Table 2.

Formulas for the various similarity and distance metrics

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[\mathit{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	$D_{A,B}=\left[{\displaystyle \sum _{j=1}^n}\left|x_{jA}-x_{jB}\right|\right]/\left[{\displaystyle \sum _{j=1}^n}\mathit{max}\left(x_{jA},x_{jB}\right)\right]$	$D_{A,B}=1-\frac{c}{\left[a+b-c\right]}$
Substructure similarity	See Ref [24]
Superstructure similarity	See Ref [25]

^aS denotes similarities, while D denotes distances (according to the more commonly used formula for the given metric). Note that distances and similarities can be converted to one another using Equation 1. x _jA means the j-th feature of molecule A. a is the number of on bits in molecule A, b is number of on bits in molecule B, while c is the number of bits that are on in both molecules.

^bThe Soergel distance is the complement of the Tanimoto coefficient.