Stochastic process-based methods have been widely used for predicting the remaining useful life (RUL) in engineering system health management. However, real-world degradation processes often exhibit non-Markovian dynamics, nonlinear evolution patterns, and varying operating conditions, which pose significant challenges for accurate RUL prediction. To address these challenges, this study proposes an adaptive RUL prediction framework tailored for nonlinear degradation with structural variability and memory effects. The degradation process is initially modeled using fractional Brownian motion (FBM) based on a segment of historical degradation data. During operation, the model adequacy is continuously assessed through a prediction error metric. If the error exceeds a predefined threshold, a prediction error model is activated to recalibrate the model structure. Subsequently, the drift coefficient is updated using an enhanced variational Bayesian Kalman filter (VBKF). The RUL is predicted based on the first hitting time (FHT) concept, from which an approximate analytical distribution is derived. Model parameters are identified through maximum likelihood estimation (MLE). Finally, the effectiveness and adaptability of the proposed approach are demonstrated through case studies involving a blast furnace and lithium-ion batteries.

The majority of existing online remaining useful life (RUL) prediction models for rolling bearings adopt a single degradation model, and lack the capacity for real-time assessment of model matching. Furthermore, these models determine the first prediction time (FPT) using subjective thresholds, which often results in inaccuracies and considerable deviations in subsequent online RUL predictions. To overcome these limitations, this article proposes an online adaptive matching multidegradation model for RUL prediction. First, a curvature analysis incorporating a dynamic sliding window strategy is proposed. This strategy determines the first prediction time in real time by analyzing the change in curvature of the root-mean-square values within a dynamic sliding window. Second, an online parallel prediction algorithm with multiple degradation models is developed. This algorithm selects the most suitable prediction model by dynamically evaluating the degree of matching between different degradation models and the actual data, thereby ensuring the timeliness of the prediction model. Finally, the validation of the proposed approach is conducted by accelerating the degradation rolling bearing test and the IMS rolling bearing dataset. The results demonstrate that the proposed method outperforms existing approaches in accurately identifying FPT and predicting online RUL.

Uncertainty quantification of the remaining useful life (RUL) for degraded systems under the big data era has been a hot topic in recent years. A general idea is to execute two separate steps: deep-learning-based health indicator (HI) construction and stochastic process-based degradation modeling. However, there exists a critical matching defect between the constructed HI and a degradation model, which seriously affects the RUL prediction accuracy. Toward this end, this article proposes an interactive prognosis framework between deep learning and a stochastic process model for the RUL prediction. First, we resort to stacked contractive autoencoders to fuse multiple sensor information of historical systems for constructing the HI in a typical unsupervised manner. Then, considering the nonlinear characteristic of the constructed HI, an exponential-like degradation model is introduced to construct its degradation evolving model, and theoretical expressions of the prediction results are derived under the concept of the first hitting time. Furthermore, we design an optimization objective function by integrating the HI construction and degradation modeling for the RUL prediction. To minimize the designed objective function of the proposed interactive prognosis framework, a gradient descent algorithm is employed to update the model parameters. Based on the well-trained interactive prognosis model, we can obtain the HI of a field system from stacked contractive autoencoders with sensor data and the probability density function (pdf) of the predicted RUL on the basis of the estimated parameters. Finally, the effectiveness and superiority of the proposed interactive prognosis method are verified by two case studies associated with turbofan engines.

Deep learning has become a promising tool for processing the massive data and attracted an increasing attention in the fields of degradation modeling and remaining useful life (RUL) prediction. The existing deep-learning-based methods are generally faced with the two aspects of problems. On the one hand, the prediction results are represented by the point estimates instead of the probabilistic distribution and, thus, the prognostic uncertainty in RUL prediction cannot be characterized. On the other hand, there exist plenty of the engineering assets without the label data related to lifetime, posing a great challenge for training the deep learning network. Toward this end, we propose a prognostic model under the framework of Bayesian deep learning for equipment lacking the label data related to lifetime. First, the monitoring data of the historical equipment and the historical data of field equipment in the database are preprocessed to generate the samples regarding degradation information as a label. Second, the bidirectional recurrent neural network (RNN) is employed as the candidate network for the advantages in handling the sequential monitoring data. On the basis of this, the idea of Bayesian deep learning is incorporated into the bidirectional RNN; thus, we can characterize the uncertainty of the predicted degradation level at any future time via utilizing the variational inference technique in the Bayesian neural networks. Furthermore, the failure probability for the concerned equipment at any time can be determined, by which the degradation uncertainty can be converted into the RUL uncertainty from the point of the reliability theory. Finally, we provide the case study associated with lithium-ion batteries to verify the proposed prognostic model.

Degradation-model-based remaining useful life (RUL) estimation is essential for effective prognostic and health management; this method can provide information for enabling effective maintenance decisions pertaining to degrading systems to avoid or mitigate loss due to impending failures. According to existing studies that estimate degradation-model-based RUL, stochastic-process-based methods are widely advocated by many researchers because they can capture stochastic dynamics within the degradation processes of systems. However, most existing studies primarily utilize the first passage time (FPT) to define the lifetime/RUL. The definition is generally conservative; thus, the lifetime/RUL estimation may be underestimated. This is particularly true for nonmonotonic stochastic degradation processes. In some extreme cases, with strong fluctuations within the degradation processes, the estimated lifetime/RUL under the FPT can be significantly less than the actual lifetime/RUL. To address this limitation, this study investigates prognostic issues, based on the stochastic degradation process, from the last exit time (LET) perspective. In contrast to the FPT, the lifetime/RUL of the degrading system is defined as the LET of its degradation process, i.e., the instant at which the degradation process passes the failure threshold for the last time. Given the new definition, we consider the most widely used degradation process model (i.e., Wiener-process-based model) as an example to demonstrate how the lifetime/RUL is estimated. Two general methods of lifetime estimation for the Wiener-process-based model are given, and some examples with associated exact and closed-form solutions are also provided for better illustration. Finally, numerical examples and a practical case study are presented to substantiate the theoretical results and illustrate the superiority of the proposed method. The results imply that the proposed method exhibits the potential to prevent premature maintenance and resource wastage because the lifetime/RUL estimation from the LET perspective can help avoid conservative results from being obtained.

Recurrent neural networks (RNNs) have been widely adopted in research areas concerned with sequential data, such as text, audio, and video. However, RNNs consisting of sigma cells or tanh cells are unable to learn the relevant information of input data when the input gap is large. By introducing gate functions into the cell structure, the long short-term memory (LSTM) could handle the problem of long-term dependencies well. Since its introduction, almost all the exciting results based on RNNs have been achieved by the LSTM. The LSTM has become the focus of deep learning. We review the LSTM cell and its variants to explore the learning capacity of the LSTM cell. Furthermore, the LSTM networks are divided into two broad categories: LSTM-dominated networks and integrated LSTM networks. In addition, their various applications are discussed. Finally, future research directions are presented for LSTM networks.

The tracking control of a multi-input multioutput nonlinear nonminimum phase system in general form is discussed. This system is assumed to be suffering from parameter uncertainties and unmodeled dynamics, and the priori information of them is unknown. By considering both the exact model and uncertain model, the sliding mode-based learning controller is proposed. By designing an appropriate sliding surface and a learning controller, the stability of the closed-loop system is guaranteed for both the exact model and uncertain model. To overcome the disadvantage caused by parameter uncertainties and unmodeled dynamics, a fuzzy logical system is adopted here. A numerical simulation result carried on vertical takeoff and landing aircraft is taken as an example to validate the effectiveness of the presented controller.

This paper is concerned with estimating remaining useful life (RUL) for a class of stochastic degrading systems with surviving degradation paths and uncertain measurements. The motivation comes from two engineering facts: the system's degradation state is discretely monitored and the measured degradation signals are taken from a survival degradation path, i.e., its lifetime is greater than the latest observation time, and the underlying degradation state cannot be perfectly observed due to the noise, disturbance, nonideal instruments, etc. Thus, the measured degradation signals are uncertain, but related to the underlying degradation state. Toward this end, we first present a general stochastic degradation modeling approach to account for the abovementioned facts. Then, a non-Gaussian degradation state transition equation is derived considering the constraint of the survival path and thus the particle filtering algorithm is used to estimate the underlying degradation state in real time from uncertain measurements. Furthermore, we derive the RUL distribution based on the first-passage time concept which incorporates the uncertainty of the estimation for the degradation state and can be real-time updated based on the available surviving yet uncertain measurements. The novelty of this paper is to allow us to exclude the probability of failure between discrete monitoring times and to account for the impacts of surviving and uncertain measurements on estimating both the degradation state and the RUL. To apply the presented methodology, a maximum likelihood estimation framework is provided to determine the model parameters based on the expectation maximization algorithm together with particle filtering and smoothing methods. As a special case fallen into the presented framework, a linear degradation modeling and prognostic realization is discussed. Finally, we demonstrate the proposed approach by a case study.

Due to the inner deteriorating mechanism or the mutant environmental stress, the degradation systems with multi-phase features have frequently been encountered in engineering practice. The key issue for prognostics of such systems is to account for the impact of the changing-point variability and the associated degradation state at this point on the progression of the degradation process. However, current studies usually treat the degradation state at the change point as a fixed value rather a random variable. Thus, it is still challenging to predict the lifetime of such multi-phase degrading systems. To this end, we first formulate a general degradation modeling framework based on a two-phase Wiener process. In prognostics, we take into full account the uncertainty of the degradation state at the changing point and then derive the analytical expressions of the lifetime and remaining useful life under the concept of the first passage time. The derived results are distinguished from existing results limited to the fixed state at the changing point. Furthermore, we extend our approach and results to cases with unit-to-unit variability and multiple phases. To facilitate the model implementation, we propose both offline and online methods for parameter identification, which make full use of the historical data and the in-service data. Finally, a numerical simulation and a practical case study are provided for illustration.

Degradation data obtained through inspections have been widely used to estimate the remaining useful life (RUL) of deteriorating systems. At the same time, such inspections may introduce external stress or release interior stress on the degradation processes of systems. However, current studies have paid little attention to the influence of such inspections during degradation modeling and RUL estimation. In this paper, we incorporate the inspection influence into Wiener-process-based degradation modeling and develop an approach to estimate the RUL of deteriorating systems. In the proposed method, the lifetime and RUL distribution with the consideration of the inspection influence on degrading systems are derived under the concept of first hitting time. To achieve an adaptive estimation of the RUL when new observations arrive, we constructed a state-space model by augmenting the drift coefficient and the inspection influence as state variables and treating the degradation increments as the observation variables. To do so, a Kalman filter is employed to update the state estimation, and then, an analytical result of the estimated RUL distribution for periodic-inspected systems is achieved. To implement the proposed prognostic method, we apply the expectation maximization algorithm to estimate unknown parameters in the constructed state-space model based on the historical observations of inspections. Finally, the proposed approach is illustrated by a numerical example and demonstrated by a case study using the mechanical gyroscopes.

Condition-based maintenance for a degrading system has been attached great importance in reducing unexpected failures and enabling the safe operation of the system. In this paper, we consider a condition-based replacement problem with observed degradation signals for the determination of the optimal replacement time of the system. The observed degradation signals contain some health related information of the system so that the system's health state may be more accurately estimated and the future degradation progression of the system can be predicted. Based on the predicted degradation state, the concerned replacement problem is formulated in the framework of the Markov decision process and then a new degradation-modeling framework based on maximum likelihood estimation is proposed to analyze the degradation signals and to predict the future degradation state of the system. To solve the proposed condition-based replacement decision problem, we analyze structural properties of the optimal replacement policy and an optimal monotonic control limit solution policy is developed for condition-based replacement. Finally, a case study is provided to illustrate the proposed optimal replacement method.

Reliability estimation is central to enhance safety, availability, and effectiveness of phased-mission systems (PMSs). With the development of information and sensing technologies, condition monitoring (CM) data are now available in many real-world PMSs, and then a more interesting question: how can we dynamically estimate the reliability of PMSs using the in-situ CM data, is of considerable significance to industrial practitioners. In this paper, using the CM data and degradation data of PMS, we present a novel condition-based approach to resolve this question under dynamic operating scenarios. This paper differs from most existing methods which only consider the static scenario without using real-time information, and estimate the reliability only for a population of PMSs but not for an individual PMS in service. To establish a linkage between the historical data and real-time data of the individual PMS, a stochastic filtering model is first utilized to model the phase duration. As such, the updated estimation of the mission time can be obtained by Bayesian law at each phase. To account for the dependency of the degradation progression of PMS on the mission process, the degradation process of PMS is modeled by a Brownian motion with a mission phase-dependent drift coefficient. The corresponding lifetime is derived and the lifetime distribution of PMS can be updated under Bayesian framework once new information is available. Unique to this paper is the union of the CM data and degradation data of PMS to real-time estimate the mission reliability through the estimated distribution of the mission time in conjunction with the estimated lifetime distribution, in which the estimated lifetime considers the dependency of the degradation rate of PMS on mission phase. The effectiveness of the proposed approach is verified by a numerical simulation and a case study.

Generalized eigendecomposition problem has been widely employed in many signal processing applications. In this paper, we propose a unified and self-stabilizing algorithm, which is able to extract the first principal and minor generalized eigenvectors of a matrix pencil of two vector sequences adaptively. Furthermore, we extend the proposed algorithm to extract multiple generalized eigenvectors. The performance analysis shows that only the desired equilibrium point of the proposed algorithm is stable and all others are (unstable) repellers or saddle points. Convergence analysis based on the deterministic discrete-time approach shows that, for a step size within a certain range, the norm of the principal/minor state vector converges to a fixed value that relates to the corresponding principal/minor generalized eigenvalue. Thus, the proposed algorithm is a generalized eigenpairs (eigenvectors and eigenvalues) extraction algorithm. Finally, the simulation experiments are carried to further demonstrate the efficiency of the proposed algorithm.

Many industrial systems inevitably suffer performance degradation. Thus, predicting the remaining useful life (RUL) for such degrading systems has attracted significant attention in the prognostics community. For some systems like batteries, one commonly encountered phenomenon is that the system performance degrades with usage and recovers in storage. However, almost all of the current prognostic studies do not consider such a recovery phenomenon in stochastic degradation modeling. In this paper, we present a prognostic model for deteriorating systems experiencing a switching operating process between usage and storage, where the system degradation state recovers randomly after the storage process. The possible recovery from the current time to the predicted future failure time is incorporated in the prognosis. First, the degradation state evolution of the system is modeled through a diffusion process with piecewise but time-dependent drift coefficient functions. Under the concept of first hitting time, we derived the lifetime and RUL distributions for systems with specific constant working mode. Further, we extended the results of RUL distribution in specific constant working mode to the case of stochastic working mode, which is modeled through a flexible two-state semi-Markov model (SMM) with phase-type distributed interval times. The unknown parameters in the present model are estimated based on the observed condition monitoring data of the system, and the SMM model is identified on the basis of the operating data. A numerical study and a case study of Li-ion batteries are carried out to illustrate and demonstrate the proposed prognostic method. Experimental results indicate that the presented method can improve the accuracy of lifetime and RUL estimation for systems with state recovery.

Repeated degradation testing data have been widely used to assess lifetime of highly-reliable products or components with scarce failures. In this paper, we consider the problem of planning repeated degradation testing for products exhibiting three-source variability in their degradation characteristics, including the temporal variability, unit-to-unit variability, and measurement variability. The primary objective of this paper is centered on deciding the amount of units and the measurement schedule to achieve required estimation precision for some important statistics of interest. In such a planning process, the testing budget is limited and used as a constraint of an optimization model. To do so, a kind of Wiener degradation process, which has a random drift coefficient and a constant volatility coefficient, is used to model the repeated degradation testing data, where the measurement errors are considered and described as additive zero-mean random variables. Under the presented modeling framework, the lifetime distribution is formulated under the concept of the first passage time of the stochastic degradation process. Then, the large-sample approximate standard errors of the maximum likelihood estimations for the mean failure time and the quantile of the degradation distribution are derived, respectively. Furthermore, the constrained optimization model, which incorporates the cost of each degradation measurement, is proposed to plan the degradation testing by minimizing the testing cost under the condition of a maximum acceptable approximate standard error. Finally, an example is provided to illustrate the procedure and advantages of the presented method.

Remaining useful life (RUL) estimation plays a vital role in the prognostics and health management of degrading systems. For complicated degrading systems, the associated degradation processes are not only subjected to the nonlinearity in the degradation evolving paths but are also influenced by three important sources of variability, i.e., temporal variability, unit-to-unit variability, and measurement variability. However, current studies do not consider the above key factors jointly. Toward this end, this paper presents a general nonlinear degradation model to characterize the degradation nonlinearity and the three-source variability simultaneously. By constructing a state-space model and applying the Kalman filtering technique, we present the method of the RUL estimate with three-source variability and derive the analytical form of the probability density function of the RUL with three-source variability and the degradation nonlinearity approximately, which can be real-time updated with the available observations. As such, the effects of the degradation nonlinearity and three-source variability are propagated into the RUL estimate. In addition, the unknown parameters of the presented nonlinear model are estimated using the maximum likelihood estimation approach. For demonstrating the presented approach, comparative studies are conducted. The results verify that the proposed approach improves the model fitting and the accuracy of the RUL estimate.

This paper proposes a prognostic-information-based joint order-replacement policy for a non-repairable critical system in service. The primary difference from existing work is to take the online condition monitoring data into consideration during the joint decision-making process. Towards this end, the system’s degradation trajectory is modeled by a Wiener process whose parameters are real-time estimated based on the newly obtained condition monitoring data by utilizing the expectation maximization algorithm and Bayesian inference. By doing so, the remaining useful life distribution of the system of interest can be predicted in real-time, which is then used as the prognostic information to dynamically update the optimal ordering and replacement times jointly. This process makes the jointly obtained order-replacement decisions rely on the prognostic information available from the system’s degradation monitoring. Finally, a practical case study of the inertial navigation system in aircraft is provided to validate the proposed joint decision policy.

Nonlinearity and stochasticity are two important factors contributing to the degradation processes of complicated systems, and thus have to be taken into account in stochastic degradation modeling based prognostics. However, current studies almost always focus on age-dependent stochastic degradation models, most of which are linear, or can be transformed into linear models. In this paper, we propose a general age- and state-dependent nonlinear degradation model for prognostics. In the presented model, a diffusion process with age- and state-dependent nonlinear drift and volatility coefficients is utilized to characterize the dynamics and nonlinearity of the degradation progression. To derive the estimated remaining useful life distribution, the considered diffusion process is first converted into a diffusion process with age- or state-dependent nonlinear drift but constant volatility through Lamperti transformation. Then, based on a well-known time-space transformation, we obtain an analytical approximated remaining useful life distribution in the concept of the first passage time. Furthermore, a maximum likelihood estimation method for unknown parameters in the concerned model is presented on the basis of closed-form approximated degradation state transition density functions by the Hermite-expansion method. An illustrative example is provided to show how the obtained results can be applied to a specific age- and state-dependent nonlinear degradation model. Finally, the presented model is fitted to bearing degradation data. Comparative results suggest the necessity of age- and state-dependent nonlinear degradation modeling in prognostics.

Reliability estimation based on degradation model is a feasible and low-cost alternative used to estimate reliability for highly reliable systems when the failure-time data are rare. Based on reliability estimation by degradation modeling, preventive maintenance work orders need to be timely triggered to minimize unscheduled downtime. In Trans. Autom. Sci. Eng., vol. 9, no. 1, pp. 209–212, Jan. 2012, Sun et al., an approach to dynamically extract maintenance threshold is presented for maintenance scheduling, in which the reliability threshold for maintenance is determined by maximizing the expected availability and the reliability estimation is achieved by a modified two-stage degradation modeling approach. Although this approach is novel and useful, its reliability estimation is an asymptotic solution in long time scale. In this paper, we generalize the above result by considering a general degradation path model and provide the exact and explicit formulation for reliability estimation. Additionally, a maximum-likelihood estimation method for parameters in the presented model is proposed based on the historical degradation observations. Finally, an example is provided for illustration.

The use of the observed degradation data of a system can help to estimate its remaining useful life (RUL). However, the degradation progression of the system is typically stochastic, and thus the RUL is also a random variable, resulting in the difficulty to estimate the RUL with certainty. In general, there are three sources of variability contributing to the uncertainty of the estimated RUL: 1) temporal variability, 2) unit-to-unit variability, and 3) measurement variability. In this paper, we present a relatively general degradation model based on a Wiener process. In the presented model, the above three-source variability is simultaneously characterized to incorporate the effect of three-source variability into RUL estimation. By constructing a state-space model, the posterior distributions of the underlying degradation state and random effect parameter, which are correlated, are estimated by employing the Kalman filtering technique. Further, the analytical forms of not only the probability distribution but also the mean and variance of the estimated RUL are derived, and can be real-time updated in line with the arrivals of new degradation observations. We also investigate the issues regarding the identifiability problem in parameter estimation of the presented model, and establish the according results. For verifying the presented approach, a case study for gyros in an inertial platform is provided, and the results indicate that considering three-source variability can improve the modeling fitting and the accuracy of the estimated RUL.

Hybrid deteriorating systems, which are made up of both linear and nonlinear degradation parts, are often encountered in engineering practice, such as gyroscopes which are frequently utilized in ships, aircraft, and weapon systems. However, little reported literature can be found addressing the degradation modeling for a system of this type. This paper proposes a general degradation modeling framework for hybrid deteriorating systems by employing an additive Wiener process model that consists of a linear degradation part and a nonlinear part. Furthermore, we derive the analytical solution of the remaining useful life distribution approximately for the presented model. For a specific system in service, the posterior estimates of the stochastic parameters in the model are updated recursively by using the condition monitoring observations based on a Bayesian framework with the consideration that the stochastic parameters in the linear and nonlinear deteriorating parts are correlated. Thereafter, the posterior distribution of stochastic parameters is used to update in real-time the distribution of the remaining useful life where the uncertainties in the estimated stochastic parameters are incorporated. Finally, a numerical example and a practical case study are provided to verify the effectiveness of the proposed method. Compared with two existing methods in literature, our proposed degradation modeling method increases the one-step prediction accuracy slightly in terms of mean squared error, but gains significant improvements in the estimated remaining useful life.

Hidden or partially observable degradation state of the equipment is frequently encountered in many engineering practices. This may encourage the state space modeling technique as a feasible way to estimate equipment's remaining useful life (RUL). However, most of the existing state space models falling into this category are based on the assumptions that the degradation process is linear or can be linearized. Therefore, modeling the hidden degradation process under a general nonlinear function and deriving the corresponding analytical form of the RUL distribution are still challenging and have not been well solved in literature. In this paper, we present a state-space-based prognostic model to address the above issues, in which the nonlinearity is characterized by an age-dependent general nonlinear function. Specifically, we model the degradation process as the unobservable nonlinear drift-based Brownian motion (BM) and apply extended Kalman filter (EKF) and expectation-maximization (EM) algorithm to estimate and update the degradation state and the unknown parameters of the established model jointly. Furthermore, we derive the analytical form of the RUL distribution approximately which incorporates the uncertainty of the estimation for hidden state and can be real-time updated based on the available observations. For verifying our approach, a numerical example and a case study for a NASA battery are provided, and the results show that both the parameters and the RUL are estimated accurately. We also consider several different nonlinear functions and compare them with the linear model. The comparative results demonstrate our approach is better than the results in the linear case.

Remaining useful life estimation is central to the prognostics and health management of systems, particularly for safety-critical systems, and systems that are very expensive. We present a non-linear model to estimate the remaining useful life of a system based on monitored degradation signals. A diffusion process with a nonlinear drift coefficient with a constant threshold was transformed to a linear model with a variable threshold to characterize the dynamics and nonlinearity of the degradation process. This new diffusion process contrasts sharply with existing models that use a linear drift, and also with models that use a linear drift based on transformed data that were originally nonlinear. Both existing models are based on a constant threshold. To estimate the remaining useful life, an analytical approximation to the distribution of the first hitting time of the diffusion process crossing a threshold level is obtained in a closed form by a time-space transformation under a mild assumption. The unknown parameters in the established model are estimated using the maximum likelihood estimation approach, and goodness of fit measures are applied. The usefulness of the proposed model is demonstrated by several real-world examples. The results reveal that considering nonlinearity in the degradation process can significantly improve the accuracy of remaining useful life estimation.

In engineering practice, a system's behavior constantly changes over time. To predict the behavior of a complex engineering system, a model can be built and trained using historical data. This paper addresses the forecasting problems with a belief rule base (BRB) to trace and predict system performance in a more interpretable and transparent way. More precisely, it extends the BRB method to handle a system's behavior prediction, and a new prediction model based on BRB is presented, which can model and analyze prediction problems using not only numerical data but human judgmental information as well. The proposed forecasting model includes some unknown parameters that can be manually tuned and trained. To build an effective BRB forecasting model, a multiple-objective optimization model is provided to locally train the BRB prediction model by minimizing the mean square error (MSE). Finally, a practical case study is provided to illustrate the detailed implementation procedures and examine the feasibility of the proposed approach in engineering application. Furthermore, the comparative studies with other state-of-the-art prediction methods are carried out. It is shown that the proposed model is effective and can generate better prediction in terms of accuracy, as well as comprehensibility.

Recently, a novel prediction model based on the evidential reasoning (ER) approach is developed to forecast reliability in engineering systems. In order to determine the parameters of the ER-based prediction model, some optimization models have been proposed to train the ER-based prediction model. However, these models are implemented in an offline fashion and thus it is very expensive to train and retrain them when new information is available. This correspondence paper is concerned with developing the recursive algorithms for updating the ER-based prediction model from the probability-based point of view. Using the recursive expectation maximization algorithm, two recursive algorithms are proposed for updating the parameters of the ER-based prediction model under judgmental and numerical outputs, respectively. As such, the proposed algorithms can be used to fine tune the ER-based prediction model online once new information becomes available. We verify the proposed method via a realistic example with missile reliability data.