Question
Apply the perception algorithm to the following pattern classes:,
.
Let and
Also, Sketch the decision surface.
Answer :
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Skip in 10
We are asked to apply the perceptron learning algorithm manually to the given pattern classes. Let's solve step by step. Given: * Class ( W_1 = {(0,0,0)^T, (1,0,0)^T, (1,0,1)^T, (1,1,0)^T} ) * Class ( W_2 = {(0,0,1)^T, (0,1,1)^T, (0,1,0)^T, (1,1,1)^T} ) * Augmented vector: we add a bias term ( x_0 = 1 ) to each pattern. * Learning rate ( C = 1 ) * Initial weight: ( W(1) = [-1, -2, -2, 0]^T ) Step 1: Augment the vectors (add ( x_0 = 1 )) * For ( W_1 ) (positive class), augmented vectors ( X ) are: [ X_1 = [1, 0,0,0]^T,\quad X_2 = [1,1,0,0]^T,\quad X_3 = [1,1,0,1]^T,\quad X_4 = [1,1,1,0]^T ] * For ( W_2 ) (negative class), we take negative of the augmented vectors to unify perceptron rule: [ -X_5 = [-1,0,0,1]^T,\quad -X_6 = [-1,0,1,1]^T,\quad -X_7 = [-1,0,1,0]^T,\quad -X_8 = [-1,1,1,1]^T ] Step 2: Perceptron learning rule [ W(k+1) = W(k) + C \cdot X \quad \text{if misclassified} ] We test each vector in order with the current weight. If ( W^T X \le 0 ), update weight. --- Iteration 1: Start with ( W = [-1, -2, -2, 0]^T ) 1. Test ( X_1 = [1,0,0,0]^T ): [ W^T X_1 = (-1)(1) + (-2)(0) + (-2)(0) + (0)(0) = -1 \le 0 \quad \Rightarrow \text{misclassified} ] Update: ( W = W + X_1 = [-1+1, -2+0, -2+0, 0+0] = [0, -2, -2, 0]^T ) 2. Test ( X_2 = [1,1,0,0]^T ): [ W^T X_2 = 0*1 + (-2)*1 + (-2)*0 +
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