Clustering is a technique in machine learning that attempts to find groups or *clusters* of observations within a dataset.

The goal is to find clusters such that the observations within each cluster are quite similar to each other, while observations in different clusters are quite different from each other.

Clustering is a form of unsupervised learning because we’re simply attempting to find structure within a dataset rather than predicting the value of some response variable.

Clustering is often used in marketing when companies have access to information like:

- Household income
- Household size
- Head of household Occupation
- Distance from nearest urban area

When this information is available, clustering can be used to identify households that are similar and may be more likely to purchase certain products or respond better to a certain type of advertising.

One of the most common forms of clustering is known as k-means clustering.

Unfortunately, this method can be influenced by outliers so an alternative that is often used is **k-medoids clustering**.

**What is K-Medoids Clustering?**

K-medoids clustering is a technique in which we place each observation in a dataset into one of *K* clusters.

The end goal is to have *K *clusters in which the observations within each cluster are quite similar to each other while the observations in different clusters are quite different from each other.

In practice, we use the following steps to perform K-means clustering:

**1. Choose a value for K.**

- First, we must decide how many clusters we’d like to identify in the data. Often we have to simply test several different values for
*K*and analyze the results to see which number of clusters seems to make the most sense for a given problem.

**2. Randomly assign each observation to an initial cluster, from 1 to K.**

**3. Perform the following procedure until the cluster assignments stop changing.**

- For each of the
*K*clusters, compute the cluster*centroid.*This is the vector of the*p*feature**medians**for the observations in the*k*th cluster. - Assign each observation to the cluster whose centroid is closest. Here,
*closest*is defined using Euclidean distance.

Technical Note:

Because k-medoids computes cluster centroids using medians instead of means, it tends to be more robust to outliers compared to k-means.

In practice, if there are no extreme outliers in the dataset then k-means and k-medoids will produce similar results.

**K-Medoids Clustering in R**

The following tutorial provides a step-by-step example of how to perform k-medoids clustering in R.

**Step 1: Load the Necessary Packages**

First, we’ll load two packages that contain several useful functions for k-medoids clustering in R.

library(factoextra) library(cluster)

**Step 2: Load and Prep the Data**

For this example we’ll use the *USArrests* dataset built into R, which contains the number of arrests per 100,000 residents in each U.S. state in 1973 for *Murder*, *Assault*, and *Rape *along with the percentage of the population in each state living in urban areas, *UrbanPop*.

The following code shows how to do the following:

- Load the
*USArrests*dataset - Remove any rows with missing values
- Scale each variable in the dataset to have a mean of 0 and a standard deviation of 1

#load data df <- USArrests #remove rows with missing valuesdf <- na.omit(df) #scale each variable to have a mean of 0 and sd of 1df <- scale(df) #view first six rows of dataset head(df) Murder Assault UrbanPop Rape Alabama 1.24256408 0.7828393 -0.5209066 -0.003416473 Alaska 0.50786248 1.1068225 -1.2117642 2.484202941 Arizona 0.07163341 1.4788032 0.9989801 1.042878388 Arkansas 0.23234938 0.2308680 -1.0735927 -0.184916602 California 0.27826823 1.2628144 1.7589234 2.067820292 Colorado 0.02571456 0.3988593 0.8608085 1.864967207

**Step 3: Find the Optimal Number of Clusters**

To perform k-medoids clustering in R we can use the **pam()** function, which stands for “partitioning around medians” and uses the following syntax:

**pam(data, k, metric = “euclidean”, stand = FALSE)**

where:

**data:**Name of the dataset.**k:**The number of clusters.**metric:**The metric to use to calculate distance. Default is*euclidean*but you could also specify*manhattan*.**stand:**Whether or not to standardize each variable in the dataset. Default is FALSE.

Since we don’t know beforehand how many clusters is optimal, we’ll create two different plots that can help us decide:

**1. Number of Clusters vs. the Total Within Sum of Squares**

First, we’ll use the **fviz_nbclust()** function to create a plot of the number of clusters vs. the total within sum of squares:

fviz_nbclust(df, pam, method = "wss")

The total within sum of squares will typically always increase as we increase the number of clusters, so when we create this type of plot we look for an “elbow” where the sum of squares begins to “bend” or level off.

The point where the plot bends is typically the optimal number of clusters. Beyond this number, overfitting is likely to occur.

For this plot it appear that there is a bit of an elbow or “bend” at k = 4 clusters.

**2. Number of Clusters vs. Gap Statistic**

Another way to determine the optimal number of clusters is to use a metric known as the gap statistic, which compares the total intra-cluster variation for different values of k with their expected values for a distribution with no clustering.

We can calculate the gap statistic for each number of clusters using the **clusGap()** function from the *cluster* package along with a plot of clusters vs. gap statistic using the **fviz_gap_stat()** function:

#calculate gap statistic based on number of clusters gap_stat <- clusGap(df, FUN = pam, K.max = 10, #max clusters to consider B = 50) #total bootstrapped iterations #plot number of clusters vs. gap statistic fviz_gap_stat(gap_stat)

From the plot we can see that gap statistic is highest at k = 4 clusters, which matches the elbow method we used earlier.

**Step 4: Perform K-Medoids Clustering with Optimal ***K*

*K*

Lastly, we can perform k-medoids clustering on the dataset using the optimal value for *k* of 4:

#make this example reproducible set.seed(1) #perform k-medoids clustering with k = 4 clusters kmed <- pam(df, k = 4) #view results kmed ID Murder Assault UrbanPop Rape Alabama 1 1.2425641 0.7828393 -0.5209066 -0.003416473 Michigan 22 0.9900104 1.0108275 0.5844655 1.480613993 Oklahoma 36 -0.2727580 -0.2371077 0.1699510 -0.131534211 New Hampshire 29 -1.3059321 -1.3650491 -0.6590781 -1.252564419 Clustering vector: Alabama Alaska Arizona Arkansas California 1 2 2 1 2 Colorado Connecticut Delaware Florida Georgia 2 3 3 2 1 Hawaii Idaho Illinois Indiana Iowa 3 4 2 3 4 Kansas Kentucky Louisiana Maine Maryland 3 3 1 4 2 Massachusetts Michigan Minnesota Mississippi Missouri 3 2 4 1 3 Montana Nebraska Nevada New Hampshire New Jersey 3 3 2 4 3 New Mexico New York North Carolina North Dakota Ohio 2 2 1 4 3 Oklahoma Oregon Pennsylvania Rhode Island South Carolina 3 3 3 3 1 South Dakota Tennessee Texas Utah Vermont 4 1 2 3 4 Virginia Washington West Virginia Wisconsin Wyoming 3 3 4 4 3 Objective function: build swap 1.035116 1.027102 Available components: [1] "medoids" "id.med" "clustering" "objective" "isolation" [6] "clusinfo" "silinfo" "diss" "call" "data"

Note that the four cluster centroids are actual observations in the dataset. Near the top of the output we can see that the four centroids are the following states:

- Alabama
- Michigan
- Oklahoma
- New Hampshire

We can visualize the clusters on a scatterplot that displays the first two principal components on the axes using the **fivz_cluster()** function:

#plot results of final k-medoids model fviz_cluster(kmed, data = df)

We can also append the cluster assignments of each state back to the original dataset:

#add cluster assignment to original data final_data <- cbind(USArrests, cluster = kmed$cluster) #view final data head(final_data) Murder Assault UrbanPop Rape cluster Alabama 13.2 236 58 21.2 1 Alaska 10.0 263 48 44.5 2 Arizona 8.1 294 80 31.0 2 Arkansas 8.8 190 50 19.5 1 California 9.0 276 91 40.6 2 Colorado 7.9 204 78 38.7 2

You can find the complete R code used in this example here.