# Articles - Principal Component Methods in R: Practical Guide

## Principal Component Analysis in R: prcomp vs princomp

This R tutorial describes how to perform a Principal Component Analysis (PCA) using the built-in R functions prcomp() and princomp(). You will learn how to predict new individuals and variables coordinates using PCA. We’ll also provide the theory behind PCA results.

Learn more about the basics and the interpretation of principal component analysis in our previous article: PCA - Principal Component Analysis Essentials. Contents:

## General methods for principal component analysis

There are two general methods to perform PCA in R :

• Spectral decomposition which examines the covariances / correlations between variables
• Singular value decomposition which examines the covariances / correlations between individuals

The function princomp() uses the spectral decomposition approach. The functions prcomp() and PCA()[FactoMineR] use the singular value decomposition (SVD).

According to the R help, SVD has slightly better numerical accuracy. Therefore, the function prcomp() is preferred compared to princomp().

## prcomp() and princomp() functions

The simplified format of these 2 functions are :

``````prcomp(x, scale = FALSE)
princomp(x, cor = FALSE, scores = TRUE)``````
1. Arguments for prcomp():
• `x`: a numeric matrix or data frame
• `scale`: a logical value indicating whether the variables should be scaled to have unit variance before the analysis takes place
1. Arguments for princomp():
• `x`: a numeric matrix or data frame
• `cor`: a logical value. If TRUE, the data will be centered and scaled before the analysis
• `scores`: a logical value. If TRUE, the coordinates on each principal component are calculated

The elements of the outputs returned by the functions prcomp() and princomp() includes :

prcomp() name princomp() name Description
sdev sdev the standard deviations of the principal components
center center the variable means (means that were substracted)
scale scale the variable standard deviations (the scaling applied to each variable )
x scores The coordinates of the individuals (observations) on the principal components.

In the following sections, we’ll focus only on the function prcomp()

## Package for PCA visualization

We’ll use the factoextra R package to create a ggplot2-based elegant visualization.

• You can install it from CRAN:
``install.packages("factoextra")``
• Or, install the latest developmental version from github:
``````if(!require(devtools)) install.packages("devtools")
devtools::install_github("kassambara/factoextra")``````
``library(factoextra)``

## Demo data sets

We’ll use the data sets `decathlon2` [in factoextra], which has been already described at: PCA - Data format.

Briefly, it contains:

• Active individuals (rows 1 to 23) and active variables (columns 1 to 10), which are used to perform the principal component analysis
• Supplementary individuals (rows 24 to 27) and supplementary variables (columns 11 to 13), which coordinates will be predicted using the PCA information and parameters obtained with active individuals/variables. Load the data and extract only active individuals and variables:

``````library("factoextra")
data(decathlon2)
decathlon2.active <- decathlon2[1:23, 1:10]
``````##           X100m Long.jump Shot.put High.jump X400m X110m.hurdle
## SEBRLE     11.0      7.58     14.8      2.07  49.8         14.7
## CLAY       10.8      7.40     14.3      1.86  49.4         14.1
## BERNARD    11.0      7.23     14.2      1.92  48.9         15.0
## YURKOV     11.3      7.09     15.2      2.10  50.4         15.3
## ZSIVOCZKY  11.1      7.30     13.5      2.01  48.6         14.2
## McMULLEN   10.8      7.31     13.8      2.13  49.9         14.4``````

## Compute PCA in R using prcomp()

In this section we’ll provide an easy-to-use R code to compute and visualize PCA in R using the prcomp() function and the factoextra package.

``library(factoextra)``
1. Compute PCA
``res.pca <- prcomp(decathlon2.active, scale = TRUE)``
1. Visualize eigenvalues (scree plot). Show the percentage of variances explained by each principal component.
``fviz_eig(res.pca)`` 1. Graph of individuals. Individuals with a similar profile are grouped together.
``````fviz_pca_ind(res.pca,
col.ind = "cos2", # Color by the quality of representation
repel = TRUE     # Avoid text overlapping
)`````` 1. Graph of variables. Positive correlated variables point to the same side of the plot. Negative correlated variables point to opposite sides of the graph.
``````fviz_pca_var(res.pca,
col.var = "contrib", # Color by contributions to the PC
repel = TRUE     # Avoid text overlapping
)`````` 1. Biplot of individuals and variables
``````fviz_pca_biplot(res.pca, repel = TRUE,
col.var = "#2E9FDF", # Variables color
col.ind = "#696969"  # Individuals color
)`````` ``````library(factoextra)
# Eigenvalues
eig.val <- get_eigenvalue(res.pca)
eig.val

# Results for Variables
res.var <- get_pca_var(res.pca)
res.var\$coord          # Coordinates
res.var\$contrib        # Contributions to the PCs
res.var\$cos2           # Quality of representation
# Results for individuals
res.ind <- get_pca_ind(res.pca)
res.ind\$coord          # Coordinates
res.ind\$contrib        # Contributions to the PCs
res.ind\$cos2           # Quality of representation ``````

## Predict using PCA

In this section, we’ll show how to predict the coordinates of supplementary individuals and variables using only the information provided by the previously performed PCA.

### Supplementary individuals

1. Data: rows 24 to 27 and columns 1 to to 10 [in decathlon2 data sets]. The new data must contain columns (variables) with the same names and in the same order as the active data used to compute PCA.
``````# Data for the supplementary individuals
ind.sup <- decathlon2[24:27, 1:10]
ind.sup[, 1:6]``````
``````##         X100m Long.jump Shot.put High.jump X400m X110m.hurdle
## KARPOV   11.0      7.30     14.8      2.04  48.4         14.1
## WARNERS  11.1      7.60     14.3      1.98  48.7         14.2
## Nool     10.8      7.53     14.3      1.88  48.8         14.8
## Drews    10.9      7.38     13.1      1.88  48.5         14.0``````
1. Predict the coordinates of new individuals data. Use the R base function predict():
``````ind.sup.coord <- predict(res.pca, newdata = ind.sup)
ind.sup.coord[, 1:4]``````
``````##            PC1    PC2   PC3    PC4
## KARPOV   0.777 -0.762 1.597  1.686
## WARNERS -0.378  0.119 1.701 -0.691
## Nool    -0.547 -1.934 0.472 -2.228
## Drews   -1.085 -0.017 2.982 -1.501``````
1. Graph of individuals including the supplementary individuals:
``````# Plot of active individuals
p <- fviz_pca_ind(res.pca, repel = TRUE) The predicted coordinates of individuals can be manually calculated as follow:

1. Center and scale the new individuals data using the center and the scale of the PCA
2. Calculate the predicted coordinates by multiplying the scaled values with the eigenvectors (loadings) of the principal components.

The R code below can be used :

``````# Centering and scaling the supplementary individuals
ind.scaled <- scale(ind.sup,
center = res.pca\$center,
scale = res.pca\$scale)
# Coordinates of the individividuals
apply(r, 2, sum)
}
ind.sup.coord[, 1:4]``````
``````##            PC1    PC2   PC3    PC4
## KARPOV   0.777 -0.762 1.597  1.686
## WARNERS -0.378  0.119 1.701 -0.691
## Nool    -0.547 -1.934 0.472 -2.228
## Drews   -1.085 -0.017 2.982 -1.501``````

### Supplementary variables

#### Qualitative / categorical variables

The data sets `decathlon2` contain a supplementary qualitative variable at columns 13 corresponding to the type of competitions.

Qualitative / categorical variables can be used to color individuals by groups. The grouping variable should be of same length as the number of active individuals (here 23).

``````groups <- as.factor(decathlon2\$Competition[1:23])
fviz_pca_ind(res.pca,
col.ind = groups, # color by groups
palette = c("#00AFBB",  "#FC4E07"),
addEllipses = TRUE, # Concentration ellipses
ellipse.type = "confidence",
legend.title = "Groups",
repel = TRUE
)`````` Calculate the coordinates for the levels of grouping variables. The coordinates for a given group is calculated as the mean coordinates of the individuals in the group.

``````library(magrittr) # for pipe %>%
library(dplyr)   # everything else
# 1. Individual coordinates
res.ind <- get_pca_ind(res.pca)
# 2. Coordinate of groups
coord.groups <- res.ind\$coord %>%
as_data_frame() %>%
select(Dim.1, Dim.2) %>%
mutate(competition = groups) %>%
group_by(competition) %>%
summarise(
Dim.1 = mean(Dim.1),
Dim.2 = mean(Dim.2)
)
coord.groups``````
``````## # A tibble: 2 x 3
##   competition Dim.1  Dim.2
##
## 1    Decastar -1.31 -0.119
## 2    OlympicG  1.20  0.109``````

#### Quantitative variables

Data: columns 11:12. Should be of same length as the number of active individuals (here 23)

``````quanti.sup <- decathlon2[1:23, 11:12, drop = FALSE]
``````##           Rank Points
## SEBRLE       1   8217
## CLAY         2   8122
## BERNARD      4   8067
## YURKOV       5   8036
## ZSIVOCZKY    7   8004
## McMULLEN     8   7995``````

The coordinates of a given quantitative variable are calculated as the correlation between the quantitative variables and the principal components.

``````# Predict coordinates and compute cos2
quanti.coord <- cor(quanti.sup, res.pca\$x)
quanti.cos2 <- quanti.coord^2
# Graph of variables including supplementary variables
p <- fviz_pca_var(res.pca) ## Theory behind PCA results

### PCA results for variables

Here we’ll show how to calculate the PCA results for variables: coordinates, cos2 and contributions:

• `var.coord` = loadings * the component standard deviations
• `var.cos2` = var.coord^2
• `var.contrib`. The contribution of a variable to a given principal component is (in percentage) : (var.cos2 * 100) / (total cos2 of the component)
``````# Helper function
#::::::::::::::::::::::::::::::::::::::::
}
# Compute Coordinates
#::::::::::::::::::::::::::::::::::::::::
sdev <- res.pca\$sdev
``````##                 PC1     PC2    PC3     PC4
## X100m        -0.851  0.1794 -0.302  0.0336
## Long.jump     0.794 -0.2809  0.191 -0.1154
## Shot.put      0.734 -0.0854 -0.518  0.1285
## High.jump     0.610  0.4652 -0.330  0.1446
## X400m        -0.702 -0.2902 -0.284  0.4308
## X110m.hurdle -0.764  0.0247 -0.449 -0.0169``````
``````# Compute Cos2
#::::::::::::::::::::::::::::::::::::::::
var.cos2 <- var.coord^2
``````##                PC1      PC2    PC3      PC4
## X100m        0.724 0.032184 0.0909 0.001127
## Long.jump    0.631 0.078881 0.0363 0.013315
## Shot.put     0.539 0.007294 0.2679 0.016504
## High.jump    0.372 0.216424 0.1090 0.020895
## X400m        0.492 0.084203 0.0804 0.185611
## X110m.hurdle 0.584 0.000612 0.2015 0.000285``````
``````# Compute contributions
#::::::::::::::::::::::::::::::::::::::::
comp.cos2 <- apply(var.cos2, 2, sum)
contrib <- function(var.cos2, comp.cos2){var.cos2*100/comp.cos2}
var.contrib <- t(apply(var.cos2,1, contrib, comp.cos2))
``````##                PC1     PC2   PC3     PC4
## X100m        17.54  1.7505  7.34  0.1376
## Long.jump    15.29  4.2904  2.93  1.6249
## Shot.put     13.06  0.3967 21.62  2.0141
## High.jump     9.02 11.7716  8.79  2.5499
## X400m        11.94  4.5799  6.49 22.6509
## X110m.hurdle 14.16  0.0333 16.26  0.0348``````

### PCA results for individuals

• `ind.coord` = res.pca\$x
• Cos2 of individuals. Two steps:
• Calculate the square distance between each individual and the PCA center of gravity: d2 = [(var1_ind_i - mean_var1)/sd_var1]^2 + …+ [(var10_ind_i - mean_var10)/sd_var10]^2 + …+..
• Calculate the cos2 as ind.coord^2/d2
• Contributions of individuals to the principal components: 100 * (1 / number_of_individuals)*(ind.coord^2 / comp_sdev^2). Note that the sum of all the contributions per column is 100
``````# Coordinates of individuals
#::::::::::::::::::::::::::::::::::
ind.coord <- res.pca\$x
``````##              PC1    PC2    PC3     PC4
## SEBRLE     0.191 -1.554 -0.628  0.0821
## CLAY       0.790 -2.420  1.357  1.2698
## BERNARD   -1.329 -1.612 -0.196 -1.9209
## YURKOV    -0.869  0.433 -2.474  0.6972
## ZSIVOCZKY -0.106  2.023  1.305 -0.0993
## McMULLEN   0.119  0.992  0.844  1.3122``````
``````# Cos2 of individuals
#:::::::::::::::::::::::::::::::::
# 1. square of the distance between an individual and the
# PCA center of gravity
center <- res.pca\$center
scale<- res.pca\$scale
getdistance <- function(ind_row, center, scale){
return(sum(((ind_row-center)/scale)^2))
}
d2 <- apply(decathlon2.active,1,getdistance, center, scale)
# 2. Compute the cos2. The sum of each row is 1
cos2 <- function(ind.coord, d2){return(ind.coord^2/d2)}
ind.cos2 <- apply(ind.coord, 2, cos2, d2)
``````##               PC1    PC2     PC3     PC4
## SEBRLE    0.00753 0.4975 0.08133 0.00139
## CLAY      0.04870 0.4570 0.14363 0.12579
## BERNARD   0.19720 0.2900 0.00429 0.41182
## YURKOV    0.09611 0.0238 0.77823 0.06181
## ZSIVOCZKY 0.00157 0.5764 0.23975 0.00139
## McMULLEN  0.00218 0.1522 0.11014 0.26649``````
``````# Contributions of individuals
#:::::::::::::::::::::::::::::::
contrib <- function(ind.coord, comp.sdev, n.ind){
100*(1/n.ind)*ind.coord^2/comp.sdev^2
}
ind.contrib <- t(apply(ind.coord, 1, contrib,
res.pca\$sdev, nrow(ind.coord)))
``````##              PC1    PC2    PC3     PC4