In a second paper we explore how copulas are used to model abundance data from patchy populations in the hypercube. A copula is a unique function that relates the joint distribution of two random variables to their marginal distributions. Even though the application of copulas to population modeling is relatively new, they appear to have great potential. We use the logit-normal copula (LN), which is a generalization of the logistic distribution, for modeling uniform abundance data. Non-uniform data are modeled by the logit-normal copula with Gaussian margins. For modeling abundance data with variation among observations, we introduce the Mixture of Logit-Normals (MLN) which is also a generalization of the Poisson distribution.
In this paper, we present a Bayesian modeling framework for joint estimation of abundance and detection for patchy populations when detection is imperfect. In the model, we assume a logistic detection model for abundance where detection is an increasing function of abundance and where the detection model is stochastic and spatially autocorrelated. We further assume a binomial sampling model for abundance, which is based on the assumption that detection is independent of abundance. We present a joint posterior distribution of population abundance and detection. Our approach is based on the use of a hinged latent variable to represent abundance, which enables the joint posterior to be factorized into a pair of conditional distributions. Our approach provides a computationally efficient alternative to capture-recapture models for estimating abundance and uses a latent variable to represent abundance that avoids over-fitting. We also provide a simulation study that assesses the accuracy of posterior inference, including mean square error and coverage of true parameter values. We use our approach to estimate the abundance of two species of voles (Microtus spp.) in Montana, USA. We find that the joint likelihood can be efficiently simulated and accurately approximated, which allow the inference to be carried out in a Bayesian framework. We also assess the frequentist properties of our approach when detection is imperfect under different patterns in abundance and detection, and for different design criteria. Our joint modeling approach provides efficient estimation of abundance for species or populations with patchy distribution, and can be readily extended to model other habitats and/or other species.
In the fourth paper, we study the problem of selecting a sequence of states. The Bayesian approach provides a natural way to construct a sequence of estimates of the size of the state space under study. One might think that the solution of this problem should simply be the sequence of the number of states of the smaller hypercubes to the power of the dimension. However, the obtained hypercubes are not well connected in the sense that they are not convex in the sense of [Frech
et al. (2007) Proc. Royal Soc. B: Biological Sciences, 274, 461-470]. Here, we consider the size of the state space to be a random quantity, and we construct a sequence of estimates of the density of this random quantity. For the particular case of the state space of a Markov chain, we show how to perform a Dirichlet process regression of the sizes of its boxes. Thus, the proposed approach can be used to build up a robust estimator of the state space of Markov chains. 827ec27edc