Ryan Leadbetter, Gabriel González Cáceres, Aloke Phatak
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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.</p>","PeriodicalId":55495,"journal":{"name":"Applied Stochastic Models in Business and Industry","volume":"41 3","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/asmb.70014","citationCount":"0","resultStr":"{\"title\":\"Bayesian Hierarchical Modeling of Noisy Gamma Processes: Formulation and Extensions for Unit-To-Unit Variability\",\"authors\":\"Ryan Leadbetter, Gabriel González Cáceres, Aloke Phatak\",\"doi\":\"10.1002/asmb.70014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The gamma process is a natural model for monotonic degradation processes. 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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.
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.