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To solve this problem using Fisher's linear discriminant, we need to go step by step, as it involves several key concepts of pattern recognition. Here's how to approach it:

### Given:
- Class \( \omega_1 \) points: \( (1, 2), (2, 3), (3, 3) \)
- Class \( \omega_2 \) points: \( (4, 5), (5, 6), (6, 5) \)
- The goal is to project the point \( (1, 1) \) onto Fisher's discriminant line.

### Fisher's Linear Discriminant Procedure:

1. **Step 1: Calculate the mean vectors for each class.**
   - Mean of class \( \omega_1 \) (denoted as \( \mu_1 \)):
     \[
     \mu_1 = \frac{1}{3} \left( \begin{pmatrix} 1 \\ 2 \end{pmatrix} + \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 3 \\ 3 \end{pmatrix} \right)
     \]
   - Mean of class \( \omega_2 \) (denoted as \( \mu_2 \)):
     \[
     \mu_2 = \frac{1}{3} \left( \begin{pmatrix} 4 \\ 5 \end{pmatrix} + \begin{pmatrix} 5 \\ 6 \end{pmatrix} + \begin{pmatrix} 6 \\ 5 \end{pmatrix} \right)
     \]

2. **Step 2: Calculate the scatter matrices for each class.**
   - For class \( \omega_1 \), the scatter matrix \( S_1 \) is calculated as:
     \[
     S_1 = \sum_{i=1}^{3} (x_i - \mu_1)(x_i - \mu_1)^T
     \]
   - Similarly, for class \( \omega_2 \), the scatter matrix \( S_2 \) is:
     \[
     S_2 = \sum_{i=1}^{3} (x_i - \mu_2)(x_i - \mu_2)^T
     \]

3. **Step 3: Compute the within-class scatter matrix \( S_W = S_1 + S_2 \).**

4. **Step 4: Compute the optimal projection vector \( w \), which maximizes the separation between the classes:**
   \[
   w = S_W^{-1} (\mu_2 - \mu_1)
   \]

5. **Step 5: Project the point \( (1, 1) \) onto the discriminant line:**
   \[
   y = w^T (x - \mu)
   \]
   where \( x \) is the point \( (1, 1) \) and \( \mu \) is the mean vector.

I will now perform the necessary calculations.### Solution Breakdown:

1. **Mean vectors:**
   - For class \( \omega_1 \):
     \[
     \mu_1 = \left( 2, 2.67 \right)
     \]
   - For class \( \omega_2 \):
     \[
     \mu_2 = \left( 5, 5.33 \right)
     \]

2. **Within-class scatter matrix \( S_W \):**
   \[
   S_W = \begin{pmatrix} 4 & 1 \\ 1 & 1.33 \end{pmatrix}
   \]

3. **Optimal projection vector \( w \):**
   \[
   w = \begin{pmatrix} 0.31 \\ 1.77 \end{pmatrix}
   \]

4. **Projection of the point \( (1, 1) \) onto Fisher's discriminant:**
   \[
   \text{Projection} = -3.26
   \]

Thus, the projection of the point \( (1,

For a two-class problem. (1,2), (2,3), (3,3) ∈ ω1 and (4,5), (5,6), (6,5) ∈ ω2, find the projection of the point (1,1) on W using Fisher's linear discriminator.

Solve the Fisher's linear discriminant problem for a two-class scenario. Points from two classes are given, and the projection of the point (1,1) on the discriminant line is computed step-by-step.

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