Capítulo 8 HCPC: Hierarchical Clustering on Principal Components
Marcela Díaz
2023-05-10
La agrupación de clústeres es uno de los métodos más importantes para descubrir conocimiento en conjunto de datos multivariantes; el objetivo es identificar grupos de objetos similares dentro de un conjunto de datos de interés. En otras palabras, Clustering consiste en agrupar ítems en grupos con características similares. Dentro del área de las geociencias, es utilizado para determinar patrones climáticos.
En este capítulo se describe el porqué y cómo combinar componentes principales y métodos clustering; se hace un enfoque a HCPC (Hierarchical Clustering on Principal Components), el cual nos permite combinar los tres métodos estándar utilizados en los análisis de datos multivariados: -Métodos de componentes principales (PCA, CA, MCA, FAMD, MFA), -Agrupación jerárquica y -Agrupación en clústeres de particiones, particularmente el método k-means.
La combinación de métodos de componentes principales y métodos de agrupación resulta útil en al menos tres situaciones. Estas situaciones son variables continuas, agrupación de datos categóricos y agrupación de datos mixtos.
Empezamos instalando los paquetes necesarios. Usamos -FactoMineR para computar HCPC y - factoextra para visualizar los resultados.
install.packages(c("FactoMineR", "factoextra"), repos = "http://cran.us.r-project.org")## Installing packages into 'C:/Users/Marcela Diaz/AppData/Local/R/win-library/4.2'
## (as 'lib' is unspecified)## package 'FactoMineR' successfully unpacked and MD5 sums checked## Warning: cannot remove prior installation of package 'FactoMineR'## Warning in file.copy(savedcopy, lib, recursive = TRUE): problema al copiar
## C:\Users\Marcela
## Diaz\AppData\Local\R\win-library\4.2\00LOCK\FactoMineR\libs\x64\FactoMineR.dll
## a C:\Users\Marcela
## Diaz\AppData\Local\R\win-library\4.2\FactoMineR\libs\x64\FactoMineR.dll:
## Permission denied## Warning: restored 'FactoMineR'## package 'factoextra' successfully unpacked and MD5 sums checked
##
## The downloaded binary packages are in
## C:\Users\Marcela Diaz\AppData\Local\Temp\RtmpGwd9xd\downloaded_packagesCargamos las librerías.
library(factoextra)## Warning: package 'factoextra' was built under R version 4.2.3## Loading required package: ggplot2## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBalibrary(FactoMineR)## Warning: package 'FactoMineR' was built under R version 4.2.3Case of continuous variables
Primero calculamos el análisis de componentes principales (pca), se mantienen los primeros 3 componentes con ncp y HCPC se aplica al resultado de pca.
library(FactoMineR)
# Compute PCA with ncp = 3
res.pca <- PCA(USArrests, ncp = 3, graph = FALSE)
# Compute hierarchical clustering on principal components
res.hcpc <- HCPC(res.pca, graph = FALSE)La visualización del dendograma es generado con la función fviz_dend() que está dentro del paquete “factoextra”.
fviz_dend(res.hcpc,
cex = 0.7, # Label size
palette = "jco", # Color palette see ?ggpubr::ggpar
rect = TRUE, rect_fill = TRUE, # Add rectangle around groups
rect_border = "jco", # Rectangle color
labels_track_height = 0.8 # Augment the room for labels
)## Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
## of ggplot2 3.3.4.
## ℹ The deprecated feature was likely used in the factoextra package.
## Please report the issue at <]8;;https://github.com/kassambara/factoextra/issueshttps://github.com/kassambara/factoextra/issues]8;;>.
Adicionalmente, se pueden visualizar individuos en el mapa de componentes principales y colorearlos según el clúster al que pertenecen. Para visualizar grupos individuales se utiliza fviz_cluster().
fviz_cluster(res.hcpc,
repel = TRUE, # Avoid label overlapping
show.clust.cent = TRUE, # Show cluster centers
palette = "jco", # Color palette see ?ggpubr::ggpar
ggtheme = theme_minimal(),
main = "Factor map"
)
También podemos dibujar una gráfica tridimensional combinando la agrupación jerárquica y el mapa factoral utilizando plot().
# Principal components + tree
plot(res.hcpc, choice = "3D.map")
La función devuelve una lista que contiene:
- data.clust: datos originales de una columna suplementaria llamada “clase”.
- desc.var: variables que describen los clústeres.
- desc.ind: individuos más típicos de cada grupo.
- desc.axes: ejes que describen los clústeres.
Para visualizar los datos originales con asignación de clúster, usamos el siguiente código:
head(res.hcpc$data.clust, 10)## Murder Assault UrbanPop Rape clust
## Alabama 13.2 236 58 21.2 3
## Alaska 10.0 263 48 44.5 4
## Arizona 8.1 294 80 31.0 4
## Arkansas 8.8 190 50 19.5 3
## California 9.0 276 91 40.6 4
## Colorado 7.9 204 78 38.7 4
## Connecticut 3.3 110 77 11.1 2
## Delaware 5.9 238 72 15.8 2
## Florida 15.4 335 80 31.9 4
## Georgia 17.4 211 60 25.8 3En la tabla anterior, la última columna contiene las asignaciones de los clústers.
Para observar las variables cuantitativas que describen cada clúster, usamos:
res.hcpc$desc.var$quanti## $`1`
## v.test Mean in category Overall mean sd in category Overall sd
## UrbanPop -3.898420 52.07692 65.540 9.691087 14.329285
## Murder -4.030171 3.60000 7.788 2.269870 4.311735
## Rape -4.052061 12.17692 21.232 3.130779 9.272248
## Assault -4.638172 78.53846 170.760 24.700095 82.500075
## p.value
## UrbanPop 9.682222e-05
## Murder 5.573624e-05
## Rape 5.076842e-05
## Assault 3.515038e-06
##
## $`2`
## v.test Mean in category Overall mean sd in category Overall sd
## UrbanPop 2.793185 73.87500 65.540 8.652131 14.329285
## Murder -2.374121 5.65625 7.788 1.594902 4.311735
## p.value
## UrbanPop 0.005219187
## Murder 0.017590794
##
## $`3`
## v.test Mean in category Overall mean sd in category Overall sd
## Murder 4.357187 13.9375 7.788 2.433587 4.311735
## Assault 2.698255 243.6250 170.760 46.540137 82.500075
## UrbanPop -2.513667 53.7500 65.540 7.529110 14.329285
## p.value
## Murder 1.317449e-05
## Assault 6.970399e-03
## UrbanPop 1.194833e-02
##
## $`4`
## v.test Mean in category Overall mean sd in category Overall sd
## Rape 5.352124 33.19231 21.232 6.996643 9.272248
## Assault 4.356682 257.38462 170.760 41.850537 82.500075
## UrbanPop 3.028838 76.00000 65.540 10.347798 14.329285
## Murder 2.913295 10.81538 7.788 2.001863 4.311735
## p.value
## Rape 8.692769e-08
## Assault 1.320491e-05
## UrbanPop 2.454964e-03
## Murder 3.576369e-03De estos datos se puede analizar que las variables UrbanPop, Asesinato, Violación y Asalto están más asociadas al grupo 1. Un ejemplo de esto es “Assault” ya que su media es de 78,53 valor que está por debajo de la media general (170,76), por ende, se concluye que el grupo 1 se caracteriza por tener una baja tasa de asalto en comparación con elresto de los grupos. De esta misma manera podemos analizar el resto de los clúster.
Para mostrar las dimensiones principales asociadas a los clústeres, usamos:
res.hcpc$desc.axes$quanti## $`1`
## v.test Mean in category Overall mean sd in category Overall sd
## Dim.1 -5.175764 -1.964502 -5.639933e-16 0.6192556 1.574878
## p.value
## Dim.1 2.269806e-07
##
## $`2`
## v.test Mean in category Overall mean sd in category Overall sd
## Dim.2 3.585635 0.7428712 -5.369316e-16 0.6137936 0.9948694
## p.value
## Dim.2 0.0003362596
##
## $`3`
## v.test Mean in category Overall mean sd in category Overall sd
## Dim.1 2.058338 1.0610731 -5.639933e-16 0.5146613 1.5748783
## Dim.3 2.028887 0.3965588 3.535366e-17 0.3714503 0.5971291
## Dim.2 -4.536594 -1.4773302 -5.369316e-16 0.5750284 0.9948694
## p.value
## Dim.1 3.955769e-02
## Dim.3 4.246985e-02
## Dim.2 5.717010e-06
##
## $`4`
## v.test Mean in category Overall mean sd in category Overall sd
## Dim.1 4.986474 1.892656 -5.639933e-16 0.6126035 1.574878
## p.value
## Dim.1 6.149115e-07Si queremos saber cuáles son los individuos representativos de cada grupo, podemos utilizar el siguiente código:
res.hcpc$desc.ind$para## Cluster: 1
## Idaho South Dakota Maine Iowa New Hampshire
## 0.3674381 0.4993032 0.5012072 0.5533105 0.5891145
## ------------------------------------------------------------
## Cluster: 2
## Ohio Oklahoma Pennsylvania Kansas Indiana
## 0.2796100 0.5047549 0.5088363 0.6039091 0.7100820
## ------------------------------------------------------------
## Cluster: 3
## Alabama South Carolina Georgia Tennessee Louisiana
## 0.3553460 0.5335189 0.6136865 0.8522640 0.8780872
## ------------------------------------------------------------
## Cluster: 4
## Michigan Arizona New Mexico Maryland Texas
## 0.3246254 0.4532480 0.5176322 0.9013514 0.9239792En la tabla se nos muestran los 5 individuos que están más cerca al centro del cluster.
Case of categorical variables
Aquí calcularemos MCA, manteniendo los primeros 20 ejes del MCA que contienen el 87% de la información.
# Loading data
library(FactoMineR)
data(tea)
# Performing MCA
res.mca <- MCA(tea,
ncp = 20, # Number of components kept
quanti.sup = 19, # Quantitative supplementary variables
quali.sup = c(20:36), # Qualitative supplementary variables
graph=FALSE)Aplicamos la agrupación jerárquica en los resultados del MCA.
res.hcpc <- HCPC (res.mca, graph = FALSE, max = 3)Para visualizar los datos utilizamos:
# Dendrogram
fviz_dend(res.hcpc, show_labels = FALSE)
# Individuals facor map
fviz_cluster(res.hcpc, geom = "point", main = "Factor map")
Los clústeres pueden ser descritos por:
- Variables y/o categorías.
- Ejes principales.
- Individuos.
A continuación se muestran unos ejemplos con un subconjunto de resultados.
# Description by variables
res.hcpc$desc.var$test.chi2## p.value df
## where 8.465616e-79 4
## how 3.144675e-47 4
## price 1.862462e-28 10
## tearoom 9.624188e-19 2
## pub 8.539893e-10 2
## friends 6.137618e-08 2
## resto 3.537876e-07 2
## How 3.616532e-06 6
## Tea 1.778330e-03 4
## sex 1.789593e-03 2
## frequency 1.973274e-03 6
## work 3.052988e-03 2
## tea.time 3.679599e-03 2
## lunch 1.052478e-02 2
## dinner 2.234313e-02 2
## always 3.600913e-02 2
## sugar 3.685785e-02 2
## sophisticated 4.077297e-02 2# Description by variable categories
res.hcpc$desc.var$category## $`1`
## Cla/Mod Mod/Cla Global p.value
## where=chain store 85.937500 93.750000 64.000000 2.094419e-40
## how=tea bag 84.117647 81.250000 56.666667 1.478564e-25
## tearoom=Not.tearoom 70.661157 97.159091 80.666667 1.082077e-18
## price=p_branded 83.157895 44.886364 31.666667 1.631861e-09
## pub=Not.pub 67.088608 90.340909 79.000000 1.249296e-08
## friends=Not.friends 76.923077 45.454545 34.666667 2.177180e-06
## resto=Not.resto 64.705882 81.250000 73.666667 4.546462e-04
## price=p_private label 90.476190 10.795455 7.000000 1.343844e-03
## tea.time=Not.tea time 67.938931 50.568182 43.666667 4.174032e-03
## How=alone 64.102564 71.022727 65.000000 9.868387e-03
## work=Not.work 63.380282 76.704545 71.000000 1.036429e-02
## sugar=sugar 66.206897 54.545455 48.333333 1.066744e-02
## always=Not.always 63.959391 71.590909 65.666667 1.079912e-02
## price=p_unknown 91.666667 6.250000 4.000000 1.559798e-02
## frequency=1 to 2/week 75.000000 18.750000 14.666667 1.649092e-02
## frequency=1/day 68.421053 36.931818 31.666667 1.958790e-02
## age_Q=15-24 68.478261 35.795455 30.666667 2.179803e-02
## price=p_cheap 100.000000 3.977273 2.333333 2.274539e-02
## lunch=Not.lunch 61.328125 89.204545 85.333333 2.681490e-02
## SPC=senior 42.857143 8.522727 11.666667 4.813710e-02
## lunch=lunch 43.181818 10.795455 14.666667 2.681490e-02
## always=always 48.543689 28.409091 34.333333 1.079912e-02
## sugar=No.sugar 51.612903 45.454545 51.666667 1.066744e-02
## work=work 47.126437 23.295455 29.000000 1.036429e-02
## tea.time=tea time 51.479290 49.431818 56.333333 4.174032e-03
## How=lemon 30.303030 5.681818 11.000000 5.943089e-04
## resto=resto 41.772152 18.750000 26.333333 4.546462e-04
## How=other 0.000000 0.000000 3.000000 2.952904e-04
## price=p_variable 44.642857 28.409091 37.333333 1.595638e-04
## frequency=+2/day 45.669291 32.954545 42.333333 9.872288e-05
## friends=friends 48.979592 54.545455 65.333333 2.177180e-06
## how=unpackaged 19.444444 3.977273 12.000000 4.328211e-07
## pub=pub 26.984127 9.659091 21.000000 1.249296e-08
## where=tea shop 6.666667 1.136364 10.000000 4.770573e-10
## price=p_upscale 18.867925 5.681818 17.666667 9.472539e-11
## how=tea bag+unpackaged 27.659574 14.772727 31.333333 1.927326e-13
## tearoom=tearoom 8.620690 2.840909 19.333333 1.082077e-18
## where=chain store+tea shop 11.538462 5.113636 26.000000 1.133459e-23
## v.test
## where=chain store 13.307475
## how=tea bag 10.449142
## tearoom=Not.tearoom 8.826287
## price=p_branded 6.030764
## pub=Not.pub 5.692859
## friends=Not.friends 4.736242
## resto=Not.resto 3.506146
## price=p_private label 3.206448
## tea.time=Not.tea time 2.864701
## How=alone 2.580407
## work=Not.work 2.563432
## sugar=sugar 2.553408
## always=Not.always 2.549133
## price=p_unknown 2.418189
## frequency=1 to 2/week 2.397866
## frequency=1/day 2.334149
## age_Q=15-24 2.293869
## price=p_cheap 2.277684
## lunch=Not.lunch 2.214202
## SPC=senior -1.976156
## lunch=lunch -2.214202
## always=always -2.549133
## sugar=No.sugar -2.553408
## work=work -2.563432
## tea.time=tea time -2.864701
## How=lemon -3.434198
## resto=resto -3.506146
## How=other -3.619397
## price=p_variable -3.775692
## frequency=+2/day -3.893709
## friends=friends -4.736242
## how=unpackaged -5.053925
## pub=pub -5.692859
## where=tea shop -6.226471
## price=p_upscale -6.475138
## how=tea bag+unpackaged -7.353743
## tearoom=tearoom -8.826287
## where=chain store+tea shop -10.029275
##
## $`2`
## Cla/Mod Mod/Cla Global p.value
## where=tea shop 90.000000 84.375 10.00000 3.703402e-30
## how=unpackaged 66.666667 75.000 12.00000 5.346850e-20
## price=p_upscale 49.056604 81.250 17.66667 2.392655e-17
## Tea=green 27.272727 28.125 11.00000 4.436713e-03
## sophisticated=sophisticated 13.488372 90.625 71.66667 8.080918e-03
## sex=M 16.393443 62.500 40.66667 9.511848e-03
## resto=Not.resto 13.122172 90.625 73.66667 1.587879e-02
## dinner=dinner 28.571429 18.750 7.00000 1.874042e-02
## escape.exoticism=Not.escape-exoticism 14.556962 71.875 52.66667 2.177458e-02
## how=tea bag+unpackaged 5.319149 15.625 31.33333 3.876799e-02
## escape.exoticism=escape-exoticism 6.338028 28.125 47.33333 2.177458e-02
## dinner=Not.dinner 9.318996 81.250 93.00000 1.874042e-02
## resto=resto 3.797468 9.375 26.33333 1.587879e-02
## Tea=Earl Grey 7.253886 43.750 64.33333 1.314753e-02
## sex=F 6.741573 37.500 59.33333 9.511848e-03
## sophisticated=Not.sophisticated 3.529412 9.375 28.33333 8.080918e-03
## where=chain store+tea shop 2.564103 6.250 26.00000 3.794134e-03
## price=p_variable 3.571429 12.500 37.33333 1.349384e-03
## age_Q=15-24 2.173913 6.250 30.66667 6.100227e-04
## price=p_branded 2.105263 6.250 31.66667 4.024289e-04
## how=tea bag 1.764706 9.375 56.66667 5.537403e-09
## where=chain store 1.562500 9.375 64.00000 1.664577e-11
## v.test
## where=tea shop 11.410559
## how=unpackaged 9.156781
## price=p_upscale 8.472945
## Tea=green 2.845318
## sophisticated=sophisticated 2.648670
## sex=M 2.593088
## resto=Not.resto 2.411690
## dinner=dinner 2.350655
## escape.exoticism=Not.escape-exoticism 2.294277
## how=tea bag+unpackaged -2.066641
## escape.exoticism=escape-exoticism -2.294277
## dinner=Not.dinner -2.350655
## resto=resto -2.411690
## Tea=Earl Grey -2.479748
## sex=F -2.593088
## sophisticated=Not.sophisticated -2.648670
## where=chain store+tea shop -2.894789
## price=p_variable -3.205264
## age_Q=15-24 -3.427119
## price=p_branded -3.538486
## how=tea bag -5.830161
## where=chain store -6.732775
##
## $`3`
## Cla/Mod Mod/Cla Global p.value
## where=chain store+tea shop 85.897436 72.826087 26.00000 5.730651e-34
## how=tea bag+unpackaged 67.021277 68.478261 31.33333 1.382641e-19
## tearoom=tearoom 77.586207 48.913043 19.33333 1.252051e-16
## pub=pub 63.492063 43.478261 21.00000 1.126679e-09
## friends=friends 41.836735 89.130435 65.33333 1.429181e-09
## price=p_variable 51.785714 63.043478 37.33333 1.572243e-09
## resto=resto 54.430380 46.739130 26.33333 2.406386e-07
## How=other 100.000000 9.782609 3.00000 1.807938e-05
## frequency=+2/day 41.732283 57.608696 42.33333 4.237330e-04
## tea.time=tea time 38.461538 70.652174 56.33333 8.453564e-04
## work=work 44.827586 42.391304 29.00000 9.079377e-04
## sex=F 37.078652 71.739130 59.33333 3.494245e-03
## lunch=lunch 50.000000 23.913043 14.66667 3.917102e-03
## How=lemon 51.515152 18.478261 11.00000 8.747530e-03
## sugar=No.sugar 36.129032 60.869565 51.66667 3.484061e-02
## home=home 31.615120 100.000000 97.00000 3.506563e-02
## home=Not.home 0.000000 0.000000 3.00000 3.506563e-02
## sugar=sugar 24.827586 39.130435 48.33333 3.484061e-02
## price=p_private label 9.523810 2.173913 7.00000 2.370629e-02
## how=unpackaged 13.888889 5.434783 12.00000 1.645107e-02
## How=alone 25.128205 53.260870 65.00000 5.300881e-03
## lunch=Not.lunch 27.343750 76.086957 85.33333 3.917102e-03
## sex=M 21.311475 28.260870 40.66667 3.494245e-03
## Tea=green 9.090909 3.260870 11.00000 2.545816e-03
## frequency=1 to 2/week 11.363636 5.434783 14.66667 1.604219e-03
## work=Not.work 24.882629 57.608696 71.00000 9.079377e-04
## tea.time=Not.tea time 20.610687 29.347826 43.66667 8.453564e-04
## where=tea shop 3.333333 1.086957 10.00000 1.466234e-04
## price=p_branded 14.736842 15.217391 31.66667 2.746948e-05
## resto=Not.resto 22.171946 53.260870 73.66667 2.406386e-07
## friends=Not.friends 9.615385 10.869565 34.66667 1.429181e-09
## pub=Not.pub 21.940928 56.521739 79.00000 1.126679e-09
## how=tea bag 14.117647 26.086957 56.66667 1.082059e-12
## tearoom=Not.tearoom 19.421488 51.086957 80.66667 1.252051e-16
## where=chain store 12.500000 26.086957 64.00000 1.711522e-19
## v.test
## where=chain store+tea shop 12.150084
## how=tea bag+unpackaged 9.053653
## tearoom=tearoom 8.278053
## pub=pub 6.090345
## friends=friends 6.052158
## price=p_variable 6.036775
## resto=resto 5.164845
## How=other 4.287379
## frequency=+2/day 3.524844
## tea.time=tea time 3.337500
## work=work 3.317602
## sex=F 2.920541
## lunch=lunch 2.884762
## How=lemon 2.621767
## sugar=No.sugar 2.110206
## home=home 2.107600
## home=Not.home -2.107600
## sugar=sugar -2.110206
## price=p_private label -2.261856
## how=unpackaged -2.398752
## How=alone -2.788157
## lunch=Not.lunch -2.884762
## sex=M -2.920541
## Tea=green -3.017842
## frequency=1 to 2/week -3.155139
## work=Not.work -3.317602
## tea.time=Not.tea time -3.337500
## where=tea shop -3.796720
## price=p_branded -4.193490
## resto=Not.resto -5.164845
## friends=Not.friends -6.052158
## pub=Not.pub -6.090345
## how=tea bag -7.119644
## tearoom=Not.tearoom -8.278053
## where=chain store -9.030332Descripción por componentes principales (ejes):
res.hcpc$desc.axes##
## Link between the cluster variable and the quantitative variables
## ================================================================
## Eta2 P-value
## Dim.2 0.66509105 2.828937e-71
## Dim.1 0.63497903 1.009707e-65
## Dim.4 0.11231020 2.073924e-08
## Dim.14 0.03141943 8.732913e-03
## Dim.6 0.02358138 2.890373e-02
##
## Description of each cluster by quantitative variables
## =====================================================
## $`1`
## v.test Mean in category Overall mean sd in category Overall sd
## Dim.6 2.647552 0.03433626 -2.721637e-17 0.2655618 0.2671712
## Dim.2 -7.796641 -0.13194656 -6.207419e-17 0.1813156 0.3486355
## Dim.1 -12.409741 -0.23196088 -1.678981e-16 0.2143767 0.3850642
## p.value
## Dim.6 8.107689e-03
## Dim.2 6.357699e-15
## Dim.1 2.314001e-35
##
## $`2`
## v.test Mean in category Overall mean sd in category Overall sd
## Dim.2 13.918285 0.81210870 -6.207419e-17 0.2340345 0.3486355
## Dim.4 4.350620 0.20342610 7.249699e-19 0.3700048 0.2793822
## Dim.14 2.909073 0.10749165 4.512196e-17 0.2161509 0.2207818
## Dim.13 2.341566 0.08930402 1.266782e-17 0.1606616 0.2278809
## Dim.3 2.208179 0.11087544 -4.161891e-17 0.2449710 0.3000159
## Dim.11 -2.234447 -0.08934293 6.504924e-17 0.2066708 0.2389094
## p.value
## Dim.2 4.905356e-44
## Dim.4 1.357531e-05
## Dim.14 3.625025e-03
## Dim.13 1.920305e-02
## Dim.3 2.723180e-02
## Dim.11 2.545367e-02
##
## $`3`
## v.test Mean in category Overall mean sd in category Overall sd
## Dim.1 13.485906 0.45155993 -1.678981e-16 0.2516544 0.3850642
## Dim.6 -2.221728 -0.05161581 -2.721637e-17 0.2488566 0.2671712
## Dim.4 -4.725270 -0.11479621 7.249699e-19 0.2924881 0.2793822
## p.value
## Dim.1 1.893256e-41
## Dim.6 2.630166e-02
## Dim.4 2.298093e-06Descripción por individuos:
res.hcpc$desc.ind$para## Cluster: 1
## 285 152 166 143 71
## 0.5884476 0.6242123 0.6242123 0.6244176 0.6478185
## ------------------------------------------------------------
## Cluster: 2
## 31 95 53 182 202
## 0.6620553 0.7442013 0.7610437 0.7948663 0.8154826
## ------------------------------------------------------------
## Cluster: 3
## 172 33 233 18 67
## 0.7380497 0.7407711 0.7503006 0.7572188 0.7701598Conclusiones
Con este capítulo se logró comprender cómo se calculan las HCPC (agrupaciones jerárquicas en los componentes principales). Este enfoque es útil cuando: - Se tiene un conjunto de datos grande que contiene variables continuas y queremos reducir la dimensión de los datos antes del análisis de agrupación jerárquica. - Se tiene un conjunto de variables categóricas y queremos convertirlas en componentes principales continuos que se puedan utilizar como entrada del análisis de conglomerados.