Jaccard Similarity

Jaccard Similarity, also known as the Jaccard Index or Jaccard Coefficient, measures the similarity between two sets. Commonly used in:

• Text and document similarity
• Recommendation systems
• Duplicate and plagiarism detection
• Clustering and classification
• Binary data and image comparison

It is calculated as the size of the intersection divided by the size of the union of the two sets.

Formula

J(A, B) = \frac{|A \cap B|}{|A \cup B|}

where:

• ∣A∩B∣ is the number of common elements between sets
• ∣A∪B∣ is the total number of unique elements in both sets.

Value of Jaccard similarity ranges from 0 to 1:

• 1 means both sets are identical.
• 0 means the sets have no elements in common.
• Value between 0 and 1 represents partial similarity.

Jaccard Similarity Between Two Sets

Python sets provide built-in operations for finding intersections and unions, which makes them suitable for calculating Jaccard Similarity. The & operator gives the intersection, while the | operator gives the union.

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

intersection = A & B
union = A | B

jaccard_similarity = len(intersection) / len(union)

print("Jaccard Similarity:", jaccard_similarity)

Jaccard Similarity: 0.3333333333333333

Explanation:

• A & B finds the elements present in both sets.
• A | B combines all unique elements from both sets.
• len(intersection) gives the number of common elements.
• len(union) gives the number of unique elements across both sets.
• Dividing these values gives the Jaccard Similarity.

Jaccard Similarity Between Binary Vectors

Jaccard Similarity can also be calculated for binary vectors containing only 0 and 1. For binary vectors, only positions containing at least one 1 contribute to the calculation.

Formula:

J(A, B) = \frac{M_{11}}{M_{01} + M_{10} + M_{11}}

where:

• M11 is the number of positions where both vectors contain 1.
• M10 is the number of positions where A contains 1 and B contains 0.
• M01 is the number of positions where A contains 0 and B contains 1.

Numerical Example

Consider the binary vectors:

A = [1, 1, 0, 1, 0, 1, 0]
B = [1, 0, 0, 1, 1, 1, 0]

Step by step calculations:

• M₁₁ = 3 (positions: 1st, 4th, 6th)
• M₁₀ = 1 (position: 2nd)
• M₀₁ = 1 (position: 5th)

J(A, B) = \frac{3}{3 + 1 + 1} = \frac{3}{5} = 0.6

Python Implementation with Visualization

import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import jaccard_score

A = np.array([1, 1, 0, 1, 0, 1, 0])
B = np.array([1, 0, 0, 1, 1, 1, 0])


similarity = jaccard_score(A, B)

print(f"Jaccard Similarity: {similarity:.2f}")

Output

Jaccard Similarity: 0.60

Calculate Using Scikit-learn

Scikit-learn library provides the jaccard_score() function for calculating jaccard similarity.

from sklearn.metrics import jaccard_score

y_true = [1, 1, 0, 1, 0, 1, 0]
y_pred = [1, 0, 0, 1, 1, 1, 0]

score = jaccard_score(y_true, y_pred)

print("Jaccard Similarity:", score)

Output

Jaccard Similarity: 0.6

Explanation:

• y_true contains the actual binary values.
• y_pred contains the corresponding predicted values.
• jaccard_score() compares both sequences.
• Resulting value represents their Jaccard Similarity.

Jaccard Similarity for Strings

It can also be used to compare strings by treating their words as sets.

A = set("Python is easy to learn".split())
B = set("Python is easy and powerful".split())

similarity = len(A & B) / len(A | B)

print("Jaccard Similarity:", similarity)

Jaccard Similarity: 0.42857142857142855

Handle Empty Sets

When both sets are empty, the Jaccard Similarity formula results in division by zero because their union has no elements.

def jaccard_similarity(A, B):
    union = A | B

    if not union:
        return 1.0

    return len(A & B) / len(union)


A = set()
B = set()

print(jaccard_similarity(A, B))

1.0

Explanation:

• Both of the sets are empty, so they contain no differences.
• Function returns 1.0 to represent identical sets.