Abstract
Biological diversity (biodiversity) is an essential concept and plays an important role in ecology, genetics and many other disciplines. Biodiversity generally refers to the variety and variability of life at the levels of genes, individuals, species, populations, communities, landscapes, etc, and therefore is inherently under a hierarchical structure. Nowadays, because of rapid advancement of technology, collecting data becomes more convenient and faster, most biological data are typically collected at various levels of multiple-level hierarchical structures, e.g., a simplest 3-level hierarchical structure is that an region includes several subregions and each subregion includes several communities. A unifying framework for the measurement of biodiversity across hierarchical levels is thus required. Based on two measures (Tsallis entropy and Hill number) and two types of decompositions (additive and multiplicative), this thesis presents a framework for the measurement of species alpha, beta and gamma entropies/diversities across hierarchical levels for both abundance data under individual sampling and incidence data under quadrat sampling. Standardized dissimilarity measures are also derived to quantify differences among multiple regions or communities. Statistical method is developed to estimate the hierarchical entropies/diversities and dissimilarity measures with estimated bootstrap variances. Simulation results are used to show that the proposed estimators outperform the maximum likelihood estimators in terms of bias and root mean squared error (RMSE). The unifying framework is applied to real data sets to illustrate the proposed estimators and interpret the numerical results. Furthermore, an online application for computing the proposed measures and estimators is developed using R language and Shiny package.