Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and mathematical tractability. However, this assumption seems to be restrictive when dealing with counting data. To deal with this situation, we propose a random field with a Poisson marginal distribution by considering a sequence of independent copies of a random field with an exponential marginal distribution as 'inter-arrival times' in the counting renewal processes framework. Our proposal can be viewed as a spatial generalization of the Poisson process. Unlike the classical hierarchical Poisson Log-Gaussian model, our proposal generates a (non)-stationary random field that is mean square continuous and with Poisson marginal distributions. For the proposed Poisson spatial random field, analytic expressions for the covariance function and the bivariate distribution are provided. In an extensive simulation study, we investigate the weighted pairwise likelihood as a method for estimating the Poisson random field parameters. Finally, the effectiveness of our methodology is illustrated by an analysis of reindeer pellet-group survey data, where a zero-inflated version of the proposed model is compared with zero-inflated Poisson Log-Gaussian and Poisson Gaussian copula models. Supplementary materials for this article, include technical proofs and R code for reproducing the work, are available as an online supplement.
翻译:随机字段在空间和(或)时间上代表复杂的依赖结构的自然现象,是有用的数学工具。特别是,高森随机字段由于具有吸引力的特性和数学可移动性而经常使用。然而,这一假设在处理计算数据时似乎具有限制性。为处理这种情况,我们建议了一个随机字段,带有Poisson边际分布的Poisson边际分布,方法是在计算更新进程框架中考虑一个随机字段独立副本的序列,该随机字段以指数性边际分布为“抵达时间 ” 。我们的建议可以被视为普瓦松进程的空间概括化。与古典的Poisson Log-Gaussian 模式不同,我们的提案产生了一个(非)固定性随机字段,该字段以正方位持续和Poisson边际分布为平均值。对于拟议的Poisson空间随机字段,我们建议使用一个随机边际字段,用于计算“抵达时间”的边际边际分布。在广泛的模拟研究中,我们把加权配对的可能性作为估计Poisson随机字段参数的一种方法。最后,我们的方法的有效性通过对制模型进行分析,我们的方法通过对制模型进行分析,对制模型进行说明,对制模型对制版的模型,用于为制版的版本的版本的版本,用于为零制版版版版版版的版本的版本的版本的版本的版本的版本的版本,用于制版制版的版本的版本的版本,用于制制版制制制制制制制版的版本的版本,用于为制版的版本,用于制制版的版本,用于制制制版制版制版制制制制制制版制版制制制制版制版制版的版本,用于制制制制制版制版的版本,用于制制制制制版制制制制制制版制版制版的纸制的纸制制制制制制制制制制制制制制制版的纸制制制制制制制版制版制版制制制制版的纸制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制制