生物多樣性在生態上的定義為所有生態系中生物體的多樣性及變異性，而在生態上，該如何量化生物多樣性一度成為最具爭議的議題，至今多數人終於達成了共識，利用Hill number來衡量物種多樣性。然而在過去大部分的研究中，我們主要基於Hill number探討兩層次區域的物種多樣性（如：一個子區域由數個群落組成），然而，一個完整的區域通常是由層層的架構所組成，如：一個三層級區域包含數個子區域，而每個子區域是由數個群落所組成，因此如何透過多層次架構下量化各層級內的多樣性及在不同層級間進行多樣性分解尤為重要。

本文透過現今常見的兩種乘法分解基於Hill指標族（分別由Routledge（1979）及Chiu等人（2014）提出）衡量多樣性，並考慮各物種間的演化歷史所建構的系統演化樹，建構本文多層次架構下各層級的系統演化多樣性 、 及 指標。此外，利用標準化轉換系統演化多樣性 指標，來衡量各層級內（子區域或群落）的差異程度，並以統計方法提出對上述多種多樣性測度合適的估計量及拔靴法標準差估計，透過電腦模擬發現，不論在偏誤或是均方根誤差上，都較優於傳統的最大概似估計法。最後利用台灣動態樣區林木資料，進行多層次架構多樣性分析與估計，來說明此方法在實際資料之運用，並利用過去文獻中的盧瓦爾河無脊椎動物資料及墨西哥蝙蝠資料將本文提出之方法與過去之方法進行比較。此外，將本文提出之方法及估計利用R語言及內建套件Shiny撰寫成線上應用軟體，hiDIP（hierarchical Diversity Partitioning），可透過此線上軟體計算本文提及之方法與估計。

Biodiversity is defined as the variety and variability of life among organisms in the ecosystem. How to qualify biodiversity had once been one of the most controversial issues in ecology. A consensus seems to have achieved that Hill numbers should be used to quantify species diversity for many people. However, most studies on Hill numbers were focused on two-level hierarchy（e.g., a region is composed of several communities）and often include multi-level hierarchical structure. For an example, a three-level hierarchy is that a region includes several subregions and each subregion includes several communities. Therefore, how to quantify diversity in a hierarchical structure and decompose diversity across different levels is urgently needed.
Based on Hill numbers for two types of multiplicative decompositions（proposed respectively by Routledge（1979）and Chiu et al.（2014））and a phylogenetic tree spanned by the evolutionary history of the focal species, this thesis proposes formulas for phylogenetic alpha, beta and gamma diversities for each level in a hierarchical structure. Standardized dissimilarity measures using transformations of phylogenetic beta diversities are also developed to measure the difference among aggregates（subregions or communities）at each level. Additionally, statistical estimators for various hierarchical diversity measures are presented and their variances are assessed by the bootstrapping method. Simulation results are reported to show that the proposed estimators have better performance than the traditional maximum likelihood estimators in terms of bias and root mean square error（RMSE）. Real woody plant data of two Taiwan dynamics plots are used for illustrating the application of the proposed hierarchical analyses and estimation. Also, we compare our framework with previous methods using macroinvertebrates and bat data available in the literature. In addition, an online application hiDIP（hierarchical Diversity Partitioning）for computing the proposed phylogenetic measures and estimators is developed using R language and Shiny package.

[1] Allen, B., Kon, M. & 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] Chao, A.（1984）Nonparametric estimation of the number of classes in a population. Scandinavian Journal of Statistics, 11, 265-270.
[3] Chao, A. & Shen, T.-J.（2003）Nonparametric Estimation of Shannon’s index of diversity when there are unseen species. Environmental and Ecological Statistics, 10, 429-443.
[4] Chao, A., Chiu C.-H. & Jost, L.（2010）Phylogenetic diversity measures based on Hill numbers. Philosophical Transactions of the Royal Society B., 365, 3599-3609.
[5] Chao, A. & Jost. L.（2012）Coverage-based rarefaction and extrapolation: standardizing samples by completeness rather than size. Ecology, 93, 2533-2547.
[6] Chao, A., Wang, Y. T. & 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.
[7] Chao, A., Gotelli, N. G., Hsieh, T. C., Sander, E. L., Ma, K. H., Colwell, R. K. & 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.
[8] Chao, A., Chiu, C.-H., Hsieh, T. C., Davis, T., Nipperess, D. & Faith, D.（2015）Rarefaction and extrapolation of phylogenetic diversity. Methods in Ecology and Evolution, 6, 380-388.
[9] Chao, A. & Jost, L.（2015）Estimating diversity and entropy profiles via discovery rates of new species. Methods in Ecology and Evolution, 6, 873-882.
[10] Chiu, C.-H., Jost, L. & Chao, A.（2014）Phylogenetic beta diversity, similarity, and differentiation measures based on Hill numbers. Ecological Monographs, 84, 21-44.
[11] Efron, B.（1983）Bootstrap Methods: Another Look at the Jackknife. The Annals of Statistics, 7,1-26.
[12] Hill, M.（1973）Diversity and evenness: a unifying notation and its consequences. Ecology, 54, 427-432.
[13] Ivol, J. M., Guinand, B., Richoux, P., & Tachet, H.（1997）Longitudinal changes in Trichoptera and Coleoptera assemblages and environmental conditions in the Loire River (France). Archiv für Hydrobiologie, 138, 525-557.
[14] Jost, L.（2006）Entropy and diversity. Oikos, 113, 363–375.
[15] Jost, L.（2007）Partitioning diversity into independent alpha and beta components. Ecology, 88, 2427–2439.
[16] MacArthur, R., Recher, H. & Cody, M.（1966）On the relation between habitat selection and species diversity. The American Naturalist, 100, 319-332.
[17] Medellín, R. A., Equihua, M. & Amin, M. A.（2000）Bat diversity and abundance as indicators of disturbance in Neotropical rainforests. Conservation Biology, 14, 1666-1675.
[18] Pavoine, S., Love, M. S. & Bonsall, M. B.（2009）Hierarchical partitioning of evolutionary and ecological patterns in the organization of phylogenetically structured species assemblages: application to rockfish（genus: Sebastes） in the Southern California Bight. Ecology Letters, 12, 898–908.
[19] Pavoine, S., Marcon, E. & Ricotta, C.（2016）‘Equivalent numbers’ for species, phylogenetic or functional diversity in a nested hierarchy of multiple scales. Methods in Ecology and Evolution, 7, 1152-1163
[20] Pielou, E. C.（1975）Ecological Diversity. New York: Wiley.
[21] Rao, C. R.（1982）Diversity and dissimilarity coefficients: A unified approach. Theoretical Population Biology, 21, 24–43.
[22] Shannon, C. E.（1948）The mathematical theory of communication. Bell System Technical Journal, 27, 379–423.
[23] Simpson, E. H.（1949）Measurement of diversity. Nature, 163, 688.
[24] Tsallis, C.（1988）Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52, 479–487.
[25] Whittaker, R.H.（1972）Evolution and measurement of species diversity. Taxon, 12, 213-251.
[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.