Based on sampling data, we propose a rigorous standardization method to measure and compare beta diversity across datasets. Here beta diversity, which quantifies the extent of among-assemblage differentiation, relies on Whittaker's original multiplicative decomposition scheme, but we use Hill numbers for any diversity order q ≥ 0. Richness-based beta diversity (q = 0) quantifies the extent of species identity shift, whereas abundance-based (q > 0) beta diversity also quantifies the extent of difference among assemblages in species abundance. We adopt and define the assumptions of a statistical sampling model as the foundation for our approach, treating sampling data as a representative sample taken from an assemblage. The approach makes a clear distinction between the theoretical assemblage level (unknown properties/parameters of the assemblage) and the sampling data level (empirical/observed statistics computed from data). At the assemblage level, beta diversity for N assemblages reflects the interacting effect of the species abundance distribution and spatial/temporal aggregation of individuals in the assemblage. Under independent sampling, observed beta (= gamma/alpha) diversity depends not only on among-assemblage differentiation but also on sampling effort/completeness, which in turn induces dependence of beta on alpha and gamma diversity. How to remove the dependence of richness-based beta diversity on its gamma component (species pool) has been intensely debated. Our approach is to standardize gamma and alpha based on sample coverage (an objective measure of sample completeness). For a single assemblage, the iNEXT method was developed, through interpolation (rarefaction) and extrapolation with Hill numbers, to standardize samples by sampling effort/completeness. Here we adapt the iNEXT standardization to alpha and gamma diversity, that is, alpha and gamma diversity are both assessed at the same level of sample coverage, to formulate standardized, coverage-based beta diversity. This extension of iNEXT to beta diversity required the development of novel concepts and theories, including a formal proof and simulation-based demonstration that the resulting standardized beta diversity removes the dependence of beta diversity on both gamma and alpha values, and thus reflects the pure among-assemblage differentiation. The proposed standardization is illustrated with spatial, temporal, and spatiotemporal datasets, while the freeware iNEXT.beta3D facilitates all computations and graphics.

中文翻译：

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Hill 数框架下的 beta 多样性稀疏和外推：iNEXT.beta3D 标准化

基于采样数据，我们提出了一种严格的标准化方法来衡量和比较跨数据集的 beta 多样性。此处，β 多样性（量化组合间分化的程度）依赖于 Whittaker 的原始乘法分解方案，但我们对任何多样性阶 q ≥ 0 使用希尔数。基于丰富度的 β多样性 ( q  = 0) 量化物种的范围身份转变，而基于丰度 ( q  > 0) 的 beta 多样性也量化了物种丰度组合之间的差异程度。我们采用并定义统计抽样模型的假设作为我们方法的基础，将抽样数据视为从集合中获取的代表性样本。该方法明确区分了理论组合级别（组合的未知属性/参数）和采样数据级别（根据数据计算的经验/观察到的统计数据）。在组合水平上，N组合的β多样性反映了组合中物种丰度分布和个体时空聚集的相互作用。在独立采样下，观察到的 beta (= gamma/alpha) 多样性不仅取决于组合间的差异，还取决于采样工作/完整性，这反过来又导致 beta 对 alpha 和 gamma 多样性的依赖。如何消除基于丰富度的β多样性对其伽玛成分（物种池）的依赖一直存在激烈争论。我们的方法是根据样本覆盖率（样本完整性的客观衡量标准）对 gamma 和 alpha 进行标准化。对于单个组合，开发了 iNEXT 方法，通过插值（稀疏）和希尔数外推，通过采样努力/完整性来标准化样本。在这里，我们将 iNEXT 标准化应用于 alpha 和 gamma 多样性，即 alpha 和 gamma 多样性都在相同的样本覆盖水平上进行评估，以制定标准化的、基于覆盖范围的 beta 多样性。iNEXT 向 beta 多样性的扩展需要开发新的概念和理论，包括正式的证明和基于模拟的演示，即由此产生的标准化 beta 多样性消除了 beta 多样性对 gamma 和 alpha 值的依赖，从而反映了纯粹的- 组合分化。所提出的标准化通过空间、时间和时空数据集进行说明，而免费软件 iNEXT.beta3D 则有助于所有计算和图形。