Question 1: SVM Classifier - Support Vectors

The figure shows a two-class classification problem. The red points represent Class 1, while the blue points represent Class 2. The goal is to identify the support vectors for a Support Vector Machine (SVM) classifier.

Support Vectors:

• In SVM, support vectors are the data points that are closest to the decision boundary (the separating hyperplane). These points directly influence the decision boundary's placement.

Let’s analyze the position of points closest to the boundary:

1. For Class 1 (Red Points):

   • Points (3, 4) and (7, 2) seem to be the closest to the decision boundary.
   • Another potential support vector is (6, 6), which also seems close to the boundary.

2. For Class 2 (Blue Points):

   • The blue point (6, 6) appears to be a critical support vector because it is on the boundary.

Thus, the correct support vectors should include:

• (3, 4), (7, 2) from Class 1
• (6, 6) from Class 2

This matches option d), which lists (3, 4), (7, 3), and (6, 6).

Question 2: Hierarchical Clustering - Linkage Method

The given formula for the distance between two clusters is: $D(c_i, c_j) = \min d(a, b) \quad \text{where} \ a \in c_i \ \text{and} \ b \in c_j$ This means that the distance between two clusters is defined as the minimum distance between any pair of points, where one point belongs to one cluster, and the other point belongs to the other cluster.

Linkage Types:

• Single linkage: This method defines the distance between clusters as the minimum distance between any point in one cluster and any point in the other cluster. This matches the formula given.
• Complete linkage: This defines the distance between clusters as the maximum distance between points.
• Average linkage: This defines the distance between clusters as the average of all pairwise distances between points in different clusters.

Since the formula defines the minimum distance, the correct answer is a) Single linkage.

Summary of Correct Answers:

1. d) (3, 4), (7, 3), (6, 6)
2. a) Single linkage

Would you like more details on how SVM and hierarchical clustering work, or do you have any other questions?

5 Related Questions:

1. What are the mathematical steps to calculate the decision boundary in SVM?
2. How does the SVM algorithm decide which points are support vectors?
3. What are the advantages and disadvantages of using single linkage in hierarchical clustering?
4. How do different linkage methods affect the shape of clusters in hierarchical clustering?
5. Can the SVM classifier handle nonlinear data, and how?

Tip:

In SVM, tuning the kernel function (linear, polynomial, RBF, etc.) can greatly improve classification performance, especially when the data is not linearly separable.