Approximate Bayesian Computation and Model Assessment for Repulsive Spatial Point Processes

In many applications involving spatial point patterns, we find evidence of inhibition or repulsion. The most commonly used class of models for such settings are the Gibbs point processes. A recent alternative, at least to the statistical community, is the determinantal point process. Here, we examine model fitting and inference for both of these classes of processes in a Bayesian framework. While usual MCMC model fitting can be available, the algorithms are complex and are not always well behaved. We propose using approximate Bayesian computation (ABC) for such fitting. This approach becomes attractive because, though likelihoods are very challenging to work with for these processes, generation of realizations given parameter values is relatively straightforward. As a result, the ABC fitting approach is well-suited for these models. In addition, such simulation makes them well-suited for posterior predictive inference as well as for model assessment. We provide details for all of the above along with some simulation investigation and an illustrative analysis of a point pattern of tree data exhibiting repulsion. R code and datasets are included in the supplementary material.

在诸多涉及空间点模式的应用场景中，我们常能观测到抑制或排斥效应的相关证据。针对这类场景，最常用的模型类别是吉布斯点过程（Gibbs point process）。而在统计学界中近年兴起的一种替代模型，则是行列式点过程（determinantal point process）。本文基于贝叶斯框架（Bayesian framework），对这两类点过程模型的拟合与推断方法展开研究。尽管常规的马尔可夫链蒙特卡洛（Markov Chain Monte Carlo, MCMC）模型拟合方法具备可行性，但此类算法复杂度较高，且稳定性欠佳。对此，我们提出采用近似贝叶斯计算（approximate Bayesian computation, ABC）开展模型拟合。该方法具备显著优势：尽管这类点过程的似然函数计算极具挑战性，但基于给定参数值生成点模式实现的过程却相对简便。因此，近似贝叶斯计算拟合方法十分适配这类模型。此外，这类模拟过程也使得该方法适用于后验预测推断与模型评估任务。针对上述所有内容，我们给出了详细推导过程，并辅以若干模拟实验与一例针对呈现排斥效应的树木点模式数据的实例分析。相关R语言代码与数据集已附于补充材料中。