Table 1 Metrics used to compute the “distance” between two atoms of a molecule
Metrics | Formulaa | Rangeb | Average | Range |
|---|---|---|---|---|
Minkowski (M1–M7) p = 0.25, 0.5, 1, 1.5, 2, 2.5, 3, and ∞ [where, when p = 1 it is the Manhattan, city-block or taxi distance (also known as Hamming distance between binary vectors) and p = 2 is Euclidean distance) | \(d_{XY} = \left( {\mathop \sum \limits_{j = 1}^{h} \left| {x_{j} - y_{j} } \right|^{p} } \right)^{{\frac{1}{p}}}\) | [0, ∞) | \(\bar{d} = \frac{{d_{XY} }}{{n^{1/p}}}\) | [0, ∞) |
Chebyshev/Lagrange (M8) (Minkowski formula when p = ∞) | \(d_{XY} = max\left\{ {\left| {x_{j} - y_{j} } \right|} \right\}\) | |||
Canberra (M10) | \(d_{XY} = \mathop \sum \limits_{j = 1}^{h} \frac{{\left| {x_{j} - y_{j} } \right| }}{{\left| {x_{j} } \right| + \left| {y_{j} } \right|}}\) | [0, n] | \(\bar{d} = \frac{{d_{XY} }}{n}\) | [0, 1] |
Lance–Williams/Bray–Curtis (M11) | \(d_{XY} = \frac{{\mathop \sum \nolimits_{j = 1}^{h} \left| {x_{j} - y_{j} } \right| }}{{\mathop \sum \nolimits_{j = 1}^{h} \left( {\left| {x_{j} } \right| + \left| {y_{j} } \right|} \right) }}\) | [0, 1] | \(\bar{d} = \frac{{d_{XY} }}{n}\) | \(\left[ {0,\frac{1}{n}} \right]\) |
Clark/coefficient of divergence (M12) | \(d_{XY} = \sqrt {\mathop \sum \limits_{j = 1}^{h} \left( {\frac{{x_{j} - y_{j} }}{{\left| {x_{j} } \right| + \left| {y_{j} } \right|}}} \right)^{2} }\) | [0, n] | \(\bar{d} = \frac{{d_{XY} }}{\sqrt n }\) | \(\left[ {0,\sqrt n } \right]\) |
Soergel (M13) | \(d_{XY} = \frac{1}{n}\mathop \sum \limits_{j = 1}^{h} \frac{{\left| {x_{j} - y_{j} } \right| }}{{max\left\{ {x_{j} ,y_{j} } \right\}}}\) | [0, 1] | \(\bar{d} = \frac{{d_{XY} }}{n}\) | \(\left[ {0,\frac{1}{n}} \right]\) |
Bhattacharyya (M14) | \(d_{XY} = \sqrt {\mathop \sum \limits_{j = 1}^{h} \left( {\sqrt {x_{j} } - \sqrt {y_{j} } } \right)^{2} }\) | [0, ∞) | \(\bar{d} = \frac{{d_{XY} }}{\sqrt n }\) | [0, ∞) |
Wave–Edges (M15) | \(d_{XY} = \mathop \sum \limits_{j = 1}^{h} \left( {1 - \frac{{min\left\{ {x_{j} ,y_{j} } \right\} }}{{max\left\{ {x_{j} ,y_{j} } \right\}}}} \right)\) | [0, n] | \(\bar{d} = \frac{{d_{XY} }}{n}\) | [0, 1] |
Angular separation/[1 − Cosine (Ochiai)] (M16) | d XY = 1−Cos XY where, \(Cos_{XY} = \frac{{\varvec{XY}}}{{\varvec{XY}}} = \frac{{\mathop \sum \nolimits_{j = 1}^{h} x_{j} y_{j} }}{{\sqrt {\mathop \sum \nolimits_{j = 1}^{h} x_{j}^{2} \mathop \sum \nolimits_{j = 1}^{h} y_{j}^{2} } }}\) | [0, 2] |