untitled


A publication of 

CCHHEEMMIICCAALL EENNGGIINNEEEERRIINNGG TTRRAANNSSAACCTTIIOONNSS

VOL. 33, 2013
The Italian Association 

of Chemical Engineering 
Online at: www.aidic.it/cet

Guest Editors: Enrico Zio, Piero Baraldi 
Copyright © 2013, AIDIC Servizi S.r.l., 
ISBN 978-88-95608-24-2; ISSN 1974-9791                                                                                    

RUL Estimation Using an Adaptive Inverse Gaussian Model 
Wenjia Xua, Wenbin Wangb 

aSalford Business School, University of Salford, Salford, M5 4WT, UK 
bDongling School of Economics and Management, University of Science and Technology Beijing, 30 Xueyuan 
Road  Haidian, Beijing, China 
wangwb@ustb.edu.cn 

In this paper, an adaptive inverse Gaussian stochastic process is developed to characterize the 
degradation process of condition monitored components. The knowledge of the degradation process is 
updated through the parameter of the process when new observations are available. The updating is 
performed through a general Bayesian filtering process within a state space model setting. The proposed 
adaptive model is history-dependent and can adjust itself to the sudden changes in degradation signals. 
The numerical case study shows that the variance of the RUL distribution obtained from the adaptive 
model is less than that of the conventional inverse Gaussian model and the predictive accuracy is 
improved by using the adaptive model in terms of TMSE. To validate our adaptive model further, we 
conduct a model prediction accuracy test. Our test result enables us to conclude that our model is stable, 
robust and beneficial for the application in prognostics and health management of systems. 

1. Introduction 
With the appearance of large industrial systems covering wide areas or large populations, such as 
electrical power systems, marine and aviation systems, prognostics has become increasingly important for 
optimising the operation of these systems and their critical components in terms of higher reliability, less 
cost, higher production quality, better inventory planning and more effective reuse of industrial resources 
(Jardine et al. (2006)). A timely and correct prognostic method could assist the selection of effective 
condition based strategies for industrial purposes, especially for vital or critical systems (Pecht (2008)). An 
effective prognostic strategy usually involves the accurate prediction of residual useful life (RUL) for a 
system under investigation.  
Stochastic processes have been used widely for modelling degradation because of uncertainty relating to 
the nature of system deterioration (Engel et al., 2000). The RUL of such a system can be solved through 
finding the first passage time (FPT) of the degradation process. The inverse Gaussian (IG) process is best 
known as corresponding to the first passage time of a Brownian motion with drift. Nevertheless, the 
utilization of the IG process to model degradation has not drawn much attention. Only Wang and Xu 
(2010) appeared to use an IG process to describe component degradation for the purpose of time to 
failure prediction. However, in their paper, they considered predicting the RUL using only current 
degradation information and did not investigate the impact of past CM information on the prediction of 
RUL. This issue has been addressed both in Si et al. (2011a) and Wang et al. (2011), but not in the 
context of IG processes. However they have demonstrated that using both past and current CM 
information is critical to the accuracy of prognosis. Therefore, we believe it is necessary to make the IG 
degradation process adaptive to past information for the purpose of RUL prognosis.  

2. Adaptive degradation modelling using an inverse Gaussian process 

2.1 Inverse Gaussian process 
An inverse Gaussian process is a stochastic process with independent inverse Gaussian distributed 
increments . We now define an inverse Gaussian process as described in Chhikara and Folks (1989), with 

 
 
 
 
 
 
 
 
 
 
 
 

 
 

DOI: 10.3303/CET1333056 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Xu W., Wang W., 2013, Rul estimation using an adaptive inverse gaussian model, Chemical Engineering
Transactions, 33, 331-336  DOI: 10.3303/CET1333056

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mean parameter bt  and scale parameter 2tλ , as a stochastic process { , 0}tX t ≥  with the following 

properties: 

1. 0 0X =  with probability one; 

2. tX  has independent increments, so that for every pair of disjoint intervals 1 2( , )t t  and 3 4( , )t t  with 

1 2 3 4t t t t< < < , the random variables
2 1t tX X− and 

4 3t tX X− are independent; 

3. Each increment tX Xτ −  has an inverse Gaussian distribution with mean ( )b tτ −  and scale 

parameter 2( )tλ τ − , for all tτ > . 
Let L be the threshold of an inverse Gaussian process, and suppose that we are interested in the 
distribution of the time T that the degradation first exceeds L, assuming it starts from 0. Here we use FT(t) 
to denote the distribution function of FPT. Since it is a non-decreasing process, the definition of FTP 

{ }inf : tT t X L= ≥  is equivalent to { }: tT t X L= ≥ . It follows that the distribution of FPT for an 

inverse Gaussian process is derived in the following way (Wang and Xu, 2010),  

( ) P 2ex( ) pT t
L Lt tF t X L
b b

t
L L b
λ λ λ= Φ − − − Φ − += ≥ (1) 

where ( )Φ ⋅  is the CDF of standard normal distribution. By differentiating Equation (1) with respect to t, 
we obtain a formula for the PDF of the first passage time T. 

2.2 Adaptive degradation modelling 
An inverse Gaussian process tΓ  with mean parameter 

it tb b t=  and scale parameter 2tλ  can be used 

for describing the evolution of the degradation based on obtained condition monitoring (CM) information. 

We use some additional notation: m is the number of CM points before failure; it  is the time of the ith CM 

point; tY  is a random variable representing the CM information at time t; ty  is the observed realisation of 

tY ; 
0 1

{ , ,..., }
i it t t ty y y=Y  is the history of degradation observations for the plant up to time it . At each 

CM point it , we observe the current degradation observation 
it
y . The future degradation ty  , it t>  is 

regarded as a three-parameter inverse Gaussian  random variable with location parameter 
it
y , mean 

parameter ( )
it ib t t−  and scale parameter 2( )it tλ − . The probability density function of ty  is given as.  

22
2

3

( )( ) 1( | ; , ) exp{ ( ( ) ) }
2 2

i

i i i

i i

i ti
t t t t i

t t

t t vt tp y y b b t t
v v

λλλ
π

−−= − − − (2)

2.3 Updating the mean parameter 
From section 2.2, it is noted that 

it
b  is not a constant and regarded as a random variable. Therefore, we 

construct a state space model for the evolvement of 
it
b . In this state space setting, 

it
b  can be updated 

through a Bayesian filtering process at each CM point once new information is obtained. It is worth noting 

that the updating process for 
it
b  is difficult as an analytical form is not generally available. As such, we 

make a few assumptions and attempt to obtain a closed form for 
it
b . First, we assume that CM 

information is collected based on a fixed time interval. This is a mild and reasonable assumption as, in 
practice, fixed time sampling is widely used and is also convenient. For notational simplicity, we will use 

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tΔ  to denote the equal interval. The increment 
it
v  is the value between the current checking point 

it
y  

and the last point 
1it

y
−

, 
1i i it t tv y y

−
= − . In the following work, we use 

1 2
{ , ,..., }

i it t t tv v v=V  as the 

observed information which is equivalent to 
0 1

{ , ,..., }
i it t t ty y y=Y . According to the properties of an 

inverse Gaussian process, at each CM point it , the increment follows an inverse Gaussian distribution, 

that is 2( | ) ( ; , )
i i i it t t tp v b IG v b t tλ= Δ Δ . The mean parameter 

it
b  before considering the observed 

it
y  

is assumed to satisfy 
1i it tb b

−
= . Then the semi-deterministic state space equation (Equation (3)) is 

constructed for the updating of 
it
b . Other state space equations can also be considered, and the reasons 

we choose this particular type lie in: 1). An explicit solution can be obtained in this model setting; 2. it is 
reasonable to assume that in between two checking points, the mean parameter does not change 
dramatically. In other cases, a particle filtering method is needed to get an approximate solution. 

1

22
2

3
1( | ) exp{ ( ) }

2 2

i i

i

i i i

i i

t t

t
t t t

t t

b b

t vtp v b b t
v v

λλ
π

−
=

ΔΔ= − Δ −
(3)

Then, under a Bayesian filtering process, we have  from Equation (3) that  

1 1

1

1
10

( | ) ( | ) ( | )
( | ) ( | , )

( | ) ( | ) ( | )
i i i i i i i

i i i i i

i i
i i i i i

t t t t t t t
t t t t t

t t t t t t t

p v b p v b p b
p b p b v

p v p v b p b dk
− −

−

−
−

∞= = =
V V

V V
V V

 (4) 

Recursively applying the Bayes’ rule, we have the probability distribution of 
it
b  conditional on the collected 

information up to time it  

1 1

1 1

0

00

( | ) ( | )... ( | ) ( | )
( | )

( | ) ( | )... ( | ) ( | )
i i i i i i

i i

i i i i i i i

t t t t t t t
t t

t t t t t t t t

p v b p v b p v b p b V
p b

p v b p v b p v b p b V dk
−

−

∞=V  (5) 

As 
1i it tb b

−
= is assumed,

1 00 0 0( | ) ( | ) ( | )
i it t tp b V p b V p b V

−
= =  holds. This largely decreases the 

computational burden of the estimation process and this is another reason why we choose the 

deterministic form 
1i it tb b

−
= . Then the distribution of 

0 0( | )tp b V  is essential to get an explicit solution of 

Equation (5). In Banerjee and Bhattacharyya (1979) and Seshadri (1993), it has been established that if 
the prior distribution of a random variable follows a truncated normal distribution, left truncated at 0, then 
the posterior distribution should be from the gamma family. Now we assume that the distribution of 

0 0( | )tp b V  follows a left truncated normal distribution with parameters 01 / β  and 01 /τ :  

2
00

2
00 0

0

2 20 0 00 0

0 0

( 1/ )
( 1/ )2/

2/
0

0 ( 1/ ) ( 1/ )
2/ 2/

0 00 0
0

1
2 /

( | )
1

2 /

t

t

t t

b
b

t b b

e
ep b V

e db e db

β
βτ

τ

β β
τ τ

π τ

π τ

−
−

−
−

− −
− −∞ ∞

= =  (6) 

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Substituting the PDF of 
0 0( | )tp b V  into Equation (5), we have the distribution of the mean parameter 

it
b  

detailed in equation (7) through several manipulations: 

 

2

2

0

( 1 / )
exp{ }

2( | )
[ 1 / ]

exp{ }
2

i

i i

i

i

i t i

t t
i t i

t

b

p b
b

db

τ β

τ β∞

− −

=
− −

V (7) 

where 2
0

1
j

i

i t
j

t vτ λ τ
=

= Δ + , 0 0

2
0

1

/ ( )1 /
/

j

i i

t
j

i t

t v

τ λββ
τ λ

=

Δ +=
Δ +

. 

For simplicity, we let 0 0τ αλβ=
, then the parameters iβ

 and iτ
 in the PDF 

( | )
i it tp b V

 can be 

expressed in terms of parameters λ , α  and 0β
 as. 

2
0

1 2
0

1

,
j

j

i

t i
j

i i t
j

t v
t v

i t

αβ
β τ λ αλβ

α
=

=

Δ +
= = Δ +

Δ +
 (8) 

Having the density function of ( | )
i it tp b V  available, we can substitute the mean of 

it
b  into Equation (1) 

in order to get the time to failure or RUL at each monitored point. Alternatively, we could utilize the whole 

distribution of ( | )
i it tp b V  for the calculation of RUL as discussed in Si et al. (2013). 

3. Numerical example 
In this section, we provide a numerical example of crack growth to demonstrate the performance of an 
adaptive inverse Gaussian process based degradation model and the conventional inverse Gaussian 
model. The crack growth data are from Lu and Meeker (1993).The expectation maximization (EM) 
algorithm (Dempster et al., 1977) is applied for parameter estimation as hidden variable is used in our 
model. The parameters in the conventional inverse Gaussian model are estimated using the maximum 
likelihood estimation algorithm (Kendall and Alan, 1973). The estimated parameters for both models are 
presented in Table 1. 

Table 1 Estimated parameter values for both models 

Parameter Adaptive  model Parameter Conventional model 
 
0 

31.6860 
61.7136 b 0.0602 

 0.1863  0.3240 
Using the estimated increments, the tracking ability of the adaptive model can be demonstrated. Figure 1 
compares the length of a real crack and the predicted results of the crack length. It can be seen that the 
predicted results from the adaptive model are closer to the real data. It should be noted that the real crack 
length increases at a higher rate after check time eight. Since the adaptive model adjusts its parameters 
from new observed information, the new crack length information is utilized in the model to update the 
parameters for future prediction after each check time. From Figure 1, it is shown that the estimated crack 
length agrees with the suddenly increasing rate of the real crack growth at check point eight and the 
adaptive model quickly catches up with this change, by adjusting itself to follow the real situation.  
Using the estimated parameters, we can also derive the PDF of the failure time or RUL using Equation (1). 
For comparison purposes, we plot the RUL PDFs for both models, the adaptive inverse Gaussian based 
model and the conventional inverse Gaussian based model, in Figure 2. The real RULs are lying in 95 % 
confidence interval of RUL distribution obtained from the adaptive model at each CM point. While for the 

334



RUL distribution obtained from conventional model, the real RUL at the last CM point falls out of the 95 % 
confidence interval. In addition, the real RULs are noticeably much closer to the modes of the RUL 
distributions obtained from the adaptive model as time progresses. Furthermore, the variances of the 
RULs produced by the adaptive model are apparently smaller than those of the conventional model, which 
shows that the predictive accuracy of the former is better than that of the latter. 
 

0 2 4 6 8 10
0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Time

C
ra

ck
 le

ng
th

 

 
real crack length
estimated mean crack length

Figure 1 Crack length prediction Figure 2 Residual life PDFs of adaptive inverse 
Gaussian model (blue) and conventional inverse 
Gaussian model (red bold) 

In addition, we compare, for each testing data group, the probabilities of late prediction of both models in 
Table 2, which are the probabilities of predicting the RUL later than the real RUL. In asset management, 
late predictions might cause catastrophic consequences and late alarms will be punished heavily. Thus the 
probability of late prediction is usually used as a popular criterion for many safety critical systems. A 
smaller probability of late prediction commonly ensures a higher reliability of the system. 

Table 2 Average late prediction probability 

Average late prediction probability Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 
Adaptive inverse Gaussian 0.5949 0.5785 0.5722 0.5781 0.5704 0.5714 
Conventional inverse Gaussian 0.6018 0.5837 0.5786 0.5839 0.5784 0.5768 
From Table 2, we note that the difference between total late prediction probabilities produced from both 
models is not great, but the outcomes from the conventional inverse Gaussian process based model are 
still slightly larger than those from the adaptive inverse Gaussian model for all the six groups of testing 
data. 
Next, the total mean square errors (TMSE) of these two models are calculated and the results are 
presented in Table 3. TMSE is a key criterion for model comparison and selection, and a model with 
smaller TMSE is usually regarded as a better model. The results show that the TMSE of the adaptive 
inverse Gaussian model is about 35 % smaller than that of the conventional inverse Gaussian model, 
which provides further evidence in support of the conclusion that the proposed adaptive model is better 
than the conventional model. 

Table 3 TMSE comparison 

Model TMSE 
Adaptive inverse Gaussian model 321.1910 
Conventional inverse Gaussian model 498.6807 

To validate our model further, we conduct a model prediction accuracy test. We choose six groups at 
random from the twelve available; then we train the six groups and test with the other six groups. The 
process is repeated a further three times, so that we conduct a total of four independent experiments. 

335



Throughout the parameter estimation and testing procedures for both the adaptive Gaussian model and 
the conventional Gaussian model, we get the prediction results shown in Table 4. By comparing these 
model outputs, we can assess whether the model is sensitive to the data change. From Table 4, we can 
see that our model performs reasonably consistently for the given different datasets and outperforms the 
conventional Gaussian model respectively in terms of TMSE. Therefore, our prediction accuracy test 
enables us to conclude that our model is stable, robust and beneficial for the analysis of different datasets. 

Table 4 TMSE results of four trials for conventional and adaptive Gaussian models 

Model TMSE Trial 1 Trial 2 Trial 3 Trial 4 
Adaptive Gaussian model 276.9428 269.572 343.4941 290.4350 
Conventional Gaussian model 451.907 449.7512 511.4516 468.3932 

4. Conclusions 
In this paper, an adaptive inverse Gaussian stochastic process is developed to characterize the 
degradation process of monitored components. The knowledge of the degradation process is updated 
through the mean parameter of the inverse Gaussian process when new observations are available. The 
updating is performed through a general Bayesian filtering process within a state space model setting. The 
proposed adaptive model is history-dependent and could adjust itself to the sudden changes in 
degradation signals. The numerical case study shows that the variance of the RUL distribution obtained 
from the adaptive model is less than that of the conventional inverse Gaussian model and the predictive 
accuracy is improved by using an adaptive model in terms of TMSE. Furthermore, a model prediction 
accuracy test is carried out to validate our model. The results show that the adaptive Gaussian model is 
robust and can be applied to different datasets with desirable results. 

Acknowledgement 

This research is partially supported by NSFC under grant number 71231001, by the MOE PhD 
supervisor’s fund and by the Fundamental Research Funds for the Central Universities of China FRF-
SD-12-020A. 

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