A log contrast of a positive vector 
is a linear
combination of natural logarithms of its
components of the form
 |
where the coefficients 
are real numbers satisfying
. The zero-sum condition
makes the value unchanged when every component is multiplied by the same positive
constant, since
 |
for every 
.
Equivalently, 
,
so the log contrast depends only on ratios among the components.
For example, choosing 
,
, and 
for 
gives 
.
For compositional data, a linear regression model with compositional independent
variables can be written 
, where 
. The zero-sum restriction makes the fitted value independent of the total to which the components
are normalized.
