This article describes *k-means* **clustering example** and provide a step-by-step guide summarizing the different steps to follow for conducting a cluster analysis on a real data set using R software.

We’ll use mainly two R packages:

- cluster: for cluster analyses and
- factoextra: for the visualization of the analysis results.

Install these packages, as follow:

`install.packages(c("cluster", "factoextra"))`

A rigorous cluster analysis can be conducted in 3 steps mentioned below:

Here, we provide quick R scripts to perform all these steps.

Contents:

#### Related Book

Practical Guide to Cluster Analysis in R## Data preparation

We’ll use the demo data set USArrests. We start by standardizing the data using the *scale*() function:

```
# Load the data set
data(USArrests)
# Standardize
df <- scale(USArrests)
```

## Assessing the clusterability

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

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

`## [1] 0.656`

```
# Visualize the dissimilarity matrix
print(res$plot)
```

The value of the 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).

## 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() [cluster package] to compute gap statistics for estimating the optimal number of clusters . The function *fviz_gap_stat*() [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 = 100)
# 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.

## 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)
```

## 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 0.4858
## Alaska 3 4 0.0583
## Arizona 3 2 0.4155
## Arkansas 4 2 0.1187
## California 3 2 0.4356
## Colorado 3 2 0.3265
```

`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 -0.0732
```

## eclust(): Enhanced clustering analysis

The function eclust()[factoextra package] 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

### K-means clustering using eclust()

```
# Compute k-means
res.km <- eclust(df, "kmeans", nstart = 25)
```

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

```
# Silhouette plot
fviz_silhouette(res.km)
```

### Hierachical clustering using eclust()

```
# Enhanced hierarchical clustering
res.hc <- eclust(df, "hclust") # compute hclust
```

```
## Clustering k = 1,2,..., K.max (= 10): .. done
## Bootstrapping, b = 1,2,..., B (= 100) [one "." per sample]:
## .................................................. 50
## .................................................. 100
```

`fviz_dend(res.hc, rect = TRUE) # dendrogam`

The R code below generates the silhouette plot and the scatter plot for hierarchical clustering.

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

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hi，is this a typo?

res$hopkins_stat

## [1] 0.656

the hopkins stat value equals 0.656… but your comments is “The value of the Hopkins statistic is significantly 0.5” ?