Tversky index

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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

,

Here, denotes the relative complement of Y in X.

Further, are parameters of the Tversky index. Setting produces the Tanimoto coefficient; setting produces Dice's coefficient.

If we consider X to be the prototype and Y to be the variant, then corresponds to the weight of the prototype and corresponds to the weight of the variant. Tversky measures with 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] .

,

,

,

This formulation also re-arranges parameters and . Thus, controls the balance between and in the denominator. Similarly, controls the effect of the symmetric difference versus in the denominator.

Notes[edit]

  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.