環境保育與生物多樣性為二十一世紀全球重大議題之一，為有助於了解生態的變化，許多量化多樣性的指標因應而生。傳統僅考慮物種相對豐富度的物種多樣性指標 (species diversity index) 已被廣泛使用，以及考慮物種演化歷史的系統演化多樣性指標 (phylogenetic diversity index) 也已發展成熟。然而，近年來利用物種特徵或性狀來量化多樣性成為熱門的研究主題，此稱為功能多樣性指標 (functional diversity index)。功能多樣性被認為是了解生態系統變遷與環境永續的關鍵，若功能多樣性高則表示此地區物種性狀差異大，且對於人為干擾有較大的穩定性。在另一方面，取樣問題也是生態學家們所關心的議題，由於抽樣限制，在現實中難以觀察到所有的物種，因此在估計生物多樣性的過程中，樣本涵蓋率 (sample coverage) 的概念用以量化樣本完整的程度扮演重要的角色。
因此本篇論文的研究主題可以分為兩個部分，第一部分為在個體抽樣下，利用統計推論方法估計單一群落的功能多樣性指標族，以及稀釋與預測曲線，稀釋與預測曲線為將各地區的數據標準化到相同取樣基準來進行客觀比較。第二部分為在個體抽樣下，利用統計推論方法將Good-Turing定義的樣本涵蓋率，推廣至包含考慮物種相對豐富度、考慮物種演化歷史以及考慮物種特徵的樣本涵蓋率指標族。
本文對於上述所提出的估計方法經由電腦模擬驗證其為可靠的估計，並與最大概似估計量比較，結果顯示本文建議之估計量明顯在偏誤、均方根誤差有較佳的表現。此外本文採用巴西雨林資料進行實例分析，並透過R語言將本文提及之多樣性指標撰寫成互動式網頁iNEXT3D-online。

Global climate warming and conservation of biodiversity are two major issues in the 21th century. In order to quantify the change of biodiversity, various diversity measures and their sampling properties have been discussed. The traditional species diversity measures only incorporate species richness and species relative abundances without considering the differences between species. The phylogenetic diversity measures which take into account species evolutionary history and their sampling properties have also been discussed in the literature. Recently, using species characteristics or traits to quantify community diversity has become a popular research topics. The associated measures are referred to as “functional diversity index”. However, the sampling issue of functional diversity measures has not been discussed in the literature.
This thesis includes two parts. The first part focuses on the estimation of the functional diversity profile on the basis of Hill numbers. Sample-sized- and coverage-based functional rarefaction and extrapolation methods are developed based on abundance data from each community. The second part of this study focuses on the extension of Good-Turing sample coverage to three sample coverage profiles and their estimators: species sample coverage profile, phylogenetic sample coverage profile, and functional sample coverage profile.
Simulation results are reported to compare the proposed estimators with the conventional empirical method; the new proposed estimator exhibits substantial improvement in bias and RMSE. The proposed estimators in this study are applied to the analysis of the Brazilian rainforest data. Relevent online software via R code (iNEXT3D) is developed to implement all diversity estimators.

[1] Allen, B., Kon, M. and Bar-Yam, Y. (2009). A new phylogenetic diversity measure generalizing the Shannon index and its application to phyllostomid bats. The American Naturalist, 174, 236-243.
[2] Basharin, G. P. (1959). On a statistical estimate for the entropy of a sequence of independent random variables. Theory of Probability & Its Applications, 4, 333-336.
[3] Chao, A. (1984). Nonparametric estimation of the number of classes in a population. Scandinavian Journal of Statistics, 11, 265-270.
[4] Chao, A. (2005). Species estimation and applications. Encyclopedia of Statistical Sciences, 12, 7907-7916.
[5] Chao, A., Chiu, C. H. and Jost, L. (2010). Phylogenetic diversity measures based on Hill numbers. Philosophical Transactions of the Royal Society B., 365, 3599-3609.
[6] Chao, A. and Jost, L. (2012). Coverage-based rarefaction: standardizing samples by completeness rather than by size. Ecology, 93, 2533-2547.
[7] Chao, A., Wang, Y. T. and Jost, L. (2013). Entropy and the species accumulation curve: a novel estimator of entropy via discovery rates of new species. Methods in Ecology and Evolution, 4, 1091-1110.
[8] Chiu, C. H., Wang, Y. T., Walther, B. A. and Chao, A. (2014). An improved non-parametric lower bound of species richness via the Good-Turing frequency formulas. Biometrics, 70, 671-682.
[9] Chao, A., Gotelli, N. G., Hsieh, T. C., Sander, E. L., Ma, K. H., Colwell, R. K. and Ellison, A. M. (2014). Rarefaction and extrapolation with Hill numbers: a framework for sampling and estimation in species biodiversity studies. Ecological Monographs, 84, 45-67.
[10] Chao, A. and Jost, L. (2015). Estimating diversity and entropy profiles via discovery rates of new species. Methods in Ecology and Evolution, 6, 873-882.
[11] Chao, A., Chiu, C. H., Hsieh, T. C., Davis, T., Nipperess, D. and Faith, D. (2015). Rarefaction and extrapolation of phylogenetic diversity. Methods in Ecology and Evolution, 6, 380-388.
[12] Chao, A. (2016). Quantifying sample completeness of a biological survey: a generalization of Good-Turing’s concept of sample coverage. Under review.
[13] Chao. A., Chiu, C. H., Colwell, R. K., Chazdon, R. L. and Gotelli, N. J. (2016). Deciphering the enigma of undetected biodiversity: The Good-Turing frequency formula and its generalizations. Under review.
[14] Chiu, C. H. and Chao, A. (2014). Distance-based functional diversity measures and their decomposition: a framework based on Hill numbers. PloS one, 9, e100014.
[15] Colwell, R. K., Chao, A., Gotelli, N. J., Lin, S. Y., Mao, C. X., Chazdon, R. L. and Longino, J. T. (2012). Models and estimators linking individual-based and sample-based rarefaction, extrapolation and comparison of assemblages. Journal of Plant Ecology, 5, 3–21.
[16] Efron, B. (1979). Bootstrap Methods: Another look at the jackknife. The Annals of Statistics, 1-26.
[17] Faith, D. P. (1992). Conservation evaluation and phylogenetic diversity. Biological conservation, 61, 1-10.
[18] Good, I. J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika, 40, 237-264.
[19] Hill, M. O. (1973). Diversity and evenness: A unifying notation and its consequences. Ecology, 54, 427-432.
[20] Magnago, L. F. S., Edwards, D. P., Edwards, F. A., Magrach, A., Martins, S. V. and Laurance, W. F. (2014). Functional attributes change but functional richness is unchanged after fragmentation of Brazilian Atlantic forests. Journal of ecology, 102, 475-485.
[21] Pielou, E. C. (1975). Ecology Diversity. J. Wiley and Sons, New York.
[22] Rao, C. R. (1982). Diversity and dissimilarity coefficients: a unified approach. Theoretical population biology, 21, 24-43.
[23] Shannon, C. E. (1948). The mathematical theory of communication. Bell System Technical Journal, 27, 379-423.
[24] Shen, T, J., Chao, A. and Lin, J. F. (2003). Predicting the number of new species in a further taxonomic sampling. Ecology, 84, 798-804.
[25] Simpson, E. H. (1949). Measurement of diversity. Nature, 163, 688-688.
[26] 趙蓮菊, 邱春火, 王怡婷, 謝宗震, 馬光輝 (2013). 仰觀宇宙之大, 俯察品類之盛：如何量化生物多樣性. Journal of the Chinese Statistical Association, 51, 8-53.
[27] 謝宗震 (民 102). 生物多樣性稀釋與預測 趙蓮菊指導 新竹市國立清華大學統計學研究所博士論文