Lecture Notes: SVM Classifier and Hyperplane Calculation

Introduction

• Focus: Finding the equation for a hyperplane with maximal margin using SVM classifier
• Previous videos covered SVM classifier algorithm and examples

Given Data Set

• 3 input vectors with 2 features
  • Input vector 1: (2, 2), Target = -1
  • Input vector 2: (4, 5), Target = +1
  • Input vector 3: (7, 4), Target = +1
• Objective: Apply SVM algorithm to find the hyperplane

SVM Classifier Key Concepts

• Need to find values for the weight vector and bias
• Calculate the alpha vector (set of variables based on number of input vectors):
  • Alpha values: α1, α2, α3
  • Conditions:
    • ∑(αᵢyᵢ) = 0
    • αᵢ > 0

Steps to Calculate Alphas

1. Calculate Alpha Values

   • Maximize the function φ(α vector)
   • Replace n = 3 (number of vectors)
   • Expand the equation considering combinations (i, j)
   • Calculate dot products for combinations, e.g., x₁•x₂

2. Simplify the Equation

   • Utilize constraint: α1 = α2 + α3
   • Replace α1 with α2 + α3, simplify equation
   • Differentiate φ(α vector) w.r.t α2 and α3, equate to 0

3. Solve for Alpha Values

   • α2 = 26/121
   • α3 = -6/121
   • α1 = α2 + α3 = 20/121

Calculate Weight Vector

• Formula: Weight vector = ∑(αᵢyᵢxᵢ) for i = 1 to n
• Substitute α and y values:
  • Final weight vector: (2/11, 6/11)

Calculate Bias

• Equation: 1/2(min(W•xᵢ) where yᵢ = +1 + max(W•xᵢ) where yᵢ = -1)
• Solve for positive and negative class examples
• Bias = 27/11

SVM Classifier Equation

• f(x vector) = Weight vector • x vector - Bias
• Maximal margin hyperplane: set f(x vector) = 0

Support Vectors and Hyperplane

• Alpha values indicate support vectors (positive alphas)
• Only x1 and x2 are support vectors
• Hyperplane passes through the midpoint of support vectors

Conclusion

• Successfully applied SVM classifier to find hyperplane with maximum margin
• Hyperplane should pass through midpoint of support vectors and be perpendicular

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