Table I.

Metrics for comparison of metabolic profiles and their characteristics

Metrics	Characteristics	Characteristics
Minkowski distance	$d(X,Y)={(\sum _{i=1}^N{|x_i-y_i|}^m)}^{\frac{1}{m}}$	A general metric, here implemented with m = 3.
Euclidean distance	$d(X,Y)={(\sum _{i=1}^N{|x_i-y_i|}^2)}^{\frac{1}{2}}$	Commonly used; increases influences of errors from large components on distance to some extent
Manhattan distance	$d(X,Y)=(\sum _{i=1}^N|x_i-y_i|)$	Each component has the same influence on distance
Jeffreys & Matusita distance	$d(X,Y)={(\sum _{i=1}^N{|\sqrt{x_i}-\sqrt{y_i}|}^2)}^{\frac{1}{2}}$	Based on Euclidean distance; increases influences of errors from small components on distance to some extent
Canberra distance	$d(X,Y)=(\sum _{i=1}^N|\frac{x_i-y_i}{x_i+y_i}|)$	A metric considering relative magnitudes of errors in components
Relative distance	$d(X,Y)={(\sum _{i=1}^N{(\frac{x_i-y_i}{y_i})}^2)}^{\frac{1}{2}}$	Similar to Euclidean distance but uses relative distance instead
Cosine of angle	$\mathit{\text{similarity}}(X,Y)=\frac{\sum _{i=1}^N{x}_i•y_i}{{(\sum _{i=1}^N{x_i}^2•\sum _{i=1}^N{y_i}^2)}^{\frac{1}{2}}}$	A similarity metric using the cosine of the angle between two vectors; not affected by absolute values of components
Dice’s coefficient	$\mathit{\text{similarity}}(X,Y)=\frac{2•\sum _{i=1}^N{x}_i•y_i}{\sum _{i=1}^N{x_i}^2+\sum _{i=1}^N{y_i}^2}$	A similarity metric comparable to the cosine similarity, using arithmetic averages instead of geometric averages
Jaccard similarity coefficient	$\mathit{\text{similarity}}(X,Y)=\frac{\sum _{i=1}^N{x}_i•y_i}{\sum _{i=1}^N{x_i}^2+\sum _{i=1}^N{y_i}^2-\sum _{i=1}^N{x}_i•y_i}$	A similarity metric similar to general Dice’s similarity