The Movie Night Decision-Maker Example

Let's explore a complete example of deciding whether to watch a movie based on two factors:

• Review Score (how well-rated the movie is)
• Friend Interest (how excited your friends are to see it)

Step 1: Set Up the Problem

Let's say we have data from past movie nights:

Movie	Review Score	Friend Interest	Decision
A	0.8	0.4	1 (Watched)
B	0.1	0.7	1 (Watched)
C	-0.5	-0.3	-1 (Skipped)
D	0.6	-0.8	-1 (Skipped)

• Review Score ranging from -1 to 1
• Friend Interest ranging from -1 to 1

Step 2: Start with Zero Weights and Bias

Let's start with:

• Weight for Review Score (w₁) = 0.0
• Weight for Friend Interest (w₂) = 0.0
• Bias (b) = 0.0
• Learning rate = 1.0

Step 3: Training Iterations

First training example (Movie A):

1. Calculate weighted sum: $$(0.0 \times 0.8) + (0.0 \times 0.4) + 0.0 = 0.0$$
2. Since 0 = 0, our prediction is 0 (Skip)
3. Actual decision was 1 (Watch)
4. Prediction was wrong! We need to update weights using this formula $$w_{new} = w_{old}+\alphayx$$

• New $w_1 = 0.0 + (1.0 \times 1 \times 0.8) = 0.0 + 0.8 = 0.8$
• New $w_2 = 0.0 + (1.0 \times 1 \times 0.4) = 0.0 + 0.4 = 0.4$

5. Calculating the bias is very similar we just alway place 1 instead of the value of x:

• New $\text{bias} = 0.0 + (1.0 \times 1 \times 1) = 1.0$

Let's see what our decision boundary looks like so far!

1. $w_1 x + w_2 y + 1 = 0$
2. $0.8x + 0.4y + 1 = 0$
3. $y = -2x - 2.5$

Second training example (Movie B):

1. Calculate weighted sum: $$(0.8 \times 0.1) + (0.4 \times 0.7) + 1.0 = 0.08 + 0.28 + 1.0 = 1.36$$
2. Since 1.36 > 0, our prediction is 1 (Watch)
3. Actual decision was 1 (Watch)
4. Prediction was correct, so no weight updates needed

Third training example (Movie C):

1. Calculate weighted sum: $$(0.8 \times -0.5) + (0.4 \times -0.3) + 1.0 = -0.4 - 0.12 + 1.0 = 0.48$$
2. Since 0.48 > 0, our prediction is 1 (Watch)
3. Actual decision was 0 (Skip)
4. Prediction was wrong! We need to update weights

• New $w_1 = 0.8 + (1.0 \times -1 \times -0.5) = 0.8 + 0.5 = 1.3$
• New $w_2 = 0.4 + (1.0 \times -1 \times -0.3) = 0.4 + 0.3 = 0.7$
• New $\text{bias} = 1.0 + (1.0 \times -1 \times 1) = 0.0$

Let's see what our updated decision boundary looks like so far!

1. $w_1 x + w_2 y + 0 = 0$
2. $1.3x + 0.7y + 0 = 0$
3. $y = -1.86x$

Fourth training example (Movie D):

1. Calculate weighted sum: $$(1.3 \times 0.6) + (0.7 \times -0.8) + 0.0 = 0.78 - 0.56 + 0.0 = 0.22$$
2. Since 0.22 > 0, our prediction is 1 (Watch)
3. Actual decision was 0 (Skip)
4. Prediction was wrong! We need to update weights

• New $w_1 = 1.3 + (1.0 \times -1 \times 0.6) = 1.3 - 0.6 = 0.7$
• New $w_2 = 0.7 + (1.0 \times -1 \times -0.8) = 0.7 + 0.8 = 1.5$
• New $\text{bias} = 0.0 + (1.0 \times -1) = -1.0$

Step 4: Continue Training

We would continue this process, going through all examples again with our updated weights (0.7, 1.5, -1.0) and repeating until predictions stabilize.

Let's check if our current weights correctly classify all examples:

• Movie A: $(0.7 \times 0.8) + (1.5 \times 0.4) + (-1.0) = 0.56 + 0.6 - 1.0 = 0.16 > 0$ → Watch ✓
• Movie B: $(0.7 \times 0.1)+(1.5 \times 0.7) + (-1.0) = 0.07 + 1.05 - 1.0 = 0.12 > 0$ → Watch ✓
• Movie C: $(0.7 \times -0.5)+(1.5 \times -0.3) + (-1.0) = -0.35 - 0.45 - 1.0 = -1.8 < 0$ → Skip ✓
• Movie D: $(0.7 \times 0.6) + (1.5 \times -0.8) + (-1.0) = 0.42 - 1.2 - 1.0 = -1.78 < 0$ → Skip ✓

Great! After just one pass through all examples, our perceptron has already learned to correctly classify all our movie data!

Step 5: The Final Decision Boundary

When a perceptron learns, it creates what's called a "decision boundary" - a line (or plane in higher dimensions) that separates the two categories.

With our final weights, our decision rule becomes:

• Calculate: 0.7 × (Review Score) + 1.5 × (Friend Interest) - 1.0
• If result > 0, watch the movie; otherwise, skip it

The final decision boundary is:

1. $0.7x + 1.5y - 1 = 0$
2. $y = -0.47x + 0.67$