噪声伽马过程的贝叶斯分层建模:单位到单位可变性的公式和扩展

IF 1.3 4区 数学 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Ryan Leadbetter, Gabriel González Cáceres, Aloke Phatak
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引用次数: 0

摘要

伽马过程是单调降解过程的自然模型。在实践中,我们希望将单个伽马过程扩展到包含测量误差,并为几个名义上相同的单元的退化构建模型。在本文中,我们将展示如何通过贝叶斯分层建模框架轻松地促进这些扩展。遵循贝叶斯统计工作流的规则,我们展示了一个有噪声的伽马过程模型的原则构造。我们还重新参数化了伽马过程,以简化先验的说明,并使单个伽马过程模型如何扩展到包括单位间的可变性或协变量变得明显。我们首先将噪声过程模型拟合到单个模拟退化轨迹上。在这样做的过程中,我们发现当只有少量噪声退化观测时,伽马过程的挥发性和测量误差之间存在可识别性问题。然而,这种可识别性的缺乏可以通过在分析中包含额外的信息来解决,通过更强的先验或通知一个不可识别参数的额外数据,或者通过从多个单元借用信息。然后,我们探索模型的扩展,以解释单位间的可变性,并使用带有附加测量误差的裂纹传播数据集来演示它们。最后,我们通过交叉验证在完全贝叶斯框架中进行模型选择,以近似新观测的期望对数概率密度。我们还展示了如何为新单元或当前正在测试但尚未失效的单元计算具有不确定间隔的故障时间分布。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Bayesian Hierarchical Modeling of Noisy Gamma Processes: Formulation and Extensions for Unit-To-Unit Variability

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 modeling 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 a 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 have yet to fail.

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来源期刊
CiteScore
2.70
自引率
0.00%
发文量
67
审稿时长
>12 weeks
期刊介绍: ASMBI - Applied Stochastic Models in Business and Industry (formerly Applied Stochastic Models and Data Analysis) was first published in 1985, publishing contributions in the interface between stochastic modelling, data analysis and their applications in business, finance, insurance, management and production. In 2007 ASMBI became the official journal of the International Society for Business and Industrial Statistics (www.isbis.org). The main objective is to publish papers, both technical and practical, presenting new results which solve real-life problems or have great potential in doing so. Mathematical rigour, innovative stochastic modelling and sound applications are the key ingredients of papers to be published, after a very selective review process. The journal is very open to new ideas, like Data Science and Big Data stemming from problems in business and industry or uncertainty quantification in engineering, as well as more traditional ones, like reliability, quality control, design of experiments, managerial processes, supply chains and inventories, insurance, econometrics, financial modelling (provided the papers are related to real problems). The journal is interested also in papers addressing the effects of business and industrial decisions on the environment, healthcare, social life. State-of-the art computational methods are very welcome as well, when combined with sound applications and innovative models.
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