Comparison with FactoMineR

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booklet was designed to return same result as FactoMineR but with a more modern and tidyverse-friendly syntax. This document aims to show the differences between the two packages.

Eigs

# Get eigvalues and eigvectors with FactoMineR
X <- decathlon[, -c(11:13)]
res_pca <- PCA(X, quanti.sup = 10, ind.sup = 1, graph = FALSE)
head(res_pca$eig)
#>        eigenvalue percentage of variance cumulative percentage of variance
#> comp 1  3.2817473              36.463859                          36.46386
#> comp 2  1.6893625              18.770694                          55.23455
#> comp 3  1.0648909              11.832121                          67.06667
#> comp 4  0.8572889               9.525432                          76.59211
#> comp 5  0.6349252               7.054725                          83.64683
#> comp 6  0.4677351               5.197057                          88.84389

# Get eigvalues and eigvectors with booklet
X_active <- X[-1, -10]
X_active_scaled <- pca_standardize(X_active, scale = TRUE)

eigs <- pca_weighted_eigen(X_active_scaled)

df_eigs <- data.frame(
  eigenvalue = eigs$values,
  `percentage of variance` = eigs$values / sum(eigs$values) * 100,
  `cumulative percentage of variance` = cumsum(eigs$values / sum(eigs$values)) * 100
)

rownames(df_eigs) <- paste0("comp ", 1:nrow(df_eigs))

df_eigs |> head()
#>        eigenvalue percentage.of.variance cumulative.percentage.of.variance
#> comp 1  3.2817473              36.463859                          36.46386
#> comp 2  1.6893625              18.770694                          55.23455
#> comp 3  1.0648909              11.832121                          67.06667
#> comp 4  0.8572889               9.525432                          76.59211
#> comp 5  0.6349252               7.054725                          83.64683
#> comp 6  0.4677351               5.197057                          88.84389

PCA

Individuals

# Get principal components with FactoMineR
head(res_pca$ind$coord)
#>                Dim.1      Dim.2      Dim.3        Dim.4       Dim.5
#> CLAY       1.3100544 -0.4159492  0.5208056  0.369856466 -2.17766124
#> KARPOV     1.4371854 -0.4206194 -0.9485451  2.055098128 -0.08739926
#> BERNARD   -0.6048025 -1.1039234  2.2461329 -0.308551094  0.52147527
#> YURKOV    -0.5701719  2.4598460  0.6145443  0.002631539  1.26970612
#> WARNERS    0.3739047 -1.8369980 -0.4725537  0.728066088  0.69493980
#> ZSIVOCZKY  0.2692284 -0.4794763 -1.9969562 -0.258190426 -0.13749027

# Get principal components with booklet
ind_coords <- pca_ind_coords(eigs)
head(ind_coords[, 1:5])
#>                Dim.1      Dim.2      Dim.3        Dim.4       Dim.5
#> CLAY       1.3100544 -0.4159492  0.5208056  0.369856466 -2.17766124
#> KARPOV     1.4371854 -0.4206194 -0.9485451  2.055098128 -0.08739926
#> BERNARD   -0.6048025 -1.1039234  2.2461329 -0.308551094  0.52147527
#> YURKOV    -0.5701719  2.4598460  0.6145443  0.002631539  1.26970612
#> WARNERS    0.3739047 -1.8369980 -0.4725537  0.728066088  0.69493980
#> ZSIVOCZKY  0.2692284 -0.4794763 -1.9969562 -0.258190426 -0.13749027

# Get individual cos2 with FactoMineR
head(res_pca$ind$cos2)
#>                Dim.1      Dim.2      Dim.3        Dim.4       Dim.5
#> CLAY      0.20619906 0.02078684 0.03258812 1.643518e-02 0.569755684
#> KARPOV    0.25723191 0.02203321 0.11205086 5.259741e-01 0.000951294
#> BERNARD   0.04764941 0.15874795 0.65720548 1.240179e-02 0.035424009
#> YURKOV    0.03581693 0.66664184 0.04160860 7.629506e-07 0.177616436
#> WARNERS   0.02392091 0.57739535 0.03820834 9.069796e-02 0.082632386
#> ZSIVOCZKY 0.01324414 0.04200650 0.72865053 1.218043e-02 0.003454030

# Get individual cos2 with booklet
ind_cos2 <- pca_ind_cos2(ind_coords)
head(ind_cos2[, 1:5])
#>                Dim.1      Dim.2      Dim.3        Dim.4       Dim.5
#> CLAY      0.20619906 0.02078684 0.03258812 1.643518e-02 0.569755684
#> KARPOV    0.25723191 0.02203321 0.11205086 5.259741e-01 0.000951294
#> BERNARD   0.04764941 0.15874795 0.65720548 1.240179e-02 0.035424009
#> YURKOV    0.03581693 0.66664184 0.04160860 7.629506e-07 0.177616436
#> WARNERS   0.02392091 0.57739535 0.03820834 9.069796e-02 0.082632386
#> ZSIVOCZKY 0.01324414 0.04200650 0.72865053 1.218043e-02 0.003454030

# Get individual contributions with FactoMineR
head(res_pca$ind$contrib)
#>                Dim.1     Dim.2      Dim.3        Dim.4       Dim.5
#> CLAY      1.30741518 0.2560340  0.6367752 3.989140e-01 18.67231141
#> KARPOV    1.57347720 0.2618157  2.1122769 1.231623e+01  0.03007689
#> BERNARD   0.27865195 1.8034123 11.8442024 2.776304e-01  1.07074211
#> YURKOV    0.24765466 8.9543282  0.8866277 2.019447e-05  6.34780926
#> WARNERS   0.10650172 4.9938388  0.5242485 1.545804e+00  1.90156779
#> ZSIVOCZKY 0.05521748 0.3402134  9.3620727 1.943986e-01  0.07443229

# Get individual contributions with booklet
ind_contrib <- pca_ind_contrib(ind_coords, eigs)
head(ind_contrib[, 1:5])
#>                Dim.1     Dim.2      Dim.3        Dim.4       Dim.5
#> CLAY      1.30741518 0.2560340  0.6367752 3.989140e-01 18.67231141
#> KARPOV    1.57347720 0.2618157  2.1122769 1.231623e+01  0.03007689
#> BERNARD   0.27865195 1.8034123 11.8442024 2.776304e-01  1.07074211
#> YURKOV    0.24765466 8.9543282  0.8866277 2.019447e-05  6.34780926
#> WARNERS   0.10650172 4.9938388  0.5242485 1.545804e+00  1.90156779
#> ZSIVOCZKY 0.05521748 0.3402134  9.3620727 1.943986e-01  0.07443229

Supplementary individuals

# Get supplementary individuals with FactoMineR
res_pca$ind.sup$coord
#>            Dim.1     Dim.2    Dim.3     Dim.4    Dim.5
#> SEBRLE 0.8344973 0.3186264 1.214558 -0.201484 1.281883

# Get supplementary individuals with booklet
weights <- rep(1, nrow(X_active)) / nrow(X_active)
center <- colMeans(X_active)
std <- sqrt(as.vector(crossprod(weights, as.matrix(X_active^2)) - center^2))

X_sup <- X[1, -10]
X_sup_scaled <- (X_sup - center) / std
ind_sup_coords <- as.data.frame(as.matrix(X_sup_scaled) %*% eigs$vectors)
rownames(ind_sup_coords) <- rownames(X_sup)
ind_sup_coords[, 1:5]
#>            Dim.1     Dim.2    Dim.3     Dim.4    Dim.5
#> SEBRLE 0.8344973 0.3186264 1.214558 -0.201484 1.281883

# Get supplementary individuals cos2 with FactoMineR
res_pca$ind.sup$cos2
#>            Dim.1      Dim.2     Dim.3       Dim.4     Dim.5
#> SEBRLE 0.1505617 0.02194968 0.3189345 0.008776992 0.3552722

# Get supplementary individuals cos2 with booklet
ind_sup_cos2 <- pca_ind_cos2(ind_sup_coords)
ind_sup_cos2[, 1:5]
#>            Dim.1      Dim.2     Dim.3       Dim.4     Dim.5
#> SEBRLE 0.1505617 0.02194968 0.3189345 0.008776992 0.3552722

Variables

# Get variable coordinates with FactoMineR
head(res_pca$var$coord)
#>                  Dim.1      Dim.2       Dim.3       Dim.4       Dim.5
#> 100m        -0.7802087  0.2474540 -0.07802294  0.15739065  0.10348090
#> Long.jump    0.7418925 -0.4073126  0.05696965 -0.10200465  0.12853454
#> Shot.put     0.6230146  0.5713999  0.28623659  0.22239225 -0.02547125
#> High.jump    0.5704197  0.4191676 -0.24210464  0.09178538  0.63728214
#> 400m        -0.6756524  0.4720254  0.12379407  0.12444012 -0.02216888
#> 110m.hurdle -0.7510101  0.2498147  0.23834558 -0.06295776  0.22508937

# Get variable coordinates with booklet
var_coords <- pca_var_coords(eigs)
head(var_coords[, 1:5])
#>                  Dim.1      Dim.2       Dim.3       Dim.4       Dim.5
#> 100m        -0.7802087  0.2474540 -0.07802294  0.15739065  0.10348090
#> Long.jump    0.7418925 -0.4073126  0.05696965 -0.10200465  0.12853454
#> Shot.put     0.6230146  0.5713999  0.28623659  0.22239225 -0.02547125
#> High.jump    0.5704197  0.4191676 -0.24210464  0.09178538  0.63728214
#> 400m        -0.6756524  0.4720254  0.12379407  0.12444012 -0.02216888
#> 110m.hurdle -0.7510101  0.2498147  0.23834558 -0.06295776  0.22508937

# Get variable cos2 with FactoMineR
head(res_pca$var$cos2)
#>                 Dim.1      Dim.2       Dim.3       Dim.4        Dim.5
#> 100m        0.6087256 0.06123346 0.006087579 0.024771818 0.0107082970
#> Long.jump   0.5504045 0.16590352 0.003245541 0.010404949 0.0165211277
#> Shot.put    0.3881471 0.32649781 0.081931386 0.049458311 0.0006487846
#> High.jump   0.3253787 0.17570146 0.058614658 0.008424556 0.4061285259
#> 400m        0.4565061 0.22280802 0.015324971 0.015485343 0.0004914592
#> 110m.hurdle 0.5640162 0.06240739 0.056808617 0.003963679 0.0506652236

# Get variable cos2 with booklet
var_cos2 <- pca_var_cos2(var_coords)
head(var_cos2[, 1:5])
#>                 Dim.1      Dim.2       Dim.3       Dim.4        Dim.5
#> 100m        0.6087256 0.06123346 0.006087579 0.024771818 0.0107082970
#> Long.jump   0.5504045 0.16590352 0.003245541 0.010404949 0.0165211277
#> Shot.put    0.3881471 0.32649781 0.081931386 0.049458311 0.0006487846
#> High.jump   0.3253787 0.17570146 0.058614658 0.008424556 0.4061285259
#> 400m        0.4565061 0.22280802 0.015324971 0.015485343 0.0004914592
#> 110m.hurdle 0.5640162 0.06240739 0.056808617 0.003963679 0.0506652236

# Get variable contributions with FactoMineR
head(res_pca$var$contrib)
#>                Dim.1     Dim.2     Dim.3     Dim.4       Dim.5
#> 100m        18.54882  3.624649 0.5716623 2.8895531  1.68654464
#> Long.jump   16.77169  9.820481 0.3047769 1.2137039  2.60205888
#> Shot.put    11.82745 19.326688 7.6938764 5.7691535  0.10218284
#> High.jump    9.91480 10.400459 5.5042879 0.9826975 63.96478233
#> 400m        13.91046 13.188882 1.4391119 1.8063156  0.07740427
#> 110m.hurdle 17.18646  3.694139 5.3346892 0.4623504  7.97971527

# Get variable contributions with booklet
var_contrib <- pca_var_contrib(var_cos2, eigs)
head(var_contrib[, 1:5])
#>                Dim.1     Dim.2     Dim.3     Dim.4       Dim.5
#> 100m        18.54882  3.624649 0.5716623 2.8895531  1.68654464
#> Long.jump   16.77169  9.820481 0.3047769 1.2137039  2.60205888
#> Shot.put    11.82745 19.326688 7.6938764 5.7691535  0.10218284
#> High.jump    9.91480 10.400459 5.5042879 0.9826975 63.96478233
#> 400m        13.91046 13.188882 1.4391119 1.8063156  0.07740427
#> 110m.hurdle 17.18646  3.694139 5.3346892 0.4623504  7.97971527

Supplementary variables

# Get supplementary variables coordinates with FactoMineR
res_pca$quanti.sup$coord
#>             Dim.1     Dim.2     Dim.3     Dim.4      Dim.5
#> 1500m -0.05231146 0.1142921 0.1396336 0.4621371 -0.1964609

# Get supplementary CONTINUOUS variables coordinates with booklet
X_sup <- X[-1, 10, drop = FALSE]
X_sup_scaled <- pca_standardize(X_sup, scale = TRUE)

var_sup_coords <- as.data.frame(t(X_sup_scaled * weights) %*% eigs$U)
rownames(var_sup_coords) <- colnames(X_sup)
var_sup_coords[, 1:5]
#>             Dim.1     Dim.2     Dim.3     Dim.4      Dim.5
#> 1500m -0.05231146 0.1142921 0.1396336 0.4621371 -0.1964609

# Get supplementary variables cos2 with FactoMineR
res_pca$quanti.sup$cos2
#>             Dim.1      Dim.2      Dim.3     Dim.4      Dim.5
#> 1500m 0.002736489 0.01306268 0.01949755 0.2135707 0.03859689

# Get supplementary variables cos2 with booklet
var_sup_cos2 <- pca_var_cos2(var_sup_coords)
var_sup_cos2[, 1:5]
#>             Dim.1      Dim.2      Dim.3     Dim.4      Dim.5
#> 1500m 0.002736489 0.01306268 0.01949755 0.2135707 0.03859689

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