Aitchison geometry is the geometry of compositional data, whose coordinates are positive and are
considered only through their relative proportions. For a fixed total 
, its sample space
is the open simplex
![S^d=[{x=(x_1,...,x_d):x_i>0, sum_(i=1)^dx_i=kappa]}.](/images/equations/AitchisonGeometry/NumberedEquation1.svg) |
(1)
|
If 
denotes closure to this simplex, the analogs of vector
addition and scalar multiplication are
the perturbation and powering operations
 |  |  |
(2)
|
 |  |  |
(3)
|
The centered logratio coordinates 
, where 
is the geometric mean
of the coordinates, identify 
with a 
-dimensional vector space.
Each centered logratio coordinate is a log contrast.
In particular, the Aitchison distance is the distance induced by the Euclidean
metric between centered logratio coordinate vectors.
