Day 22: Set Theory and Venn Diagrams for Comparisons 🎯🔵

Measure overlap between sets with precision. Learn set theory fundamentals and visualize relationships with Venn diagrams.

When comparing different versions, thresholds, or classifications, understanding set overlap is crucial. Set theory provides the mathematical foundation, while Venn diagrams offer intuitive visualizations.

💡 Note: This article uses technical terms and abbreviations. For definitions, check out the Key Terms & Glossary page.

The Problem: Measuring Overlap 📊

Scenario: You're comparing two fraud detection models (Model A and Model B).

Questions:

• How many cases did both models catch? (Overlap)
• How many cases did only Model A catch? (Unique to A)
• How many cases did only Model B catch? (Unique to B)
• How similar are the two models? (Similarity metric)

Simple counting isn't enough. You need set theory to answer these questions systematically! 🎯

Set Theory Fundamentals: Building Blocks 🧱

What is a Set?

A set is a collection of distinct objects (elements).

Examples:

• Set A: All fraud cases detected by Model A = {case1, case2, case3, ...}
• Set B: All fraud cases detected by Model B = {case2, case3, case5, ...}
• Universal Set U: All fraud cases = {case1, case2, case3, case4, case5, ...}

Cardinality

Cardinality is the number of elements in a set.

Notation:

• |A| = size of set A (number of elements)
• |B| = size of set B (number of elements)

Example:

|A| = 1000  (Model A detected 1000 fraud cases)
|B| = 1200  (Model B detected 1200 fraud cases)

Visual Example:

Set Operations: Intersection and Union 🔄

Intersection (A ∩ B)

Definition: The intersection of two sets contains elements that are in both sets.

Notation: A ∩ B

Example:

A = {1, 2, 3, 4, 5}
B = {3, 4, 5, 6, 7}
A ∩ B = {3, 4, 5}

Cardinality: |A ∩ B| = number of elements in both sets

Interpretation:

• For fraud detection: Cases caught by both models
• For thresholds: Events captured by both threshold settings
• For versions: Items changed in both versions

Union (A ∪ B)

Definition: The union of two sets contains all elements that are in either set (or both).

Notation: A ∪ B

Example:

A = {1, 2, 3, 4, 5}
B = {3, 4, 5, 6, 7}
A ∪ B = {1, 2, 3, 4, 5, 6, 7}

Cardinality: |A ∪ B| = number of elements in either set

Interpretation:

• For fraud detection: All cases caught by either model
• For thresholds: All events captured by either threshold
• For versions: All items changed in either version

Visual Example:

Inclusion-Exclusion Principle: Computing Union Size 📐

The Formula

Inclusion-Exclusion Principle:

|A ∪ B| = |A| + |B| - |A ∩ B|

Why subtract the intersection?

• When adding |A| and |B|, elements in the intersection are counted twice
• Subtract |A ∩ B| to correct for the double-counting

Example:

|A| = 1000
|B| = 1200
|A ∩ B| = 700

|A ∪ B| = 1000 + 1200 - 700 = 1500

Interpretation:

• Model A caught 1000 cases
• Model B caught 1200 cases
• Both models caught 700 cases (overlap)
• Total unique cases caught: 1500

Visual Example:

Venn Diagrams: Visualizing Set Relationships 📊

What is a Venn Diagram?

A Venn diagram is a visual representation of set relationships using overlapping circles.

Basic 2-Set Venn Diagram:

• Two circles represent two sets
• Overlap represents intersection
• Non-overlapping regions represent unique elements

Visual Example:

Venn Diagram with Counts

To make Venn diagrams useful, we annotate them with cardinalities (counts):

Example:

|A| = 1000
|B| = 1200
|A ∩ B| = 700

Venn Diagram Annotations:
- Only A: |A| - |A ∩ B| = 1000 - 700 = 300
- Only B: |B| - |A ∩ B| = 1200 - 700 = 500
- Both (Intersection): |A ∩ B| = 700
- Total (Union): |A ∪ B| = 1500

Visual Example:

Jaccard Index: Measuring Set Similarity 🎯

What is the Jaccard Index?

The Jaccard index (also called Jaccard similarity coefficient) measures the similarity between two sets.

Formula:

J(A, B) = |A ∩ B| / |A ∪ B|

Properties:

• Range: 0 to 1
• J = 0: No overlap (disjoint sets)
• J = 1: Complete overlap (identical sets)
• Higher J: More similar sets

Example:

|A| = 1000
|B| = 1200
|A ∩ B| = 700
|A ∪ B| = 1500

J(A, B) = 700 / 1500 = 0.467 ≈ 47%

Interpretation: The two sets share 47% of their combined elements.

Visual Example:

When to Use Jaccard Index

Perfect for:

• Comparing model outputs
• Measuring threshold overlap
• Evaluating version differences
• Clustering similarity

Not ideal for:

• When one set is much larger than the other (skewed towards larger set)
• When absolute counts matter more than ratios

Overlap Coefficient: Alternative Similarity Metric 📈

What is the Overlap Coefficient?

The overlap coefficient (also called Szymkiewicz–Simpson coefficient) measures overlap relative to the smaller set.

Formula:

Overlap(A, B) = |A ∩ B| / min(|A|, |B|)

Properties:

• Range: 0 to 1
• Overlap = 0: No overlap
• Overlap = 1: Smaller set is completely contained in larger set
• More sensitive to the smaller set

Example:

|A| = 1000
|B| = 1200
|A ∩ B| = 700

Overlap(A, B) = 700 / min(1000, 1200) = 700 / 1000 = 0.70 = 70%

Interpretation: 70% of the smaller set (A) is covered by the larger set (B).

Jaccard vs Overlap Coefficient

Comparison:

|A| = 100
|B| = 1000
|A ∩ B| = 80

Jaccard = 80 / (100 + 1000 - 80) = 80 / 1020 = 0.078 ≈ 8%
Overlap = 80 / min(100, 1000) = 80 / 100 = 0.80 = 80%

Key Difference:

• Jaccard: Normalized by union (both sets matter equally)
• Overlap: Normalized by smaller set (focuses on coverage of smaller set)

When to use each:

• Jaccard: When both sets are equally important
• Overlap: When you care about how much of the smaller set is covered

Visual Example:

Real-World Application: show_venn_diagram Function 🔧

Purpose

The show_venn_diagram function compares "events captured" across threshold sets or versions.

Use Cases:

• Compare fraud cases across two model versions
• Analyze events captured by different thresholds
• Measure overlap between rule sets
• Evaluate classification consistency

Code Concept:

def show_venn_diagram(set_a, set_b, label_a="Set A", label_b="Set B"):
    """
    Visualize overlap between two sets using Venn diagram
    
    Parameters:
    - set_a: First set (list or set)
    - set_b: Second set (list or set)
    - label_a: Label for first set
    - label_b: Label for second set
    
    Returns:
    - Venn diagram visualization with counts and Jaccard index
    """
    # Calculate cardinalities
    cardinality_a = len(set_a)
    cardinality_b = len(set_b)
    intersection = len(set_a & set_b)
    union = len(set_a | set_b)
    
    # Calculate Jaccard index
    jaccard = intersection / union if union > 0 else 0
    
    # Create Venn diagram
    # - Draw two circles
    # - Annotate with counts
    # - Show Jaccard index
    # - Color-code regions

Visual Example:

Exercise: Computing Jaccard and Overlap Coefficient 🎓

The Problem

Given triplet: (a, b, ab) where:

• a = |A| = 1000 (cardinality of set A)
• b = |B| = 1200 (cardinality of set B)
• ab = |A ∩ B| = 700 (intersection size)

Compute:

1. Jaccard index
2. Overlap coefficient

Solution

Step 1: Calculate Union Size

Using inclusion-exclusion principle:

|A ∪ B| = |A| + |B| - |A ∩ B|
         = 1000 + 1200 - 700
         = 1500

Step 2: Calculate Jaccard Index

J(A, B) = |A ∩ B| / |A ∪ B|
         = 700 / 1500
         = 0.467
         ≈ 46.7%

Step 3: Calculate Overlap Coefficient

Overlap(A, B) = |A ∩ B| / min(|A|, |B|)
              = 700 / min(1000, 1200)
              = 700 / 1000
              = 0.70
              = 70%

Step 4: Interpret Results

• Jaccard = 46.7%: The two sets share about 47% of their combined elements
• Overlap = 70%: 70% of the smaller set (A) is covered by set B

Visual Example:

Python Implementation

def compute_similarity_metrics(a, b, ab):
    """
    Compute Jaccard index and overlap coefficient
    
    Parameters:
    - a: |A| (cardinality of set A)
    - b: |B| (cardinality of set B)
    - ab: |A ∩ B| (intersection size)
    
    Returns:
    - dict with Jaccard and Overlap coefficients
    """
    # Calculate union
    union = a + b - ab
    
    # Calculate Jaccard index
    jaccard = ab / union if union > 0 else 0
    
    # Calculate overlap coefficient
    overlap = ab / min(a, b) if min(a, b) > 0 else 0
    
    return {
        'jaccard': jaccard,
        'overlap': overlap,
        'union': union
    }

# Example usage
result = compute_similarity_metrics(a=1000, b=1200, ab=700)
print(f"Jaccard Index: {result['jaccard']:.3f} ({result['jaccard']*100:.1f}%)")
print(f"Overlap Coefficient: {result['overlap']:.3f} ({result['overlap']*100:.1f}%)")

Output:

Jaccard Index: 0.467 (46.7%)
Overlap Coefficient: 0.700 (70.0%)

Advanced: Three-Set Venn Diagrams 🔵🔵🔵

Three Sets

When comparing three sets (A, B, C), the relationships become more complex:

Regions:

• Only A: A - (B ∪ C)
• Only B: B - (A ∪ C)
• Only C: C - (A ∪ B)
• A and B (not C): (A ∩ B) - C
• A and C (not B): (A ∩ C) - B
• B and C (not A): (B ∩ C) - A
• All three: A ∩ B ∩ C

Inclusion-Exclusion for Three Sets:

|A ∪ B ∪ C| = |A| + |B| + |C| 
             - |A ∩ B| - |A ∩ C| - |B ∩ C| 
             + |A ∩ B ∩ C|

Visual Example:

Best Practices for Set Comparison Analysis ✅

1. Always Annotate Venn Diagrams

Include cardinalities for:

• Each set
• Intersections
• Unique regions
• Total union

2. Calculate Multiple Metrics

Don't rely on a single metric. Calculate:

• Jaccard index
• Overlap coefficient
• Absolute counts
• Relative percentages

3. Visualize Before Analyzing

Always create a Venn diagram to:

• Understand relationships visually
• Spot patterns quickly
• Communicate findings effectively

4. Consider Set Sizes

Be aware of:

• Size differences between sets
• Impact on similarity metrics
• Whether absolute or relative measures matter

5. Use Appropriate Metrics

Choose metrics based on your question:

• Jaccard: When both sets matter equally
• Overlap: When smaller set coverage matters
• Absolute counts: When volumes are critical

6. Document Your Sets Clearly

For each set, document:

• What it represents
• How it was created
• When it was captured
• Any filters applied

Summary Table 📋

Final Thoughts 🌟

Set theory and Venn diagrams are powerful tools for understanding relationships between sets. They help you visualize overlaps, calculate similarity metrics, and make informed decisions when comparing versions, thresholds, or classifications.

Key Takeaways:

✅ Cardinality measures set size ✅ Intersection identifies overlap ✅ Union captures combined coverage ✅ Inclusion-Exclusion calculates union size ✅ Jaccard index measures normalized similarity ✅ Overlap coefficient measures smaller set coverage ✅ Venn diagrams visualize relationships intuitively

Visualize your sets to understand your data! 📊

Tomorrow's Preview: Day 23 - Label Post-Processing: Partitioning ATL vs BTL Mathematically 🏷️📊