退化数据分析与剩余寿命预测：基于Wiener过程方法的综述

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"本文是一篇关于退化数据分析和剩余使用寿命预测的综述文章，重点讨论了基于Wiener过程的方法。作者包括Zhengxin Zhang、Xiaosheng Si、Changhua Hu和Yaguo Lei，分别来自中国西安的自动化研究所和西安交通大学的制造系统工程国家重点实验室。文章在2017年1月收到，2018年2月接受，并于同年2月在线发布。关键词涉及可靠性、退化建模、剩余使用寿命、Wiener过程、预测与健康管理以及基于状态的维护。" 退化数据分析是研究设备或系统性能随时间逐渐下降的过程，这种下降可能由磨损、疲劳、腐蚀等因素引起。在许多领域，如航空航天、电力系统、机械工程等，对设备的健康状态进行实时监控和准确预测其剩余使用寿命（RUL）至关重要，因为这有助于实现可靠、安全且经济的操作。 Wiener过程是一种广泛应用于随机过程理论的连续时间随机过程，它描述了物体性能随时间的连续随机变化。在退化数据分析中，Wiener过程常被用来模拟设备性能的平滑退化路径，尤其是在没有突然故障的情况下。这种方法可以提供一个数学框架，用于估计设备的未来状态并预测其可能的失效时间。文章首先介绍了退化建模的基本概念和重要性，强调了它在预测和健康管理（PHM）中的核心地位。然后，详细探讨了基于Wiener过程的不同退化模型，这些模型考虑了各种因素，如初始退化水平、退化速度以及随机扰动。作者还可能分析了这些模型在处理不完整或有噪声的数据（退化数据）时的适用性和挑战。接下来，文章可能会深入到如何利用这些模型进行RUL预测，包括数据采集、预处理、参数估计和预测模型的选择。在这一部分，作者可能讨论了不同的统计方法，如滤波理论（如卡尔曼滤波）、贝叶斯方法和机器学习算法，以及它们在处理退化数据中的应用。此外，文章还会涉及基于状态的维护策略，这是利用退化数据预测来优化维护决策的一种方式。通过预测设备的RUL，可以安排预防性维护，避免不必要的早期维护成本，同时减少因设备突然失效导致的损失。最后，作者可能会总结当前研究的局限性，并提出未来的研究方向，比如改进模型的复杂性和适应性，处理非线性退化，或者开发更有效的数据处理技术来应对现实世界中的退化数据分析问题。这篇综述提供了对退化数据分析和基于Wiener过程的RUL预测方法的全面理解，对于从事预测性维护和健康管理领域的研究人员和工程师来说，是一份宝贵的参考资料。通过深入研究这些方法，可以提高设备的运行效率，降低维护成本，保障系统的稳定性和安全性。

Z. Zhang et al. / European Journal of Operational Research 271 (2018) 775–796 779

erated degradation model for the geometric BM degradation path

after converting it into a linear Wiener process through logarith-

mic function ( Park, Padgett, 2005 ; Parkand & Padgett, 2005 ). How-

ever, in these works, the proposed methods do not use the in-situ

degradation data during lifetime inference for an in-service system

of interest. To overcome this limitation, Gebraeel, Lawley, Li, and

Ryan (2006) proposed two exponential-like degradation models. In

their models, stochastic parameters were updated via a Bayesian

approach to incorporate real-time information. Following Gebraeel

et al. (2006) , many variants have been reported in prognostics (see,

e.g., Elwany & Gebraeel, 2009; Elwany & Gebraeel, 2008; H.Elwany

& Gebraeel, 2009; Kaiser & Gebraeel, 2009 ). However, even when

the exponential-type degradation processes have been linearized

to a linear degradation model through the logarithm transforma-

tion in Gebraeel, Lawley, Li, and Ryan (2005) , Elwany and Ge-

braeel (2009) , Elwany and Gebraeel (2008) , Gebraeel et al. (2006) ,

the corresponding RUL in these works are just deﬁned as L

{ l

: Y (t

+ l

) ≥ g(w ) } . As such, the RUL distribution was formu-

lated through Pr (L

≤ l

) = Pr (Y (t

+ l

) ≥ g(w )) , where Y (t

+ l

)

and g(w ) are the degradation value and the failure threshold af-

ter the transformation, respectively. Therefore, the obtained RUL

is just the hitting time that the transformed degradation process

hits the transformed threshold, rather than the FHT. Because the

Wiener-process based model is non-monotonous and the degrada-

tion signal may have already crossed the failure threshold during

, t

+ l

) , signifying failure, prior to estimating the RUL. Thus, the

associated solution in the above works is just an approximate to

the FHT. As shown in Si et al. (2013) , such approximate estimations

may lead to a decision delay when applied to CBM. In addition,

in above works, the obtained RUL distributions belong to a family

of Bernstein distributions. Consequently, the moments of the RUL

do not exist. But in maintenance practice, the expectation of the

RUL is required to be existent sometimes. Also, the stochastic co-

eﬃcients in Gebraeel et al. (2005) , Elwany and Gebraeel (2009) ,

Elwany and Gebraeel (2008) , Gebraeel et al. (2006) had some prior

distributions but no elaborated method is presented to select the

hyperparameters of the prior distributions. Typically, several sys-

tems historical degradation data of the same type are required to

determine the deterministic coeﬃcient and the unknown param-

eters in the prior distributions of the stochastic coeﬃcients. But

the scarcity of such historical degradation data of multiple sys-

tems is a commonly encountered case in practice, particularly for

newly armed systems. To overcome these limitations, Si et al. re-

considered an exponential degradation model based on Wiener

process, and transformed the exponential-type degradation paths

into linear paths and presented a degradation path dependent ap-

proach with an exact and a closed-form solution to the RUL esti-

mation in the sense of the FHT ( Si et al., 2013 ). At the meanwhile,

the deterministic coeﬃcient and the unknown hyperparameters in

the prior distributions of the stochastic coeﬃcients can be updated

when the new degradation observation is available through the

combination of Bayesian updating and EM algorithm. Following the

work in Si et al. (2013) , Li et al. (2015) applied the exponential-

type model in Si et al. (2013) to the rolling bearing degradation

modeling and RUL estimation, with an improved method in the

model parameter estimation by introducing the particle ﬁltering

algorithm. Besides the above log-transformation techniques, an-

other kind of the state transformation is the well-known Box–Cox

transformation ( Box & Cox, 1964 ). As for the Box–Cox transforma-

tion, the transformation function is deﬁned as Y (t) = g(X (t) ; λ) =

X(t )

−1

for λ  = 0; otherwise Y (t) = g(X(t) ; λ) = ln X (t ) , where λ is

the transformation parameter. It has been proved that the transfor-

mation function g ( X ( t ); λ) is continuous in λ = 0 and the Box–Cox

transformation includes the log-transformation as a special case.

It is noted that the Box–Cox transformation has been recently in-

troduced into the degradation modeling ﬁeld to transform nonlin-

ear degradation paths into nearly linear paths. For example, Liu,

Zhou, Liao, Peng, and Peng (2015) and Zhou, Huang, Chen, and Y

(2016) respectively presented several new health indicators, and

then the Box–Cox transformation was used to linearize nonlinear

paths where the transformation parameter λ was determined by

the MLE method based on the historical data. After the transforma-

tion, the relevance vector machine was utilized to learn the degra-

dation progression of the health indicator and thus the accuracy

of the RUL estimation could be improved. It is noted here that

the works in Liu et al. (2015) and Zhou et al. (2016) are not un-

der the framework of Wiener-process based degradation modeling,

but these two works promisingly indicate that the Box–Cox trans-

formation has the great potential to be applied to Wiener-process

based degradation modeling framework in handling the nonlinear-

ity. Therefore, we discuss the Box–Cox transformation as a state

transformation technique for interested readers. In these works, it

is found that, for nonlinear degradation process which can be lin-

earized through the above introduced state-transformation, several

important results of linear Wiener degradation models including

the PDF of lifetime, online updating of the framework for the RUL

estimate, and the degradation test or accelerated degradation test

(ADT) design procedures can be directly inherited. Thus, imple-

mentation of these types of methods is relatively easy. However,

the key to the methods using the state-transformations lies in se-

lecting the appropriate transformation function g ( · ), which is usu-

ally case-dependent and challenging in practice.

The time-scale transformation technique is another line han-

dling nonlinearity in degradation processes. In this case, the degra-

dation model in (4) will be reconstructed as

dX (t ) = dY ((t)) = μd(t) + σ

dB ((t)) , (7)

where μ and σ

are the constant parameters, ( t ) is the

time-scale transformation function and its speciﬁc form is case-

dependent and will be determined by the prior knowledge of the

concerned system. According to the deﬁned degradation process in

(7) , we can ﬁnd that the transformed degradation process { Y ( ( t )),

t ≥ 0} evolves linearly in the time scale ( t ).

The above time-scale transformation technique was ﬁrst pro-

posed in Whitmore (1995) . In Whitmore (1995) , the original

accelerated degradation data of a self-regulating heating cable

was linearized through using the time-scale functions (t) = 1 −

exp

(−λt

) and (t) = t

to extrapolate the lifetime under normal

working conditions. Huang et al. recently developed a Wiener pro-

cess model with an adaptive drift in the form of (7) , and applied

it to achieve an online RUL estimation of rotating bearings ( Huang,

Xu, Wang, & Sun, 2015 ). The parameters estimation and updating

method was inherited from Si et al. (2013) by using the stochastic

ﬁltering technique and the EM algorithm. Note that the transfor-

mation of X ( t ) into Y ( t ) is not required in the time-scale transfor-

mation.

Together with the above discussions, it can be observed that

the conventional Wiener-process-based degradation models have

long been developed and applied to many practical systems. How-

ever, as mentioned in Section 1 , the associated degradation pro-

cesses of complex systems are usually affected by many factors

including nonlinearity, multi-source variability, covariates, and bi-

variate or multivariate. Particularly, the models described in (3) or

(4) , which assume that the model parameters are deterministic

and the degradation path is simply a linear or nearly linear func-

tion of time, have the limited ability in handling the complicated

phenomena encountered in the industry. Though two kinds of the

transformation techniques have been developed to handle nonlin-

earity, the above reviewed works are limited to the cases in which

such transformations exist. Unfortunately, only a few nonlinear

processes can be transformed in these manners. The issues of how

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