1 Required packages

The following packages will be used:

• cluster for clustering analyses
• factoextra for visualizing clusters using ggplot2 plotting system

Install factoextra package as follow:

if(!require(devtools)) install.packages("devtools")
devtools::install_github("kassambara/factoextra")

The cluster package can be installed using the code below:

install.packages("cluster")

Load packages:

library(cluster)
library(factoextra)

2 Data preparation

We’ll use the built-in R data set USArrests, which can be loaded and prepared as follow:

# Load the data set
data(USArrests)

# Remove any missing value (i.e, NA values for not available)
# That might be present in the data
USArrests <- na.omit(USArrests)

# View the firt 6 rows of the data
head(USArrests, n = 6)

##            Murder Assault UrbanPop Rape
## Alabama      13.2     236       58 21.2
## Alaska       10.0     263       48 44.5
## Arizona       8.1     294       80 31.0
## Arkansas      8.8     190       50 19.5
## California    9.0     276       91 40.6
## Colorado      7.9     204       78 38.7

In this data set, columns are variables and rows are observations (i.e., samples).

To inspect the data before the K-means clustering we’ll compute some descriptive statistics such as the mean and the standard deviation of the variables.

The apply() function is used to apply a given function (e.g : min(), max(), mean(), …) on the data set. The second argument can take the value of:

• 1: for applying the function on the rows
• 2: for applying the function on the columns

desc_stats <- data.frame(
  Min = apply(USArrests, 2, min), # minimum
  Med = apply(USArrests, 2, median), # median
  Mean = apply(USArrests, 2, mean), # mean
  SD = apply(USArrests, 2, sd), # Standard deviation
  Max = apply(USArrests, 2, max) # Maximum
  )
desc_stats <- round(desc_stats, 1)
head(desc_stats)

##           Min   Med  Mean   SD   Max
## Murder    0.8   7.2   7.8  4.4  17.4
## Assault  45.0 159.0 170.8 83.3 337.0
## UrbanPop 32.0  66.0  65.5 14.5  91.0
## Rape      7.3  20.1  21.2  9.4  46.0

Note that the variables have a large different means and variances. They must be standardized to make them comparable.

Standardization consists of transforming the variables such that they have mean zero and standard deviation one. The scale() function can be used as follow:

df<- scale(USArrests)

3 Assessing the clusterability

The function get_clust_tendency() [in factoextra] can be used. It computes Hopkins statistic and provides a visual approach.

library("factoextra")
res <- get_clust_tendency(df, 40, graph = FALSE)
# Hopskin statistic
res$hopkins_stat

## [1] 0.3440875

# Visualize the dissimilarity matrix
res$plot

## NULL

The value of Hopkins statistic is significantly < 0.5, indicating that the data is highly clusterable. Additionally, It can be seen that the ordered dissimilarity image contains patterns (i.e., clusters).

4 Estimate the number of clusters in the data

As k-means clustering requires to specify the number of clusters to generate, we’ll use the function clusGap() [in cluster] to compute gap statistics for estimating the optimal number of clusters . The function fviz_gap_stat() [in factoextra] is used to visualize the gap statistic plot.

library("cluster")
set.seed(123)
# Compute the gap statistic
gap_stat <- clusGap(df, FUN = kmeans, nstart = 25, 
                    K.max = 10, B = 500) 
# Plot the result
library(factoextra)
fviz_gap_stat(gap_stat)

The gap statistic suggests a 4 cluster solutions.

It’s also possible to use the function NbClust() [in NbClust] package.

5 Compute k-means clustering

K-means clustering with k = 4:

# Compute k-means
set.seed(123)
km.res <- kmeans(df, 4, nstart = 25)
head(km.res$cluster, 20)

##     Alabama      Alaska     Arizona    Arkansas  California    Colorado 
##           4           3           3           4           3           3 
## Connecticut    Delaware     Florida     Georgia      Hawaii       Idaho 
##           2           2           3           4           2           1 
##    Illinois     Indiana        Iowa      Kansas    Kentucky   Louisiana 
##           3           2           1           2           1           4 
##       Maine    Maryland 
##           1           3

# Visualize clusters using factoextra
fviz_cluster(km.res, USArrests)

6 Cluster validation statistics: Inspect cluster silhouette plot

Recall that the silhouette measures (\(S_i\)) how similar an object \(i\) is to the the other objects in its own cluster versus those in the neighbor cluster. \(S_i\) values range from 1 to - 1:

• A value of \(S_i\) close to 1 indicates that the object is well clustered. In the other words, the object \(i\) is similar to the other objects in its group.
• A value of \(S_i\) close to -1 indicates that the object is poorly clustered, and that assignment to some other cluster would probably improve the overall results.

sil <- silhouette(km.res$cluster, dist(df))
rownames(sil) <- rownames(USArrests)
head(sil[, 1:3])

##            cluster neighbor  sil_width
## Alabama          4        3 0.48577530
## Alaska           3        4 0.05825209
## Arizona          3        2 0.41548326
## Arkansas         4        2 0.11870947
## California       3        2 0.43555885
## Colorado         3        2 0.32654235

fviz_silhouette(sil)

##   cluster size ave.sil.width
## 1       1   13          0.37
## 2       2   16          0.34
## 3       3   13          0.27
## 4       4    8          0.39

It can be seen that there are some samples which have negative silhouette values. Some natural questions are :

Which samples are these? To what cluster are they closer?

This can be determined from the output of the function silhouette() as follow:

neg_sil_index <- which(sil[, "sil_width"] < 0)
sil[neg_sil_index, , drop = FALSE]

##          cluster neighbor   sil_width
## Missouri       3        2 -0.07318144

7 eclust(): Enhanced clustering analysis

The function eclust() [in factoextra] provides several advantages compared to the standard packages used for clustering analysis:

• It simplifies the workflow of clustering analysis
• It can be used to compute hierarchical clustering and partitioning clustering in a single line function call
• The function eclust() computes automatically the gap statistic for estimating the right number of clusters.
• It automatically provides silhouette information
• It draws beautiful graphs using ggplot2

# Compute k-means
res.km <- eclust(df, "kmeans")

# Gap statistic plot
fviz_gap_stat(res.km$gap_stat)

# Silhouette plot
fviz_silhouette(res.km)

##   cluster size ave.sil.width
## 1       1   13          0.27
## 2       2   13          0.37
## 3       3    8          0.39
## 4       4   16          0.34

 # Enhanced hierarchical clustering
res.hc <- eclust(df, "hclust") # compute hclust
fviz_dend(res.hc, rect = TRUE) # dendrogam

fviz_silhouette(res.hc) # silhouette plot
fviz_cluster(res.hc) # scatter plot

8 Infos

This analysis has been performed using R software (ver. 3.2.3)