Our method extends the application of random spanning trees to cases where the response variable belongs to the exponential family, making it suitable for a wide range of real-world scenarios, including non-Gaussian likelihoods. The proposed model addresses the limitations of previous spatial clustering methods by allowing all within-cluster model parameters to be cluster-specific, thus offering greater flexibility. Additionally, we propose a Bayesian inference algorithm that overcomes the computational challenges associated with the reversible jump Markov chain Monte Carlo (RJ-MCMC) algorithm by employing composition sampling and the integrated nested Laplace approximation (INLA) to compute the marginal distribution necessary for the acceptance probability. This enhancement improves the mixing and feasibility of Bayesian inference for complex models. We demonstrate the effectiveness of our approach through simulation studies and apply it to real-world disease mapping applications: COVID-19 in the United States of America, and dengue fever in the states of Minas Gerais and S\~ao Paulo, Brazil. Our results highlight the model's capability to uncover meaningful spatial patterns and temporal dynamics in disease outbreaks, providing valuable insights for public health decision-making and resource allocation.

翻译：我们的方法将随机生成树的应用扩展至响应变量属于指数族的情形，使其适用于包括非高斯似然在内的广泛现实场景。所提出的模型通过允许所有簇内模型参数具有簇特异性，克服了先前空间聚类方法的局限性，从而提供了更大的灵活性。此外，我们提出了一种贝叶斯推断算法，该算法通过采用组合抽样和集成嵌套拉普拉斯近似（INLA）来计算接受概率所需的边际分布，克服了可逆跳转马尔可夫链蒙特卡洛（RJ-MCMC）算法相关的计算挑战。这一增强提高了复杂模型贝叶斯推断的混合性与可行性。我们通过模拟研究证明了该方法的有效性，并将其应用于现实世界的疾病制图案例：美国的新型冠状病毒肺炎（COVID-19）疫情，以及巴西米纳斯吉拉斯州和圣保罗州的登革热疫情。我们的结果凸显了该模型在揭示疾病暴发中有意义的空间模式与时间动态方面的能力，为公共卫生决策和资源分配提供了重要见解。