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 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 ) = | X ∩ Y | | X ∩ Y | + α | X − Y | + β | Y − X | ,

Here, X − Y denotes the relative complement of Y in X.

Further, α , β ≥ 0 are parameters of the Tversky index. Setting α = β = 1 produces the Tanimoto coefficient; setting α = β = 0.5 produces Dice's coefficient.

If we consider X to be the prototype and Y to be the variant, then α corresponds to the weight of the prototype and β corresponds to the weight of the variant. Tversky measures with α + β = 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 functions .

S ( X , Y ) = | X ∩ Y | | X ∩ Y | + β ( α a + ( 1 − α ) b ) ,

a = min ( | X − Y | , | Y − X | ) ,

b = max ( | X − Y | , | Y − X | ) ,

This formulation also re-arranges parameters α and β . Thus, α controls the balance between | X − Y | and | Y − X | in the denominator. Similarly, β controls the effect of the symmetric difference | X △ Y | versus | X ∩ Y | in the denominator.