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We will compute the Linear Discriminant Projection (LDA) manually for the given two-class dataset $X_1$ and $X_2$. --- ### Step 1: Given Data Let * Class 1: $X_1 = \{(4,2), (2,1), (2,4), (3,5), (4,5)\}$ * Class 2: $X_2 = \{(9,9), (6,9), (9,6), (8,7), (10,9)\}$ --- ### Step 2: Compute Class Means Let $\mu_1$ and $\mu_2$ be the means of $X_1$ and $X_2$, respectively. #### Mean of $X_1$: $$ \mu_1 = \frac{1}{5}[(4+2+2+3+4),\ (2+1+4+5+5)] = \frac{1}{5}[15,\ 17] = (3,\ 3.4) $$ #### Mean of $X_2$: $$ \mu_2 = \frac{1}{5}[(9+6+9+8+10),\ (9+9+6+7+9)] = \frac{1}{5}[42,\ 40] = (8.4,\ 8) $$ --- ### Step 3: Compute Within-Class Scatter Matrices Let the within-class scatter matrix be: $$ S_W = S_1 + S_2 $$ We compute each scatter matrix as: $$ S_i = \sum (x - \mu_i)(x - \mu_i)^T $$ #### Compute $S_1$: Let’s calculate differences from mean $\mu_1 = (3, 3.4)$ for each point in _______ ____ _______ ______ _____ ________ _________ __________.
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