利用 Good-Turing 頻度方程式之 Bray-Curtis 相異性指標估計方法

Abstract

Beta多樣性指標主要是用來量化不同群落間物種組成的差異，文獻中已發展出眾多的Beta多樣性定量指標，其中Bray-Curtis相異性指標是最廣為引用的指標之一，然而樣本中觀測的Bray-Curtis相異性指標常呈現嚴重偏誤，因此本文將透過無母數統計方法推導出Bray-Curtis相異性指標估計式以修正觀測Bray-Curtis相異性指標的偏誤。 首先，將Bray-Curtis相異性指標分為三個部份：群落一特有種的相對豐富度總和，群落二特有種的相對豐富度總和，以及兩群落共有種相對豐富度的絕對差異總和，根據Good-Turing frequency formula 理論分別估計上述三個部分，進而推導Bray-Curtis相異性指標估計式，本文同時根據Horvitz-Thompson理論的推導Bray-Curtis相異性指標估計式。利用電腦模擬和實例分析進行上述三種估計方法的評估，並以拔靴法(Bootstrap Method)進行估計式標準差之估計，及建構對應之95%信賴區間，並評估各方法之信賴區間涵蓋到真值的比例。 模擬結果呈現，Good-Turing估計法在電腦模擬以及實例分析下，都可修正觀測指標值的偏誤；從樣本均方根誤差來看，本文提出的Good-Turing估計法總體而言較其他方法更為穩定；並且相較其他兩種估計法，Good-Turing 估計值的95%信賴區間可提供較精確的真值涵蓋率，因此總結，Good-Turing估計方法能夠有效地修正觀測指標值的偏誤，對於Bray-Curtis相異性指標的估計提供一個可行的方法。

Beta diversity is an important biodiversity component, measuring compositional change between plots and assemblages. Beta diversity can be measured in different ways. Among these, Bray-Curtis dissimilarity index is one of the most common method to measure the difference between sites. However, it appears to be significantly bias if we use Empirical abundance data to estimate the Bray-Curtis dissimilarity index. This paper put emphasize on developing a nonparametric estimation of Bray-Curtis dissimilarity index which can adjust the bias considering there might be unseen species in the sample. First, the Bray-Curtis index is divided into three components: (1) the sum of relative abundance of species abundance in community 1, (2) the sum of relative abundance of species abundance in community 2, (3) and the sum of absolute differences between the relative abundance of species shared by two communities. The estimation of the Bray-Curtis dissimilarity index was based on the theory of Good-Turing frequency formula. We also derive the Bray-Curtis index estimation formula based on the Horvitz-Thompson theory. In this paper, computer simulations and case studies were used to evaluate the estimation methods mention in this article. Then we use Bootstrap method to estimate the standard deviations, and construct the corresponding 95% confidence intervals to evaluate the average probabilities that the confidence interval of each method covers the true value. Judging by the average estimates, the Good-Turing estimation method can perform a better result than other methods, and was found to be more stable than the other methods in terms of the root mean square error. In addition, the 95% CI built by the Good-Turing estimation method provides a better chance to cover the true value.