From: Why is Tanimoto index an appropriate choice for fingerprint-based similarity calculations?
Distance metric | Formula for continuous variables a | Formula for dichotomous variables a |
---|---|---|
Manhattan distance | \( {D}_{A,\ B}={\displaystyle \sum_{j=1}^n}\left|{x}_{jA}-{x}_{jB}\right| \) | D A,B = a + b − 2c |
Euclidean distance | \( {D}_{A,\ B}={\left[{\displaystyle \sum_{j=1}^n}{\left({x}_{jA}-{x}_{jB}\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \) | \( {D}_{A,B}={\left[a+b-2c\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \) |
Cosine coefficient | \( {S}_{A,B}=\left[{\displaystyle \sum_{j=1}^n}{x}_{jA}{x}_{jB}\right]/{\left[{\displaystyle \sum_{j=1}^n}{\left({x}_{jA}\right)}^2{\displaystyle \sum_{j=1}^n}{\left({x}_{jB}\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \) | \( {S}_{A,B}=\frac{c}{{\left[ ab\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} \) |
Dice coefficient | \( {S}_{A,B}=\left[2{\displaystyle \sum_{j=1}^n}{x}_{jA}{x}_{jB}\right]/\left[{\displaystyle \sum_{j=1}^n}{\left({x}_{jA}\right)}^2+{\displaystyle \sum_{j=1}^n}{\left({x}_{jB}\right)}^2\right] \) | S A,B = 2c/[a + b] |
Tanimoto coefficient | \( {S}_{A,B}=\frac{\left[{\displaystyle {\sum}_{j=1}^n}{x}_{jA}{x}_{jB}\right]}{\left[{\displaystyle {\sum}_{j=1}^n}{\left({x}_{jA}\right)}^2+{\displaystyle {\sum}_{j=1}^n}{\left({x}_{jB}\right)}^2-{\displaystyle {\sum}_{j=1}^n}{x}_{jA}{x}_{jB}\right]} \) | S A,B = c/[a + b − c] |
Soergel distanceb | \( {D}_{A,\ B}=\left[{\displaystyle \sum_{j=1}^n}\left|{x}_{jA}-{x}_{jB}\right|\right]/\left[{\displaystyle \sum_{j=1}^n} max\left({x}_{jA},{x}_{jB}\right)\right] \) | \( {D}_{A,B}=1-\frac{c}{\left[a+b-c\right]} \) |
Substructure similarity | See Ref [24] | |
Superstructure similarity | See Ref [25] |