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A stochastic deterioration process for time-dependent reliability analysis
Published in Marc A. Maes, Luc Huyse, Reliability and Optimization of Structural Systems, 2020
J.M. van Noortwijk, M.D. Pandey
In order to apply the gamma process model to practical examples, statistical methods for the parameter estimation of gamma processes are required. A typical data set consists of inspection times ti, i =1, …, n, where 0 = t0< t1< t2< … < tn, and corresponding observations of the cumulative amounts of deterioration xi, i = 1, …, n, where 0 = x0 ≤ x1 ≤ x2 ≤ … ≤ xn. Consider a gamma process with shape function υ(t) = ctb and scale parameter u. We assume that the value of the power b is known, but c and u are unknown. The two most common methods of parameter estimation, namely, Maximum Likelihood and Method of Moments, are discussed in this Section. Both methods for deriving the estimators of c and u were initially presented by Çinlar et al. (1977).
Bayesian information fusion for non-competing relationship degradation process
Published in Stein Haugen, Anne Barros, Coen van Gulijk, Trond Kongsvik, Jan Erik Vinnem, Safety and Reliability – Safe Societies in a Changing World, 2018
Junyu Guo, Hong-Zhong Huang, Yan-Feng Li, Jie Zhou, Xiang-Yu Li
The performance degradation process of a product is usually an evolutionary process. The stochastic process model can describe this process well. The gamma process is a stochastic process generally used to describe the performance degradation of mechanical products. Because the increment of the gamma process is non-negative, it is consistent with the performance evolvement of the mechanical products.
Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence
Published in IISE Transactions, 2023
Nan Zhang, Sen Tian, Kaiquan Cai, Jun Zhang
To present the characteristics and the applicability of the proposed model, some numerical illustrations are presented in this section. A Gamma process is a pure jump Lévy process that has been proved to be a well-suited candidate in modelling the temporal variability of deterioration and in developing inspection & maintenance policies (van Noortwijk, 2009; Le Son et al., 2016). A Gamma process with shape parameter a > 0 and scale parameter b > 0 possesses the following properties:For all follows a Gamma distribution with shape parameter and scale parameter b:X(t) has independent increments.
A comprehensive toolbox for the gamma distribution: The gammadist package
Published in Journal of Quality Technology, 2023
Piao Chen, Kilian Buis, Xiujie Zhao
Further development of the package could focus on incorporating more functions related to the gamma distribution. For example, goodness-of-fit test of the gamma distribution is a premise to implement the model in practice. We show in Section 4 that the use of the parest function could be the first step to facilitate graphical assessments or some commonly used tests. Nevertheless, the uncertainties in the estimators may weaken the power of those tests. If more accurate and advanced methods of testing goodness-of-fit of the gamma distribution become available, they could be integrated to the gammadist package as a separate function. As another example, the three-parameter gamma distribution with an additional scale parameter is often used to fit lifetimes of products/units that cannot fail below a threshold (see, e.g., Ye, Hong, and Xie 2013). The rGamma function can be naturally extended to generate random variables from a three-parameter gamma distribution. However, estimation of parameters is a difficult research problem, let alone construction of the statistical limits. Substantial efforts are needed in order to include the three-parameter gamma distribution in our package. At last, the package may be further extended to deal with the gamma process, which is a commonly used stochastic model. Although the gamma distribution and the gamma process have a close relationship, some fundamental issues for the gamma process, such as accurate estimation and prediction, have not been completely addressed in the literature. The authors will pay special attention to the methodology development of the gamma process and are willing to enrich the gammadist package once these methods become available.
Systems thinking approach for improving maintenance management of discrete rail assets: a review and future perspectives
Published in Structure and Infrastructure Engineering, 2023
Yue Shang, Maria Nogal, Haoyu Wang, A. R. M. (Rogier) Wolfert
Gamma process is a stochastic process with independent and non-negative gamma-distributed increments (Van Noortwijk, 2009). The feature determines its applicability to characterise monotonic degradation processes, e.g. track geometry irregularities can only grow without interventions. A gamma process model was proposed to describe the evolution of longitudinal level defects, where a cost model was linked for maintenance optimisation (Meier-Hirmer, Riboulet, Sourget, & Roussignol, 2009). Further, the work was extended by a bivariate gamma process to include alignment in the prediction (Mercier, Meier-Hirmer, & Roussignol, 2012).