We present the Probabilistic Context Neighborhood model designed for two-dimensional lattices as a variation of a Markov Random Field assuming discrete values. In this model, the neighborhood structure has a fixed geometry but a variable order, depending on the neighbors' values. Our model extends the Probabilistic Context Tree model, originally applicable to one-dimensional space. It retains advantageous properties, such as representing the dependence neighborhood structure as a graph in a tree format, facilitating an understanding of model complexity. Furthermore, we adapt the algorithm used to estimate the Probabilistic Context Tree to estimate the parameters of the proposed model. We illustrate the accuracy of our estimation methodology through simulation studies. Additionally, we apply the Probabilistic Context Neighborhood model to spatial real-world data, showcasing its practical utility.

翻译：我们提出了一种面向二维格点的概率性上下文邻域模型，作为假设离散值的马尔可夫随机场的一种变体。该模型的邻域结构具有固定几何形状但阶数可变，具体取决于邻域值。本模型扩展了最初适用于一维空间的概率性上下文树模型，保留了其优良特性，例如将依赖邻域结构表示为树形图，便于理解模型复杂度。此外，我们改进了用于估计概率性上下文树的算法，以适配所提模型的参数估计。通过模拟研究验证了估计方法的准确性，并将概率性上下文邻域模型应用于真实空间数据，展示了其实际应用价值。