Elbow Method for Optimal Cluster Number in K-Means

Clustering is an unsupervised machine-learning technique. It is the process of division of the dataset into groups in which the members in the same group possess similarities in features. The commonly used clustering techniques are K-Means clustering, Hierarchical clustering, Density-based clustering, Model-based clustering, etc. It can even handle large datasets. We can implement the K-Means clustering machine learning algorithm in the elbow method using the scikit-learn library in Python.

In this article, you will learn about the elbow method in machine learning, specifically how it applies to k-means clustering. We will explore the elbow method formula, demonstrating its use in both Python and R. Additionally, you will see an elbow method example that illustrates how to identify the optimal number of clusters for your data analysis.

Learning Objectives

• Learn the basics of clustering in unsupervised machine learning and its significance in data analysis.
• Grasp the steps and processes involved in the K-Means clustering algorithm.
• Apply the Elbow method to determine the optimal number of clusters in K-Means using Python.
• Assess the quality of K-Means clusters using Within-Cluster Sum of Squares (WCSS).
• Recognize the drawbacks of the Elbow method and K-Means, and explore alternative techniques for cluster determination.

This article was published as a part of the Data Science Blogathon.

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What is the Elbow Method ?

The Elbow Method is a technique used in data analysis and machine learning for determining the optimal number of clusters in a dataset. It involves plotting the variance explained by different numbers of clusters and identifying the “elbow” point, where the rate of variance decreases sharply levels off, suggesting an appropriate cluster count for analysis or model training.

This method is a visual technique used to determine the best K value for a k-means clustering algorithm. In this method, a graph known as the elbow graph plots the within-cluster-sum-of-square (WCSS) values against various K values. The optimal K value is identified at the point where the graph bends like an elbow.

What is the Elbow Method in K-Means Clustering?

The elbow method is a graphical representation of finding the optimal ‘K’ in a K-means clustering. It works by finding WCSS (Within-Cluster Sum of Square) i.e. the sum of the square distance between points in a cluster and the cluster centroid.

Let’s go through the steps involved in K-means clustering for a better understanding:

• Select the number of clusters for the dataset (K)
• Select the K number of centroids randomly from the dataset.
• Now we will use Euclidean distance or Manhattan distance as the metric to calculate the distance of the points from the nearest centroid and assign the points to that nearest cluster centroid, thus creating K clusters.
• Now we find the new centroid of the clusters thus formed.
• Again reassign the whole data point based on this new centroid, then repeat step 4. We will continue this for a given number of iterations until the position of the centroid doesn’t change, i.e., there is no more convergence.

Finding the optimal number of clusters is an important part of this algorithm. A commonly used method for finding the optimum K value is Elbow Method.  

K Means Clustering Using the Elbow Method

In the Elbow method, we are actually varying the number of clusters (K) from 1 – 10. For each value of K, we are calculating WCSS (Within-Cluster Sum of Square). WCSS is the sum of the squared distance between each point and the centroid in a cluster. When we plot the WCSS with the K value, the plot looks like an Elbow. As the number of clusters increases, the WCSS value will start to decrease. WCSS value is largest when K = 1. When we analyze the graph, we can see that the graph will rapidly change at a point and thus creating an elbow shape. From this point, the graph moves almost parallel to the X-axis. The K value corresponding to this point is the optimal value of K or an optimal number of clusters.

Now let’s implement K-Means clustering using Python.

Elbow method k means

The elbow method is a common technique used to determine the optimal number of clusters (k) in k-means clustering. It’s a graphical approach that relies on the idea that as you increase the number of clusters, the sum of squared distances between points and their cluster centers (WCSS) will continue to decrease. This is because you’re essentially splitting the data into increasingly finer groups.

Here’s how it works:

1. Calculate WCSS for different K values: You iterate through a range of possible k values (number of clusters). For each k, you perform k-means clustering and calculate the WCSS. This represents the total variance within each cluster.
2. Plot the WCSS vs. K: You plot the WCSS values on the y-axis and the number of clusters (k) on the x-axis.
3. Identify the elbow point: The ideal k value is identified as the “elbow” of the curve. This is the point where the WCSS starts to decrease at a much slower rate. In essence, you’re looking for the point where adding more clusters isn’t significantly improving the fit within the clusters.

Here are some things to keep in mind about the elbow method:

• It can be subjective: The “elbow” might not always be a very sharp bend, and it can be difficult to determine the exact point objectively.
• Not always the best method: The elbow method might not be suitable for all datasets, especially for those with high dimensionality or clusters of irregular shapes.
• Consider other factors: While the elbow method is a helpful starting point, it’s important to consider other factors like the interpretability of the clusters and the domain knowledge of the data when choosing the final k value.pen_spark

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Implementation of the Elbow Method

Sample Dataset

The dataset we are using here is the Mall Customers data (Download here). It’s unlabeled data that contains the details of customers in a mall (features like genre, age, annual income(k$), and spending score). Our aim is to cluster the customers based on the relevant features of annual income and spending score.

Importing Libraries

First of all, we have to import the essential libraries.

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd 
import sklearn

Now let’s import the given dataset and slice the important features.

dataset = pd.read_csv('Mall_Customers.csv')
X = dataset.iloc[:, [3, 4]].values

We have to find the optimal K value for clustering the data. Now we are using the Elbow Method to find the optimal K value.

from sklearn.cluster import KMeans
wcss = [] for i in range(1, 11): 
    kmeans = KMeans(n_clusters = i, init = 'k-means++', random_state = 42)
    kmeans.fit(X) 
    wcss.append(kmeans.inertia_)

The “init” argument is the method for initializing the centroid. We calculated the WCSS value for each K value. Now we have to plot the WCSS with the K value.

The graph will be like this:

Training the Model

The point at which the elbow shape is created is 5; that is, our K value or an optimal number of clusters is 5. Now let’s train the model on the input data with a number of clusters 5.

kmeans = KMeans(n_clusters = 5, init = "k-means++", random_state = 42)
y_kmeans = kmeans.fit_predict(X)

y_kmeans will be:

array([3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0,
       3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 1,
       3, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 2, 4, 2, 4, 2,
       1, 2, 4, 2, 4, 2, 4, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2,
       4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2,
       4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2,
       4, 2])

y_kmeans gives us different clusters corresponding to X. Now, let’s plot all the clusters using matplotlib.

plt.scatter(X[y_kmeans == 0, 0], X[y_kmeans == 0, 1], s = 60, c = 'red', label = 'Cluster1')
plt.scatter(X[y_kmeans == 1, 0], X[y_kmeans == 1, 1], s = 60, c = 'blue', label = 'Cluster2')
plt.scatter(X[y_kmeans == 2, 0], X[y_kmeans == 2, 1], s = 60, c = 'green', label = 'Cluster3)
plt.scatter(X[y_kmeans == 3, 0], X[y_kmeans == 3, 1], s = 60, c = 'violet', label = 'Cluster4')
plt.scatter(X[y_kmeans == 4, 0], X[y_kmeans == 4, 1], s = 60, c = 'yellow', label = 'Cluster5') 
plt.scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1], s = 100, c = 'black', label = 'Centroids')
plt.xlabel('Annual Income (k$)') plt.ylabel('Spending Score (1-100)') plt.legend() 

plt.show()

Graph:

Now we will visualize the clusters using the scatter plot. As you can see, there are 5 clusters in total that are visualized in different colors, and the centroid of each cluster is visualized in black color.

Full Code

# Importing the libraries
import numpy as np 
import matplotlib.pyplot as plt
import pandas as pd # Importing the dataset 

X = dataset.iloc[:, [3, 4]].values
dataset = pd.read_csv('Mall_Customers.csv') 

from sklearn.cluster import KMeans

# Using the elbow method to find the optimal number of clusters wcss = [] for i in range(1, 11): 
 wcss.append(kmeans.inertia_)
 kmeans = KMeans(n_clusters = i, init = 'k-means++', random_state = 42) kmeans.fit(X) plt.plot(range(1, 11), wcss) plt.xlabel('Number of clusters') 

y_kmeans = kmeans.fit_predict(X)

plt.ylabel('WCSS') plt.show() # Training the K-Means model on the dataset kmeans = KMeans(n_clusters = 5, init = 'k-means++', random_state = 42) y_kmeans = kmeans.fit_predict(X)

# Visualising the clusters
plt.scatter( X[y_kmeans == 1, 0], X[y_kmeans == 1, 1], s = 60, c = 'blue', label = 'Cluster2')
plt.scatter( X[y_kmeans == 0, 0], X[y_kmeans == 0, 1], s = 60, c = 'red', label = 'Cluster1') plt.scatter( X[y_kmeans == 2, 0], X[y_kmeans == 2, 1], s = 60, c = 'green', label = 'Cluster3') 
plt.scatter( kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1], s = 100, c = 'black', label = 'Centroids')
plt.scatter( X[y_kmeans == 3, 0], X[y_kmeans == 3, 1], s = 60, c = 'violet', label = 'Cluster4') plt.scatter( X[y_kmeans == 4, 0], X[y_kmeans == 4, 1], s = 60, c = 'yellow', label = 'Cluster5') plt.xlabel('Annual Income (k$)') plt.ylabel('Spending Score (1-100)') plt.legend() 

plt.show()

Elbow Method Drawbacks

The elbow method, while a useful tool for determining the optimal number of clusters in K-means clustering, has some drawbacks:

• Subjectivity: The choice of the “elbow point” can be subjective and might vary between individuals analyzing the same data.
• Non-Gaussian Data: It assumes that clusters are spherical and equally sized, which may not hold for complex datasets with irregularly shaped or differently sized clusters.
• Sensitivity to Initialization: K-means itself is sensitive to initial cluster centroids, which can affect the WCSS values and, consequently, the choice of the optimal K.
• Inefficient for Large Datasets: For large datasets, calculating WCSS for a range of K values can be computationally expensive and time-consuming.
• Unsuitable for All Distributions: The elbow method is not suitable for all data distributions, especially when clusters have varying densities or are non-convex.
• Limited to K-means: It specifically applies to K-means clustering and may not be suitable for other clustering algorithms with different objectives.

Despite these drawbacks, the elbow method remains a valuable starting point for selecting the number of clusters, and it often provides useful insights into the data’s underlying structure. However, it’s essential to complement it with other validation techniques when working with more complex datasets or different clustering algorithms.

Conclusion

In this article, we covered the basic concepts of the K-Means Clustering algorithm in Machine Learning. We used the Elbow method to find the optimal K value for clustering the data in our sample data set. We then used the matplotlib Python library to visualize the clusters as a scatterplot graph. In the upcoming articles, we can learn more about different ML Algorithms.

Key Takeaways

• K-Means is a popular unsupervised machine-learning algorithm widely used by Data Scientists on unlabeled data.
• The k-Means Elbow method is used to find the optimal value of the K in the K-Means algorithm.

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Basil Saji

Python Developer, ML Enthusiast, Blogger and an Electronics and Communication Engineering aspirant determined and motivated to finish tasks with atmost sincerity and dedication.I'am a good learner who ready to accept challenges to bring up my best even in the worst. Wish for a world with enough advancements and opportunities for the people.