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  • On close inspection the pattern shows large areas of

    2020-01-20

    On close inspection, the pattern shows large areas of very low point intensity where hardly any trees can be found. Anecdotal knowledge reveals that these regions are covered by a swamp, where the tree species is known to be very unlikely to grow, independent of local soil covariates and topography. However, data on the exact extent of the swamp is not available. We initially fitted a log Gaussian Cox process to the data, with an intensity function as in (1) using covariates, and the estimated posterior mean of the model also predicts large regions of low intensity, as plotted in Fig. 1(c). However, when a LGCP model that ignores the presence of swamp is fitted to this pattern, the swamp is likely to act as a confounding factor and this is likely to impact on inference. Hence, any conclusions on habitat preferences of the species will be heavily biased. Covariates associated with the presence of the swamp may appear to have a significant correlation with the intensity of the tree growth, or important covariates might appear non-significant as they vary independently of the presence of the swamp. The approach we take here is designed to capture sharp discontinuities in the intensity surface that result from qualitative yet unavailable covariates or environmental conditions as the one seen in this example. Further examples where this approach could be important is ecological data with several distinct types of habitat, spatial regions with different treatment regimes in medical data, or finding regions of interest in biology (Hendrix et al., 2016). Specifically, we consider a Cox process model where the intensity surface is modeled using a Bayesian level set approach. The proposed model is an extension of the log-Gaussian Cox process with increased flexibility resulting from a random segmentation of the spatial region into classes. The intensity surfaces of the regions associated with the different no sodium salt receptor can be modeled separately of each other by log-Gaussian random fields with simple covariance structures, while still maintaining flexibility. We refer to the proposed model as the level set Cox process (LSCP). Finding sharp discontinuities in the intensity surface is similar to the level set inversion problem Santosa, 1996, Burger, 2001 where the main objective is to find interfaces between geometrical regions based on observed data. Level set inversion has been used extensively for segmentation Chung, 2005, Lorentzen et al., 2012, Scheuermann and Rosenhahn, 2009, for multiphase flow modeling Barman and Bolin, 2018, Desjardins and Pitsch, 2009, and for statistical modeling of porous materials (Mourzenko, 2001). The interfaces in the level set inversion approach are modeled as level sets of an unknown level set function. Higgs and Hoeting (2010) modeled spatially correlated categorical data using a Bayesian level set approach, where the level set function was modeled as a Gaussian random field. This probabilistic approach, which Iglesias et al. (2016) and Dunlop et al. (2016) extended to more general inverse problems, has the advantage that the level sets can be estimated through the posterior distribution of the level set function given the observed data. The LSCP model could be viewed as an extension of these approaches to point pattern data. The LSCP is, like the LGCP, a process defined on a continuous domain. In order to use the model in practical inference some finite dimensional approximations are required. We show that the classical lattice approximation that is often used for LGCP models converges, in total variation distance, to the continuous model as the grid gets finer also for the LSCP models. We further propose a computationally efficient Markov chain Monte-Carlo (MCMC) algorithm for Bayesian inference on the model parameters, based on preconditioned Crank–Nicholson Langevin proposals (Cotter et al., 2013). This paper is structured as follows. A detailed model description is given in Section 2. In Section 3, we derive the MCMC algorithm for Bayesian inference. Section 4 analyses the B. Pendula point pattern of rainforest trees with the new approach. Finally, Section 5 discusses the presented material and possible future extensions of it. The theoretical results and proofs are given in two appendices.