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". 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.