Fig. 2.

a Two vectors of observations when their goodness-of-fit parameters are $r=1$ (same direction) and $\mathit{\beta }=1$ (same length), respectively. Their log-ratio variance is the squared length of the link connecting them (here, roughly of the same size in both cases). b The relative data log$X_i$, log$X_j$ (black vectors) transformed to (logged) absolute data $A_i$, $A_j$ by the normalization −logs (green vectors) and log-ratio transformed using the reference $X_d$ to $Y_i$, $Y_j$ (red vectors). Note that the log ratio between $X_i$ and $X_j$ (dashed line) remains the same under both transformations (colour figure online)