Tversky index

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 Dice's coefficient and Tanimoto coefficient.

For sets X and Y the Tversky index is a number between 0 and 1 given by

$S(X, Y) = \frac{| X \cap Y |}{| X \cap Y | + \alpha | X - Y | + \beta | Y - X |}$,

Here, $X-Y$ denotes the relative complement of Y in X.

Further, $\alpha, \beta \ge 0$ are parameters of the Tversky index. Setting $\alpha = \beta = 1$ produces the Tanimoto coefficient; setting $\alpha = \beta = 0.5$ produces Dice's coefficient.

If we consider X to be the prototype and Y to be the variant, then $\alpha$ corresponds to the weight of the prototype and $\beta$ corresponds to the weight of the variant. Tversky measures with $\alpha + \beta = 1$ 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] .

$S(X,Y)=\frac{| X \cap Y |}{| X \cap Y |+\beta\left(\alpha a+(1-\alpha)b\right)}$,

$a=\min\left(|X-Y|,|Y-X|\right)$,

$b=\max\left(|X-Y|,|Y-X|\right)$,

This formulation also re-arranges parameters $\alpha$ and $\beta$. Thus, $\alpha$ controls the balance between $|X - Y|$ and $|Y - X|$ in the denominator. Similarly, $\beta$ controls the effect of the symmetric difference $|X\,\triangle\,Y\,|$ versus $| X \cap Y |$ in the denominator.

Notes

1. Tversky, Amos (1977). "Features of Similarity" (PDF). Psychological Reviews. 84 (4): 327–352.
2. 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.