ABSTRACT
We consider a monitored system with observation Y(t) at time t modeled by a stochastic process, and where system failure is connected to the exceedance of some threshold for this process. Typically, the threshold is not exceeded under normal conditions, where the process Y(t) is supposed to show some kind of stationarity. However, due to unexpected events the process may leave the stationary behavior, in which case an early detection of an increasing Y(t) is necessary to avoid costly failures of the system. The prediction of the time T of future exceedance of a given threshold for the process Y(t) will hence be an important issue, and is the basic problem studied in the paper. We assume that Y(t) is a stochastic process with a probability mechanism depending on an unobservable underlying process S(t). The latter process has a finite state space, {0,1, …,k}, where state 0 corresponds to the normal stationary conditions for the process Y(t), while states 1,2, … in increasing order means an increasing severity of the underlying conditions which eventually will lead to system failure. In particular we consider the case when Y(t) is modeled as a Wiener process. It is then natural to assume that the drift parameter of the process equals 0 when S(t) = 0, while the drift is positive and increasing with S(t), if S(t) ≥ 1. A special case with k = 1 will be considered in detail. In this case estimation of the unobservable “switching” time τ at which the underlying process S(t) changes from state 0 to state 1 is of particular interest. A Bayesian approach will be used, wherea Markov Chain Monte Carlo approach will be needed for doing the computations in the most general cases.
