Spatiotemporal nonhomogeneous poisson model with a seasonal component applied to the analysis of extreme rainfall

This paper develops an extension of spatiotemporal models that handle count data using nonhomogeneous Poisson processes. In this new proposal, we incorporate a seasonal cycle component in the definition of the intensity function to control possible effects produced by the occurrence of the event of interest in regular periods. The seasonal cycle can cause problems in estimating the shape parameter of the Weibull and generalized Goel intensity functions. This shape parameter serves to confront the research hypothesis that seeks to identify a trend in the occurrence rate of an event of interest. In the case of the Weibull intensity function, a value significantly equal to one of the shape parameters indicates a constant rate of occurrence, less than one indicates a decreasing rate, and greater than one indicates an increasing rate. In the case of the Goel intensity function, parameter values less than or equal to one indicate a decreasing occurrence rate, and values greater than one indicate the presence of a change point. We also built a spatial model using the Musa-Okumoto intensity function as an alternative to approximate counting processes for which there is a decreasing trend in the occurrence rate of the event of interest. We estimated the parameters of the proposed method from a Bayesian perspective. Finally, we fitted the proposed model and compared it with other approximations to analyze the frequency of extreme rainfall in the northern region of the states of Maranhão and Piauí in northeastern Brazil over ten years. Among the main results, we found that (1) the proposed method has proven superior in terms of fit and prediction performance than the other models, and (2) unlike other approximations, the proposed model does not detect changes in the rate of extreme rainfall occurrences.

本研究提出了一类面向计数数据的时空模型扩展方法，该方法基于非齐次泊松过程（nonhomogeneous Poisson processes）构建。在该扩展模型中，我们在强度函数的定义中引入季节周期分量，用以控制目标事件在周期性时段内发生所带来的潜在影响。季节周期成分可能会对威布尔（Weibull）强度函数与广义戈尔（generalized Goel）强度函数的形状参数估计造成干扰。该形状参数可用于检验旨在识别目标事件发生速率趋势的研究假设。针对威布尔强度函数而言，当其形状参数显著取值为1时，事件发生速率保持恒定；形状参数小于1时，发生速率呈递减趋势；大于1时则呈递增趋势。针对戈尔强度函数而言，当参数取值小于或等于1时，事件发生速率呈递减趋势；取值大于1时，则表明存在变化点。我们还构建了基于穆萨-奥库莫托（Musa-Okumoto）强度函数的空间模型，作为近似目标事件发生速率呈递减趋势的计数过程的替代方案。我们从贝叶斯（Bayesian）视角对所提方法的参数进行了估计。最后，我们将所提模型进行拟合，并与其他近似模型展开对比，以分析巴西东北部马拉尼昂州（Maranhão）与皮奥伊州（Piauí）北部地区十年间的极端降雨发生频率。主要研究结果如下：(1) 所提方法在拟合效果与预测性能上均优于其他对比模型；(2) 与其他近似模型不同，所提模型未检测到极端降雨发生速率存在变化。