Word Count : 732

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To compute the Linear Discriminant projection (also known as Fisher’s Linear Discriminant) manually for two classes, follow these steps: --- ### Step 1: Define the datasets We are given two classes: #### Class X₁: * (4, 2), (2, 1), (2, 4), (3, 5), (4, 5) #### Class X₂: * (9, 9), (6, 9), (9, 6), (8, 7), (10, 9) Let’s denote these as: * $X_1 = \{x^{(1)}_1, x^{(1)}_2, x^{(1)}_3, x^{(1)}_4, x^{(1)}_5\}$ * $X_2 = \{x^{(2)}_1, x^{(2)}_2, x^{(2)}_3, x^{(2)}_4, x^{(2)}_5\}$ --- ### Step 2: Compute the mean vector of each class #### For Class X₁: Add each component: * $\mu_1 = \frac{1}{5} \left( \sum x_1, \sum x_2 \right) = \frac{1}{5} \left(4+2+2+3+4, 2+1+4+5+5\right)$ * $\mu_1 = \frac{1}{5} (15, 17) = (3, 3.4)$ #### For Class X₂: * $\mu_2 = \frac{1}{5} (9+6+9+8+10, 9+9+6+7+9) = \frac{1}{5} (42, 40) = (8.4, 8)$ --- ### Step 3: Compute the Within-class Scatter Matrix $S_W$ We calculate for each class: #### For class X₁: Compute each $(x_i - \mu_1)(x_i - \mu_1)^T$ Let’s define the scatter matrix: $$ S_1 = \sum_{i=1}^{5} (x_i^{(1)} - \mu_1)(x_i^{(1)} - \mu_1)^T $$ Compute deviations: 1. (4,2) - ____ ________ ______ ____ _______.
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