We build a general framework which establishes a one-to-one correspondence between species abundance distribution (SAD) and species accumulation curve (SAC). The appearance rates of the species and the appearance times of individuals of each species are modeled as Poisson processes. The number of species can be finite or infinite. Hill numbers are extended to the framework. We introduce a linear derivative ratio family of models, $\mathrm{LDR}_1$, of which the ratio of the first and the second derivatives of the expected SAC is a linear function. A D1/D2 plot is proposed to detect this linear pattern in the data. The SAD of $\mathrm{LDR}_1$ is the Engen's extended negative binomial distribution, and the SAC encompasses several popular parametric forms including the power law. Family $\mathrm{LDR}_1$ is extended in two ways: $\mathrm{LDR}_2$ which allows species with zero detection probability, and $\mathrm{RDR}_1$ where the derivative ratio is a rational function. We also consider the scenario where we record only a few leading appearance times of each species. We show how maximum likelihood inference can be performed when only the empirical SAC is observed, and elucidate its advantages over the traditional curve-fitting method.
翻译:我们建立了一个总框架,在物种丰量分布(SAD)和物种积累曲线(SAC)之间建立一对一的对应关系。物种的外观率和每个物种个人的外观时间以Poisson过程为模型。物种数量可以是有限或无限的。山数可以延伸至框架。我们引入了模型的线性衍生比率组合,$\mathrm{LDR ⁇ 1$,其中第一个和第二个预期SAC衍生物的比值是线性功能。建议绘制一个D1/D2图,以探测数据中的这一线性模式。$\mathrm{LDR ⁇ 1$是Engen的扩展负双向分布,而SAC包含几种流行的参数形式,包括权力法。我们以两种方式扩展了家族 $\mathrm{LDR ⁇ 1$的线性比率,允许检测不到检测概率的物种的第一和第二衍生物的比值为直线性功能。我们还考虑了我们只能对每个物种进行多少次前期实验性展示的概率的情景。