Quantifying and assessing changes in biological diversity are central aspects of many ecological studies, yet accurate methods of estimating biological diversity from sampling data have been elusive. Hill numbers, or the effective number of species, are increasingly used to characterize the taxonomic, phylogenetic, or functional diversity of an assemblage. However, empirical estimates of Hill numbers, including species richness, tend to be an increasing function of sampling effort and, thus, tend to increase with sample completeness. Integrated curves based on sampling theory that smoothly link rarefaction (interpolation) and prediction (extrapolation) standardize samples on the basis of sample size or sample completeness and facilitate the comparison of biodiversity data. Here we extended previous rarefaction and extrapolation models for species richness (Hill number qD, where q = 0) to measures of taxon diversity incorporating relative abundance (i.e., for any Hill number qD, q > 0) and present a unified approach for both individual‐based (abundance) data and sample‐based (incidence) data. Using this unified sampling framework, we derive both theoretical formulas and analytic estimators for seamless rarefaction and extrapolation based on Hill numbers. Detailed examples are provided for the first three Hill numbers: q = 0 (species richness), q = 1 (the exponential of Shannon's entropy index), and q = 2 (the inverse of Simpson's concentration index). We developed a bootstrap method for constructing confidence intervals around Hill numbers, facilitating the comparison of multiple assemblages of both rarefied and extrapolated samples. The proposed estimators are accurate for both rarefaction and short‐range extrapolation. For long‐range extrapolation, the performance of the estimators depends on both the value of q and on the extrapolation range. We tested our methods on simulated data generated from species abundance models and on data from large species inventories. We also illustrate the formulas and estimators using empirical data sets from biodiversity surveys of temperate forest spiders and tropical ants.

Summary1. Hill numbers (or the effective number of species) have been increasingly used to quantify the species/taxonomic diversity of an assemblage. The sample-size-and coverage-based integrations of rarefaction (interpolation) and extrapolation (prediction) of Hill numbers represent a unified standardization method for quantifying and comparing species diversity across multiple assemblages. 2. We briefly review the conceptual background of Hill numbers along with two approaches to standardization. We present an R package iNEXT (iNterpolation/EXTrapolation) which provides simple functions to compute and plot the seamless rarefaction and extrapolation sampling curves for the three most widely used members of the Hill number family (species richness, Shannon diversity and Simpson diversity). Two types of biodiversity data are allowed: individual-based abundance data and sampling-unit-based incidence data. 3. Several applications of the iNEXT packages are reviewed: (i) Non-asymptotic analysis: comparison of diversity estimates for equally large or equally complete samples. (ii) Asymptotic analysis: comparison of estimated asymptotic or true diversities. (iii) Assessment of sample completeness (sample coverage) across multiple samples. (iv) Comparison of estimated point diversities for a specified sample size or a specified level of sample coverage. 4. Two examples are demonstrated, using the data (one for abundance data and the other for incidence data) included in the package, to illustrate all R functions and graphical displays.

There have been intense debates about the decomposition of regional diversity (gamma) into its within-community component (alpha) and between-community component (beta). Although a recent Ecology Forum achieved consensus in the use of "numbers equivalents" (Hill numbers) as the proper choice of diversity measure, three related major issues were still left unresolved. (1) What is the precise meaning of the "independence" or "statistical independence" of alpha diversity and beta diversity? (2) Which partitioning (additive vs. multiplicative) should be used for a given application? (3) What is the proper formula for alpha diversity, as there are two formulas in the literature? This paper proposes a possible resolution to each of these issues. For the first issue, we clarify the definitions of "independence" and "statistical independence" from two perspectives so that confusion about this issue can be cleared up. We also discuss the causes of dependence, so that the dependence relationship between any two diversity components in both partitioning schemes can be rigorously justified by theory and also intuitively understood by simulation. For the second issue, both multiplicative and additive beta diversities based on Hill numbers are useful measures and quantify different aspects of communities. However, neither can be directly applied to compare relative compositional similarity or differentiation across multiple regions with different numbers of communities because multiplicative beta diversity depends on the number of communities, and additive beta diversity additionally depends on alpha (equivalently, on gamma). Such dependences should be removed. We propose transformations to remove these dependences, and we show that the transformed multiplicative beta and additive beta both lead to the same classes of measures, which are always in a range of [0, 1] and thus can be used to compare relative similarity or differentiation among communities across multiple regions. These similarity measures include multiple-community generalizations of the Sørenson, Jaccard, Horn, and Morisita-Horn measures. For the third issue, we present some observations including a finding about which alpha formula produces independent alpha and beta components. These may help to resolve the choice of a proper formula for alpha diversity. Some related issues are also briefly discussed.

Context. Recent years have seen building evidence that planet formation starts early, in the first ~0.5 Myr. Studying the dust masses available in young disks enables us to understand the origin of planetary systems given that mature disks are lacking the solid material necessary to reproduce the observed exoplanetary systems, especially the massive ones. Aims. We aim to determine if disks in the embedded stage of star formation contain enough dust to explain the solid content of the most massive exoplanets. Methods. We use Atacama Large Millimeter/submillimeter Array (ALMA) Band 6 (1.1–1.3 mm) continuum observations of embedded disks in the Perseus star-forming region together with Very Large Array (VLA) Ka-band (9 mm) data to provide a robust estimate of dust disk masses from the flux densities measured in the image plane. Results. We find a strong linear correlation between the ALMA and VLA fluxes, demonstrating that emission at both wavelengths is dominated by dust emission. For a subsample of optically thin sources, we find a median spectral index of 2.5 from which we derive the dust opacity index β = 0.5, suggesting significant dust growth. Comparison with ALMA surveys of Orion shows that the Class I dust disk mass distribution between the two regions is similar, but that the Class 0 disks are more massive in Perseus than those in Orion. Using the DIANA opacity model including large grains, with a dust opacity value of κ9 mm = 0.28 cm2 g−1, the median dust masses of the embedded disks in Perseus are 158 M⊕ for Class 0 and 52 M⊕ for Class I from the VLA fluxes. The lower limits on the median masses from ALMA fluxes are 47 M⊕ and 12 M⊕ for Class 0 and Class I, respectively, obtained using the maximum dust opacity value κ1.3 mm = 2.3 cm2 g−1. The dust masses of young Class 0 and I disks are larger by at least a factor of ten and three, respectively, compared with dust masses inferred for Class II disks in Lupus and other regions. Conclusions. The dust masses of Class 0 and I disks in Perseus derived from the VLA data are high enough to produce the observed exoplanet systems with efficiencies acceptable by planet formation models: the solid content in observed giant exoplanets can be explained if planet formation starts in Class 0 phase with an efficiency of ~15%. A higher efficiency of ~30% is necessary if the planet formation is set to start in Class I disks.

Based on a sample of individuals, we focus on inferring the vector of species relative abundance of an entire assemblage and propose a novel estimator of the complete species-rank abundance distribution (RAD). Nearly all previous estimators of the RAD use the conventional "plug-in" estimator Pi (sample relative abundance) of the true relative abundance pi of species i. Because most biodiversity samples are incomplete, the plug-in estimators are applied only to the subset of species that are detected in the sample. Using the concept of sample coverage and its generalization, we propose a new statistical framework to estimate the complete RAD by separately adjusting the sample relative abundances for the set of species detected in the sample and estimating the relative abundances for the set of species undetected in the sample but inferred to be present in the assemblage. We first show that P, is a positively biased estimator of pi for species detected in the sample, and that the degree of bias increases with increasing relative rarity of each species. We next derive a method to adjust the sample relative abundance to reduce the positive bias inherent in j. The adjustment method provides a nonparametric resolution to the longstanding challenge of characterizing the relationship between the true relative abundance in the entire assemblage and the observed relative abundance in a sample. Finally, we propose a method to estimate the true relative abundances of the undetected species based on a lower bound of the number of undetected species. We then combine the adjusted RAD for the detected species and the estimated RAD for the undetected species to obtain the complete RAD estimator. Simulation results show that the proposed RAD curve can unveil the true RAD and is more accurate than the empirical RAD. We also extend our method to incidence data. Our formulas and estimators are illustrated using empirical data sets from surveys of forest spiders (for abundance data) and soil ciliates (for incidence data). The proposed RAD estimator is also applicable to estimating various diversity measures and should be widely useful to analyses of biodiversity and community structure.