Fig. 4.

a Direction of the vector corresponding to $Y_k$ when it lies in the same plane as $Y_i$ and $Y_j$ and the partial proportionality coefficient is one (green) and minus one (red), repectively. If $Y_k$ is parallel to the difference (log ratio) between $Y_i$ and $Y_j,$ the linear least squares predictors ${\widehat{Y}}_i$ and ${\widehat{Y}}_j$ point in opposite directions (green labeled vectors), and the residuals $Y_i-{\widehat{Y}}_i,$ $Y_j-{\widehat{Y}}_j$ coincide in the green, unlabeled vector. If $Y_k$ is parallel to the sum of $Y_i$ and $Y_j$, the linear least squares predictors ${\widehat{Y}}_i$ and ${\widehat{Y}}_j$ point in the same direction (red labeled vectors), and the residuals $Y_i-{\widehat{Y}}_i$, $Y_j-{\widehat{Y}}_j$ have the same length and point in opposite directions (unlabeled red vectors). b Log ratios between $X_i$ and $X_j$ and between $X_k$ and $X_d$ are parallel whenever $Y_i$ and $Y_j$ are partially proportional wrt. $Y_k$ (colour figure online)