Figure 1.

Geometric interpretation of subtracting phylogenetic background. Panel A corresponds to Eq(2), where a phylogenetic vector |v> is subtracted by the background vector |u_16s>, the resulting vector |ε> is guaranteed to be orthogonal to |u_16s>. Panel B corresponds to Eq(2'), where a phylogenetic vector |p> is subtracted by the background vector |r>, the resulting vector |p'> may still have residual components along the orientation of |r>. Panel C shows that the resulting vector |p'> may become as nearly orthogonal to |r> when the length of the vector |r> is properly rescaled.