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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Analyse des CorrespondancesMultiples
François Husson & Jérôme Pagès
Laboratoire de mathématiques appliquées - AGROCAMPUS OUEST
1 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Plan
1 Données - objectifs
2 Etude des individus
3 Etude des modalités
4 Aide à l’interprétation
2 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Les données
1
i
j
Ind
ivid
us
I individusJ variables qualitatives
Exemple : enquêteI personnes interrogées
1. Données
1 JVariable qualitative (couleur)Modalité (bleu)
xij : modalité de la variable jpossédée par l’individu i
Variables qualitatives
Vij
3
I
Ind
ivid
us
I personnes interrogéesJ questions à choix multiples
Tableau de codage condensé
Exemples de questionCatégorie socio-professionnelle(ouvrier, employé, etc.)
Codage de xij
Littéral : « bleu »Numérique : 2 si « bleu » = 2
Situation de famille(célibataire, marié, etc.)
I individusJ variables qualitatives
vij : modalité de la variable jpossédée par l’individu i
Exemple : enquête où I personnessont interrogées sur J questions àchoix multiples
3 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Les données
Kj = 5
Variables qualitatives
1
i
j
Ind
ivid
us
1 J
vij
1
i
j
Ind
ivid
us
1 J
yik
1 k Kj
Modalités
K Σ
J
28
i’ 0 1 0 0 0
I
Ind
ivid
us
i’ marié
I
Ind
ivid
us
Tableau disjonctif complet (TDC)
J
IJIpk
I
4 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Objectifs – problématique
1 Etude des individusUn individu = une ligne du TDC = ensemble de ses modalitésRessemblance des individus Variabilité des individusPrincipales dimensions de la variabilité des individus
(en relation avec les modalités)2 Etude des variables
Liaisons entre variables qualitatives(en relation avec les modalités)
Visualisation d’ensemble des associations entre modalitésVariable synthétique
(Indicateur quantitatif fondé sur des variables qualitatives)
⇒ Problématique voisine de celle de l’ACP
5 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Les données loisirs
• Extrait d’une enquête de l’Insee de 2003 sur la constructiondes identités, appelée « Histoire de vie »
• 8403 individus• 2 sortes de variables :
• Parmi les loisirs suivants, indiquez ceux que vous pratiquezrégulièrement : Lecture, Ecouter de la musique, Cinéma,Spectacle, Exposition, Ordinateur, Sport, Marche, Voyage,Jouer de la musique, Collection, Activité bénévole, Bricolage,Jardinage, Tricot, Cuisine, Pêche, nombre d’heures moyen parjour à regarder la TV
• le signalétique (4 questions) : sexe, âge, profession, statutmatrimonial
6 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Les données loisirs
Activité Effectif Sexe Femme 4616
Ecouter de la musique 5947 Homme 3787
Lecture 5646 Age [15,25] 857
Marche 4175 (25,35] 1302
Cuisine 3686 (35,45] 1646
Bricolage 3539 (45,55] 1837
Voyage 3363 (55,65] 1257
Cinéma 3359 (65,75] 937
Jardinage 3356 (75,85] 482
Ordinateur 3158 (85,100] 85
Sport 3095 Statut Divorcé 792
SignalétiqueActivités pratiquées
Sport 3095 Statut Divorcé 792
Exposition 2595 matrimonial Marié 4333
Spectacle 2425 remarié 404
Jouer de la musique 1460 Seul 2140
Tricot 1413 Veuf 734
Activité.bénévole 1285 Profession agent de maîtrise 735
Pêche 945 cadre 1052
Collection 862 employé 2552
Nb d'heure à regarder la TV 0 1017 manoeuvre 792
1 1223 ouvrier 1161
2 2156 technicien 401
3 1775 autre 212
4 2232 Non réponse 1498
7 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Les données loisirs
Enquêté
1
1
18 1 4
Activités Signalétique
= oui
ou nonEnquêté
8403
ou non
ACM 1 : loisirs en actif, signalétique en supplémentaire• 1 individu = profil d’activités• Principales dimensions de variabilité des profils d’activités• Liaisons entre ces dimensions et le signalétique
ACM 2 : signalétique en actif, loisirs en supplémentaireACM 3 : loisirs et signalétique en actif
8 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Transformation du tableau disjonctif complet
Le poids d’un individu est1I
yik = 1 si i possède la modalité k de la variable j (quel que soit pk)= 0 sinon
Idée : xik = yik/pk
∑Ii=1 xikI
=1I
∑Ii=1 yikpk
=1I
I × pkpk
= 1
Centrage : xik = yik/pk − 1
9 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Plan
1 Données - objectifs
2 Etude des individus
3 Etude des modalités
4 Aide à l’interprétation
10 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Nuage des individus
modalité k
IN
)',( iid
kix '
KRI
iM
OGI =
'iM(poids = 1/I)
(poids = )Jpk
ikx
1
i
k1 K
Ind
ivid
us
Modalités
Tableau Disjonctif Complet
ikx
I
d2i,i ′ =
K∑k=1
pkJ
(xik − xi ′k)2 =
K∑k=1
pkJ
(yikpk− yi ′k
pk
)2
=1J
K∑k=1
1pk
(yik−yi ′k)2
d(i ,GI )2 =
K∑k=1
pkJ(xik)
2 =K∑
k=1
pkJ
(yikpk− 1)2
=1J
K∑k=1
yijpk− 1
Inertie(NI ) =I∑
i=1
1Id2(i ,O)︸ ︷︷ ︸
inertie de i
=I∑
i=1
(1IJ
K∑k=1
yikpk− 1
I
)=
K
J− 1
11 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Ajustement du nuage des individus
Recherche des dimensions factorielles commme pour toute méthoded’analyse factorielle
Construction séquentielle : recherche d’un axe qui maximise l’inertieet qui est orthogonal aux axes précédemment trouvés
12 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Les données loisirs
• Extrait d’une enquête de 2003 de l’Insee sur la constructiondes identités, appelée « Histoire de vie »
• 8403 individus• 2 sortes de variables :
• Parmi les loisirs suivants, indiquez ceux que vous pratiquezrégulièrement : Lecture, Ecouter de la musique, Cinéma,Spectacle, Exposition, Ordinateur, Sport, Marche, Voyage,Jouer de la musique, Collection, Activité bénévole, Bricolage,Jardinage, Tricot, Cuisine, Pêche, nombre d’heures moyen parjour à regarder la TV
• le signalétique (4 questions) : sexe, âge, profession, statutmatrimonial
13 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Diagramme des inerties
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Valeurs propres
Numéro de la dimension
Val
eurs
pro
pres
0.00
0.05
0.10
0.15
14 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation du nuage des individus
0.0
0.5
1.0
Dim
2 (6
.91%
)
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
15 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation du nuage des individus
Qu’est-ce qu’une représentation particulière ?
Effet Guttman
16 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des individus en fonction du jardinage
Idée : utiliser les mo-dalités et les variablespour interpréter legraphe des individus
Une modalité aubarycentre des individusqui possèdent cettemodalité
●
−1.0 −0.5 0.0 0.5 1.0
−1.
0−
0.5
0.0
0.5
1.0
Dim 1 (16.95%)
Dim
2 (
6.91
%)
Jardinage_NJardinage_O
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17 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des modalités dans le nuage des individusChaque modalité est au barycentre des individus qui la prennent
0.0
0.5
1.0
Dim
2 (6
.91%
)
Lecture_NLecture_O
Ecouter.musique_NCinéma_NSpectacle_N
Spectacle_OExposition_N Exposition_O
Ordinateur_NSport_N
Marche_O
Voyage_NVoyage_OJouer.musique_N Collection_N
Collection_O
Activité.bénévole_N
Activité.bénévole_OBricolage_O
Jardinage_OTricot_O
Cuisine_O
Pêche_N
Pêche_O
234
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
Lecture_N Ecouter.musique_OCinéma_O
Spectacle_OExposition_O
Ordinateur_O
Sport_N
Sport_OMarche_NVoyage_N
Jouer.musique_NJouer.musique_O
Collection_NActivité.bénévole_NBricolage_NJardinage_N
Tricot_NCuisine_NPêche_N
01
4
Activité non choisie – activité choisie 18 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des modalités dans le nuage des individus
Dim
2 (
6.91
%)
0.0
0.2
0.4
Lecture
LectureEcouter.musique Ecouter.musique
Cinéma
SpectacleExposition
Exposition
Ordinateur
Sport
Marche
VoyageVoyage
Jouer.musique
Jouer.musiqueCollection
Collection
Activité.bénévole
Activité.bénévole
Bricolage
Jardinage
Tricot
Cuisine
Pêche
2
3
4
Dim 1 (16.95%)
Dim
2 (
6.91
%)
-0.4 -0.2 0.0 0.2 0.4
-0.2
0.0
Lecture
Cinéma
SpectacleOrdinateurSportMarche
Voyage Jouer.musiqueCollectionActivité.bénévoleBricolage
Jardinage
TricotCuisine
Pêche
0
1
Activité non choisie – activité choisie
19 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des modalités dans le nuage des individus0.
00.
51.
0
Dim
2 (6
.91%
)
5938
8325
2432
Dim
2 (
6.91
%)
0.0
0.2
0.4
Lecture
LectureEcouter.musique Ecouter.musique
Cinéma
SpectacleExposition
Exposition
Ordinateur
Sport
Marche
VoyageVoyage
Jouer.musique
Jouer.musiqueCollection
Collection
Activité.bénévole
Activité.bénévole
Bricolage
Jardinage
Tricot
Cuisine
Pêche
2
3
4
Ecouter Jouer ActivitéLecture musique Ciné Spectacle Expo Ordi Sport Marche Voyage musique Collec bénévole Bricol Jardin Tricot Cuisine Pêche TV
5938 O O N O O O O O O O O O O O O O N 32432 O O O O O O N O O O O O O O O O N 28325 N N N N N N N N N N N N N N N N N 4203 N N N N N N N N N N N N N N N N N 4
-1.0 -0.5 0.0 0.5 1.0 1.5
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
8325203
Dim 1 (16.95%)
Dim
2 (
6.91
%)
-0.4 -0.2 0.0 0.2 0.4
-0.2
0.0
Lecture
Cinéma
SpectacleOrdinateurSportMarche
Voyage Jouer.musiqueCollectionActivité.bénévoleBricolage
Jardinage
TricotCuisine
Pêche
0
1
20 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des modalités dans le nuage des individus0.
00.
51.
0
Dim
2 (6
.91%
)
5676
1143
Dim
2 (
6.91
%)
0.0
0.2
0.4
Lecture
LectureEcouter.musique Ecouter.musique
Cinéma
SpectacleExposition
Exposition
Ordinateur
Sport
Marche
VoyageVoyage
Jouer.musique
Jouer.musiqueCollection
Collection
Activité.bénévole
Activité.bénévole
Bricolage
Jardinage
Tricot
Cuisine
Pêche
2
3
4
Ecouter Jouer ActivitéLecture musique Ciné Spectacle Expo Ordi Sport Marche Voyage musique Collec bénévole Bricol Jardin Tricot Cuisine Pêche TV
255 O O O O O O O O O O N O N N N N N 1 6766 O O O O O O O O O O N N N N N O N 0 5676 N N N N N N N N N N N N O O O O N 41143 O N N N N N N N N N N N O O O N N 4
-1.0 -0.5 0.0 0.5 1.0 1.5
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
2556766
Dim 1 (16.95%)
Dim
2 (
6.91
%)
-0.4 -0.2 0.0 0.2 0.4
-0.2
0.0
Lecture
Cinéma
SpectacleOrdinateurSportMarche
Voyage Jouer.musiqueCollectionActivité.bénévoleBricolage
Jardinage
TricotCuisine
Pêche
0
1
21 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des variables pour interpréter les dimensions
Idée : considérer lescoordonnées des projetésdes individus sur un axe etcalculer un indicateur deliaison entre cescoordonnées et chaquevariable qualitative
Rapport de corrélationentre la variable j et lacomposante s : η(v.j ,Fs)
η2(F2, Jardinage) = 0.453
η2(F1, Jardinage) = 0.047
●
−1.0 −0.5 0.0 0.5 1.0
−1.
0−
0.5
0.0
0.5
1.0
Dim 1 (16.95%)
Dim
2 (
6.91
%)
Jardinage_NJardinage_O
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22 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des variables pour interpréter les dimensions
Idée : considérer lescoordonnées des projetésdes individus sur un axe etcalculer un indicateur deliaison entre cescoordonnées et chaquevariable qualitative
Rapport de corrélationentre la variable j et lacomposante s : η(v.j ,Fs)
η2(F2, Jardinage) = 0.453
η2(F1, Jardinage) = 0.047
●
−1.0 −0.5 0.0 0.5 1.0
−1.
0−
0.5
0.0
0.5
1.0
Dim 1 (16.95%)
Dim
2 (
6.91
%)
Jardinage_NJardinage_O
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22 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des variables pour interpréter les dimensions
Utilisation des rapportsde corrélation au carré
L’axe s est orthogo-nal à tout axe t (t < s)et est le plus lié auxvariables qualitatives ausens du η2 :
Fs = maxF
J∑j=1
η2(F , v.j)
0.3
0.4
0.5
Dim
2 (
6.91
%)
Jardinage
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.1
0.2
Dim 1 (16.95%)
Dim
2 (
6.91
%)
LectureEcouter.musique
Cinéma
SpectacleExposition
OrdinateurSportMarche
VoyageJouer.musique
CollectionActivité.bénévole
Bricolage
Tricot
CuisinePêche
TV
23 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Plan
1 Données - objectifs
2 Etude des individus
3 Etude des modalités
4 Aide à l’interprétation
24 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Nuage des modalités
Individu i
1
i
k1 KIn
div
idu
s
ModalitésI )',( kkd
x'ikxikxGJ
Jpkpoids =KN
RI
kM'kM
Individu i
poids = 1/I
i
I
Ind
ivid
us
0Σ
ikx'ikxikx
Var(k) = d2(k,O) =I∑
i=1
1Ix2ik =
I∑i=1
(yikpk− 1)2
=1pk− 1
pk 1/2 1/5 1/10 1/101d(k,O) 1 2 3 10
(si J = 10) Inertie(k) 0.05 0.08 0.09 0.099
Inertie(k) =pkJd2(k ,O) =
1− pkJ
d2(k, k ′) =I∑
i=1
(yikpk− yik′
pk′
)2
=pk + pk′ − 2pkk′
pkpk′25 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Inertie d’une modalité et d’une variable
Inertie(k) =1− pk
J
Inertie(j) =1J
Kj∑k=1
(1− pk) =Kj − 1
J
Variable Nb de modalité Inertie nb dim. du sous-espacesexe 2 1/J 1région 21 20/J 20département 96 95/J 95
MAIS : l’inertie Kj−1J se répartit dans un ss-espace à Kj − 1 dim.
Inertie totale =J∑
j=1
Kj − 1J
=K
J− 1
26 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Ajustement du nuage des modalitésRecherche séquentielle des dimensions commme pour toute méthode d’analysefactorielle : un axe doit maximiser l’inertie et être orthogonal aux axes précédents
0.5
1.0
Dim
2 (
6.91
%)
Ecouter.musique Cinéma
Ordinateur Sport
Marche
Collection
Activité.bénévole
Bricolage
Jardinage
Tricot
Cuisine
Pêche
2
3
-1.0 -0.5 0.0 0.5 1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (
6.91
%)
Lecture
Lecture
Ecouter.musique
Cinéma
Spectacle
Spectacle
Exposition
Exposition
Ordinateur
Ordinateur
Sport
SportMarche
VoyageVoyage
Jouer.musique
Jouer.musiqueCollection
Activité.bénévole
Activité.bénévole
Bricolage
Jardinage
Tricot
Cuisine
Pêche
0
1
2
4
Activité non choisie – activité choisie
27 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Projection des individusChaque individu au barycentre des modalités qu’il possède
0.0
0.5
1.0
Dim
2 (6
.91%
)
Lecture_O
Ecouter.musique_NCinéma_N
Spectacle_NExposition_N
Exposition_O
Ordinateur_N Sport_NMarche_O
Voyage_N Voyage_OJouer.musique_N
Collection_N
Collection_O
Activité.bénévole_N
Activité.bénévole_O
Bricolage_O
Jardinage_O
Tricot_O
Cuisine_O
Pêche_O
2
3
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
Lecture_N Ecouter.musique_O
Cinéma_O
Spectacle_O
Exposition_O
Ordinateur_OSport_O
Marche_N
Voyage_N
Jouer.musique_O
Collection_NActivité.bénévole_N
Bricolage_N
Jardinage_N
Tricot_N
Cuisine_N
Pêche_N
0
14
28 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentations barycentriques – représentation simultanéeReprésentation optimale des individusModalités au barycentre :
Gs(k) =
1√λs
I∑i=1
yikIk
Fs(i)
0.0
0.5
1.0
Dim
2 (6
.91%
)
Lecture_NLecture_O
Ecouter.musique_NCinéma_NSpectacle_N
Spectacle_OExposition_N Exposition_O
Ordinateur_NSport_N
Marche_O
Voyage_NVoyage_OJouer.musique_N Collection_N
Collection_O
Activité.bénévole_N
Activité.bénévole_OBricolage_O
Jardinage_OTricot_O
Cuisine_O
Pêche_N
Pêche_O
234
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
Lecture_N Ecouter.musique_OCinéma_O
Spectacle_OExposition_O
Ordinateur_O
Sport_N
Sport_OMarche_NVoyage_N
Jouer.musique_NJouer.musique_O
Collection_NActivité.bénévole_NBricolage_NJardinage_N
Tricot_NCuisine_NPêche_N
01
4
Représentation optimale des modalitésIndividus au barycentre :
Fs(i) =
1√λs
J∑j=1
yikJGs(k)
0.0
0.5
1.0
Dim
2 (6
.91%
)
Lecture_O
Ecouter.musique_NCinéma_N
Spectacle_NExposition_N
Exposition_O
Ordinateur_N Sport_NMarche_O
Voyage_N Voyage_OJouer.musique_N
Collection_N
Collection_O
Activité.bénévole_N
Activité.bénévole_O
Bricolage_O
Jardinage_O
Tricot_O
Cuisine_O
Pêche_O
2
3
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
Lecture_N Ecouter.musique_O
Cinéma_O
Spectacle_O
Exposition_O
Ordinateur_OSport_O
Marche_N
Voyage_N
Jouer.musique_O
Collection_NActivité.bénévole_N
Bricolage_N
Jardinage_N
Tricot_N
Cuisine_N
Pêche_N
0
14
29 / 38
Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentations barycentriques – représentation simultanéeReprésentation optimale des individusModalités au pseudo-barycentre :
Gs(k) =1√λs
I∑i=1
yikIk
Fs(i)
0.0
0.5
1.0
Dim
2 (6
.91%
)
Lecture_O
Ecouter.musique_NCinéma_N
Spectacle_NExposition_N Exposition_O
Ordinateur_NSport_NMarche_O
Voyage_N Voyage_OJouer.musique_N
Collection_N
Collection_O
Activité.bénévole_O
Bricolage_O
Jardinage_O
Tricot_O
Cuisine_O
Pêche_O
2
3
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
Lecture_N Ecouter.musique_O
Cinéma_O
Spectacle_OOrdinateur_OSport_O
Marche_N
Voyage_N Voyage_O
Jouer.musique_O
Collection_NActivité.bénévole_N
Bricolage_N
Jardinage_N
Tricot_N
Cuisine_N
Pêche_N
0
14
Représentation optimale des modalitésIndividus au barycentre :
Fs(i) =
1√λs
J∑j=1
yikJGs(k)
0.0
0.5
1.0
Dim
2 (6
.91%
)
Lecture_O
Ecouter.musique_NCinéma_N
Spectacle_NExposition_N
Exposition_O
Ordinateur_N Sport_NMarche_O
Voyage_N Voyage_OJouer.musique_N
Collection_N
Collection_O
Activité.bénévole_N
Activité.bénévole_O
Bricolage_O
Jardinage_O
Tricot_O
Cuisine_O
Pêche_O
2
3
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
Lecture_N Ecouter.musique_O
Cinéma_O
Spectacle_O
Exposition_O
Ordinateur_OSport_O
Marche_N
Voyage_N
Jouer.musique_O
Collection_NActivité.bénévole_N
Bricolage_N
Jardinage_N
Tricot_N
Cuisine_N
Pêche_N
0
14
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentations barycentriques – représentation simultanéeReprésentation optimale des individusModalités au pseudo-barycentre :
Gs(k) =1√λs
I∑i=1
yikIk
Fs(i)
0.0
0.5
1.0
Dim
2 (6
.91%
)
Lecture_O
Ecouter.musique_NCinéma_N
Spectacle_NExposition_N Exposition_O
Ordinateur_NSport_NMarche_O
Voyage_N Voyage_OJouer.musique_N
Collection_N
Collection_O
Activité.bénévole_O
Bricolage_O
Jardinage_O
Tricot_O
Cuisine_O
Pêche_O
2
3
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
Lecture_N Ecouter.musique_O
Cinéma_O
Spectacle_OOrdinateur_OSport_O
Marche_N
Voyage_N Voyage_O
Jouer.musique_O
Collection_NActivité.bénévole_N
Bricolage_N
Jardinage_N
Tricot_N
Cuisine_N
Pêche_N
0
14
Représentation optimale des modalitésIndividus au pseudo-barycentre :
Fs(i) =1√λs
J∑j=1
yikJGs(k)
0.0
0.5
1.0
Dim
2 (6
.91%
)
Lecture_O
Ecouter.musique_NCinéma_N
Spectacle_NExposition_N Exposition_O
Ordinateur_NSport_NMarche_O
Voyage_N Voyage_OJouer.musique_N
Collection_N
Collection_O
Activité.bénévole_O
Bricolage_O
Jardinage_O
Tricot_O
Cuisine_O
Pêche_O
2
3
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
Lecture_N Ecouter.musique_O
Cinéma_O
Spectacle_OOrdinateur_OSport_O
Marche_N
Voyage_N Voyage_O
Jouer.musique_O
Collection_NActivité.bénévole_N
Bricolage_N
Jardinage_N
Tricot_N
Cuisine_N
Pêche_N
0
14
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Plan
1 Données - objectifs
2 Etude des individus
3 Etude des modalités
4 Aide à l’interprétation
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Inertie et pourcentage d’inertie en ACM
λs =1J
J∑j=1
η2(Fs , v.j)
⇒ λs est la moyenne des carrés des rapports de corrélation
• Individus vivent dans RK−J ⇒ pourcentages d’inertie faibles• Pourcentage maximum pour une dimension s :
λs∑K−Jt=1 λt
× 100 ≤1
K−JJ
× 100
≤J
K − J× 100
Avec K = 100, J = 10 : λs ≤ 11.1 %
• Moyenne des valeurs propres non nulles : 1K−J
×∑
t λt =1
K−J×(
KJ− 1)= 1
J
⇒ interpréter les dimensions d’inertie supérieure à 1/J
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Contribution et qualité de représentation
• Contribution et cos2 pour les individus et les modalités
⇒ Modalités extrêmes ne contribuent pas nécessairementbeaucoup (cela dépend de leur fréquence)⇒ cos2 petits ... ce qui est attendu car bcp de dimensions
• Contribution absolue d’une variable :
CTR(j) =
Kj∑k=1
CTR(k) =η2(Fs , v.j)
J
• Contribution relative : CTR(j) = η2(Fs ,v.j )Jλs
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Représentation des modalités supplémentaires
Utilisation des relations de transition pour les éléments (individus,modalités) supplémentaires
0.0
0.5
Dim
2 (
6.91
%)
F
M
(25,35]
(35,45]
(45,55]
(55,65](65,75]
(75,85]
(85,100]
divorcé
mariéremariéveuf
agent demaîtriseautre
cadre
employémanoeuvreouvrier
Non renseigné
technicien
-1.0 -0.5 0.0 0.5
-0.5
Dim 1 (16.95%)
Dim
2 (
6.91
%)
(25,35]
[15,25]
seul
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Variable quantitative supplémentaire⇒ Comment faire avec les variables quantitatives ?
• Information supplémentaire : projetée sur les dimensions,coefficient de corrélation calculé avec chaque dimension
• Information active : découper la variable en classes0.
00.
51.
0Supplementary variables on the MCA factor map
Dim
2 (6
.91%
)
Nb.activités
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.0
-0.5
0.0
Dim 1 (16.95%)
Dim
2 (6
.91%
)
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Description des dimensionsPar les variables qualitatives (test de Fisher), les modalités (test deStudent) et les variables quantitatives (corrélation)Variables quantitatives
correlation p.valueNb.activités 0.9753459 0
Variables quantitatives ModalitésR2 p.value Estimate p.value
Lecture 0.239 0.00e+00 Jouer.musique_O 0.268 0Ecouter.musique 0.275 0.00e+00 Voyage_O 0.270 0Cinéma 0.389 0.00e+00 Marche_O 0.184 0Spectacle 0.383 0.00e+00 Sport_O 0.247 0Exposition 0.399 0.00e+00 Ordinateur_O 0.263 0Ordinateur 0.327 0.00e+00 Exposition_O 0.304 0Sport 0.287 0.00e+00 Spectacle_O 0.304 0Marche 0.172 0.00e+00 Sport_N -0.247 0Voyage 0.355 0.00e+00 Ordinateur_N -0.263 0Jouer.musique 0.209 0.00e+00 Exposition_N -0.304 0Bricolage 0.135 8.82e-267 Spectacle_N -0.304 0Cuisine 0.125 9.42e-247 Cinéma_N -0.283 0Profession 0.128 7.20e-245 Ecouter.musique_N -0.257 0Activité.bénévole 0.109 2.25e-212 Lecture_N -0.231 0
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Autre présentation de l’ACM : tableau de Burt
Tableau de Burt :• Ensemble des liaisons entre variables
prises 2 à 2 (tableau analogue à lamatrice des corrélations entre variablesquantitatives)
• Analyse des correspondances sur letableau de Burt
• Résultats uniquement sur les modalités :même représentation mais avec desvaleurs propres différentesλBurts =
(λTDCs
)2• λTDC
s moyenne des carrés des rapports decorrélation
variable l variable j
varia
ble
jva
riabl
e l
⇒ L’ACM ne dépend que des liaisons entre les variables prises 2 à 2(comme l’ACP ne dépend que de la matrice des corrélations)
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Conclusion
• L’ACM est la méthode factorielle adaptée aux tableauxindividus × variables qualitatives
• Les valeurs propres s’interprètent comme des moyennes derapports de corrélation au carré
• Le carré des liaisons est précieux en particulier lorsqu’il y abeaucoup de variables
• Revenir aux données en analysant des tableaux de contingencepar AFC
• La convergence entre l’analyse du TDC et celle du tableau deBurt est un argument en faveur de l’intérêt de la méthode
• L’ACM comme pré-traitement d’une classification
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Données - objectifs Etude des individus Etude des modalités Aide à l’interprétation
Suppléments
Analyse de données avec R (2009).Husson, Lê, Pagès.Presses Universitaires de Rennes.
Package FactoMineR pour faire des ACP :http://factominer.free.fr/index_fr.html
Vidéos sur Youtube :• une chaîne Youtube : youtube.com/HussonFrancois• une playlist de vidéos en français• une playlist de vidéos en anglais
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