Bounded Transformation of the Gamma Degradation Process

Although the finiteness of physical dimensions and/or the nature of the degradation mechanism make the degradation phenomena of several technological units naturally bounded, the stochastic models used to describe these phenomena are typically unbounded. In general, this apparent contradiction does not significantly affect the effectiveness of unbounded degradation models, because degrading units are conventionally considered failed when their degradation level exceeds a threshold value that is quite far from the “natural” bounds. On the other side, however, the effectiveness of an unbounded degradation models can drastically diminish if the physical bound is slightly greater than the threshold value. The aim of this paper is then to propose a bounded transformation of the gamma process able to correctly model the bounded degradation phenomena even when the “natural” bound and the threshold have comparable values. This idea is not completely new, but, unlike what is assumed in existing models, the upper bound is here treated as an unknown parameter that must be estimated from the available data. The proposed approach is then applied to a real dataset consisting of the wear measurements of eight cylinder liners equipping a Diesel engine for marine propulsion. Model parameters are estimated by using the maximum likelihood method. The fitting ability of the proposed bounded process is compared with that of the unbounded transformed gamma process, previously adopted to analyze these wear data. A condition-based maintenance policy is also applied to the above wear data in order to highlight the need to correctly model the degradation phenomena for avoiding unnecessary maintenance costs. Potentiality of the proposed approach are critically discussed in the paper

About Bounded Transformations of the Gamma Degradation Process

Although the degradation processes of technological units are naturally bounded, due to the finiteness of their physical dimensions and/or the nature itself of the degradation mechanism, the models adopted to describe degradation phenomena are typically unbounded. In general, this apparent contradiction does not significantly affect the effectiveness of unbounded degradation models, because degrading units are conventionally considered failed when their degradation level exceeds a threshold value that is quite far from the “natural” bounds. On the other side, however, the effectiveness of an unbounded degradation models can drastically diminish if the physical bound and threshold have comparable values. The aim of this paper is then to investigate the potentiality of the transformed gamma process in modelling bounded degradation phenomena. This idea is not new. Yet, differently than in other existing models, here the upper bound is treated as an unknown parameter and is estimated from the available degradation data. The proposed approach, which led to the definition of a bounded (state-dependent) transformed gamma process, is illustrated starting with a motivating example, which is developed on the basis of a real set of wear data of cylinder liners equipping a diesel engine for marine propulsion. Model parameters are estimated by using the maximum likelihood method. Fitting ability of the innovative proposed bounded process is compared with those of the unbounded gamma process, previously adopted to analyze these wear data. Potentiality of the proposed approach are critically discussed in the paper

A Bayesian estimation approach for the age- and state-dependent transformed Wiener degradation process

Very recently, a new age and state-dependent degradation process, named the Transformed Wiener (TW) process, has been proposed to describe degradation phenomena when the degradation growth of the units under study is not necessarily monotonically increasing and depends stochastically on the current degradation level. This paper suggests a Bayesian estimation approach for such a process, based on informative priors of its parameters, which allows one to incorporate into the estimation procedure the prior information on meaningful physical characteristics of the observed degradation process that is generally available to the analyst. Several different prior distributions are proposed, reflecting different degrees of knowledge on the observed phenomenon. A Monte Carlo Markov Chain technique is adopted for estimating the TW process parameters and some functions thereof. Finally, in order to show the feasibility of the proposed Bayesian estimation procedure and the flexibility of the TW process an example of application is developed