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### **Question 6:**

For a two-class problem, \( \left( \begin{matrix} 1 \\ 2 \end{matrix} \right), \left( \begin{matrix} 2 \\ 3 \end{matrix} \right), \left( \begin{matrix} 3 \\ 3 \end{matrix} \right) \in \omega_1 \) and \( \left( \begin{matrix} 4 \\ 5 \end{matrix} \right), \left( \begin{matrix} 5 \\ 6 \end{matrix} \right), \left( \begin{matrix} 6 \\ 5 \end{matrix} \right) \in \omega_2 \), find the projection of the point \( \left( \begin{matrix} 1 \\ 1 \end{matrix} \right) \) on \( W \) using Fisher's linear discriminant.

#### Solution Approach:

1. **Step 1:** Calculate the mean of each class:
   - For \( \omega_1 \), take the average of the three points.
   - For \( \omega_2 \), similarly, take the average of its three points.

2. **Step 2:** Compute the within-class scatter matrix for \( S_W \).

3. **Step 3:** Determine the direction vector \( W \) for the linear discriminant using the formula:
   \[
   W = S_W^{-1} (\mu_1 - \mu_2)
   \]
   where \( \mu_1 \) and \( \mu_2 \) are the means of the two classes.

4. **Step 4:** Project the point \( \left( \begin{matrix} 1 \\ 1 \end{matrix} \right) \) onto the vector \( W \) to find its projection.

This will require some calculations based on the steps above.

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### **Question 7:**

For the same two-class problem, find the separation (absolute) of means (Euclidean distance) before and after projection using Fisher's linear discriminant.

#### Solution Approach:

1. **Separation before projection:**
   - Compute the Euclidean distance between the two means, \( \mu_1 \) and \( \mu_2 \), before the projection.

2. **Separation after projection:**
   - Project the means \( \mu_1 \) and \( \mu_2 \) onto the Fisher's linear discriminant direction vector \( W \).
   - Calculate the Euclidean distance between the projected means.

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### **Question 8:**

Which of the following could be a weight update rule in perceptron criterion using gradient descent procedure?

- **Perceptron learning rule** adjusts the weights based on misclassified points. The correct weight update rule typically takes the form:
  \[
  a(k+1) = a(k) + \eta \sum \text{(for all misclassified y)} \cdot y
  \]
  This means that we update the weights for the misclassified examples, which helps to minimize classification error.

Thus, **Option b** seems to be the most correct.

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Let me know if you need more detailed explanations or clarifications! Here are five questions related to the above problems:

1. How is Fisher's linear discriminant different from other linear classifiers?
2. What is the role of the within-class scatter matrix in Fisher's discriminant?
3. How is the Euclidean distance calculated between two points in \( \mathbb{R}^n \)?
4. What is the significance of the projection in Fisher's discriminant analysis?
5. How does the learning rate \( \eta \) affect the weight update rule in gradient descent?

**Tip:** When solving classification problems, Fisher's linear discriminant maximizes the ratio of the between-class variance to the within-class variance, optimizing the class separation.

For a two-class problem, find the projection of the point using Fisher's linear discriminant and calculate the separation of means before and after projection.

In this problem, find the projection of a point onto Fisher's linear discriminant for a two-class problem, and calculate the separation of means using Euclidean distance. The problem involves understanding LDA and the significance of within-class scatter matrices.

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