How to Find Optimal Clusters in R?

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How to Find Optimal Clusters in R, K-means clustering is one of the most widely used clustering techniques in machine learning.

With the K-means clustering technique, each observation in a dataset is assigned to one of K clusters.

The ultimate goal is to have K clusters in which the observations are relatively similar to one another within each cluster and considerably dissimilar from one another within different clusters.

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The first stage in k-means clustering is to decide on a value for K or the number of clusters we want to group the observations into.

The elbow method is one of the most popular approaches to choosing a value for K.

It entails plotting the total inside the sum of squares on the y-axis and the number of clusters on the x-axis to locate the plot’s “elbow” or bend.

The best number of clusters to utilize in the k-means clustering algorithm is indicated by the location on the x-axis where the “elbow” occurs.

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The elbow method in R is demonstrated in the example that follows.

How to Find Optimal Clusters in R

We’ll use the USArrests dataset from R for this example, which includes the proportion of the population residing in urban areas in each state, or UrbanPop, as well as the number of murder, assault, and rape arrests made per 100,000 citizens in each state of the United States in 1973.

The dataset may be loaded using the code below, which also demonstrates how to delete rows with blank values and scale each variable in the dataset to have a mean and standard deviation of 0 and 1, respectively.

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Now let’s load the data

df <- USArrests

Then we can remove rows with missing values

df <- na.omit(df)

As you know before clustering we need to scale the data frame. Scale each variable to have a mean of 0 and sd of 1.

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df <- scale(df)

Let’s see the first six rows of the 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

We’ll use the fviz_nbclust() function from the factoextra package to make a plot of the number of clusters vs. the total inside the sum of squares in order to determine the ideal number of clusters to use in the k-means algorithm.

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library(cluster)
library(factoextra)

Plot the number of clusters relative to the total within the sum of squares

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

At k = 4 clusters, it appears like there are an “elbow” or bends in the plot. The sum of the total of the squares starts to level out at this point.

This indicates that using four clusters is the ideal amount to employ when using the k-means method.

Although employing more clusters might result in a lower sum of squares, we would probably be overfitting the training data, which would cause the k-means algorithm to perform worse on the testing data.

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We can now run k-means clustering on the dataset using the kmeans() function from the cluster package and the recommended value of k of 4.

We can make this example reproducible

set.seed(1234)

Now perform k-means clustering with k = 4 clusters

km <- kmeans(df, centers = 4, nstart = 25)

Let’s view the output

km
K-means clustering with 4 clusters of sizes 13, 13, 8, 16
Cluster means:
      Murder    Assault   UrbanPop        Rape
1  0.6950701  1.0394414  0.7226370  1.27693964
2 -0.9615407 -1.1066010 -0.9301069 -0.96676331
3  1.4118898  0.8743346 -0.8145211  0.01927104
4 -0.4894375 -0.3826001  0.5758298 -0.26165379
Clustering vector:
       Alabama         Alaska        Arizona       Arkansas     California
             3              1              1              3              1
      Colorado    Connecticut       Delaware        Florida        Georgia
             1              4              4              1              3
        Hawaii          Idaho       Illinois        Indiana           Iowa
             4              2              1              4              2
        Kansas       Kentucky      Louisiana          Maine       Maryland
             4              2              3              2              1
 Massachusetts       Michigan      Minnesota    Mississippi       Missouri
             4              1              2              3              1
       Montana       Nebraska         Nevada  New Hampshire     New Jersey
             2              2              1              2              4
    New Mexico       New York North Carolina   North Dakota           Ohio
             1              1              3              2              4
      Oklahoma         Oregon   Pennsylvania   Rhode Island South Carolina
             4              4              4              4              3
  South Dakota      Tennessee          Texas           Utah        Vermont
             2              3              1              4              2
      Virginia     Washington  West Virginia      Wisconsin        Wyoming
             4              4              2              2              4
Within cluster sum of squares by cluster:
[1] 19.922437 11.952463  8.316061 16.212213
 (between_SS / total_SS =  71.2 %)
Available components:
[1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
[6] "betweenss"    "size"         "iter"         "ifault"    

Additionally, we can add each state’s cluster assignments to the initial dataset.

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Now add cluster assignment to the original data

finaldata <- cbind(USArrests, cluster = km$cluster)
head(finaldata)
               Murder Assault UrbanPop Rape cluster
Alabama      13.2     236       58 21.2       3
Alaska       10.0     263       48 44.5       1
Arizona       8.1     294       80 31.0       1
Arkansas      8.8     190       50 19.5       3
California    9.0     276       91 40.6       1
Colorado      7.9     204       78 38.7       1

There are four clusters in which each observation from the first data frame has been sorted.

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