You might experience some issues with vanilla PCA on CLR coordinates. There are two major problems with compositional data:
- they are strictly non-negative
- they have a sum constraint
Various compositional transforms address one or both of these issues. In particular, CLR transforms your data by taking the log of the ratio between observed frequencies ${\bf x}$ and their geometric mean $G({\bf x})$, i.e.
$$
\hat{\bf{x}} = \left \{ \log \left (\frac{{x}_{1}}{G({\bf x})} \right), \dots, \log \left (\frac{{x}_{n}}{G({\bf x})} \right) \right \} =
\left\{ \log ({x}_{1}) - \log( G({\bf x}) ) , \dots ,\log({x}_{n}) - \log( G({\bf x})) \right\}
$$
Now, consider that
$$
\log (G({\bf x} ))=\log \left( \exp \left[ \frac { 1 }{ n } \sum _{ i=1 }^{ n }{ \log ({ x }_{ i }) } \right] \right) = \mathop{\mathbb{E}}\left[ \log({\bf x}) \right]
$$
This effectively means that
$$
\sum{\hat{{\bf x}}} = \sum{ \left [ \log({\bf x}) - \mathop{\mathbb{E}}\left[ \log({\bf x}) \right] \right ]} = 0
$$
In other words CLR removes the value-range restriction (which is good for some applications), but does not remove the sum constraint, resulting in a singular covariance matrix, which effectively breaks (M)ANOVA/linear regression/... and makes PCA sensitive to outliers (because robust covariance estimation requires a full-rank matrix). As far as I know, of all compositional transforms only ILR addresses both issues without any major underlying assumptions. The situation is a bit more complicated, though. SVD of CLR coordinates gives you an orthogonal basis in the ILR space (ILR coordinates span a hyperplane in CLR), so your variance estimations will not differ between ILR and CLR (that is of course obvious, because both ILR and CLR are isometries on the simplex). There are, however, methods for robust covariance estimation on ILR coordinates [2].
Update I
Just to illustrate that CLR is not valid for correlation and location-dependant methods. Let's assume we sample a community of three linearly independent normally distributed components 100 times. For the sake of simplicity, let all components have equal expectations (100) and variances (100):
In [1]: import numpy as np
In [2]: from scipy.stats import linregress
In [3]: from scipy.stats.mstats import gmean
In [4]: def clr(x):
...: return np.log(x) - np.log(gmean(x))
...:
In [5]: nsamples = 100
In [6]: samples = np.random.multivariate_normal(
...: mean=[100]*3, cov=np.eye(3)*100, size=nsamples
...: ).T
In [7]: transformed = clr(samples)
In [8]: np.corrcoef(transformed)
Out[8]:
array([[ 1. , -0.59365113, -0.49087714],
[-0.59365113, 1. , -0.40968767],
[-0.49087714, -0.40968767, 1. ]])
In [9]: linregress(transformed[0], transformed[1])
Out[9]: LinregressResult(
...: slope=-0.5670, intercept=-0.0027, rvalue=-0.5936,
...: pvalue=7.5398e-11, stderr=0.0776
...: )
Update II
Considering the responses I've received, I find it necessary to point out that at no point in my answer I've said that PCA doesn't work on CLR-transformed data. I've stated that CLR can break PCA in subtle ways, which might not be important for dimensionality reduction, but is important for exploratory data analysis. The paper cited by @Archie covers microbial ecology. In that field of computational biology PCA or PCoA on various distance matrices are used to explore sources of variation in the data. My answer should only be considered in this context. Moreover, this is highlighted in the paper itself:
... The compositional biplot [note: referring to PCA] has several
advantages over the principal co-ordinate (PCoA) plots for β-diversity
analysis. The results obtained are very stable when the data are
subset (Bian et al., 2017), meaning that exploratory analysis is not
driven simply by the presence absence relationships in the data nor by
excessive sparsity (Wong et al., 2016; Morton et al., 2017).
Gloor et al., 2017
Update III
Additional references to published research (I thank @Nick Cox for the recommendation to add more references):
- Arguments against using CLR for PCA
- Arguments against using CLR for correlation-based methods
- Introduction to ILR
