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. 2020 Jul 14;11:3514. doi: 10.1038/s41467-020-17041-7

Table 1.

Summary of notations.

Notation	Description
i	Taxon index, i = 1, 2, …, m.
j	Group index, j = 1, 2, …, g.
k	Sample index, k = 1, 2, …, n_j.
θ_ij^a	Expected absolute abundance of ith taxon in a unit volume of ecosystem in the jth group.
A_ijk^b	Unobserved absolute abundance of ith taxon in a unit volume of ecosystem of kth sample in the jth group.
A._jk^b	Microbial load in a unit volume of ecosystem of kth sample in the jth group. $A_{\cdot j k} = \sum_{i = 1}^{m} A_{i j k}$ .
γ_ijk^b	Unobserved relative abundance of ith taxon in a unit volume of ecosystem of kth sample in the jth group. $γ_{i j k} = \frac{A_{i j k}}{A_{\cdot j k}}$ .
O_ijk^b	Observed absolute abundance of ith taxon in a random specimen taken from a unit volume of ecosystem of kth sample in the jth group.
O._jk^b	Library size of a random specimen taken from a unit volume of ecosystem of kth sample in the jth group. $O_{\cdot j k} = \sum_{i = 1}^{m} O_{i j k}$ .
r_ijk^b	Observed relative abundance of ith taxon in a random specimen taken from a unit volume of ecosystem of kth sample in the jth group. $r_{i j k} = \frac{O_{i j k}}{O_{\cdot j k}}$ .
c_jk^a	For kth sample from the jth group, c_jk represents the proportion of its ecosystem (unobserved absolute abundance) in a random sample (observed absolute abundance), thus $c_{j k} = \frac{E (O_{i j k} ∣ A_{i j k})}{A_{i j k}}$ . We shall refer to this constant as “sampling fraction”.
y_ijk^b	$\log (O_{i j k})$ .
μ_ij^a	$\log (θ_{i j})$ .
d_jk^a	$\log (c_{j k})$ .

^aParameter.

^bRandom variable.