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

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

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

Further, $\alpha ,\beta \geq 0$ 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 functions .

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

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

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

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.