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How to Find the Optimal 𝑘 Using the Elbow Method?

Overview

This document provides a detailed explanation of how to determine the optimal number of clusters (𝑘) in clustering analysis using the elbow method. Clustering is a fundamental technique in data analysis, allowing for the partitioning of data into meaningful groups based on similarities.

Contents

• Understanding 𝑘 in Clustering
• Steps to Determine 𝑘
• Elbow Point Interpretation
• Interpreting the Elbow Plot
• Decision on 𝑘
• Choosing 𝑘
• What does it look like after deciding 𝑘 = 4?
• Implementation Considerations
• Additional Considerations

Understanding 𝑘 in Clustering:

• In clustering analysis, determining the appropriate number of clusters 𝑘 is crucial for interpreting and understanding patterns in your data.
• The elbow method is a popular technique to help identify this optimal 𝑘.
• Clustering algorithms, such as K-means, partition the data into 𝑘 clusters where each data point belongs to the cluster with the nearest mean (centroid).

Steps to Determine 𝑘

1. Elbow Point Interpretation:

• The elbow method suggests choosing 𝑘 where the decrease in distortion (inertia or SSE - sum of squared errors) slows down significantly, forming an elbow-like bend in the plot. Distortion represents the sum of squared distances between data points and their assigned cluster centroids.
• Distortion represents the sum of squared distances between data points and their assigned cluster centroids.

from sklearn.cluster import KMeans
import matplotlib.pyplot as plt

# Determine optimal number of clusters using the elbow method
distortions = []
for k in range(1, 11):
    kmeans = KMeans(n_clusters=k, random_state=42, n_init=10)  # Explicitly set n_init to suppress warning (optional)
    kmeans.fit(X_scaled)
    distortions.append(kmeans.inertia_)

# Plotting the elbow curve
plt.figure(figsize=(10, 6))
plt.plot(range(1, 11), distortions, marker='o')
plt.title('Elbow Method for Optimal k')
plt.xlabel('Number of Clusters')
plt.ylabel('Distortion')

# Save the elbow plot as PNG in your Jupyter repository
plt.savefig('elbow_plot.png')
plt.show()

2. Interpreting the Elbow Plot:

• Plot the distortion values for different 𝑘 values.
• The plot typically shows a decrease in distortion as 𝑘 increases.
• Identify the point where the decrease in distortion starts to flatten out (the "elbow" point).

In my example:

Example:
At 𝑘 = 1, distortion is highest (1.2).
At 𝑘 = 2, distortion decreases to around 0.85.
At 𝑘 = 3, distortion further decreases to around 0.5.
At 𝑘 = 4, it decreases to around 0.3.
At 𝑘 = 5, it decreases to around 0.17.
At 𝑘 = 6, it decreases to around 0.1.
Beyond 𝑘 = 6, the decrease in distortion becomes less pronounced.

3. Decision on 𝑘: Based on the elbow plot:

• Choose 𝑘 where the decrease in distortion starts to slow down noticeably.
• Adding more clusters beyond this point does not significantly reduce distortion.
• Balance model complexity (higher 𝑘) and the interpretability of clusters (lower distortion).

Based on the plot interpretation, 𝑘 = 4 or 𝑘 = 5 could be optimal because these points show a significant decrease in distortion, and adding more clusters beyond this point provides diminishing returns in reducing distortion.

The optimal 𝑘 is typically where the distortion starts to flatten out, indicating that additional clusters do not explain much more of the variance in the data.

4. Choosing 𝑘:

• Select 𝑘 based on your specific objectives and interpretation of the elbow plot.
• Consider the balance between model complexity (higher 𝑘) and the explanatory power of the clusters (lower distortion).

For example:
If the elbow plot shows that distortion decreases sharply up to 𝑘 = 4,  
and then the decrease becomes less pronounced, 𝑘 = 4 might be a suitable choice.

5. So, what does it look like after deciding 𝑘 = 4?

# Based on the elbow method, choose the optimal number of clusters (k)
k = 4  # Adjust this based on your analysis from the elbow plot

# Perform K-means clustering with the chosen k
kmeans = KMeans(n_clusters=k, n_init=10, random_state=42)
kmeans.fit(X_scaled)
df['Cluster'] = kmeans.labels_

# Visualize clusters
plt.figure(figsize=(12, 8))
sns.scatterplot(x='Quantity', y='TotalAmount', hue='Cluster', data=df, palette='viridis', s=100, alpha=0.8)
plt.title('Customer Segmentation by K-means Clustering')
plt.xlabel('Quantity')
plt.ylabel('Total Amount')
plt.legend(title='Cluster')

# Save the scatter plot as PNG in your Jupyter repository
plt.savefig('customer_segmentation.png')
plt.show()

After implementation, it looks something like this:

Purple dots (Cluster 0): As there are many purple dots are clustered around the origin (0,0) and in the middle, it suggests that Cluster 0 represents customers who might have low to moderate 'Quantity' and 'TotalAmount'. These customers are neither the highest spenders nor the lowest, but somewhere in between.

More details on this can be found in Customer Segmentation for E-commerce file

6. Implementation Considerations:

• Use Python libraries such as scikit-learn to perform K-means clustering and compute distortions for different 𝑘 values - can be found in scikit-learn-projects.
• Ensure reproducibility by setting random seeds (e.g., random_state in scikit-learn).

7. Additional Considerations:

• If the elbow in the plot is not distinct, consider using other metrics like silhouette score to validate 𝑘.
• Domain knowledge or business context can also provide insights into choosing 𝑘 based on what makes sense for interpreting the clusters.
• Experimenting with different 𝑘 values and evaluating their impact on your data analysis can further refine your choice.

By following these steps, you can effectively determine the optimal number of clusters 𝑘 using the elbow method, ensuring a balance between model complexity and explanatory power. Data interpretation can also provide insights into the optimal number of clusters.