 TBii 9 # (+* # #!! # * [ ] # " ## # ( ! * #J * #J ! # ##! +* # ! '' 8% 9 TBii TRii = 1 ! #!! #!! * " ! ** TBij 0 ≤ TRij ≤ 1 3 * * # * # ! " # TR = TRij = " ( # # ! * [ #! TE = TE ij ! " ! # #!! # +# ] # !J !#! J TE ij = 1 si TRij ≥ seuil TE ij = 0 si TRij < seuil 8% 9 TE ii = 0 # ! * ( #J " ( ' # # ! ** ! # ! ! =) ( . * 6 1 *! (! * # # ( # J ' 3 #!! # J # " !# ! # J # * * *! " # 3! J . " *! ! * # # # * ! '!! # ! # #!! # *J # * # *! (! -# #! . ! '! ! " # 5 # # 3 ! ! !# # " J # # # * 6 #,# -( # # " ! ** ! ## . Y # TBij j 6 # # ! # # 3! ( " 1 # #! # # ( ! * ! # # * ( "! * ' 3 ! #J #!! # (+ ' ! # 3 ! [ ] TB ji TA = TA ji = 5 * ## # # #! . ! '! ! " # ! # ( # ! " * 6 #!! # 1 5 # ! # * * ## # 3 ! J # * ## # 5 # 3! ## ! J & != . # ( ! # # " # 6 *[ ( '' TE ! "! # ! # ( , !' ( ! R ! *! - # !! 3 # (( 6! # * ## # J #! * # ! # * #3 ** # * = '* 6! +! *! # 8%$9 TB jj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' ! # ( ! 3 e " *! ! ?! . % # * # *S ( * # ' ! S*! ( ! 3! # !* # +* S !# ** !* # * ! # ! 3 # $ ! #! # # )( # # ! # ! S S*! ( ! 5) 3 # ) '' . #! ( * ( ! 3 ( * $ XTJ ! * # ! " KA * ( 8!@9 ( * **-* 8 P) J J ) (! 3 ! ' ) '' " ) . ! ## ** . " **-* ( ! T? # * # ! # ## ( ( , (! 3 # 3S # " 4 ! J " ! * #4#,# -( # • " ! ( * 6 J # ! ** # # # # ! ** ) ** ! 8" ! ?! . # ) ( . ! •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b F +# >.Z ) l l 4 4 4 $ 4$ 4 4 4 $ %4 $ 4 4 4 0( 4 % / $ $: : $ :% 4 ##! 83N9 # * * ) '' #! # & + % ! ** # . * ( ! !* 5) ! ##! n . * + #" # * !# ' " # ** * J (R( J * ! " .! ## # n * ! * 8 S9 ( *! # # # #4#,# -( # ( * ( +! ! ( ! # ! J '! # (+* # ! & * # 83<9 ! ! " #! . 1 " # ) . 8B%J B J B 9 -* " ! # ( ! 3 * # !** # ! ! # @ ' ! n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'' !** # 8 ' ! - # * ! * # " ' 5) ! *! ! # ! +G ! ' . # $ 1 / 6 & ( # *! # 8 # 8 ∆p 9 1 * # (! #! * # * # # " ##! +* ∆p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( %/J # ! )4 * 0I # # ,J J= ) . 4+ # ## = ;;; H * # :: #* J (+ 5 ! ## . ! ]J c E . ! * 40 # # 0d 45 8 H C; //I c E )! 3 . ! * 40 * 7 *!- ! 6 ( * # = H C; : I & " J; ! " #! . ! * 40 & -" " ! ## # ; ?J # #,(+ *! 3 ( * 6 #= ( #J #) >* b " & %4/ J // (! 7 +*! #) #J //: * #CJ= ( * = !7 * # * # " " ! # *) b XJ ; ' ## # // J $/4 "# J $ F! +*!. HF L: I F E X ]5 J *. HF / I HF /$I F %I . !! " 3"'# 5J > " # > ##J 7 H5 !)(# = 3# F! #) 7 =7 b V XJ ; )( ]J + # #= & ##J 7 $ % J b!) ' EE, 3 +G ! " J b V XJ /: 8 J A6' )! ' ! ; ! " #!, // / J = ) ## $# & * ' X b* . ! # " ,! J Z 0 //$ 5 #! (! J " * )-# *@ * 5 * , J % H5 / I 5 ## ] 7J #b! X!0 J = ' ! * ! ! 79!39 ? J 7 /%I 5) !] H5 / I 5) .] J ' H5 0I H5 A I ! 7 ) # 3 ( 70; I ! ' 4 J / 4 ! # ! +* # 5) b 5 > J p A H5A I 5 * ( : ) ! # I # 5 (+ I & . * # 0J F "! #! ( ! ' 7J = ) # ( #J 7 ! #J " ' J //% ## Q7 $J ' (! *, ' ? ? #7 # ; ) .Z 4 ''Jq ;;; D $J ! '' / ! # + # " J =7 !. K" # . #! # # ' * " * + # * #,# J K8 ?GG?J (= % 4 G! J 5) + ! #,# * .! # !#J 057 J C !. . !! & 8 4 ! !. ' ## ! #J G 7J L ( ! ! * .! 3 ) ! . !! *! ! % J * ) F *' J & T !. #J F = H& C 4 : # U % H5A0 # Z Z bbb ) (4 ## E ]J C ( <R ' 3 # 3 # # ! ( ! =3 ]J 0 * #! . J ( , // E ]J = # #3 J (#= " ( b X= 7 ! ( ( ]J 0 * *, (!* * ) ( #J 7 *J " * 5 * ( #,# # * #CJ= . H5A ! '! ! * + ( * J )' // J B* * 7 % /4%% J H5 # # !1 + # ## 7 J >) (# = ( +* J ? ( + . ( J -* #= 5) ! J F! +*!. H& )! %I & #) J . #b (, * ##! '! ! 7 H& : I & # ' J B X ' * #; )# 7 J !. , ) * H&;F/ I # ! # " & + ! ## M # ! B J =? EE,4* .! + # ) (! * % ## # = 7 4%$ J F* " % # "# 5 ( #! # ' )B* *# # + & "# #J # ) . J " (+ )J /: FJ . "& ## # ) ! &!. # ! 0 ! J ( #J 7 ! #J // H&;F/%I & + ! ## F J 0 ## X b* . + 0J = # ! # ! * * ## #J= 3 ! #! 8 * b!) ! J " * ( * $J 4 $ J //% H&;& I H&;& H +I & J 3 7# & J 3 7# /:I * ' 7J X& J =5) J 7J X & J =5) J J ?! . ( !! J T! * ,4 J= b * * + *!. = ;;; . H 5 I % J //: + J # . # 7 H? V/$I ' * %$I # 7 ; J 7! > b* . #X,4 ) ! &! # " , :/I ? X7 0 J Tr +# 7 " + # ) ( #=J J I C *! % ## CJ ]J =? 7 . ]J = + # ' & * ]J ? * V J 6 $J (+ 0!!. & * //$ (! #! # #!. X 7 0 J 5* X # ' *# = ;" $# !J ; * 4 ! ( & % !. * 4 (,) 7 J =? ##J 5 *! ' J # * ( . *J "!b = ##Z 0 ! +* (#= X b ! 8 7 # 9 ! "# ! **J /:/ 0J . ! * 40 ! EG U J 7 7 * HC " * # " # J $ % ) B * /J J2 # ! 0 ) #=J # ! J= ) # ' ( * !* ( # ( #! ;" JB* J / D ::J /%$ # H? 4( 3 *! ! " ;77 G< ) 7 ? ,, ?! #) ! * + # ) H? J )! 3 #=J J b X' " # " 40 ## ,-# +! : !! J T! * ,4 ! ! # ! ! ]J = 0& F;8 J "! ** J ! c # *! ! c G? # T . J * !. c# ! ' " (+ !* 7 J # #! # * .s = # : HC F! +*!. /:I C * ]J ! "# & XX J HC AT I /I & H " Z Z C # '# & ;;;7 7J * ]J ] 5 J =7 !( /%I # #! . ( H / I # H] //I J= H] / I " #E 0 J B X ; H> 0 %I %J ! BJ = # > '( J #* J T! * ,J >( ## b#X, J " # ! J J * # 4:% J //% #! #0 ) #D $%/4$ !. J // "!b = 7 /// ( ' # ! ( ! * &!. !. ! * ! ## ) B* " . ! * 40 ! ]J ! "!, + b#X, J +!# ' 3 *! ! " J !. #! #= J % /4 +!# ., ' ! J # #J B * % J )C J J ' . ! * 40 * :/4%%:J ! # &5J; ! ]J = !. # *'4 !. J ]! % "! #! #! #= 3 ' ( * 6 73 ?GGA) . " J " (+ 4 !. !# ! ' ( ! // $ J > ! # F J =2 6 ! "# )-# " 0 B J =5* ##! '! ! ?GG@J T #)!. 5)! * J 5)! * J %4 >) , J b 7 !. H>; :$I " b V XJ // + # >( > ' b 7 ]J ! 7 H>A /%I " : ? (+ # 7#" ## # # H>A / I ! / /. ) * .!#J J 7 ]J =& )! 3 # = !7 I " // * H> 0 # 8 ( /J )B* # %%/J H> ;/ I ( ) #= ! 3 ## # $ D $$J )! # "! * & J ?*, # + ' ' EE, J " ** # )B* "& * ) B * % J # !! ! +* // " #; ,0 !. b!) # #! # ' ( "! #! J ? >J 0 ] !. ! =J 7 ]! * 6 ) (* &!. J ; ! ( #J ##!.=J ( 7 * . ! #J *J J =? X b* 0 ! # b )Z ! # ( * ! " H )Z . 3 J > ## * T 5 J =? EE, * # ! 6=3 / I $ ! Z ((Z # '# ( * b V XJ //: 7 ) HC; ; )! b! ## ? *! ! " #! ( * /:$ * # ! ! =J $ ' ) * .,J # ## ## ! ## J ) B* # //% /J %J F! +*!. H ; / )! I 5J ; # * J ) "! # 4 %/4 ) ' I ( % 0 ( ! J % *!. ! # ! #,# ! . b X=J 3 4 ! " ), ## ) B* ' ' EE, * .! (# = !"44* $ % ) J G! ** J *! ! # *! ! > " # > "! # ! !. # ( ! # ! ' & ! 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nnexe B. Manuel d’utilisation de SALSA User's Manual SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 144 Annexe B. Manuel d’utilisation de SALSA Annexe B. Manuel d’utilisation de SALSA 145 Contents 1. ABOUT THIS USER MANUAL 149 2. SOFTWARE INSTALLATION 149 2.1 INTRODUCTION 2.2 TECHNICAL DATA 2.3 SALSA SETUP 149 150 150 3. GETTING STARTED 150 3.1 SALSA MAIN SCREEN 3.2 SALSA – OFFLINE 3.2.1 FUNCTIONS OF THE OFFLINE STAGE 3.2.1.1 Loading of the training data file in the specified format 3.2.1.2 Selection of the Classification Algorithm - Quantitative descriptors 3.2.1.3 Selection of the connectives and the exigency level 3.2.1.4 Selection of the classification mode 3.2.1.5 Data Processing, Treatment and Visualisation 3.2.1.6 Classification Results and Visualisation 3.2.1.7 Mapping Classes to States 3.3 SALSA – ONLINE 3.3.1 ONLINE CONFIGURATION 3.3.2 ONLINE CONNECTION 3.3.3 RECEIVING ONLINE DATA 3.3.3.1 Visualizing incoming data 3.3.3.2 Visualizing the Current State 150 151 152 152 154 154 155 156 158 162 164 165 165 167 167 168 4. APPENDIX 168 4.1 ONLINE DATA GENERATOR 168 5. REFERENCES 170 SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 146 Annexe B. Manuel d’utilisation de SALSA 147 Annexe B. Manuel d’utilisation de SALSA Figures Table Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 1 - SALSA - Main Screen: Options .................................................................. 151 2 – SALSA - OFFLINE Main Screen.................................................................. 151 3 – Example of the .DAT file .......................................................................... 152 4 – New Context & New Population buttons ..................................................... 153 5 - Loading a New Context: Induce or Retrieve ................................................ 153 6 – Loading a ‘New’ population ...................................................................... 153 7 - View/Edit File & Imposed Classification Windows ......................................... 154 8 - Presence Function selection ...................................................................... 154 9 – Connectives & Exigency selection.............................................................. 155 10 – Classification Mode selection .................................................................. 155 11 – Max. Variability & Iterations selectors ...................................................... 155 12 – Supervised Learning Options window....................................................... 156 13 – Recognition Options window................................................................... 156 14 - Quantitative & Qualitative Descriptor windows........................................... 157 15 – Context Description window ................................................................... 157 16 – Active quantitative descriptors graph....................................................... 158 17 – Active qualitative descriptors graph ......................................................... 158 18 – Application ready for classification........................................................... 158 19 – Classification Results using Supervised Learning ....................................... 159 20 – Classification visualizing buttons ............................................................. 159 21 – Profile for each Class............................................................................. 159 22 – Class Details window............................................................................. 160 23 - Recognition Results Window ................................................................... 160 24 - Reference Classification Graph ................................................................ 161 25 - Analyse Window.................................................................................... 161 26 - Membership Graph ................................................................................ 162 27 - Mapping Classes to States Window .......................................................... 162 28 - Creating the list of States ....................................................................... 163 29 - Assignment of Class to State .................................................................. 163 30 - View/Edit Mapping & States buttons......................................................... 163 31 - Current States Graph............................................................................. 164 32 - SALSA Online Main Screen...................................................................... 164 33 - SALSA Online Configuration .................................................................... 165 34 – List of Online Quantitative Descriptors ..................................................... 165 35 – Ready to Send Online configuration ......................................................... 166 36 – Ready to Receive Online data ................................................................. 166 37 – Path for data transfer ............................................................................ 166 38 – Format for the online data file ................................................................ 166 39 – Format for the output file by SALSA ........................................................ 166 40 – Stop receiving data ............................................................................... 167 41 - Online Quantitative Descriptors ............................................................... 167 42 - Online Qualitative Descriptors ................................................................. 167 43 - Current State ....................................................................................... 168 44 - Bar Graph with States............................................................................ 168 45 - Table with change of State occurrences .................................................... 168 46 - Online Data Generator setup window ....................................................... 169 47 - Online Data generator Main Screen.......................................................... 169 SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 148 Annexe B. Manuel d’utilisation de SALSA Annexe B. Manuel d’utilisation de SALSA 149 1. ABOUT THIS USER MANUAL This manual is a reference for the qualitative situation assessment of a given industrial process. It provides the guidelines for the construction of classification systems for the offline characterisation and online identification of the functional states of a process by means of a learning classification technique using qualitative and/or quantitative information of the process variables and the expert's knowledge. The manual is divided into two main chapters as follows: Chapter 2: It covers some technical information, the installation of the application and the system requirements. Chapter 3: This section provides sufficient information to use the application. It shows an overview of the basic features of SALSA as well as the steps that must be followed. SALSA application as well as this user manual were developed by the DISCO group from the LAASCNRS. All rights reserved. For further information and details please contact: Joseph AGUILAR-MARTIN ([email protected]), Tatiana KEMPOWSKY ([email protected]). This is a January 2004 version 1.1 of SALSA User's Manual. Updates will be constantly made. 2. SOFTWARE INSTALLATION 2.1 Introduction The main function of SALSA is to identify in a qualitative way the current functional state of a supervised process using the online symbolic and/or numeric information from the process variables. To do this SALSA has two main stages. The first, offline stage corresponds to a learning stage which, from a given data set, generates classes and assigns significant functional states to these classes by a constant dialogue with the expert. The learning stage includes: • Class Generation: a clustering classifier with the capacity to perform a supervised or unsupervised learning procedure. Interaction with the expert is necessary in order to tune up the classifier parameters. • Mapping Classes to States: a dialog tool with the expert, for the assignment of classes, or groups of classes, to meaningful functional states. The online stage is the recognition stage, which will determine de functional state of the process from online data of plant sensors and actuators or other information from the process variables. During plant operation SALSA generates the current functional state of the supervised process by using real-time (online) data. SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 150 Annexe B. Manuel d’utilisation de SALSA 2.2 Technical Data • • • • This version of SALSA was programmed in plain ANSI-C, although the underlying philosophy is object-oriented. SALSA is a standalone tool, it can operate on its own without need of another software tool. The selected platform for the application’s development was LabWindows/CVI 6: - Graphic User Interface - Source Code programming The software may be installed in Windows 98/NT/2000/XP 2.3 SALSA Setup 1. From the installation Folder InstallSALSA04vbat double click at the ‘setup.exe’ file and let you guide by the installation wizard. 2. Once SALSA application has been installed verify that the directory SALSA04v1 has been created. 3. Verify that the sub-folder \DataTr has been created in the SALSA04v1 directory. This folder is required for the online data transfer between SALSA and another application running online. 3. GETTING STARTED 3.1 SALSA MAIN SCREEN When SALSA is launched, Figure 1 shows its principal window. There are three main options to choose: • OFFLINE: Learning Stage. This concerns the design and construction of a classification system, from a set of experimental data. It consists of two main stages: a learning stage (unsupervised or supervised) and the assignment of the resulting classes into significant functional states. Active participation of the process expert is required. • ONLINE: Recognition Stage. Using a previously generated classification system, the current functional state of the process will be identified. • QUIT: This option exits the toolbox. To exit the application the user must first close (quit) the option he is working on (one of the above) an exit by clicking the QUIT button in the principal MENU window. Annexe B. Manuel d’utilisation de SALSA Figure 1 - SALSA - Main Screen: Options 3.2 SALSA – OFFLINE Figure 2 – SALSA - OFFLINE Main Screen SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 151 152 Annexe B. Manuel d’utilisation de SALSA Figure 2 shows the main screen for SALSA offline (learning) stage. The following sections give some guidelines of the features and use of this stage. 3.2.1 Functions of the Offline stage 3.2.1.1 Loading of the training data file in the specified format • Format Specification: The format for the input (training) files (the context data base and the population to be classified) is a text file with a .DAT extension. Lines can be blank ones, comments (MatLab style, always preceded by “%”) or observations. These observations or individuals present values for the context descriptors in columns separated by ‘blank spaces’ and may have an additional field if we are interested in supervised learning (i.e. the additional field correspond to the pre-defined class). - Type of Data: a descriptor is assumed to be quantitative if it starts with a number, a point or a minus sign, and qualitative otherwise. - The name or tag: for each descriptor must be a line with the “&” sign at the beginning of the line, and they should be also separated by a ‘blank space’. - Pre-defined classes: the additional field for supervised learning will indicate the class where the individual is to be assigned. The algorithm expects these numbers to appear in a semi-ordered way, that is, the class “1” has to appear at least once before the class “2”, there has to be at least one “2” before any “3” and so on. When these numbers are greater than 0, they create a class. The zero (0) means that the individual is part of a population that later will undergo pattern recognition. Figure 3 shows an example of the training file format for the case of supervised learning. Figure 3 – Example of the .DAT file Annexe B. Manuel d’utilisation de SALSA 153 • Loading a Training File: To load a file the user must select either the ‘New Context’ or the ‘New Population’ buttons (Figure 4). Although it is possible to start with any of the two options it is recommended to first load the context data file and then the population file. Figure 4 – New Context & New Population buttons - Loading a new context: The context file normally contains the number of descriptors and its type (numeric or symbolic). It may be induced from a population (or plain text) file with a .DAT extension or retrieved from a binary file previously saved with a .CON extension (Figure 5). If the context has been ‘induced’ from a population the application will demand the user if he wants also to load the corresponding population to be used for classification. Figure 5 - Loading a New Context: Induce or Retrieve - Loading a new population: This file contains the individuals that will be used for training. It must have the same number and type of descriptors as in the context and it may contain an additional field corresponding to the pre-defined classes (to be used for the supervised learning case). A dialog window will appear asking the user if he wants to load the corresponding context (Figure 6). If changes have previously been made (e.g.: a descriptor has been disabled) to the context, they will be lost if the user accepts loading the new context also. Figure 6 – Loading a ‘New’ population • Because of the presence of commented lines (with a ‘%’ symbol) and a possible supervised learning field, there exist a couple of dialog windows that ask the user if the population file is to be looked at or edited with a Windows text editor, and if the last field is to be treated like an imposed classification value. See Figure 7. SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 154 Annexe B. Manuel d’utilisation de SALSA Figure 7 - View/Edit File & Imposed Classification Windows 3.2.1.2 Selection of the Classification Algorithm - Quantitative descriptors To make the different calculations required in the learning and classification procedures it is necessary to choose the type of algorithm to be used for the quantitative descriptors (Figure 8): Figure 8 - Presence Function selection • Lamda1 : ρ (1− ρ) • Lamda2 : Kρ (1− ρ) • Lamda3 : ρ • Lamda4 : Kρ 1− x x 1−x x 1− x − c (1− ρ) 1− x − c ( ) ρ 1− ρ 2ρ − 1 x −c (1− ρ) (x−µ) • Gauss1 : e 2σ 2 2 (x−µ) − • Gauss2 : Ke 2σ 2 K= log x −c ρ log ρ − 1 K= c 1− c 2 ρ − ρ c (1 − ρ ) − ρ 1−c (1 − ρ ) 2 − K= σ0 σ 3.2.1.3 Selection of the connectives and the exigency level • Connectives: The user may choose between two different families of logic operators (T-norms & dual T-conorms functions) in order to aggregate all the Marginal Adequacy Degrees of an individual to a class (Figure 9). These operators are: Product-Probabilistic Sum (T-norm & Tconorm) and the Minimum-Maximum (T-norm & T-conorm). Annexe B. Manuel d’utilisation de SALSA 155 • Exigency Level: The user may also use mixed connectives of the same family by choosing an exigency level (α) between 0 and 1 (Figure 9). In recognition we call α the exigency level and in self-learning it is called selectivity level. By changing the value of α different partitions based on the same data used may be obtained. Thus, as the value of α increases more classes will be created or in the case of recognition a greater adequacy is required of a measurement to thee assigned to a pre-established class. Figure 9 – Connectives & Exigency selection 3.2.1.4 Selection of the classification mode When a file has been loaded for the first time the classification mode will be selected according to the characteristics of the file. Figure 10 – Classification Mode selection • Self – Learning: Unsupervised learning (No pre-defined classes). There is no preliminary knowledge. SALSA creates a group of classes using as base the values of the descriptors from the individuals of the data file that has been loaded. There are two selectors that may be changed by the user only when unsupervised learning has been chosen. These are the maximum desired variation percentage and the maximum allowed iterations number (see Figure 11). Figure 11 – Max. Variability & Iterations selectors They are necessary because in unsupervised learning the class parameters vary, so to obtain some stability and to overcome in a certain way the effect of the observations ordering, the same population is classified several times until a maximum percentage of individuals changes from one iteration to another. However the time to reach stability could be very long, so a maximum number of iterations has also been introduced. This means that the population will be presented to the existing classes once and again until either no more than the specified percentage of individuals varies its assignment from the previous iteration or the cycles limit has been reached. SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 156 Annexe B. Manuel d’utilisation de SALSA • Supervised Learning: The loaded population has a class imposition field. This option allows performing a different number of choices, like learning from an initial set of classes, which can be modified by adding new classes or by updating their parameters or both (Figure 12). Figure 12 – Supervised Learning Options window • Pattern Recognition: This option has two alternatives, either the user allows unclassified individuals, meaning that an individual has not been recognized in any class (its adequacy degree is lower than the minimum threshold) and has been placed in the NIC class, or force every individual to be assign to a class, in this last case the Non-Informative Class is not taken into account for recognition (Figure 13). Figure 13 – Recognition Options window 3.2.1.5 Data Processing, Treatment and Visualisation • Once the context and the population have been loaded, SALSA identifies the number of descriptors and its nature, and it internally rescales quantitative values between 0 and 1 (“Normalisation” process). • To make possible a direct confrontation between individuals and classes, it is necessary that the descriptors used for observations are the same (same order) than the concepts used for classes (context). However, whenever the number of descriptors, or its nature, or the actual values exhibited in the population do not fit the ones stored in the context, the problem is reported in an error file, named “…\error.out”. The error is explained in this file with an indication of its nature type and location (number of individual and number of descriptor if necessary). A message window is displayed to notify the user. If the population can still be loaded, for instance, if a value is lower than the minimum value SALSA will take into account a 0 or if a value is greater than the maximum value SALSA will replace it with a 1, the process will continue. • The user may view each context descriptor separately and in a detailed way, as well as the values of each individual (Figure 14). It is possible for the user to change the descriptors he wants to use for learning and classification; he can enable and disable a descriptor for classification. Annexe B. Manuel d’utilisation de SALSA 157 Figure 14 - Quantitative & Qualitative Descriptor windows For qualitative descriptors, a histogram of the number of individuals for each modality is shown in Figure 14 as well as a small window that gives the name of the modality and the exact number of individuals. For quantitative descriptors a graph representing each individual with its normalized value is shown. There is also a text window with all the information about the loaded context (see Figure 15). Figure 15 – Context Description window • By clicking the ‘View’ Population button all active descriptors are graphically displayed. There is a graph for quantitative descriptors where for each individual its normalized value is shown, and at the left part of the graph the names of the active quantitative descriptors are displayed. For qualitative descriptors, modalities corresponding to each individual are shown in another graph. The user may change from one graph to another using the corresponding option: ‘View Qualitative Descriptors” or “View Quantitative Descriptors”, which is located at the right bottom corner of each graph. (See Figure 16 & Figure 17). SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 158 Annexe B. Manuel d’utilisation de SALSA Figure 16 – Active quantitative descriptors graph Figure 17 – Active qualitative descriptors graph For qualitative descriptors we represent each modality with a similar but different colour. 3.2.1.6 Classification Results and Visualisation • After successfully loading context and population (green LEDs for Context and Population and name of the loaded file), it is possible to put the algorithm to work by pressing the Classify button. See Figure 18 Figure 18 – Application ready for classification Annexe B. Manuel d’utilisation de SALSA 159 • Once a classification has been made, SALSA graphically displays the class assigned to each individual (Figure 19), and by clicking on the Class Profile button (Figure 20)the profile of each class will be displayed. Figure 19 – Classification Results using Supervised Learning Figure 20 – Classification visualizing buttons Figure 21 – Profile for each Class SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 160 Annexe B. Manuel d’utilisation de SALSA Figure 21 shows the profile for each class, each descriptor is represented with a different colour, for quantitative descriptors their mean value is shown, for qualitative descriptors each modality is represented with a different colour, and their occurrence frequency is shown (the sum of all modalities for a qualitative descriptor must be 1). It is also possible to view the complete information for each class in text mode (Class details button). See Figure 22. Figure 22 – Class Details window The Class Details window shows all the parameters values for each class. Depending on the type of algorithm used for classification different information will appear. Also the number of elements (individuals) assigned to each class are shown. • When Recognition is launched, and if the loaded population has pre-defined classes a 'Results Window' is displayed with different information (Figure 23): number (and percentage) of individuals not recognised, number of individuals with pre-defined classes, number (and percentage) of individuals badly recognised, and the list of the individuals badly recognised. Figure 23 - Recognition Results Window Annexe B. Manuel d’utilisation de SALSA 161 • When a population with pre-defined classes has been loaded the "View Prof CL" button is enabled. By clicking this button the pre-defined classes for each individual are displayed (Figure 24). Figure 24 - Reference Classification Graph • SALSA automatically creates a result file where each individual has its corresponding class. This file is a text file and has a .RES extension. It is possible to save the classification result as the ‘reference’ one by clicking on the "Save Ref. CL" button. The user may choose the name of the classification, and he can decide to set it as the reference for comparison with other classification results. • The ANALYSE button shows the resulting membership matrix of each individual for each existing class (Figure 25). This matrix can also be graphically viewed by clicking on the View Membership Graph button (Figure 26). The user may save this matrix in a *.CSV format by clicking on the EXPORT button. Figure 25 - Analyse Window SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 162 Annexe B. Manuel d’utilisation de SALSA Figure 26 - Membership Graph 3.2.1.7 Mapping Classes to States • When a suitable classification is obtained the user must assign the different classes into representative states. This is done using the ‘Mapping to States’ button, located at the right bottom side of the ‘Current Classification’ graph. • The user must first of all create the list of possible significant states, by clicking at the ‘Create List of States’ button. (See Figure 27). Figure 27 - Mapping Classes to States Window • The user can add and remove the number of representative states of the process. Once the list is completed the user can close the window by clicking the OK button (See Figure 28). Annexe B. Manuel d’utilisation de SALSA 163 Figure 28 - Creating the list of States • After the list of states is created the table of classes and states must be filled up. A class can be mapped into a state from the list by double-clicking into the cell of the corresponding class (Figure 29). Figure 29 - Assignment of Class to State • The table may be viewed or edited with the "View/Edit Mapping" button. The user can also view the assignment of states to the individuals by clicking the States button (Figure 30). Figure 30 - View/Edit Mapping & States buttons SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 164 Annexe B. Manuel d’utilisation de SALSA Figure 31 - Current States Graph • SALSA allows saving the different contexts used (active descriptors, presence function for quantitative descriptors, connectives and exigency level), as well as the classification results for future use. Context will be saved in a binary format with a .CON extension. Whereas, a classification parameters (classes and states) will be saved with in a binary format with a .CLA extension (See Figure 18 for location of save buttons for context and classification). 3.3 SALSA – ONLINE Figure 32 - SALSA Online Main Screen These are the steps the user must follow for online recognition. Annexe B. Manuel d’utilisation de SALSA 165 3.3.1 Online Configuration The online configuration button is the only enabled button when the user chooses the online stage of SALSA in the main menu (Figure 32). Figure 33 - SALSA Online Configuration • The user must load up the context that will be used for recognition, i.e. the number and type of descriptors used on the offline stage for the creation of classes (Figure 33). This file is a binary file with a .CON extension (this file should have been previously created during the offline stage). • Once the context is loaded the user must choose a classification file with .CLA extension. This classification must have the same number and type of descriptors as the loaded context. Once the user has loaded the classification, the learning parameters used for this classification will appear under the labels: “Presence Function”, “Connectives”, and “Exigency Index”. • Once both files have been loaded the user may close the configuration window by clicking on the OK button. 3.3.2 Online Connection Once the configuration window is closed the user will see the list of the quantitative descriptors (or the qualitative descriptors) that will be received online. See Figure 34 Figure 34 – List of Online Quantitative Descriptors SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 166 Annexe B. Manuel d’utilisation de SALSA Figure 35 – Ready to Send Online configuration The Send Configuration button is enabled (Figure 35). The user must click this button for SALSA to get ready to receive online data. If the connection is successful SALSA will start checking for a new message to arrive. Figure 36 – Ready to Receive Online data The Get Data button is enabled (Figure 36). SALSA will display the path where the data transfer will take place (Figure 37). The "DataTr" folder must contain the file with the current information message. This file has the name: d_in.dat. It contains one single line in the same format as for the training files (see Figure 38 ). Once SALSA reads the d_in.dat file it destroys it, and when recognition is made an output file is generated (d_out.dat see Figure 39) the application that is sending the online data must destroy this file and create the d_in.dat file to tell SALSA that a new message has arrived. Figure 37 – Path for data transfer Figure 38 – Format for the online data file Figure 39 – Format for the output file by SALSA Annexe B. Manuel d’utilisation de SALSA 167 Figure 40 – Stop receiving data The user may stop receiving messages with the Stop Data button. 3.3.3 Receiving Online Data The online screen offers the visualization of the incoming data, the current state, the state of the plant at the instance of an arriving message it has a historical table with the time at which there was a change of state, with its associated class and Global Adequacy Degree. 3.3.3.1 Visualizing incoming data The user may choose to visualise either the quantitative or the qualitative data. Figure 41 - Online Quantitative Descriptors Figure 42 - Online Qualitative Descriptors SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 168 Annexe B. Manuel d’utilisation de SALSA 3.3.3.2 Visualizing the Current State • The user may visualise what the current state of the plant is, with the rectangle at the top left corner of the screen: Figure 43 - Current State • A bar graph with the state of the plant each time a new message data arrives is also available: Figure 44 - Bar Graph with States • A Table with the time of the change of state, the state, its associated class and the membership of the message to that class for the last 10 states is shown. The current state is highlighted with the corresponding colour of the state: Figure 45 - Table with change of State occurrences To EXIT the online screen the user must FIRST STOP the online connection with the STOP button and then EXIT the screen with the EXIT button. 4. APPENDIX 4.1 ONLINE DATA GENERATOR This application permits to generate online data from stored data files. It can be used in order to validate SALSA Online stage. The Online Data Generator is a standalone executable that may be installed in the following way: 1. In the InstallationSALSA04v1 Folder look for the sub-folder: Package. Click the 'setup.exe' file. Annexe B. Manuel d’utilisation de SALSA 169 Figure 46 - Online Data Generator setup window 2. Once the installation is completed it is possible to generate online data from a file. Figure 47 - Online Data generator Main Screen 3. First of all choose the data file with extension *.DAT and click the Set Data File button (see Figure 47). 4. Choose the folder where the online data file must be created (it must be in the \DataTr folder created by SALSA: …\SALSA04v1\DataTr) and click the Set Folder button 5. The RUN button is enabled. When clicking this button the program will create the d_in.dat file, the information of this file will be displayed in the 'current datum' window. Online Data is generated every two seconds. 6. When SALSA creates the output file d_out.dat the program will display its contents in the 'Current Class' window, and will destroy this file in order to generate a new d_in.dat file with a new datum. 7. To quit the application click on the EXIT button. SALSA – User's Manual February 2004 v1.2 Copyright © 2004 LAAS-CNRS 170 Annexe B. Manuel d’utilisation de SALSA 5. REFERENCES [Kempowsky, Aguilar-Martin, Le Lann, Subias] Learning Methodology for a Supervision System using LAMDA Classification Method. IBERAMIA’02 - VIII Iberoamerican Conference on Artificial Intelligence, Sevilla (Spain), Nov. 12-15 2002 [Kempowsky, Aguilar-Martin, Subias, Le Lann] Classification Tool based on Interactivity between Expertise and Self-Learning Techniques. IFAC-Safeprocess 2003, Washington D.C. (USA), June 9-11 2003