HCPC Exemple
T.Mellah
5/12/2018
Contents
1 Présentation de la Base de données
1.1 Résumé statistique des données . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Classification hierarchique sur composantes principales (HCPC)
2.1 Preprocessing: analyse en composantes principales . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Clustering sur les outputs de l’ACP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Représentation graphique des résultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Indicateurs numériques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
1
2
2
2
4
5
6
Dans cet exemple, nous étudions le climat de diérents pays européens. Pour ce faire, les températures (en
degrés Celsius) ont été collectées mensuellement pour les principales capitales européennes et autres grandes
villes. En plus des températures mensuelles, la température annuelle moyenne et l’amplitude thermique
(diérence entre la moyenne mensuelle maximale et la moyenne mensuelle minimale d’une ville) ont été
enregistrées pour chaque ville. Nous incluons également deux variables quantitatives de positionnement
(latitude et longitude) ainsi qu’une variable catégorique: Zone (variable catégorique avec les quatre catégories
nord, sud, est et ouest de l’Europe).
L’objectif de cette analyse est de regrouper les capitales dans des classes afin que les villes d’une classe donnée
présentent toutes des températures similaires toute l’année.
On décrit les classes ainsi définies à l’aide de variables ou d’individus spécifiques. Pour déterminer le nombre
de clusters, on construit un arbre, selon une classification hiérarchique ascendante.
Une présentation détaillée de l’analyse de cette base par la méthode de HCPC est donnée par Husson, F.,
Josse, J., & Pages, J. (2010). Principal component methods-hierarchical clustering-partitional clustering: why
would we need to choose for visualizing data. Applied Mathematics Department.
1 Présentation de la Base de données
data.frame:
$ January
: num
$ February : num
: num
$ March
: num
$ April
: num
$ May
: num
$ June
: num
$ July
: num
$ August
$ September: num
$ October
: num
$ November : num
$ December : num
: num
$ Annual
$ Amplitude: num
$ Latitude : num
35 obs. of 17 variables:
2.9 9.1 -0.2 3.3 -1.1 -0.4 4.8 -5.8 -5.9 -3.7 ...
2.5 9.7 0.1 3.3 0.8 -0.4 5 -6.2 -5 -2 ...
5.7 11.7 4.4 6.7 5.5 1.3 5.9 -2.7 -0.3 1.9 ...
8.2 15.4 8.2 8.9 11.6 5.8 7.8 3.1 7.4 7.9 ...
12.5 20.1 13.8 12.8 17 11.1 10.4 10.2 14.3 13.2 ...
14.8 24.5 16 15.6 20.2 15.4 13.3 14 17.8 16.9 ...
17.1 27.4 18.3 17.8 22 17.1 15 17.2 19.4 18.4 ...
17.1 27.2 18 17.8 21.3 16.6 14.6 14.9 18.5 17.6 ...
14.5 23.8 14.4 15 16.9 13.3 12.7 9.7 13.7 13.7 ...
11.4 19.2 10 11.1 11.3 8.8 9.7 5.2 7.5 8.6 ...
7 14.6 4.2 6.7 5.1 4.1 6.7 0.1 1.2 2.6 ...
4.4 11 1.2 4.4 0.7 1.3 5.4 -2.3 -3.6 -1.7 ...
9.9 17.8 9.1 10.3 10.9 7.8 9.3 4.8 7.1 7.7 ...
14.6 18.3 18.5 14.4 23.1 17.5 10.2 23.4 25.3 22.1 ...
52.2 37.6 52.3 50.5 47.3 55.4 53.2 60.1 50.3 50 ...
1
$ Longitude: num
$ Area
4.5 23.5 13.2 4.2 19 12.3 6.1 25 30.3 19.6 ...
: Factor w/ 4 levels "East","North",..: 4 3 4 4 1 2 2 2 1 1 ...
1.1 Résumé statistique des données
February
Min.
:-7.900
1st Qu.:-0.150
Median : 1.900
Mean
: 2.217
3rd Qu.: 5.800
:11.800
Max.
March
Min.
:-3.700
1st Qu.: 1.600
Median : 5.400
Mean
: 5.229
3rd Qu.: 8.500
:14.100
Max.
April
Min.
: 2.900
1st Qu.: 7.250
Median : 8.900
Mean
: 9.283
3rd Qu.:12.050
:16.900
Max.
June
Min.
: 9.30
1st Qu.:15.40
Median :16.90
:17.41
Mean
3rd Qu.:19.80
:24.50
Max.
October
: 4.50
Min.
1st Qu.: 8.65
Median :10.20
Mean
:11.00
3rd Qu.:13.30
:19.40
Max.
Amplitude
Min.
:10.20
1st Qu.:14.90
Median :18.50
Mean
:18.32
3rd Qu.:21.45
:27.60
Max.
July
Min.
:11.10
1st Qu.:17.30
Median :18.90
:19.62
Mean
3rd Qu.:21.75
:27.40
Max.
November
:-1.100
Min.
1st Qu.: 3.200
Median : 5.100
Mean
: 6.066
3rd Qu.: 7.900
:14.900
Max.
Latitude
Min.
:37.20
1st Qu.:43.90
Median :50.00
Mean
:49.04
3rd Qu.:53.35
:64.10
Max.
August
Min.
:10.60
1st Qu.:16.65
Median :18.30
:18.98
Mean
3rd Qu.:21.60
:27.20
Max.
December
:-6.00
Min.
1st Qu.: 0.25
Median : 1.70
Mean
: 2.88
3rd Qu.: 5.40
:12.00
Max.
Longitude
Min.
: 0.00
1st Qu.: 5.05
Median :10.50
Mean
:13.01
3rd Qu.:19.30
:37.60
Max.
May
January
September
##
Min.
:-9.300
1st Qu.:-1.550
Median : 0.200
Mean
: 1.346
3rd Qu.: 4.900
Max.
:10.700
##
Min.
: 6.50
1st Qu.:12.15
Median :13.80
:13.91
Mean
3rd Qu.:16.35
Max.
:20.90
##
: 7.90
Min.
1st Qu.:13.00
Median :14.80
Mean
:15.63
3rd Qu.:18.25
Max.
:24.30
##
Min.
: 4.50
1st Qu.: 7.75
Median : 9.70
Mean
:10.27
3rd Qu.:12.65
Max.
:18.20
##
East : 8
North: 8
South:10
West : 9
##
##
Annual
Publicité
Area
2 Classification hierarchique sur composantes principales (HCPC)
2.1 Preprocessing: analyse en composantes principales
##
Call:
PCA(X = temperature[1:23, ], scale.unit = TRUE, ncp = Inf, quanti.sup = 13:16,
##
##
##
quali.sup = 17, graph = FALSE)
2
Dim.1
9.948
Dim.2
1.848
82.898 15.397
82.898 98.295
Dim.8
0.002
0.017
99.961 99.978
Dim.7
0.006
0.049
Dim.3
0.126
1.052
Dim.4
0.038
0.319
99.347 99.666
Dim.9
0.001
0.009
0.001
0.008
99.986 99.994
Dim.5
0.017
0.139
Dim.6
0.013
0.107
99.805 99.912
Dim.10 Dim.11 Dim.12
0.000
0.001
0.001
0.004
99.999 100.000
ctr
cos2
cos2
Dim.2
Dim.1
ctr
0.025 | -1.371 4.426 0.904 |
0.227 0.023
0.930 2.037 0.015 |
7.601 25.249 0.978 |
0.328 |
0.001 0.001 |
0.016
0.216 | -1.177 3.261 0.753 |
0.463 |
6.903 0.489 |
0.806 | -0.492 0.570 0.091 |
0.034 | -2.673 16.819 0.955 |
0.502 0.013 |
0.957 |
9.484 0.574 |
0.417 |
1.802 0.312 |
0.645 |
| 1.443 |
| 7.684 |
| 0.503 | -0.288 0.036
0.631 0.174
| 1.357 |
| 2.450 |
1.668 1.216
| 1.629 | -1.462 0.935
| 2.736 | -0.505 0.112
| 4.127 | -4.036 7.121
| 2.650 | -1.712 1.281
| 1.567 | -1.259 0.692
ctr
-0.104 0.375
0.462
2.008
0.875
1.713
Dist
Dim.3
Eigenvalues
##
Variance
% of var.
Cumulative % of var.
##
Variance
% of var.
Cumulative % of var.
##
Individuals (the 10 first)
##
Amsterdam
Athens
Berlin
Brussels
Budapest
Copenhagen
Dublin
Elsinki
Kiev
Krakow
##
Amsterdam
Athens
Berlin
Brussels
Budapest
Copenhagen
Dublin
Elsinki
Kiev
Krakow
##
Variables (the 10 first)
##
January
February
March
April
May
June
July
August
September
October
##
January
February
March
April
May
June
July
Dim.1
| 0.842
| 0.884
| 0.945
| 0.974
| 0.870
| 0.833
| 0.844
| 0.909
| 0.986
| 0.992
cos2
0.005 |
0.000 |
0.015 |
0.039 |
0.024 |
0.002 |
0.024 |
-0.291
-0.152 0.796
-0.499 8.574
0.440 6.678
-0.171
-0.274
-0.179
cos2
0.005 |
0.561 10.855 0.005 |
2.908 0.334 |
0.013 |
0.041 |
0.073 |
1.097 0.004 |
0.593 12.117 0.021 |
1.003 0.004 |
2.585 0.031 |
ctr
cos2
cos2
Dim.3
Dim.2
ctr
ctr
0.282 | 0.068 3.637
7.135 0.710 | -0.531 15.281
0.782 | -0.456 11.246 0.208 | -0.003 0.010
7.861
4.468 0.083 | -0.121 11.557
8.978 0.893 | -0.287
9.534 0.948 | 0.100 0.537
0.010 | -0.198 31.132
7.606 0.757 | 0.458 11.344 0.210 | -0.156 19.296
6.981 0.694 | 0.545 16.095 0.297 | 0.050 1.944
7.164 0.713 | 0.509 14.004 0.259 | 0.154 18.708
0.162 | 0.088 6.064
8.311 0.827 | 0.402 8.743
1.259
9.766
0.023 | 0.023 0.405
0.153
0.971 |
0.389 0.007 | 0.000 0.000
9.885 0.983 | -0.085
3
cos2
0.008 |
0.001 |
0.000 |
Dim.1
| 0.998
| -0.314
| -0.910
| -0.364
August
September
October
##
Supplementary continuous variables
##
Annual
Amplitude
Latitude
Longitude
##
Supplementary categories
##
East
North
South
West
##
East
North
South
West
cos2 v.test
0.021 -2.246 |
0.010 2.346 |
0.001 0.804 |
0.025 -0.995 |
Dim.3
-0.257
0.269
Publicité
0.116
-0.164
Dim.1
Dist
cos2
Dim.2
0.995 | -0.068 0.005 |
0.099 | 0.944
0.892 |
0.828 | -0.215 0.046 |
0.133 | 0.645
Dim.3
0.005 0.000 |
0.039
0.002 |
0.182 0.033 |
0.416 | -0.036 0.001 |
cos2
cos2 v.test
| 1.784 | -1.099 0.380 -1.081 |
| 2.650 | -2.444 0.850 -2.404 | -0.988
| 4.566 |
0.137
| 1.032 |
Dim.2
1.380 0.599 3.150 |
0.139 -2.256 |
3.576 |
0.001 0.250 |
0.339 | -0.857 0.690 -1.358 |
4.562 0.998
0.497 0.232
cos2 v.test
2.2 Clustering sur les outputs de l’ACP
$tree
##
Call:
flashClust::hclust(d = dissi, method = method, members = weight)
##
Cluster method
Distance
Number of objects: 23
: ward
: euclidean
$nb.clust
[1] 3
$within
[1] 12.000000000
[6] 1.232214708
[11] 0.370631136
[16] 0.086935386
[21] 0.013934889
5.237349047 2.881145878 2.119286630 1.523668752
0.959979191 0.798661153 0.643435441 0.492720128
0.255438457 0.201557505 0.152512154 0.118340088
0.064962439 0.047549074 0.035692437 0.024106815
0.006904631
$inert.gain
[1] 6.762650953 2.356203170 0.761859247 0.595617878 0.291454044
[6] 0.272235517 0.161318038 0.155225712 0.150715314 0.122088991
[11] 0.115192679 0.053880952 0.049045351 0.034172066 0.031404701
[16] 0.021972947 0.017413366 0.011856636 0.011585622 0.010171927
[21] 0.007030258 0.006904631
$quot
4
[1] 0.5501153 0.7355707 0.7189536 0.8087156 0.7790681 0.8319567 0.8056426
[8] 0.7657647
2.3 Représentation graphique des résultats
Cluster Dendrogram
5.0
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5
Cluster plot
Moscow
Kiev
Minsk
Budapest
Krakow
Sofia
Prague
Elsinki
Oslo
Stockholm
Sarajevo
Berlin
Copenhagen
Paris
Brussels
Amsterdam
London
Dublin
2
1
0
−1
−2
Publicité
)
%
4
.
5
1
(
2
m
D
i
Athens
Madrid
Rome
cluster
a
a
a
1
2
3
Lisbon
Reykjavik
−3
−5.0
−2.5
0.0
2.5
Dim1 (82.9%)
5.0
7.5
Hierarchical clustering on the factor map
t
h
g
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h
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7
6
5
4
3
2
1
0
cluster 1
cluster 2
cluster 3
−6
−4
Reykjavik
Moscow
Minsk
Elsinki
Oslo
Stockholm
−2
Kiev
Krakow
Sofia
Berlin Prague
Sarajevo
Copenhagen
BrusselsParis
Amsterdam
London
2
0
Dublin
Budapest
4
Madrid Rome
Lisbon
6
8
Dim 1 (82.9%)
−3−2−1 0 1 2 3
Athens
)
%
4
.
5
1
(
2
m
D
i
2.4 Indicateurs numériques
##
6
80
Eta2
Global
p.value
Cla/Mod Mod/Cla
p.value df
6
v.test
100 21.73913 0.0005646527 3.448048
Link between the cluster variable and the categorical variables (chi-square test)
=================================================================================
##
Area 0.001195843
##
Description of each cluster by the categories
=============================================
$1
NULL
##
$2
NULL
##
$3
##
Area=South
##
##
Link between the cluster variable and the quantitative variables
================================================================
P-value
##
0.9177156 1.422898e-11
Annual
0.8989724 1.107640e-10
October
0.8864732 3.556182e-10
March
November
0.8707485 1.301232e-09
September 0.8559706 3.841596e-09
0.8353281 1.466202e-08
April
0.8246130 2.754070e-08
February
0.7730136 3.630759e-07
December
0.7476664 1.046528e-06
January
0.7378117 1.535089e-06
Latitude
0.7159888 3.414722e-06
August
0.6309204 4.690301e-05
July
May
0.5860137 1.478632e-04
0.5752594 1.910900e-04
June
Longitude 0.3720657 9.531078e-03
##
Description of each cluster by quantitative variables
=====================================================
$1
##
3.173406
Latitude
2.858559
Longitude
2.142021
Amplitude
-1.994799
July
-2.058250
June
-2.479950
August
May
-2.549115
September -3.137128
-3.255703
January
-3.269173
December
-3.359278
November
-3.373602
Annual
-3.387118
April
v.test Mean in category Overall mean sd in category
4.2809974
8.1046220
4.8430889
2.4549824
2.5172061
2.2585890
2.4289075
1.6685568
2.6250753
1.8302448
0.9396048
0.7671841
1.5489298
50.2956522
15.4521739
18.8000000
18.9260870
16.7652174
18.3043478
13.2739130
14.7086957
0.1739130
1.8434783
5.0782609
9.3739130
8.3782609
57.485714
24.328571
21.985714
16.785714
14.728571
15.485714
10.842857
10.985714
-5.142857
-2.914286
0.600000
5.500000
4.671429
7
-4.600000
5.757143
-1.142857
0.9565217
10.0652174
4.0608696
2.3366643
0.9194053
1.1043513
p.value
7.029223 0.0015066157
9.633631 0.0042557039
4.614062 0.0321917637
3.328822 0.0460648072
3.069855 0.0395661536
3.526111 0.0131400659
2.958733 0.0107996531
3.681790 0.0017061164
5.066447 0.0011311207
4.515079 0.0010786223
4.135841 0.0007814643
3.562512 0.0007419150
3.395260 0.0007063103
5.008152 0.0005771764
3.869795 0.0005527723
4.392854 0.0002377646
Overall sd
-3.442119
-3.453788
Publicité
-3.675091
Overall sd
February
October
March
##
Latitude
Longitude
Amplitude
July
June
August
May
September
January
December
November
Annual
April
February
October
March
##
$2
##
Longitude -2.060647
##
Longitude
##
$3
##
Annual
September
October
August
November
July
April
March
February
June
December
January
May
Latitude
##
Annual
September
October
August
November
July
April
March
February
June
3.851788
3.808964
3.717610
3.705134
3.692832
3.603579
3.531688
3.448552
3.434969
3.389407
3.387321
3.292484
3.183059
-3.328226
Overall sd
v.test Mean in category Overall mean sd in category
7.449161
15.45217
11.4
p.value
9.633631 0.03933669
v.test Mean in category Overall mean sd in category
1.393736
1.536839
1.911151
1.883315
2.264260
2.089258
1.175798
1.274755
1.744276
1.864135
2.337199
2.076505
1.553222
1.523770
9.3739130
14.7086957
10.0652174
18.3043478
5.0782609
18.9260870
8.3782609
4.0608696
0.9565217
16.7652174
1.8434783
0.1739130
13.2739130
50.2956522
15.750
21.225
16.750
24.375
12.175
24.500
13.950
11.100
8.950
21.600
8.950
7.925
17.650
39.425
p.value
3.562512 0.0001172584
3.681790 0.0001395501
3.869795 0.0002011160
3.526111 0.0002112792
4.135841 0.0002217705
3.328822 0.0003138648
3.395260 0.0004129164
4.392854 0.0005636006
5.008152 0.0005926217
3.069855 0.0007004406
8
December
January
May
Latitude
4.515079 0.0007057887
5.066447 0.0009930662
2.958733 0.0014572796
7.029223 0.0008740097
v.test Mean in category Overall mean sd in category Overall sd
3.154005
9.98899e-16
##
Link between the cluster variable and the quantitative variables
================================================================
##
P-value
Eta2
Dim.1 0.9086821 4.032342e-11
Dim.3 0.2691961 4.345188e-02
##
Description of each cluster by quantitative variables
=====================================================
$1
##
Dim.1 -3.317358
p.value
##
Dim.1 0.0009087321
##
$2
##
Dim.3 -2.414108
##
p.value
Dim.3 0.01577378
##
$3
##
Dim.1 3.863084
##
p.value
Dim.1 0.0001119644
-3.372519
0.8494564
1.264335
v.test Mean in category Overall mean sd in category Overall sd
3.154005
5.661507 9.98899e-16
v.test Mean in category Overall mean sd in category Overall sd
0.2178523 0.3553249
-0.1750962 -2.289835e-16
Rome
Oslo
Minsk
Berlin
Brussels
Amsterdam
0.8838369
Elsinki Stockholm
Sarajevo
0.7163471
Prague
1.0377318 1.0555319 1.1242019
Moscow
0.9222205 0.9652667 1.7664186
$para
Cluster: 1
##
0.3393856
--------------------------------------------------------
Cluster: 2
##
0.5764625
--------------------------------------------------------
Cluster: 3
##
0.3598845 1.7370025 1.8352756 2.1670401
##
$dist
Cluster: 1
Reykjavik
##
--------------------------------------------------------
Cluster: 2
##
##
--------------------------------------------------------
Cluster: 3
Paris
4.381195
Brussels
4.352044
Moscow
4.339979
Dublin
4.284426
Elsinki
4.280308
Minsk
3.738541
Amsterdam
Budapest
4.374324
5.474055
4.077437
3.484728
Madrid
Lisbon
Athens
Oslo
9
Athens
7.665256 5.660918 5.350595 4.216158
Lisbon
Madrid
Rome
10