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 (aka Jaccard 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 (aka Jaccard index). 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 | {displaystyle S(X,Y)={frac {|Xcap Y|}{|Xcap Y|+alpha |X-Y|+eta |Y-X|}}} , Here, X − Y {displaystyle X-Y} denotes the relative complement of Y in X. Further, α , β ≥ 0 {displaystyle alpha ,eta geq 0} are parameters of the Tversky index. Setting α = β = 1 {displaystyle alpha =eta =1} produces the Tanimoto coefficient; setting α = β = 0.5 {displaystyle alpha =eta =0.5} produces Dice's coefficient. If we consider X to be the prototype and Y to be the variant, then α {displaystyle alpha } corresponds to the weight of the prototype and β {displaystyle eta } corresponds to the weight of the variant. Tversky measures with α + β = 1 {displaystyle alpha +eta =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 ) {displaystyle S(X,Y)={frac {|Xcap Y|}{|Xcap Y|+eta left(alpha a+(1-alpha )b ight)}}} , a = min ( | X − Y | , | Y − X | ) {displaystyle a=min left(|X-Y|,|Y-X| ight)} ,