Bayesian Hierarchical Modelling of Noisy Gamma Processes: Model Formulation, Identifiability, Model Fitting, and Extensions to Unit-to-Unit Variability
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含噪 Gamma 过程的贝叶斯层级建模：识别、拟合与个体差异扩展
- 研究问题
- 如何在单调退化的 Gamma 过程模型中同时处理测量误差和个体差异，以及少量含噪观测导致的过程波动与测量误差难以区分？
- 方法与思路
- 作者构建并重新参数化贝叶斯层级模型，通过先验、额外数据或多单位信息处理识别困难，并用交叉验证比较模型和预测失效时间。
- 材料陈述的贡献
- 摘要称，研究揭示稀少观测下的参数识别问题及缓解途径，展示个体差异扩展，并给出新单位和在测单位的失效时间不确定性区间。
- 阅读关注点
- 应核对识别困难的具体表现、强先验对后验的影响、多单位共享信息的假设，以及交叉验证与失效时间预测的校准方式。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未说明先验强度、模型限制和预测精度，不能认定层级结构自动解决识别问题。
原文摘要
The gamma process is a natural model for monotonic degradation processes. In practice, it is desirable to extend the single gamma process to incorporate measurement error and to construct models for the degradation of several nominally identical units. In this paper, we show how these extensions are easily facilitated through the Bayesian hierarchical modelling framework. Following the precepts of the Bayesian statistical workflow, we show the principled construction of a noisy gamma process model. We also reparameterise the gamma process to simplify the specification of priors and make it obvious how the single gamma process model can be extended to include unit-to-unit variability or covariates. We first fit the noisy gamma process model to a single simulated degradation trace. In doing so, we find an identifiability problem between the volatility of the gamma process and the measurement error when there are only a few noisy degradation observations. However, this lack of identifiability can be resolved by including extra information in the analysis through a stronger prior or extra data that informs one of the non-identifiable parameters, or by borrowing information from multiple units. We then explore extensions of the model to account for unit-to-unit variability and demonstrate them using a crack-propagation data set with added measurement error. Lastly, we perform model selection in a fully Bayesian framework by using cross-validation to approximate the expected log probability density of new observation. We also show how failure time distributions with uncertainty intervals can be calculated for new units or units that are currently under test but are yet to fail.