. 2013 Mar 22;4:39. doi: 10.3389/fpls.2013.00039

Table 1.

Sequential binary partition (SBP) elaborated to compute balances between groups of nutrients as isometric log-ratios (ilr).

ilr	SBP contrasts						Balance designation	r^†	s^†	Ilr computation^‡
	N	P	K	Ca	Mg	F_v
ilr₁	+1	+1	+1	−1	−1	0	[N, P, K \| Ca,Mg]	3	2	$\sqrt{\frac{3 \times 2}{3 + 2}} ln \frac{g (c_{N} c_{P} c_{K})}{g (c_{c a} c_{M g})}$
ilr₂	+1	+1	−1	0	0	0	[N, P \| K]	2	1	$\sqrt{\frac{1 \times 2}{1 + 2}} ln \frac{g (c_{N} c_{P})}{g (c_{K})}$
ilr₃	+1	−1	0	0	0	0	[N \| P]	1	1	$\sqrt{\frac{1 \times 1}{1 + 1}} ln \frac{g (c_{N})}{g (c_{P})}$
ilr₄	0	0	0	+1	−1	0	[Ca \| Mg]	1	1	$\sqrt{\frac{1 \times 1}{1 + 1}} ln \frac{g (c_{c a})}{g (c_{M g})}$
Optional	1	1	1	1	1	−1	[N, P, K, Ca, Mg \| F_υ]	5	1	$\sqrt{\frac{5 \times 1}{5 + 1}} \ln \frac{g (c_{N} c_{P} c_{K} c_{c a} c_{M g})}{g (c_{F υ})}$

An optional balance is a computed as a contrast between the geometric mean of nutrient analyses and the filling value, computed by difference.

^†r and s are counts of +1 and −1 elements in the balance, respectively, ^‡g is geometric mean.