Principal Component Analysis with R Programming

Principal component analysis(PCA) in R programming is analysis on the linear components of all existing attributes. Principal components are linear combination(orthogonal transformation) of the original predictor in the dataset. It is a useful technique for EDA(Exploratory data analysis) and allowing you to better visualize the variations present in a dataset with many variables.

First principal component captures the maximum variance in dataset. It determines direction in of higher variability. Second principal component captures the remaining variance in data and is uncorrelated with PC1. The correlation between PC1 and PC2 should be zero. So, all succeeding principal components follows the same concept. They capture the remaining variance without being correlated to previous principal component.

The Dataset

The dataset mtcars(motor trend car road test) comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles. It comes pre-installed with dplyr package in R.

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# Installing required package
install.packages("dplyr")
  
# Loading the package
library(dplyr)
  
# Importing excel file
str(mtcars)

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Performing PCA using dataset

We perform Principal component analysis on mtcars which consists of 32 car brands and 10 variables.



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# Loading Data
data(mtcars)
   
# Apply PCA using prcomp function
# Need to scale / Normalize as 
# PCA depends on distance measure
my_pca <- prcomp(mtcars, scale = TRUE, 
                 center = TRUE, retx = T)
names(my_pca)
   
# Summary
summary(my_pca)
my_pca
   
# View the principal component loading
# my_pca$rotation[1:5, 1:4]
my_pca$rotation
   
# See the principal components
dim(my_pca$x)
my_pca$x
   
# Plotting the resultant principal components
# The parameter scale = 0 ensures that arrows
# are scaled to represent the loadings 
biplot(my_pca, main = "Biplot", scale = 0)
   
# Compute standard deviation
my_pca$sdev
   
# Compute variance
my_pca.var <- my_pca$sdev ^ 2
my_pca.var
   
# Proportion of variance for a scree plot
propve <- my_pca.var / sum(my_pca.var)
propve
   
# Plot variance explained for each principal component
plot(propve, xlab = "principal component",
             ylab = "Proportion of Variance Explained",
             ylim = c(0, 1), type = "b"
             main = "Scree Plot")
   
# Plot the cumulative proportion of variance explained
plot(cumsum(propve), 
     xlab = "Principal Component",
     ylab = "Cumulative Proportion of Variance Explained",
     ylim = c(0, 1), type = "b")
   
# Find Top n principal component 
# which will atleast cover 90 % variance of dimension
which(cumsum(propve) >= 0.9)[1]
   
# Predict mpg using first 4 new Principal Components
# Add a training set with principal components
train.data <- data.frame(disp = mtcars$disp, my_pca$x[, 1:4])
   
# Running a Decision tree algporithm
## Installing and loading packages
install.packages("rpart")
install.packages("rpart.plot")
library(rpart)
library(rpart.plot)
   
rpart.model <- rpart(disp ~ ., 
                     data = train.data, method = "anova")
   
rpart.plot(rpart.model)

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Output:

  • Bi plot

    The resultant principal components are plotted as Biplot. Scale value 0 represents that arrows are scaled representing loadings.

  • Variance explained for each principal component

    Scree Plot represents the proportion of variance and principal component. Below 2 principal components, there is maximum proportion of variance as clearly seen in the plot.
  • Cumulative proportion of variance

    Scree Plot represents the Cumulative proportion of variance and principal component. Above 2 principal components, there is maximum cumulative proportion of variance as clearly seen in the plot.
  • Decision tree model

    Decision tree model was build to predict disp using other variables in the dataset and using anova method. The decision tree plot is plotted and displays the information.




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