The Tversky index, named after Amos Tversky,[1] is an asymmetric similarity measure on sets that compares a variant to a prototype. The Tversky index can be seen as a generalization of the Sørensen–Dice coefficient and the Tanimoto coefficient (aka Jaccard index). For sets X and Y the Tversky index is a number between 0 and 1 given by

[math]\displaystyle{ S(X, Y) = \frac{| X \cap Y |}{| X \cap Y | + \alpha | X \setminus Y | + \beta | Y \setminus X |} }[/math]

Here, [math]\displaystyle{ X \setminus Y }[/math] denotes the relative complement of Y in X.

Further, [math]\displaystyle{ \alpha, \beta \ge 0 }[/math] are parameters of the Tversky index. Setting [math]\displaystyle{ \alpha = \beta = 1 }[/math] produces the Tanimoto coefficient; setting [math]\displaystyle{ \alpha = \beta = 0.5 }[/math] produces the Sørensen–Dice coefficient.

If we consider X to be the prototype and Y to be the variant, then [math]\displaystyle{ \alpha }[/math] corresponds to the weight of the prototype and [math]\displaystyle{ \beta }[/math] corresponds to the weight of the variant. Tversky measures with [math]\displaystyle{ \alpha + \beta = 1 }[/math] are of special interest.[2]

Because of the inherent asymmetry, the Tversky index does not meet the criteria for a similarity metric. However, if symmetry is needed a variant of the original formulation has been proposed using max and min functions[3] .

[math]\displaystyle{ S(X,Y)=\frac{| X \cap Y |}{| X \cap Y |+\beta\left(\alpha a+(1-\alpha)b\right)} }[/math]

[math]\displaystyle{ a=\min\left(|X \setminus Y|,|Y \setminus X|\right) }[/math],

[math]\displaystyle{ b=\max\left(|X \setminus Y|,|Y \setminus X|\right) }[/math],

This formulation also re-arranges parameters [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math]. Thus, [math]\displaystyle{ \alpha }[/math] controls the balance between [math]\displaystyle{ |X \setminus Y| }[/math] and [math]\displaystyle{ |Y \setminus X| }[/math] in the denominator. Similarly, [math]\displaystyle{ \beta }[/math] controls the effect of the symmetric difference [math]\displaystyle{ |X\,\triangle\,Y\,| }[/math] versus [math]\displaystyle{ | X \cap Y | }[/math] in the denominator.

## Notes

1. ↑ Tversky, Amos (1977). "Features of Similarity". Psychological Review 84 (4): 327–352. doi:10.1037/0033-295x.84.4.327. http://www.cogsci.ucsd.edu/~coulson/203/tversky-features.pdf.
2. ↑ "Daylight Theory: Fingerprints". http://www.daylight.com/dayhtml/doc/theory/theory.finger.html.
3. ↑ Jimenez, S., Becerra, C., Gelbukh, A. SOFTCARDINALITY-CORE: Improving Text Overlap with Distributional Measures for Semantic Textual Similarity. Second Joint Conference on Lexical and Computational Semantics (*SEM), Volume 1: Proceedings of the Main Conference and the Shared Task: Semantic Textual Similarity, p.194-201, June 7–8, 2013, Atlanta, Georgia, USA.

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