Zeta diversity

The most basic measure of community diversity, alpha diversity (α-diversity), can be described as the average number of distinct taxonomic groups (e.g. unique genera or operational taxonomic unit) present, independent of their abundances, on a per sample basis. In the ζ-diversity framework this can be described as ζ₁, the number of unique taxa present in one sample.

Beta diversity (β-diversity) is a measure to allow for a comparison between the diversity of local communities (α-diversity). The greater the similarity in community composition between multiple communities, the lower the value of β-diversity for that set of communities. Using the number of distinct taxonomic groups per community as a measure of α-diversity one can then describe β-diversity between two communities in terms of distinct number taxonomic groups held in common between them. Given two communities, A and B, a measure β-diversity between both communities can be described in terms of their overlap A ∩ B (ζ₂) as well as the average number of unique taxonomic categories found in A and B (ζ₁). In the framework of ζ-diversity we can then describe the average β-diversity, as described by the Jaccard index, for a set of samples as ${\displaystyle {\frac {\zeta _{2}}{2\zeta _{1}-\zeta _{2}}}}$

The framework then for ζ-diversity can then be extended beyond measures of diversity in one (α-diversity), or between two communities (β-diversity), to describing diversity across sets of three or more communities. If ζ₁ describes the number of distinct taxa in community A, and ζ₂ describes the number of distinct taxa held in common between communities A and B, then ζ_n describes the number of distinct taxa held in common across n communities.