Lévy, Non-Gaussian Ornstein-Uhlenbeck, and Markov Additive Processes in Reliability Analysis
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Unavoidable degradation is one of the major failure mechanisms of many systems due to internal properties (mechanical, thermal, electrical, or chemical) and external influences (temperature, humidity, or vibration). Such degradation in critical engineering systems (e.g., pipelines, wind turbines, power/smart grids, and mechanical devices, etc.) takes the form of corrosion, erosion, fatigue crack, deterioration or wear that may lead to the loss of structural integrity and catastrophic failure. Therefore, developing stochastic degradation models based on appropriate stochastic processes becomes imperative in the reliability and statistics research communities. This dissertation aims to develop a new research framework to integrally handle the complexities in degradation processes (the intrinsic/extrinsic stochastic properties, complex jump mechanisms and dependence) based on general stochastic processes including Lévy, non-Gaussian Ornstein-Uhlenbeck (OU), and Markov additive processes; and to develop a new systematic methodology for reliability analysis that provides compact and explicit results for reliability function and lifetime characteristics. First, to handle the intrinsic stochastic properties and complex jumps, we use Lévy subordinators and their functional extensions, Lévy driven non-Gaussian OU processes, to model the cumulative degradation with jumps that occur at random times and have random sizes. We then integrally handle the complexities of a degradation process including both intrinsic and extrinsic stochastic properties with complex jump mechanisms, by constructing general Markov additive processes. Moreover, the models are extended to multi-dimensional cases for multiple dependent degradation processes under dynamic environments, where the Lévy copulas are studied to construct Markov-modulated multi-dimensional Lévy processes. The Fokker-Planck equations for such general stochastic processes are developed, based on which we derive the explicit results for reliability function and lifetime moments, represented by the Lévy measures, the infinitesimal generator matrices and the Lévy copulas. To analyze the degradation data series from such degradation phenomena of interest, we propose a systematic statistical estimation method using linear programming estimators and empirical characteristic functions. We also construct bootstrap procedures for the confidence intervals. Simulation studies for Lévy measures of gamma processes, compound Poisson processes, positive stable processes and positive tempered stable processes are performed. The framework can be recognized as a general approach that can be used to flexibly handle stylized features of widespread classes of degradation data series such as jumps, linearity/nonlinearity, symmetry/asymmetry, and light/heavy tails, etc. The results are expected to provide accurate reliability prediction and estimation that can be used to assist the mitigation of risk and property loss associated with system failures.
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