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A prerequisite for the existing remaining useful life prediction methods based on stochastic processes is the assumption of independent increments. However, this is in sharp contrast to some practical systems including batteries and blast furnace walls, in which the degradation processes have the property of long-range dependence. Based on the fractional Brownian motion, we adopt a degradation process with long-range dependence to predict the remaining useful life of the above systems. Because the degradation process with long-range dependence is neither a Markovian process nor a semimartingale, the exact analytical first passage time is difficult to derive directly. To address this problem, a weak convergence theorem is first adopted to approximately transform a fractional Brownian motion-based degradation process into a Brownian motion-based one with a time-varying coefficient. Then, with a space-time transformation, the first passage time of the degradation process with long-range dependence can be obtained in a closed form. Unknown parameters in the degradation model can be identified using discrete dyadic wavelet transform and maximum likelihood estimation. Numerical simulations and a practical example of a blast furnace wall are given to verify the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]

For data-driven remaining useful life (RUL) prediction, an appropriate degradation model is critically important to achieve accurate prediction. The degradation processes in some practical systems are not only related to the age but also related to the current degradation state, and the degradation processes may be non-Markovian processes. However, most existing stochastic process-based degradation models only depend on the age, and simply assume that the increments are independent. In this paper, an age- and state-dependent degradation model with long-range dependence is developed, which is more general than most of the existing models based on either Brownian motions (BMs) or fractional Brownian motions (FBMs). The Radon-Nikodym derivative is utilized to obtain a likelihood ratio function of unknown parameters, and the estimates are obtained by maximizing the likelihood ratio function. A weak convergence theorem is introduced to approximate the FBM by a BM with a time-varying coefficient. A time-space transformation is further utilized to obtain an approximate explicit solution of the RUL. At last, numerical simulations and two real case studies of blast furnace walls and ball bearings are adopted to verify the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]