Jaccard Set Dissimilarity Function

A Jaccard Set Dissimilarity Function is the Inverse Function of the Jaccard Set Similarity Function.

• AKA: Jaccard Dissimilarity Function, Jaccard Dissimilarity.
• Context:
  • Input: (set [math]\displaystyle{ A }[/math], set B).
  • Output: a Rational Number in [0,1].
  • Definition: Jaccard(A,B) = Divide(Set Difference(A,B), Union(A,B)) = [math]\displaystyle{ J(A,B) = {{|A \setminus B|}\over{|A \cup B|}}. }[/math]
  • It can be converted to a Jaccard Dissimilarity Function (1-Jaccard).
• Example(s):
  • Jaccard(A,A) = 0
  • Jaccard(A,B) = 1 if A ∩ B = 0
• Counter-Example(s):
  • TF-IDF Distance Function.
  • Jaccard Similarity Function.
• See: String Distance Function.

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References

2011

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Jaccard_distance

... The Jaccard distance, which measures dissimilarity between sample sets, is complementary to the Jaccard coefficient and is obtained by subtracting the Jaccard coefficient from 1, or, equivalently, by dividing the difference of the sizes of the union and the intersection of two sets by the size of the union:

[math]\displaystyle{ J_{\delta}(A,B) = 1 - J(A,B) = { { |A \cup B| - |A \cap B| } \over |A \cup B| }. }[/math]

This distance is a proper metric (Lipkus, 1999).

1999

• Alan H. Lipkus. (1999). “A proof of the triangle inequality for the Tanimoto distance.” In: Journal Math Chem, 26(1-3)

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