Statistical modeling of dependent directional data remains relatively underexplored, particularly in high-dimensional spatial settings. Existing approaches for spatial angular data primarily rely on wrapped Gaussian process (WGP) models, which provide a coherent framework for capturing spatial dependence on the circle. However, WGP-based methods become computationally challenging when the spatial domain is large, and observations are available at high resolution. This limitation is especially relevant in the analysis of large-scale geological and climate phenomena, such as tsunamis and hurricanes, where directional measurements (e.g., wave or wind directions) may be available over an entire ocean basin. To address these challenges, we propose a wrapped Gaussian Markov random field (WGMRF) model for large spatial directional datasets. By exploiting the sparse precision structure inherent in Gaussian Markov random fields, the proposed approach achieves substantial computational gains while preserving flexible spatial dependence on the circular scale. We discuss key properties of the model, including its identifiability and dependence characteristics. The model fitting involves standard Markov chain Monte Carlo techniques. Through extensive simulation studies and an application to the wave direction data across the Indian Ocean during the 2004 Indian Ocean Tsunami, we compare the proposed method with both a non-spatial wrapped Gaussian model and a low-rank WGP alternative. The results demonstrate that the WGMRF offers improved predictive performance and scalability in large-domain applications.

翻译：方向性数据的依赖性统计建模研究仍相对不足，尤其是在高维空间场景中。现有的空间角度数据方法主要依赖于缠绕高斯过程模型，该模型为捕捉圆上的空间依赖性提供了一个连贯的框架。然而，当空间域范围较大且观测数据分辨率较高时，基于WGP的方法在计算上变得极具挑战。这一局限在分析大规模地质与气候现象（如海啸和飓风）时尤为突出，此类研究中方向性测量数据（例如波浪或风向）可能覆盖整个海盆。为应对这些挑战，我们提出了一种针对大规模空间方向性数据集的缠绕高斯马尔可夫随机场模型。通过利用高斯马尔可夫随机场固有的稀疏精度矩阵结构，所提方法在保持圆尺度上灵活空间依赖性的同时，实现了显著的计算效率提升。我们讨论了模型的关键性质，包括其可识别性与依赖性特征。模型拟合采用标准的马尔可夫链蒙特卡洛技术。通过大量模拟研究以及对2004年印度洋海啸期间印度洋全域波向数据的应用分析，我们将所提方法与无空间结构的缠绕高斯模型及一种低秩WGP替代方案进行了比较。结果表明，WGMRF在大范围应用场景中提供了更优的预测性能与可扩展性。