Condition monitoring and failure prediction in complex systems
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We consider a monitored system W(t) at time t which is modeled by a stochastic process. Failure of the system is connected to the process overreaching a certain threshold. The system is governed by an unobservable marker process in such a way that the state of the marker process is connected to the technical condition of the system. The system exhibits different traits for the different states of the marker process. The goal is to estimate the first passage time of the critical threshold. The stochastic process under study is the Wiener process, more precisely, a piecewise Wiener process with change points. The change points signify the occurrence of an event which causes a change in parameters of the Wiener process. The change points are governed by the marker process: the time between change points is the time spent in each state for the marker process. This is an unknown quantity, which is estimated by the observable Wiener process. The situation with one change point and two change points and the Wiener process parameters known and unknown are examined and numerical examples are provided for simulated data. The formulas are extended to m change points. A Bayesian approach is used, and Markov Chain Monte Carlo methods are employed to estimate the distribution of the process parameters. In order to predict the hitting time, the hitting time cumulative distribution functions are estimated through simulation of Wiener processes, straight-forward calculations and a time transformation approach.