ABSTRACT We propose a flexible approach to modeling for renewal processes. The model is built from a structured mixture of Erlang densities for the renewal process inter-arrival density. The Erlang mixture components have a common scale parameter, and the mixture weights are defined through an underlying distribution function modeled nonparametrically with a Dirichlet process (DP) prior. This model specification enables nonstandard shapes for the inter-arrival time density, including heavy tailed and multimodal densities. Moreover, the choice of the DP centering distribution controls clustering or declustering patterns for the point process, which can therefore be encouraged in the prior specification. Using the analytically available Laplace transforms of the relevant functions, we study the renewal function and the directly related K function, which can be used to infer about clustering or declustering patterns. From a computational point of view, the model structure is attractive as it enables efficient posterior simulation while properly accounting for the likelihood normalizing constant implied by the renewal process. A hierarchical extension of the model allows for the quantification of the impact of different levels of a factor. The modeling approach is illustrated with several synthetic datasets, earthquake occurrences data, and coal-mining disaster data.

中文翻译：

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更新过程的非参数贝叶斯建模和估计

摘要 我们提出了一种灵活的更新过程建模方法。该模型是由 Erlang 密度的结构化混合构建的，用于更新过程到达间密度。Erlang 混合组件有一个共同的尺度参数，混合权重是通过一个底层分布函数定义的，该函数使用 Dirichlet 过程 (DP) 先验进行非参数建模。此模型规范支持到达间隔时间密度的非标准形状，包括重尾和多模态密度。此外，DP 中心分布的选择控制了点过程的聚类或去聚类模式，因此在先前的规范中可以鼓励这种模式。使用相关函数的解析可用拉普拉斯变换，我们研究更新函数和直接相关的 K 函数，可用于推断聚类或去聚类模式。从计算的角度来看，模型结构很有吸引力，因为它可以实现有效的后验模拟，同时正确考虑更新过程所隐含的似然归一化常数。模型的分层扩展允许量化不同级别因素的影响。建模方法通过几个合成数据集、地震发生数据和煤矿灾害数据进行说明。模型的分层扩展允许量化不同级别因素的影响。建模方法通过几个合成数据集、地震发生数据和煤矿灾害数据进行说明。模型的分层扩展允许量化不同级别因素的影响。建模方法通过几个合成数据集、地震发生数据和煤矿灾害数据进行说明。