Answer:
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To implement the perceptron rule training using the given parameters, we need to follow these steps:
Initialize the weights w' as specified.
For each training pair (X, d), calculate the net value using the formula: net = X · w, where · represents the dot product.
Apply the activation function f(net) = sgn(net) to determine the output.
If the output matches the desired output d, move on to the next training pair. If not, update the weights using the formula: w_new = w_old + c * d * X, where c = 1.
Repeat steps 2-4 for all training pairs until two correct responses in a row are achieved.
Given:
Initial weights w' = [1, -1, 0.5]
Training pair 1: X1 = [1, 0, -1], d1 = 1
Training pair 2: X2 = [1, -1, 0], d2 = -1
Let's start the training process:
Iteration 1:
For training pair (X1, d1):
net1 = [1, 0, -1] · [1, -1, 0.5] = 1 - 0 - 0.5 = 0.5
f(net1) = sgn(0.5) = 1
Output matches the desired output, so no weight update is needed.
For training pair (X2, d2):
net2 = [1, -1, 0] · [1, -1, 0.5] = 1 + 1 + 0 = 2
f(net2) = sgn(2) = 1
Output does not match the desired output, so we update the weights:
w_new = [1, -1, 0.5] + 1 * (-1) * [1, -1, 0] = [0, 0, 0.5]
Iteration 2:
For training pair (X1, d1):
net1 = [1, 0, -1] · [0, 0, 0.5] = 0 - 0 - 0.5 = -0.5
f(net1) = sgn(-0.5) = -1
Output does not match the desired output, so we update the weights:
w_new = [0, 0, 0.5] + 1 * 1 * [1, 0, -1] = [1, 0, -0.5]
For training pair (X2, d2):
net2 = [1, -1, 0] · [1, 0, -0.5] = 1 + 0 + 0 = 1
f(net2) = sgn(1) = 1
Output does not match the desired output, so we update the weights:
w_new = [1, 0, -0.5] + 1 * (-1) * [1, -1, 0] = [0, 1, -0.5]
Iteration 3:
For training pair (X1, d1):
net1 = [1, 0, -1] · [0, 1, -0.5] = 0 + 0 + 0.5 = 0.5
f(net1) = sgn(0.5) = 1
Output matches the desired output, so no weight update is needed.
For training pair (X2, d2):
net2 = [1, -1, 0] · [0, 1, -0.5] = 0 - 1 + 0 = -1
f(net2) = sgn(-1) = -1
Output matches the desired output, so no weight update is needed.
The training sequence has achieved two correct responses in a row, so the training is complete.
The net values obtained during training are:
Iteration 1: net1 = 0.5, net2 = 2
Iteration 2: net1 = -0.5, net2 = 1
Iteration 3: net1 = 0.5, net2 =