Log-Ratio Analysis Is a Limiting Case of Correspondence Analysis

It is common practice in compositional data analysis to perform the log-ratio transformation in order to preserve sub-compositional coherence in the analysis. Correspondence analysis is an alternative approach to analyzing ratio-scale data and is often contrasted with log-ratio analysis. It turns out that if one introduces a power transformation into the correspondence analysis algorithm, then the limit of the power-transformed correspondence analysis, as the power parameter tends to zero, is exactly the log-ratio analysis. Depending on how the power transformation is applied, we can obtain as limiting cases either Aitchison's unweighted log-ratio analysis or the weighted form called "spectral mapping". The upshot of this is that one can come as close as one likes to the log-ratio analysis, weighted or unweighted, using correspondence analysis.