This is a simple function of the constant-sum constraint.
Although the basis correlations approach zero, the constant-sum condition (i.e., "matrix closure," although this term is no longer favored) results in highly significant correlations among variables in the compositional data [ILLUSTRATION FOR FIGURE 1 OMITTED].
For example Butler (1981) used a variety of transformations and found the transformations do not correct for the constant-sum constraint even though the intervariable or interobservation relationships may be altered.
However, the constant-sum constraint would lead us to conclude that potentially meaningful patterns exist within the data if the underlying random nature of the raw data is unknown (i.e., nontrivial axes exist).
Although it is easy for researchers to rationalize such standardizations (e.g., only the patterns of relative abundance rather than total abundance are expressed in the compositional PCA), it is often unclear from subsequent analyses just how much of this result is meaningful pattern and how much of this is an artefact (i.e., due to the constant-sum constraint; see Fig.
However, from the simulation, we recognize that these patterns are due only to the constant-sum constraint of the compositional data and not to any meaningful relationships among the variables.