Perceptron Weights Update Calculation

This solution details the step-by-step process of calculating the updated weights for a perceptron model using the provided input samples and learning parameters. We will apply the perceptron learning rule iteratively.

Perceptron Model Setup

The perceptron model is defined with the following initial parameters:

• Initial Weights: $w_1 = 0.0$, $w_2 = 0.0$
• Learning Rate ($\eta$): $0.1$
• Bias: The model does not involve a bias term.
• Activation Function: The activation function determines the observed output ($y_{observed}$) based on the net input ($y_{in}$): $y_{observed} = \begin{cases} 1 & \text{if } y_{in} \gt 0 \\ 0 & \text{if } y_{in} \lt 0 \end{cases}$

Training Data

The following table shows the sample inputs ($x_1, x_2$) and their corresponding target values ($t$):

Applying the Perceptron Learning Rule

We will now iterate through each sample (S1, S2, S3, S4) and update the weights ($w_1, w_2$) if the predicted output does not match the target.

Processing Sample S1

• Inputs ($x_1, x_2$): $(0, 0)$
• Target ($t$): $0$
• Current Weights ($w_1, w_2$): $(0.0, 0.0)$
• Calculate Net Input ($y_{in}$): $y_{in} = w_1 x_1 + w_2 x_2 = (0.0 \times 0) + (0.0 \times 0) = 0$
• Determine Observed Output ($y_{observed}$): Based on the activation function, since $y_{in} = 0$, $y_{observed} = 0$.
• Calculate Error: $error = t - y_{observed} = 0 - 0 = 0$
• Update Weights: The error is 0. No update is needed. $w_1 = 0.0$ $w_2 = 0.0$
• Weights after S1: $w_1 = 0.0, w_2 = 0.0$

Processing Sample S2

• Inputs ($x_1, x_2$): $(0, 1)$
• Target ($t$): $1$
• Current Weights ($w_1, w_2$): $(0.0, 0.0)$
• Calculate Net Input ($y_{in}$): $y_{in} = w_1 x_1 + w_2 x_2 = (0.0 \times 0) + (0.0 \times 1) = 0$
• Determine Observed Output ($y_{observed}$): Since $y_{in} = 0$, $y_{observed} = 0$.
• Calculate Error: $error = t - y_{observed} = 1 - 0 = 1$
• Update Weights: The error is non-zero. Apply the update rule: $\Delta w_1 = \eta \cdot error \cdot x_1 = 0.1 \times 1 \times 0 = 0.0$ $\Delta w_2 = \eta \cdot error \cdot x_2 = 0.1 \times 1 \times 1 = 0.1$ New weights: $w_1 = w_1 + \Delta w_1 = 0.0 + 0.0 = 0.0$ $w_2 = w_2 + \Delta w_2 = 0.0 + 0.1 = 0.1$
• Weights after S2: $w_1 = 0.0, w_2 = 0.1$

Processing Sample S3

• Inputs ($x_1, x_2$): $(1, 0)$
• Target ($t$): $1$
• Current Weights ($w_1, w_2$): $(0.0, 0.1)$
• Calculate Net Input ($y_{in}$): $y_{in} = w_1 x_1 + w_2 x_2 = (0.0 \times 1) + (0.1 \times 0) = 0$
• Determine Observed Output ($y_{observed}$): Since $y_{in} = 0$, $y_{observed} = 0$.
• Calculate Error: $error = t - y_{observed} = 1 - 0 = 1$
• Update Weights: The error is non-zero. Apply the update rule: $\Delta w_1 = \eta \cdot error \cdot x_1 = 0.1 \times 1 \times 1 = 0.1$ $\Delta w_2 = \eta \cdot error \cdot x_2 = 0.1 \times 1 \times 0 = 0.0$ New weights: $w_1 = w_1 + \Delta w_1 = 0.0 + 0.1 = 0.1$ $w_2 = w_2 + \Delta w_2 = 0.1 + 0.0 = 0.1$
• Weights after S3: $w_1 = 0.1, w_2 = 0.1$

Processing Sample S4

• Inputs ($x_1, x_2$): $(1, 1)$
• Target ($t$): $1$
• Current Weights ($w_1, w_2$): $(0.1, 0.1)$
• Calculate Net Input ($y_{in}$): $y_{in} = w_1 x_1 + w_2 x_2 = (0.1 \times 1) + (0.1 \times 1) = 0.1 + 0.1 = 0.2$
• Determine Observed Output ($y_{observed}$): Since $y_{in} = 0.2 \gt 0$, $y_{observed} = 1$.
• Calculate Error: $error = t - y_{observed} = 1 - 1 = 0$
• Update Weights: The error is 0. No update is needed. $w_1 = 0.1$ $w_2 = 0.1$
• Weights after S4: $w_1 = 0.1, w_2 = 0.1$

Conclusion

After applying the perceptron learning algorithm to all the samples sequentially (S1, S2, S3, S4), the final updated weights are $w_1 = 0.1$ and $w_2 = 0.1$.