Analyse en Composantes Principales
Husson et al.
05/09/2018
1. Importer les données
decath <- read.table("https://r-stat-sc-donnees.github.io/decathlon.csv",
sep = ";", dec = ".", header = TRUE, row.names = 1, check.names = FALSE)
summary(decath) 100m Longueur Poids Hauteur 400m 110m H
Min. :10.44 Min. :6.61 Min. :12.68 Min. :1.850 Min. :46.81 Min. :13.97
1st Qu.:10.85 1st Qu.:7.03 1st Qu.:13.88 1st Qu.:1.920 1st Qu.:48.93 1st Qu.:14.21
Median :10.98 Median :7.30 Median :14.57 Median :1.950 Median :49.40 Median :14.48
Mean :11.00 Mean :7.26 Mean :14.48 Mean :1.977 Mean :49.62 Mean :14.61
3rd Qu.:11.14 3rd Qu.:7.48 3rd Qu.:14.97 3rd Qu.:2.040 3rd Qu.:50.30 3rd Qu.:14.98
Max. :11.64 Max. :7.96 Max. :16.36 Max. :2.150 Max. :53.20 Max. :15.67
Disque Perche Javelot 1500m Classement Points
Min. :37.92 Min. :4.200 Min. :50.31 Min. :262.1 Min. : 1.00 Min. :7313
1st Qu.:41.90 1st Qu.:4.500 1st Qu.:55.27 1st Qu.:271.0 1st Qu.: 6.00 1st Qu.:7802
Median :44.41 Median :4.800 Median :58.36 Median :278.1 Median :11.00 Median :8021
Mean :44.33 Mean :4.762 Mean :58.32 Mean :279.0 Mean :12.12 Mean :8005
3rd Qu.:46.07 3rd Qu.:4.920 3rd Qu.:60.89 3rd Qu.:285.1 3rd Qu.:18.00 3rd Qu.:8122
Max. :51.65 Max. :5.400 Max. :70.52 Max. :317.0 Max. :28.00 Max. :8893
Competition
Decastar:13
JO :28
2 et 3. Paramètres de l’ACP
library(FactoMineR)
res.pca <- PCA(decath, quanti.sup = 11:12, quali.sup = 13)
4. Choisir le nombre d’axes
barplot(res.pca$eig[,2],names=paste("Dim",1:nrow(res.pca$eig)))
5. Analyser les résultats
summary(res.pca, ncp = 2, nbelements = 3)
Call:
PCA(X = decath, quanti.sup = 11:12, quali.sup = 13)
Eigenvalues
Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7 Dim.8 Dim.9
Variance 3.272 1.737 1.405 1.057 0.685 0.599 0.451 0.397 0.215
% of var. 32.719 17.371 14.049 10.569 6.848 5.993 4.512 3.969 2.148
Cumulative % of var. 32.719 50.090 64.140 74.708 81.556 87.548 92.061 96.030 98.178
Dim.10
Variance 0.182
% of var. 1.822
Cumulative % of var. 100.000
Individuals (the 3 first)
Dist Dim.1 ctr cos2 Dim.2 ctr cos2
Sebrle | 4.843 | 4.038 12.158 0.695 | 1.366 2.619 0.080 |
Clay | 4.647 | 3.919 11.451 0.711 | 0.837 0.984 0.032 |
Karpov | 5.006 | 4.620 15.911 0.852 | 0.040 0.002 0.000 |
Variables (the 3 first)
Dim.1 ctr cos2 Dim.2 ctr cos2
100m | -0.775 18.344 0.600 | 0.187 2.016 0.035 |
Longueur | 0.742 16.822 0.550 | -0.345 6.869 0.119 |
Poids | 0.623 11.844 0.388 | 0.598 20.607 0.358 |
Supplementary continuous variables
Dim.1 cos2 Dim.2 cos2
Classement | -0.671 0.450 | 0.051 0.003 |
Points | 0.956 0.914 | -0.017 0.000 |
Supplementary categories
Dist Dim.1 cos2 v.test Dim.2 cos2 v.test
Decastar | 0.946 | -0.600 0.403 -1.430 | -0.038 0.002 -0.123 |
JO | 0.439 | 0.279 0.403 1.430 | 0.017 0.002 0.123 |plot(res.pca, choix = "ind", habillage = 13, cex = 1.1,
select = "cos2 0.6", title = "Graphe des individus")
# axes 3 et 4
plot(res.pca, choix = "ind", habillage = 13, axes = 3:4, cex = 0.7)
plot(res.pca, choix = "var", habillage = 13, axes = 3:4, new.plot = TRUE)
6. Décrire de façon automatique les principales dimensions de variabilité
dimdesc(res.pca, proba = 0.2)$`Dim.1`
$`Dim.1`$`quanti`
correlation p.value
Points 0.9561543 2.099191e-22
Longueur 0.7418997 2.849886e-08
Poids 0.6225026 1.388321e-05
Hauteur 0.5719453 9.362285e-05
Disque 0.5524665 1.802220e-04
Javelot 0.2771108 7.942460e-02
Classement -0.6705104 1.616348e-06
400m -0.6796099 1.028175e-06
110m H -0.7462453 2.136962e-08
100m -0.7747198 2.778467e-09
$`Dim.1`$quali
R2 p.value
Competition 0.05110487 0.1552515
$`Dim.1`$category
Estimate p.value
JO 0.4393744 0.1552515
Decastar -0.4393744 0.1552515
$Dim.2
$Dim.2$`quanti`
correlation p.value
Disque 0.6063134 2.650745e-05
Poids 0.5983033 3.603567e-05
400m 0.5694378 1.020941e-04
1500m 0.4742238 1.734405e-03
Hauteur 0.3502936 2.475025e-02
Javelot 0.3169891 4.344974e-02
110m H 0.2287933 1.501925e-01
Longueur -0.3454213 2.696969e-02
$Dim.3
$Dim.3$`quanti`
correlation p.value
1500m 0.7821428 1.554450e-09
Perche 0.6917567 5.480172e-07
Hauteur -0.2595119 1.013160e-01
Javelot -0.3896554 1.179331e-027. Retour aux données brutes
round(scale(decath[,1:12]),2) 100m Longueur Poids Hauteur 400m 110m H Disque Perche Javelot 1500m Classement
Sebrle -0.56 1.83 2.28 1.61 -1.09 -1.18 1.30 0.85 2.53 0.08 -1.40
Clay -2.12 2.21 0.91 0.94 -0.37 -1.01 1.71 0.49 2.36 0.25 -1.28
Karpov -1.89 1.74 1.76 1.27 -2.43 -1.35 2.17 -0.58 -0.58 -0.08 -1.15
Macey -0.41 0.66 1.52 1.95 -0.56 -0.10 1.19 -1.30 0.03 -1.17 -1.03
Warners -1.44 1.52 0.00 -0.08 -1.43 -1.26 -0.18 0.49 -0.61 -0.08 -0.90
Zsivoczky -0.33 -0.38 1.01 1.61 -0.19 0.73 0.38 -0.22 1.06 -0.81 -0.77
Hernu -0.11 -0.22 0.21 0.60 -0.77 -0.75 0.12 0.14 -0.12 -1.26 -0.65
Nool -0.75 0.85 -0.26 -1.09 -0.70 0.41 -0.67 2.29 0.62 -0.23 -0.52
Bernard -1.17 0.70 0.39 1.61 -0.42 -0.92 0.13 -1.30 -0.63 -0.23 -0.39
Schwarzl -0.07 0.73 -0.57 -0.41 0.12 -0.75 -0.56 1.21 -0.41 -0.47 -0.27
Pogorelov -0.18 0.16 0.76 0.94 1.02 -0.84 0.08 0.85 -1.01 0.74 -0.14
Schoenbeck -0.37 0.13 0.36 -1.09 0.59 -0.56 0.02 0.85 0.53 -0.02 -0.02
Barras 0.54 -0.85 0.53 -0.41 -0.18 -0.50 0.15 -0.58 1.29 -1.02 0.11
Smith -0.56 -1.42 0.93 -0.75 -0.30 -1.26 1.39 -2.02 0.66 -0.54 0.24
Averyanov -1.70 0.25 -0.04 -0.41 0.09 -0.46 -1.32 0.14 -0.79 -0.69 0.36
Ojaniemi -1.21 0.76 0.60 -0.41 -0.43 0.86 -1.18 -0.58 0.20 -0.28 0.49
Smirnov -0.41 -0.60 -0.72 -0.41 -0.44 0.35 -0.55 -0.22 0.53 -1.35 0.62
Qi 0.24 0.25 -1.12 -0.08 0.03 0.37 0.24 -0.94 0.51 -0.55 0.74
Drews -0.49 0.38 -1.71 -1.09 -0.96 -1.26 -1.25 0.85 -1.41 -0.41 0.87
Parkhomenko 0.54 -2.05 1.47 0.60 1.23 0.58 -0.72 0.14 1.55 -0.09 0.99
Terek -0.30 -1.01 0.82 -0.41 -0.05 1.09 0.38 1.93 -1.59 0.97 1.12
Gomez 0.31 0.00 0.11 -1.43 -0.87 -0.42 -1.00 -1.30 0.50 -0.80 1.25
Turi 0.31 -1.11 -1.04 0.60 1.78 -0.73 -1.33 0.14 0.21 0.94 1.37
Lorenzo 0.39 -0.73 -1.52 -1.43 -0.24 1.64 -1.22 -0.94 0.01 -1.37 1.50
Karlivans 1.26 0.00 -1.43 -0.08 0.80 0.79 -0.29 -0.94 -1.12 -0.03 1.63
Korkizoglou -0.52 -0.60 0.40 -0.41 1.34 0.75 0.52 -0.22 -1.09 3.25 1.75
Uldal 0.88 -0.85 -1.15 -1.43 1.16 1.03 -0.39 -0.94 0.35 0.23 1.88
Casarsa 1.38 -1.83 0.54 -0.41 3.11 1.66 1.28 -1.30 0.06 1.46 2.01
SEBRLE 0.16 1.01 0.43 1.05 0.17 0.18 -0.17 0.93 1.01 1.09 -1.40
CLAY -0.91 0.44 -0.26 -1.31 -0.21 -1.18 1.89 0.57 0.38 1.93 -1.28
KARPOV 0.08 0.13 0.36 0.71 -1.08 -1.09 1.37 0.57 -1.66 1.81 -1.15
BERNARD 0.08 -0.09 -0.28 -0.64 -0.60 0.81 -1.02 2.01 0.92 0.09 -1.03
YURKOV 1.30 -0.54 0.86 1.38 0.70 1.49 0.57 -0.15 1.06 -0.22 -0.90
WARNERS 0.43 1.07 -0.20 0.04 -0.81 -0.80 -0.95 0.57 -1.36 -0.08 -0.77
ZSIVOCZKY 0.50 0.13 -1.21 0.37 -0.86 -0.92 0.40 -1.23 -0.61 -0.94 -0.65
McMULLEN -0.64 0.16 -0.87 1.72 0.25 -0.48 0.02 -1.23 -0.40 0.52 -0.52
MARTINEAU 2.44 -1.42 0.11 -0.30 0.45 0.69 0.97 0.57 -1.24 -1.45 -0.39
HERNU 1.41 0.95 -0.08 -1.31 1.29 0.96 0.20 0.21 -0.23 0.52 -0.27
BARRAS 1.26 -0.92 -0.47 -0.30 -0.12 -0.27 -0.66 -0.15 -0.60 0.25 -0.14
NOOL 1.26 0.03 -2.18 0.04 -0.36 1.45 -1.90 -0.51 -0.18 -1.06 -0.02
BOURGUIGNON 1.38 -1.45 -1.23 -1.31 1.34 2.26 -1.14 0.93 -0.75 1.09 0.11
Points
Sebrle 2.59
Clay 2.38
Karpov 2.10
Macey 1.19
Warners 0.99
Zsivoczky 0.82
Hernu 0.68
Nool 0.67
Bernard 0.64
Schwarzl 0.28
Pogorelov 0.23
Schoenbeck 0.21
Barras 0.18
Smith 0.05
Averyanov 0.05
Ojaniemi 0.00
Smirnov -0.04
Qi -0.21
Drews -0.23
Parkhomenko -0.26
Terek -0.33
Gomez -0.41
Turi -0.87
Lorenzo -1.21
Karlivans -1.23
Korkizoglou -1.26
Uldal -1.49
Casarsa -1.76
SEBRLE 0.62
CLAY 0.34
KARPOV 0.27
BERNARD 0.18
YURKOV 0.09
WARNERS 0.07
ZSIVOCZKY 0.00
McMULLEN -0.03
MARTINEAU -0.59
HERNU -0.80
BARRAS -0.87
NOOL -1.03
BOURGUIGNON -2.02
attr(,"scaled:center")
100m Longueur Poids Hauteur 400m 110m H Disque Perche
10.998049 7.260000 14.477073 1.976829 49.616341 14.605854 44.325610 4.762439
Javelot 1500m Classement Points
58.316585 279.024878 12.121951 8005.365854
attr(,"scaled:scale")
100m Longueur Poids Hauteur 400m 110m H Disque
0.26302300 0.31640164 0.82442781 0.08895052 1.15345081 0.47178902 3.37784476
Perche Javelot 1500m Classement Points
0.27799982 4.82682018 11.67324722 7.91894918 342.38514542 round(cor(decath[1:12,1:12]),2) 100m Longueur Poids Hauteur 400m 110m H Disque Perche Javelot 1500m Classement Points
100m 1.00 -0.78 -0.22 -0.15 0.59 0.45 -0.57 0.21 -0.17 -0.37 0.55 -0.63
Longueur -0.78 1.00 0.35 0.11 -0.57 -0.59 0.56 0.07 0.39 0.45 -0.69 0.80
Poids -0.22 0.35 1.00 0.75 -0.36 -0.19 0.85 -0.44 0.43 0.09 -0.69 0.74
Hauteur -0.15 0.11 0.75 1.00 -0.24 -0.10 0.65 -0.77 0.13 -0.17 -0.55 0.49
400m 0.59 -0.57 -0.36 -0.24 1.00 0.32 -0.50 0.26 -0.04 0.20 0.71 -0.63
110m H 0.45 -0.59 -0.19 -0.10 0.32 1.00 -0.36 0.12 0.14 -0.45 0.23 -0.38
Disque -0.57 0.56 0.85 0.65 -0.50 -0.36 1.00 -0.49 0.39 0.10 -0.76 0.85
Perche 0.21 0.07 -0.44 -0.77 0.26 0.12 -0.49 1.00 0.24 0.39 0.26 -0.17
Javelot -0.17 0.39 0.43 0.13 -0.04 0.14 0.39 0.24 1.00 0.09 -0.56 0.65
1500m -0.37 0.45 0.09 -0.17 0.20 -0.45 0.10 0.39 0.09 1.00 0.09 0.17
Classement 0.55 -0.69 -0.69 -0.55 0.71 0.23 -0.76 0.26 -0.56 0.09 1.00 -0.93
Points -0.63 0.80 0.74 0.49 -0.63 -0.38 0.85 -0.17 0.65 0.17 -0.93 1.00Factoshiny
library(Factoshiny)
res.shiny <- PCAshiny(decath)Pour aller plus loin
plotellipses(res.pca)
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