We develop an identifiable reduced-rank spatial multinomial model for categorical data with many classes. The model represents class-specific spatial effects through a low-dimensional set of shared latent factors, substantially reducing parameter dimension while preserving joint dependence across classes. Because standard conjugate and Pólya-Gamma methods fail under this factorization, we propose a Gibbs sampler using Laplace-approximation proposals within Metropolis-Hastings updates. Simulation studies examine dimension selection and the accuracy of the Laplace proposals. An application to dominant tree species mapping in the Blue Ridge Mountains demonstrates scalable inference and flexible joint predictions for individual classes, class unions, and area-level summaries.

翻译：我们针对具有大量类别的分类数据，提出一种可识别的降秩空间多项模型。该模型通过一小组共享潜在因子表示类别特定空间效应，在显著降低参数维度的同时保持类别间的联合依赖性。由于标准共轭方法和Pólya-Gamma方法在此分解框架下失效，我们提出一种Gibbs采样器，在Metropolis-Hastings更新中采用拉普拉斯近似提议。模拟研究探讨了维度选择及拉普拉斯提议的准确性。在蓝岭山脉优势树种制图的应用中，该模型展示了可扩展的推断能力，以及对单个类别、类别并集和区域级汇总量进行灵活的联合预测。