Tversky Index

The Tversky index, named after Amos Tversky, 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 Jaccard index.

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

$S(X, Y) = \frac$

Here, $X \setminus Y$ 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 Jaccard index; setting $\alpha = \beta = 0.5$ produces the Sørensen–Dice 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.

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 functionsJimenez, 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.
.

$S(X,Y)=\frac$

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

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

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

Notes

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Eponymous indices
Index numbers
Measure theory
Similarity measures
Asymmetry