Bayesian Approach Using Latent Variable for Zero Truncated Poisson Distribution: Application for Species-Area Relationship

Abstract

In ecology, understanding the species-area relationship (SARs) is extremely important to determine species diversity. SARs are fundamental to evaluate the impact in this diversity due to destruction of natural habitats, to create biodiversity maps and to determine the minimum area to preserve. In this study, the number of species is observed in different area sizes. These studies are referred in the literature through nonlinear models without assuming any distribution of the data. In this situation, it only makes sense to consider areas in which the number of species is greater than zero. As the dependent variable is a count data, we assume that this variable comes from a known distribution for discrete positive data. In this paper, we used the zero truncated poisson distribution (ZTP) to represent the probability distribution of the random variable "species diversity" and we considered some nonlinear models to describe the relationship between species diversity and habitat area. Among the proposed models in literature, we considered the Arrhenius power function, Persistence function (P1 e P2), Negative Exponential and Chapman-Richards to describe the abundance of species. In this paper, we take a Bayesian approach to fit models. With the purpose of obtaining conditional distributions, we propose the use of latent variables to implement the Gibbs Sampler. In order to progress using the best possible models for data, a comparison of performance between models referred in this paper will be verified through the criteria Extended Akaike Information Criterion (EAIC), Extended Bayesian Information Criterion (EBIC), Deviance Information Criterion (DIC) and Conditional Predictive Ordinate Criterion (CPO). In addition to selecting the best model, it will also assist to define the best selection criterion.