How to Use This Demo:
• Select different Decision Boundary Types to handle nonlinear data patterns
• Adjust the parameters shown below the boundary type (different parameters appear for different types)
• Use the "Click adds" dropdown to select which class to add, then click anywhere on the plot to add data points
• Use "Generate New Data" to create random 2D classification data
• Use "Clear All Data" to remove all points
• Find Optimal Solution: Optimizes parameters for the selected boundary type

Interactive Features:
• Click on the plot to manually add data points of the selected class
• Watch how the decision boundary changes as you adjust parameters
• Observe real-time updates to accuracy and loss metrics

Visual Elements:
• Red circles: Class 0 data points
• Blue squares: Class 1 data points
• Solid line: Decision boundary (50% probability - where the model is most uncertain)
• Dotted lines: Probability contours at 25% and 75% - show regions of high confidence for each class
• Background shading: Probability regions (redder = more likely Class 0, bluer = more likely Class 1)

Boundary Types:
• Linear: Uses w₁, w₂, b → Straight line boundary
• Quadratic: Uses w₁, w₂, b, w₁₁, w₂₂, w₁₂ → Curved boundaries (ellipses, parabolas)

Mathematical Foundation:
Binary classification uses the sigmoid function to map any real value to a probability between 0 and 1:

$$P(y=1|x)=\sigma (f(x))=\frac{1}{1+e^{-f(x)}}$$

Where $f(x)$ is the decision function that depends on the boundary type:
• Linear: $f(x)=w_1{x}_1+w_2{x}_2+b$
• Quadratic: $f(x)=w_1{x}_1+w_2{x}_2+w_{11}{x}_1^2+w_{22}{x}_2^2+w_{12}{x}_1{x}_2+b$

The decision boundary occurs where $f(x)=0$ (50% probability).

Optimization Tips:
• Start with linear boundaries for linearly separable data
• Use quadratic boundaries when data forms curved patterns
• The "Find Optimal Solution" button uses gradient descent to minimize loss

Interpreting Results:
• Accuracy: Percentage of correctly classified points (higher is better)
• Loss: Cross-entropy loss measuring prediction confidence (lower is better)
• Perfect separation: Loss approaches 0, accuracy approaches 100%
• Overfitting risk: Complex boundaries may not generalize to new data

Engineering Applications:
• Use simpler boundaries (linear) when interpretability is important
• Complex boundaries may be needed for real-world civil engineering data
• Always validate on separate test data to check generalization