隨著環境保育與生物多樣性的概念逐漸受到重視，許多量化多樣性的指標相繼被提出，透過這些指標可以了解多樣性隨著時間或空間的變化，除了僅考慮物種相對豐富度的物種多樣性和物種演化歷史的系統演化多樣性，考慮物種特徵並量化的多樣性近年來逐漸受到重視，稱為功能多樣性指標（functional diversity）。功能多樣性越高代表一地區的生態功能越健全，越不容易受到外在環境變化的影響。另一方面，大部分多樣性指標僅考慮單一群落或群落間的關係，局限於兩層次下的多樣性分析，然而隨著蒐集資料技術的進步，包含基因、個體、物種、群落、地域、地景或整個生態系的多層次結構資料越來越常見，勢必要以結構關係來衡量各層次的多樣性。
本文的研究主題分為兩部分，第一部分為Chao等人（2018）和Gregorius & Kosman（2017）的單一群落功能多樣性指標的比較，探討各指標的性質及優劣，合理指標必須滿足特定基本性質，結果顯示Chao等人（2018）的功能多樣性指標滿足較多常見的基本性質且適用範圍較無限制；第二部分為基於Chao等人（2018）功能多樣性指標，以廣義熵指標的形式及加法分解的概念，建構多層次的功能多樣性分解形式，包含多層次架構、指標分解形式及衡量各層次差異的標準化相異性指標等，本文分解形式可考量不同階次q>=0及距離差異門檻下各層次的alpha、beta、gamma指標。兩部分均透過電腦模擬驗證結果並以真實資料介紹其實際應用，此外，透過R語言及網路套件Shiny，將本文功能多層次內容撰寫成互動式網頁軟體hiDIP Online，方便不擅長程式語言的使用者進行分析。

As more people are paying attention to the issue of environmental protection and biodiversity conservation, an enormous number of diversity measures have been proposed to quantify diversity and assess diversity change across spatial or temporal scales. In addition to the species diversity and the phylogenetic diversity, functional diversity has been increasingly used to take the difference between species traits into account. The higher the functional diversity is, the more stable the ecosystem is, and thus the whole ecosystem is less affected by the change of environment. Most functional diversity measures were derived for a single community, or multiple communities under two-level structure. Nowadays, because of the rapid advances of collecting-data technique, multi-level structures are involved in most data sets, which includes various levels of genes, individuals, species, communities, regions, landscapes, and ecosystems. Therefore, hierarchical analysis across multiple levels is essential to diversity analysis.
This thesis includes two parts. The first part focuses on the comparisons between the single community functional diversity measures of Chao et al.（2018） and Gregorius & Kosman （2017）, and discusses the properties, advantages and disadvantages of each measure. The results show that the functional diversity measure of Chao et al.（2018）satisfies more good properties, and can be practically applied to a wide range of ecosystems. For the second part, based on Chao et al.’s（2018）functional diversity measure, hierarchical decomposition of functional diversity based on generalized entropies and addition decomposition is developed under a specified multi-level hierarchical structure. Standardized dissimilarity indices measuring the difference among aggregates (e.g., communities or subregions) at each level are also derived. The proposed decomposition quantifies alpha, beta and gamma diversity of each level under various diversity order q >= 0 and under varying thresholds of functional distinctiveness tau. For both parts, computer simulations are reported to show the performance of the proposed formulas; real data sets are used for illustrating practical application of the proposed hierarchical analysis. In addition, using R language and network package Shiny, an online software hiDIP Online (hierarchical Diversity Partitioning) is developed to facilitate the application of the proposed hierarchical functional analysis for users without R backgrounds.

第一章 緒論 1
第二章 符號介紹與相關文獻探討 4
2.1 模式假設與符號定義 4
2.1.1 抽樣方法與模式假設 4
2.1.2 符號介紹 4
2.2 物種多樣性相關文獻回顧 7
2.2.1 單一群落物種多樣性 7
2.2.2 多群落物種多樣性 10
2.3 功能多樣性相關文獻回顧 16
2.3.1 單一群落功能多樣性（Functional diversity） 16
2.3.2 Gregorius & Kosman功能多樣性概念 17
2.3.3 Gregorius & Kosman結構多樣性（Structural diversity）概念 19
2.3.4 Ricotta（2006）推廣至多層次分解架構 22
2.3.5 Pavoine et al.（2016）多層次分解架構 27
2.3.6 Smouse等人（2017）多層次分解架構 32
第三章 功能多樣性（Functional Diversity）相關探討 34
3.1功能多樣性指標族介紹 36
3.1.1 Chao等人（2018）功能多樣性指標族介紹 36
3.1.2 Gregorius & Kosman（2017）功能多樣性指標族介紹 38
3.2功能多樣性指標族性質比較 41
3.3 模擬研究與討論 48
3.3.1 模擬設定說明 48
3.3.2 模擬結果 49
3.4 實例分析 53
3.4.1 墾丁樣區資料 53
3.4.2 福山樣區資料 60
第四章 功能多層次分解架構 67
4.1 分解形式與架構建立 67
4.1.1 相對豐富度分解形式 67
4.1.2 絕對豐富度分解形式 73
4.2 結構與應用範圍 75
4.3 相異性指標轉換與定義 77
4.3.1 相對豐富度相異性指標 77
4.3.2 絕對豐富度相異性指標 79
4.4 與乘法分解的關係 80
4.4.1 相對豐富度 80
4.4.2 絕對豐富度 85
4.5 模擬研究與討論 90
4.5.1 模擬設定說明 90
4.5.2模擬結果 93
4.6 實例分析 101
4.6.1 羅亞爾河（Loire River）無脊椎動物資料 101
4.6.2 澳洲有袋類（marsupial）寬足袋鼩（Antechinus）基因資料 126
第五章 網頁開發與介紹 136
5.1 簡介 136
5.2 使用說明 136
5.3 輸出結果 139
第六章 結論與後續研究 146
參考文獻 148
附錄 151
附錄S1 功能多樣性指標性質證明 151
附錄S2 功能多層次 多樣性範圍證明 161
附錄S3 功能多樣性模擬 178
附錄S4 功能多層次架構模擬 184

[1] Champely, S., & Chessel, D. (2002). Measuring biological diversity using Euclidean metrics. Environmental and Ecological Statistics, 9, 167-177.
[2] Chao, A. (1984). Nonparametric estimation of the number of classes in a population. Scandinavian Journal of Statistics, 11, 265-270.
[3] Chao, A., & Jost. L. (2012). Coverage-based rarefaction and extrapolation: standardizing samples by completeness rather than size. Ecology, 93, 2533-2547.
[4] Chao, A., Chiu, C. H., & Jost, L. (2014). Unifying species diversity, phylogenetic diversity, functional diversity, and related similarity/differentiation measures through Hill numbers. Annual Reviews of Ecology, Evolution, and Systematics, 45, 297-324.
[5] Chao, A., Chiu, C. H., Villéger, S., Sun, I. F., Thorn, S., Lin, Y. C., Chiang, J. M., & Sherwin, W. B. (2018). An attribute-diversity approach to functional diversity, functional beta diversity, and related (dis)similarity measures. Under review.
[6] Chevene, F., Doléadec, S., & Chessel, D. (1994). A fuzzy coding approach for the analysis of long‐term ecological data. Freshwater Biology, 31, 295-309.
[7] Chiu, C. H., Jost, L., & Chao, A. (2014). Phylogenetic beta diversity, similarity, and differentiation measures based on Hill numbers. Ecological Monographs, 84, 21-44.
[8] Chiu, C. H., & Chao, A. (2014). Distance-based functional diversity measures and their decomposition: a framework based on Hill numbers. PLoS ONE, 9, e100014.
[9] Crist, T. O., Veech, J. A., Gering, J. C., & Summerville, K. S. (2003). Partitioning species diversity across landscapes and regions: a hierarchical analysis of α, β, and γ diversity. The American Naturalist, 162, 734-743.
[10] Crist, T. O., & Veech, J. A. (2006). Additive partitioning of rarefaction curves and species-area relationships: unifying α‐, β‐and γ‐diversity with sample size and habitat area. Ecology, 9, 923-932.
[11] Gregorius, H. R. (2014). Partitioning of diversity: the ‘within communities’ component. Web Ecology, 14, 51-60.
[12] Gregorius, H. R., & Kosman, E. (2017). On the notion of dispersion: from dispersion to diversity. Methods in Ecology and Evolution, 8, 278-287.
[13] Gregorius, H. R., & Kosman, E. (2018). Structural type diversity: measuring structuredness of communities by type diversity. Theoretical Ecology, 1-12.
[14] Horn, H. S. (1966). Measurement of ‘overlap’ in comparative ecological studies. The American Naturalist, 100, 419-424.
[15] 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.
[16] Jost, L. (2007). Partioning diversity into independent alpha and beta components. Ecology, 88, 2427-2439.
[17] Jost, L., Chao, A., & Chazdon, R. L. (2011). Compositional similarity and β (beta) diversity. Biological Diversity: frontiers in measurement and assessment, Magurran, A. E., & McGill, B. J. (eds). Oxford University Press, New York, 66-87.
[18] MacArthur, R., Recher, H., & Cody, M. (1966). On the relation between habitat selection and species diversity. The American Naturalist, 100, 319-332.
[19] Morisita, M. (1959). Measuring of interspecific association and similarity between communities. Memoirs of Faculty of Science. Kyushu University, Series E, 3, 65-80.
[20] Pavoine, S., & Dolédec, S. (2005). The apportionment of quadratic entropy: a useful alternative for partitioning diversity in ecological data. Environmental and Ecological Statistics, 12, 125-138.
[21] Pavoine, S., & Ricotta, C. (2014). Functional and phylogenetic similarity among communities. Methods in Ecology and Evolution, 5, 666-675.
[22] 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.
[23] Rao, C. R. (1982). Diversity and dissimilarity coefficients: a unified approach. Theoretical Population Biology, 21, 24-43.
[24] Rao, C. R. (1982b). Diversity: Its measurement, decomposition, apportionment and analysis. Sankhyā: The Indian Journal of Statistics, Series A, 44, 1-22.
[25] Rao, C. R. (1984). Convexity properties of entropy functions and analysis of diversity. Inequalities in Statistics and Probability: Proceedings of the Symposium on Inequalities in Statistics and Probability, Tong, Y. L. (ed). Institute of Mathematical Statistics, Hayward, California, USA, 68-77.
[26] Ricotta, C. (2005). A note on functional diversity measures. Basic and Applied Ecology, 6, 479-486.
[27] Ricotta, C., & Szeidl, L. (2006). Towards a unifying approach to diversity measures: bridging the gap between the Shannon entropy and Rao's quadratic index. Theoretical Population Biology, 70, 237-243.
[28] Routledge, R. (1979). Diversity indices: which ones are admissible? Theoretical Population Biology, 76, 503-515.
[29] Smouse, P. E., Banks, S. C., & Peakall, R. (2017). Converting quadratic entropy to diversity: Both animals and alleles are diverse, but some are more diverse than others. PLoS ONE, 12, e0185499.
[30] Wagner, H. H., Wildi, O., & Ewald, K. C. (2000). Additive partitioning of plant species diversity in an agricultural mosaic landscape. Landscape Ecology, 15, 219-227.
[31] Walker, B., Kinzig, A., & Langridge, J. (1999). Plant attribute diversity, resilience, and ecosystem function: The nature and significance of dominant and minor species. Ecosystems, 2, 95-113.
[32] Whittaker, R. H. (1960). Vegetation of the Siskiyou mountains, Oregon and California. Ecological Monographs, 30, 279-338.
[33] Whittaker, R. H. (1972). Evolution and measurement of species diversity. Taxon, 23, 213-251.
[34] 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.
[35] 伍淑惠, 許正一, 施郁庭, 孫義方, 王相華, 沈勇強 (2011). 墾丁喀斯勒森林永久樣區之樹種組成及生育地類型 林業叢刊第220號