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library(FactoMineR)
## Warning: package 'FactoMineR' was built under R version 4.2.3
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/ve3WBa
data(decanthlon2)
## Warning in data(decanthlon2): data set 'decanthlon2' not found
head(decathlon2)
## X100m Long.jump Shot.put High.jump X400m X110m.hurdle Discus
## SEBRLE 11.04 7.58 14.83 2.07 49.81 14.69 43.75
## CLAY 10.76 7.40 14.26 1.86 49.37 14.05 50.72
## BERNARD 11.02 7.23 14.25 1.92 48.93 14.99 40.87
## YURKOV 11.34 7.09 15.19 2.10 50.42 15.31 46.26
## ZSIVOCZKY 11.13 7.30 13.48 2.01 48.62 14.17 45.67
## McMULLEN 10.83 7.31 13.76 2.13 49.91 14.38 44.41
## Pole.vault Javeline X1500m Rank Points Competition
## SEBRLE 5.02 63.19 291.7 1 8217 Decastar
## CLAY 4.92 60.15 301.5 2 8122 Decastar
## BERNARD 5.32 62.77 280.1 4 8067 Decastar
## YURKOV 4.72 63.44 276.4 5 8036 Decastar
## ZSIVOCZKY 4.42 55.37 268.0 7 8004 Decastar
## McMULLEN 4.42 56.37 285.1 8 7995 Decastar
str(decathlon2)
## 'data.frame': 27 obs. of 13 variables:
## $ X100m : num 11 10.8 11 11.3 11.1 ...
## $ Long.jump : num 7.58 7.4 7.23 7.09 7.3 7.31 6.81 7.56 6.97 7.27 ...
## $ Shot.put : num 14.8 14.3 14.2 15.2 13.5 ...
## $ High.jump : num 2.07 1.86 1.92 2.1 2.01 2.13 1.95 1.86 1.95 1.98 ...
## $ X400m : num 49.8 49.4 48.9 50.4 48.6 ...
## $ X110m.hurdle: num 14.7 14.1 15 15.3 14.2 ...
## $ Discus : num 43.8 50.7 40.9 46.3 45.7 ...
## $ Pole.vault : num 5.02 4.92 5.32 4.72 4.42 4.42 4.92 4.82 4.72 4.62 ...
## $ Javeline : num 63.2 60.1 62.8 63.4 55.4 ...
## $ X1500m : num 292 302 280 276 268 ...
## $ Rank : int 1 2 4 5 7 8 9 10 11 12 ...
## $ Points : int 8217 8122 8067 8036 8004 7995 7802 7733 7708 7651 ...
## $ Competition : Factor w/ 2 levels "Decastar","OlympicG": 1 1 1 1 1 1 1 1 1 1 ...
decathlon2.active <- decathlon2[1:23, 1:10]
head(decathlon2.active[, 1:5],3) # head of 3 rows of 5 columns.
## X100m Long.jump Shot.put High.jump X400m
## SEBRLE 11.04 7.58 14.83 2.07 49.81
## CLAY 10.76 7.40 14.26 1.86 49.37
## BERNARD 11.02 7.23 14.25 1.92 48.93
res.pca <- PCA(decathlon2.active, graph=FALSE) #Standadized data.
#View(res.pca)
print(res.pca)
## **Results for the Principal Component Analysis (PCA)**
## The analysis was performed on 23 individuals, described by 10 variables
## *The results are available in the following objects:
##
## name description
## 1 "$eig" "eigenvalues"
## 2 "$var" "results for the variables"
## 3 "$var$coord" "coord. for the variables"
## 4 "$var$cor" "correlations variables - dimensions"
## 5 "$var$cos2" "cos2 for the variables"
## 6 "$var$contrib" "contributions of the variables"
## 7 "$ind" "results for the individuals"
## 8 "$ind$coord" "coord. for the individuals"
## 9 "$ind$cos2" "cos2 for the individuals"
## 10 "$ind$contrib" "contributions of the individuals"
## 11 "$call" "summary statistics"
## 12 "$call$centre" "mean of the variables"
## 13 "$call$ecart.type" "standard error of the variables"
## 14 "$call$row.w" "weights for the individuals"
## 15 "$call$col.w" "weights for the variables"
# 4.4.1 eigenvales/variances
eig.val <- get_eigenvalue(res.pca)
eig.val
## eigenvalue variance.percent cumulative.variance.percent
## Dim.1 4.1242133 41.242133 41.24213
## Dim.2 1.8385309 18.385309 59.62744
## Dim.3 1.2391403 12.391403 72.01885
## Dim.4 0.8194402 8.194402 80.21325
## Dim.5 0.7015528 7.015528 87.22878
## Dim.6 0.4228828 4.228828 91.45760
## Dim.7 0.3025817 3.025817 94.48342
## Dim.8 0.2744700 2.744700 97.22812
## Dim.9 0.1552169 1.552169 98.78029
## Dim.10 0.1219710 1.219710 100.00000
#scree plot
fviz_eig(res.pca, addlabels = TRUE, elim=c(0,50))
# 4.4.2 graph of variables
var <- get_pca_var(res.pca)
print(var)
## Principal Component Analysis Results for variables
## ===================================================
## Name Description
## 1 "$coord" "Coordinates for the variables"
## 2 "$cor" "Correlations between variables and dimensions"
## 3 "$cos2" "Cos2 for the variables"
## 4 "$contrib" "contributions of the variables"
#coordinates
head(var$coord)
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
## X100m -0.8506257 -0.17939806 0.3015564 0.03357320 -0.1944440
## Long.jump 0.7941806 0.28085695 -0.1905465 -0.11538956 0.2331567
## Shot.put 0.7339127 0.08540412 0.5175978 0.12846837 -0.2488129
## High.jump 0.6100840 -0.46521415 0.3300852 0.14455012 0.4027002
## X400m -0.7016034 0.29017826 0.2835329 0.43082552 0.1039085
## X110m.hurdle -0.7641252 -0.02474081 0.4488873 -0.01689589 0.2242200
head(var$cos2)
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
## X100m 0.7235641 0.0321836641 0.09093628 0.0011271597 0.03780845
## Long.jump 0.6307229 0.0788806285 0.03630798 0.0133147506 0.05436203
## Shot.put 0.5386279 0.0072938636 0.26790749 0.0165041211 0.06190783
## High.jump 0.3722025 0.2164242070 0.10895622 0.0208947375 0.16216747
## X400m 0.4922473 0.0842034209 0.08039091 0.1856106269 0.01079698
## X110m.hurdle 0.5838873 0.0006121077 0.20149984 0.0002854712 0.05027463
head(var$contrib)
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
## X100m 17.544293 1.7505098 7.338659 0.13755240 5.389252
## Long.jump 15.293168 4.2904162 2.930094 1.62485936 7.748815
## Shot.put 13.060137 0.3967224 21.620432 2.01407269 8.824401
## High.jump 9.024811 11.7715838 8.792888 2.54987951 23.115504
## X400m 11.935544 4.5799296 6.487636 22.65090599 1.539012
## X110m.hurdle 14.157544 0.0332933 16.261261 0.03483735 7.166193
#coordinates of varibles
head(var$coord, 4)
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
## X100m -0.8506257 -0.17939806 0.3015564 0.0335732 -0.1944440
## Long.jump 0.7941806 0.28085695 -0.1905465 -0.1153896 0.2331567
## Shot.put 0.7339127 0.08540412 0.5175978 0.1284684 -0.2488129
## High.jump 0.6100840 -0.46521415 0.3300852 0.1445501 0.4027002
#plot variables
fviz_pca_var(res.pca, col.var = "black")
# 4.4.2.3 quality of representation
head(var$cos2, 4)
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
## X100m 0.7235641 0.032183664 0.09093628 0.00112716 0.03780845
## Long.jump 0.6307229 0.078880629 0.03630798 0.01331475 0.05436203
## Shot.put 0.5386279 0.007293864 0.26790749 0.01650412 0.06190783
## High.jump 0.3722025 0.216424207 0.10895622 0.02089474 0.16216747
#let visualize the cos2
library(corrplot)
## corrplot 0.92 loaded
corrplot(var$cos2, is.corr = FALSE)
#total cos2 of variables on Dim.1 and Dim.2
fviz_cos2(res.pca, choice = "var", axes=1:2)
#color by cos2 values: quality on the factor map
fviz_pca_var(res.pca, col.var = "cos2", alpha.var = "cos2",
gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
repel = TRUE # Avoid text overlapping
)
# 4.4.2.4 contribution of variables to PCs
head(var$contrib,4)
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
## X100m 17.544293 1.7505098 7.338659 0.1375524 5.389252
## Long.jump 15.293168 4.2904162 2.930094 1.6248594 7.748815
## Shot.put 13.060137 0.3967224 21.620432 2.0140727 8.824401
## High.jump 9.024811 11.7715838 8.792888 2.5498795 23.115504
corrplot(var$contrib, is.corr = F)
# Contributions of variables to PC1
fviz_contrib(res.pca, choice = "var", axes = 1, top = 10)
# Contributions of variables to PC2
fviz_contrib(res.pca, choice = "var", axes = 2, top = 10)
#The total contribution to PC1 and PC2 is obtained with the following R code:
fviz_contrib(res.pca, choice = "var", axes = 1:2, top = 10)
#the most important plot #page 21
fviz_pca_var(res.pca, col.var = "contrib",
gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07")
)
