February 23, 2025

Week 3: Unsupervised Learning - Clustering - K-means/Kernel K-means


Introduction

Clustering is a fundamental technique in unsupervised learning, which aims to group similar data points together. In this blog, we will cover K-means and Kernel K-means clustering in depth, including their mathematical foundations, examples, and real-world applications.

What is Clustering?

Clustering is the task of dividing a dataset into groups (clusters) where objects in the same group are more similar to each other than to those in other groups. It is widely used in customer segmentation, image compression, anomaly detection, and bioinformatics.

Types of Clustering Algorithms:

  1. Partition-based Clustering (e.g., K-means, Kernel K-means)
  2. Hierarchical Clustering (Agglomerative, Divisive)
  3. Density-based Clustering (DBSCAN, Mean Shift)
  4. Model-based Clustering (Gaussian Mixture Models)

K-Means Clustering

Algorithm Steps:

  1. Select the number of clusters, K.
  2. Initialize K cluster centroids randomly.
  3. Assign each data point to the nearest centroid.
  4. Recalculate the centroids by taking the mean of all points assigned to that cluster.
  5. Repeat steps 3 and 4 until convergence (centroids no longer change significantly).

Example 1: Clustering Customers Based on Spending Behavior

Given Dataset:

Customer ID Annual Income (k$) Spending Score (1-100)
1 15 39
2 16 81
3 17 6
4 18 77
5 19 40

Step 1: Choose K = 2 and initialize centroids randomly.

Step 2: Compute distances and assign points to the closest centroid.

Step 3: Recalculate centroids.

Step 4: Repeat until convergence.

Conclusion: K-means successfully clusters customers into different spending groups, allowing businesses to tailor marketing strategies accordingly.

Kernel K-Means Clustering

Kernel K-means is an extension of K-means that maps data into a higher-dimensional space using a kernel function before performing clustering.

Common Kernel Functions:

  1. Linear Kernel: K(xi,xj)=xi⋅xjK(x_i, x_j) = x_i \cdot x_j
  2. Polynomial Kernel: K(xi,xj)=(xi⋅xj+c)dK(x_i, x_j) = (x_i \cdot x_j + c)^d
  3. Gaussian (RBF) Kernel: K(xi,xj)=e−∣∣xi−xj∣∣22σ2K(x_i, x_j) = e^{-\frac{||x_i - x_j||^2}{2\sigma^2}}

Example 2: Clustering Non-Linearly Separable Data

Dataset: Imagine we have two circular clusters that are not linearly separable.

  1. Apply the Gaussian kernel to transform the data.
  2. Use K-means on the transformed space.

Conclusion: Kernel K-means enables clustering in situations where standard K-means fails.

Advantages & Disadvantages

Method Advantages Disadvantages
K-Means Fast, easy to implement, scalable Sensitive to outliers, requires specifying K
Kernel K-Means Works on non-linear data Computationally expensive

Real-World Applications

  1. Marketing Segmentation – Group customers based on behavior.
  2. Image Segmentation – Divide images into meaningful regions.
  3. Anomaly Detection – Detect fraud in transactions.

20+ Questions and Answers for Understanding

# Question Answer
1 What is clustering in machine learning? Clustering is an unsupervised learning technique that groups similar data points together.
2 What is the primary objective of K-means? To minimize the variance within each cluster.
3 What is a centroid in K-means? The center of a cluster, computed as the mean of all points in that cluster.
4 How does Kernel K-means differ from K-means? It applies a kernel function to transform data into a higher-dimensional space before clustering.
5 What is the time complexity of K-means? O(n * k * i), where n is the number of data points, k is the number of clusters, and i is the number of iterations.
6 What happens if you choose a bad initial centroid? The algorithm may converge to a local minimum.
7 How can you determine the best value for K? Using the Elbow Method or Silhouette Score.
8 What metric is used to measure clustering performance? Inertia, Davies-Bouldin Index, Silhouette Score.
9 What type of clustering is K-means? Partition-based clustering.
10 What is an application of K-means in healthcare? Grouping patients based on medical conditions.
11 What is an outlier in clustering? A data point that does not belong to any cluster.
12 What kernel is commonly used in Kernel K-means? Gaussian (RBF) kernel.
13 Can K-means work with categorical data? No, K-means works best with numerical data.
14 What is the Silhouette Score? A metric that evaluates how well clusters are separated.
15 Why does K-means require normalization? To prevent features with large ranges from dominating the distance calculation.
16 How do you deal with outliers in K-means? Use K-medoids or remove extreme values.
17 What does the term "inertia" mean in K-means? The sum of squared distances from each point to its assigned centroid.
18 How do you speed up K-means? Use K-means++ initialization or Mini-batch K-means.
19 What is a drawback of Kernel K-means? Higher computational cost.
20 When should you use Kernel K-means over K-means? When the data is not linearly separable.

Conclusion

K-means and Kernel K-means are powerful clustering techniques that help analyze and segment data efficiently. While K-means is simple and scalable, Kernel K-means is better suited for complex datasets. Understanding these methods, along with their mathematical foundations and real-world applications, will prepare you well for exams and practical implementations.

Let me know if you need further refinements! 🚀

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