Z. Zhang et al. / European Journal of Operational Research 0 0 0 (2018) 1–22 5
ARTICLE IN PRESS
JID: EOR [m5G; March 7, 2018;12:7 ]
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
k
=
{ l
k
: Y (t
k
+ l
k
) ≥ g(w ) } . As such, the RUL distribution was formu-
lated through Pr (L
k
≤ l
k
) = Pr (Y (t
k
+ l
k
) ≥ g(w )) , where Y (t
k
+ l
k
)
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
k
, t
k
+ l
k
) , 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)
λ
−1
λ
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
B
dB ((t)) , (7)
where μ and σ
B
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
γ
) and (t) = t
γ
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
Please cite this article as: Z. Zhang et al., Degradation data analysis and remaining useful life estimation: A review on Wiener-process-
based methods, European Journal of Operational Research (2018), https://doi.org/10.1016/j.ejor.2018.02.033
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