Editing Compositional data (section)

==Simplicial sample space==
In general, [[John Aitchison]] defined compositional data to be proportions of some whole in 1982.<ref>{{cite journal|last=Aitchison|first=John|title=The Statistical Analysis of Compositional Data|journal=Journal of the Royal Statistical Society. Series B (Methodological)|volume=44|issue=2|year=1982|pages=139–177|doi=10.1111/j.2517-6161.1982.tb01195.x}}</ref> In particular, a compositional data point (or ''composition'' for short) can be represented by a real vector with positive components. The sample space of compositional data is a simplex:
:: <math> \mathcal{S}^D=\left\{\mathbf{x}=[x_1,x_2,\dots,x_D]\in\mathbb{R}^D \,\left|\, x_i>0,i=1,2,\dots,D; \sum_{i=1}^D x_i=\kappa \right. \right\}. \ </math>
[[File:Aitchison-simplex.jpg|thumb|An illustration of the Aitchison simplex.  Here, there are 3 parts, <math>x_1, x_2,  x_3</math> represent values of different proportions.  A, B, C, D and E are 5 different compositions within the simplex.  A, B and C are all equivalent and D and E are equivalent.]]

The only information is given by the ratios between components, so the information of a composition is preserved under multiplication by any positive constant. Therefore, the sample space of compositional data can always be assumed to be a standard simplex, i.e. <math>\kappa = 1</math>. In this context, normalization to the standard simplex is called '''closure''' and is denoted by <math>\scriptstyle\mathcal{C}[\,\cdot\,]</math>:

:: <math>\mathcal{C}[x_1,x_2,\dots,x_D]=\left[\frac{x_1}{\sum_{i=1}^D x_i},\frac{x_2}{\sum_{i=1}^D x_i}, \dots,\frac{x_D}{\sum_{i=1}^D x_i}\right],\ </math>

where ''D'' is the number of parts (components) and <math> [\cdot]</math> denotes a row vector.