Statistical Analysis of Modern Reliability Data

Abstract

Traditional reliability analysis has been using time to event data, degradation data, and recurrent event data, while the associated covariates tend to be simple and constant over time. Over the past years, we have witnessed the rapid development of sensor and wireless technology, which enables us to track how the product has been used and under which environmental conditions it has been used. Nowadays, we are able to collect richer information on covariates which provides opportunities for better reliability predictions. In this chapter, we first review recent development on statistical methods for reliability analysis. We then focus on introducing several specific methods that were developed for different types of reliability data with covariate information. Illustrations of those methods are also provided using examples from industry. Test planning is also an important part of reliability analysis. In addition to data analysis, we also provide a briefly review on recent developments of test planning and then focus on illustrating the sequential Bayesian design with an example of fatigue testing for polymer composites. The paper is concluded with some discussions and remarks.

Full text

Statistical Analysis of Modern Reliability Data
Yueyao Wang
Department of Statistics
Virginia Tech
Blacksburg, VA 24061
I-Chen Lee
Department of Statistics
National Cheng Kung University
Tainan, Taiwan
Lu Lu
Department of Mathematics & Statistics
University of South Florida
Tampa, FL, 33620
Yili Hong
Department of Statistics
Virginia Tech
Blacksburg, VA 24061
Abstract
Traditional reliability analysis has been using time to event data, degradation data,
and recurrent event data, while the associated covariates tend to be simple and constant
over time. Over the past years, we have witnessed the rapid development of sensor
and wireless technology, which enables us to track how the product has been used and
under which environmental conditions it has been used. Nowadays, we are able to
collect richer information on covariates which provides opportunities for better reliability
predictions. In this chapter, we first review recent development on statistical methods
for reliability analysis. We then focus on introducing several specific methods that were
developed for different types of reliability data with covariate information. Illustrations
of those methods are also provided using examples from industry. Test planning is also
an important part of reliability analysis. In addition to data analysis, we also provide a
briefly review on recent developments of test planning and then focus on illustrating the
sequential Bayesian design with an example of fatigue testing for polymer composites.
The chapter is concluded with some discussions and remarks.
Key Words: Degradation data, Dynamic covariates, Lifetime data, Recurrent
events data, Reliability prediction, Sequential test planning.
1

1 Introduction
1.1 Background
Traditional reliability data analysis mainly use time to event data, degradation data, and
recurrent event data to make reliability predictions [1]. The covariate information involved in
the reliability analysis is usually time-invariant and the number of covariates is typically small.
For time to event data, parametric models such as the Weibull and lognormal distributions are
popular and accelerated failure time models are often used to incorporate covariate information
on accelerating factors. For degradation data, the general path models and stochastic models
are the common choices and the covariate information is often incorporated through regression
type of models. The recurrent event data are often modeled by the event intensity models or
mean cumulative functions with regression type of models that are often used to incorporate
covariates.
With technological advances, new information on covariates become available. Products
and systems can be equipped with sensors and smart chips to keep track of various information
on the field usage of product units, number of transfers, and environmental conditions such as
temperature and humidity. Such covariate information often change over time, so we refer to
them as dynamic covariate information. Because the dynamic covariates often come in large
volume and variety, it presents big data opportunities and challenges in the area of reliability
analysis (e.g., [2] and [3]). Dynamic covariate data can be used for modeling and prediction
of reliability because units under heavy usage often fail sooner than those lightly used. In
recent years, more statistical methods for dynamic covariates have been being developed to
make use of this new type of covariate data.
Another important area of reliability analysis is about test planning, which focuses on
how to efficiently collect various types of data to make better prediction of reliability. For
accelerated life tests (ALTs), it is especially challenging to timely collect sufficient failure data
because the data collection is a time-consuming process and often requires using expensive
equipment for testing units under elevated stress conditions. In some laboratories, there
are typically one or two machines available for testing certain material. In this case, it is
impractical to test multiple samples simultaneously and therefore limits the total obtainable
sample size. Another challenge with traditional test planning is that it typically relies on a
single set of best guess of the parameter values, which may lead to suboptimal designs when
the specified parameter values are not accurate. Due to these challenges, sequential designs
become popular where earlier test results can be utilized to determine the test conditions for
later runs. In addition, Bayesian methods can be used to leverage prior information from the
expert’s knowledge or related historical data to inform the test planning. The objective of
2

this chapter is to review current development and then introduce the statistical methods for
dynamic covariates and sequential Bayesian design (SBD) for ALT.
1.2 Related Literature
In lifetime data analysis, product usage information has been used to improve reliability model.
Lawless et al. [4] consider warranty prediction problem using product usage information on
return units. Constant usage information are used in [5] and [6]. Averaged product use-rate
information are used in [7]. Nelson [8] and Voiculescu et al. [9] use cumulative exposure model
in ALT and reliability analysis. Hong and Meeker [10] use cumulative exposure model to
incorporate dynamic covariates and apply it to the Product D2 application.
In degradation data analysis stochastic process models are widely used. The Wiener
process ([11, 12, 13]), Gamma process ([14]), and Inverse Gaussian process ([15, 16]) are
among popular models in this class. The general path models are also widely used, which
include [17, 18, 19, 20, 21]. For accelerated destructive degradation tests, the typical work
includes [22, 23, 24]. Hong et al. [25] and Xu et al. [26] develop degradation model using the
general path model framework to incorporate dynamic covariate information.
For recurrent events data, the nonhomogeneous Poisson process (NHPP) and the renewal
process (RP) are widely used (e.g., [27, 28]). Kijima [29] introduce virtual age models which
can model imperfect repairs. Pham and Wang [30] develop a quasi-renewal process, and
Doyen and Gaudoin [31] propose models for imperfect repairs. The trend-renewal process
(TRP) proposed in [32] can include the NHPP and RP as special cases, which has been used
in [33, 34, 35] and other places. Xu et al. [36] develop a multi-level trend renewal process
(MTRP) model for recurrent event with dynamic covariates.
For test planning, the optimum designs in traditional test planning framework are devel-
oped using non-Bayesian approaches (e.g., [37, 38]) and the true parameters are assumed to
be known. Bayesian test planning for life data is developed in [39, 40, 41]. King et al. [42]
develop optimum test plans for fatigue test of polymer composites. Lee et al. [43] develop
SBD test planning for polymer composites and Lu et al. [44] extend it to test planning with
dual objectives.
1.3 Overview
The rest of this chapter is organized as follows. Section 2 describes an application on time-
to-event data with dynamic covariates. Section 3 illustrates the modeling of degradation with
dynamic covariates. Section 4 describes the MTRP model for describing recurrent event data
with dynamic covariates. Section 5 introduces SBD strategies for ALTs. Section 6 contains
3

Weeks after Installation
Unit No.
0 10 20 30 40 50 60 70
1
5
10
15
20
censored failed
0 10 20 30 40 50 60 70
0.2
0.5
1.0
2.0
5.0
10.0
Weeks after Installation
Use Rate
(a) Failure-time Data (b) Use-rate processes
Figure 1: (a) The event plot for a subset of the Product D2 failure-time data and (b) the
corresponding plot of the use-rate trajectories. Figure reproduced with permission.
some concluding remarks.
2 Time to Event Data Analysis
In this section, we briefly introduce the application of using dynamic covariates for time to
event prediction as described in Hong and Meeker [10].
2.1 Background and Data
A general method was developed by Hong and Meeker [10] to model failure-time data with
dynamic covariates. The work was motivated by the Product D2 application, which is a
machine used in office/residence. Product D2 is similar to high-end copy machine where the
number of pages used could be recorded dynamically. For this product, the use-rate data R(t)
(cycles/week) were collected weekly as a time series for those units connected to the network.
This information could be downloaded automatically in addition to the failure-time data. In
the Product D2 dataset, data were observed within a 70-week period and 69 out of 1800 units
failed during the study period. Figure 1 illustrates the event plot of the failure-time data and
the use-rate over time for a subset of the data.
4

2.2 Model for Time to Event and Parameter Estimation
Three sets of observable random variables: the failure time, censoring indicator and dynamic
covariate over time are are considered, which are denoted by {T, ∆,X(T)}. The observed
data are described by {ti, δi,xi(ti)}. Here ndenotes the number of units in the dataset, tiis
the failure time or time in service, and δiis the observed censoring indicator (i.e., it equals
to 1 if unit ifails and 0 otherwise). The xi(ti) = {xi(s) : 0 ≤s≤ti}is the observed dynamic
covariate information of unit ifrom the time 0 to ti, where xi(s) is the observed covariate
value at time sfor unit i. Particularly for Product D2, we use X(t) = log[R(t)/R0(t)] as the
form of the covariate in the model, where R0(t) is the baseline use-rate that is chosen to be a
typical constant use rate.
The cumulative exposure model in [45] is used to model the failure-time data with dynamic
covariate. The cumulative exposure u(t) is defined as,
u(t) = u[t;β, x(t)] = Zt
0
exp[βx(s)]ds,
where βrepresents the influence of the covariate on the exposure. When the cumulative
exposure of a unit reaches a random threshold Uat time T, the unit fails. This establishes a
relationship between Uand T, that is,
U=u(T) = ZT
0
exp[βx(s)]ds. (1)
Under the above model and the covariate history x(∞), the cumulative distribution function
(cdf) of the failure time Tis
F(t;β, θ0) = Pr(T≤t) = Pr{U≤u[t;β, x(t)]}=F0{u[t;β, x(t)]; θ0}
and probability density function (pdf) is f(t;β, θ0) = exp[βx(t)]f0{u[t;β, x(t)]; θ0}.Here θ0
is the parameter in the baseline cdf of the cumulative exposure threshold Uand f0(u;θ0) is
the pdf of U. In the Product D2 application, the baseline cumulative exposure distribution
F0(u;θ0) was modeled by the Weibull distribution, of which the cdf and pdf are
F0(u;θ0) = Φsev log(u)−µ0
σ0and f0(u;θ0) = 1
σ0uφsev log(u)−µ0
σ0.
In the above expression, θ0= (µ0, σ0)′, where µ0and σ0are the location and scale parameters.
Also, Φsev(z) = 1 −exp[−exp(z)], and φsev(z) = exp[z−exp(z)]. Lognormal and other log-
location-scale distributions can also be used if they are considered appropriate for certain
applications.
5

2.3 Model for Covariates
To model the covariate process, we use the linear mixed effect model. In particular, X(t) is
modeled as
Xi(tij ) = η+Zi(tij )wi+εij.(2)
In model (2), ηis the constant mean, and the term Zi(tij )wiis used to model variation at
individual level. Here Zi(tij ) = [1,log(tij )] and wiis the vector of random effects of the initial
covariate at time 0 and the changing rate for unit i. It is assumed that wi= (w0i, w1i)′∼
N(0,Σw) with the covariance matrix
Σw=σ2
1ρσ1σ2
ρσ1σ2σ2
2,
and εij ∼N(0, σ2) is the error term.
The parameter estimation is conducted in two parts since parameters in the failure-time
model θT= (θ′
0, β)′and covariates process model θX= (η, Σw, σ2) are separate, using the
maximum Likelihood (ML) method. The joint likelihood for θTand θXis
L(θT,θX) = L(θT)×L(θX).(3)
The first component of (3) is the likelihood function of the failure-time data, which is
L(θT) =
n
Y
i=1
{exp[βxi(ti)]f0(u[ti;β, xi(ti)]; θ0)}δi{1−F0(u[ti;β, xi(ti)]; θ0)}1−δi.(4)
The second component of (3) is the likelihood of covariate data, which is
L(θX) =
n
Y
i=1 Zwi

Y
tij ≤ti
f1xi(tij )−η−Zi(tij )wi;σ2

f2(wi;Σw)dwi.(5)
In the above equation, f1(·) is the pdf of a univariate normal and f2(·) is the pdf of a bivariate
normal distribution.
2.4 Reliability Prediction
In order to predict future field failures, the distribution of the remaining life (DRL) is con-
sidered in the prediction procedure. The DRL describes the distribution of Tifor unit igiven
Ti> tiand Xi(ti) = xi(ti). Particularly, the probability of unit ifailing within the next s
time units given it has survived by the time tiis
ρi(s;θ) = Pr[ti< Ti≤ti+s|Ti> ti,Xi(ti)], s > 0,(6)
6

where θdenotes all parameters. Then ρi(s;θ) can be further expressed as
ρi(s;θ) = EXi(ti,ti+s)|Xi(ti)=xi(ti){Pr[ti< Ti≤ti+s|Ti> ti,Xi(ti),Xi(ti, ti+s)]}(7)
=EXi(ti,ti+s)|Xi(ti)=xi(ti){F0(u[ti+s;β, Xi(ti+s)]; θ0)} − F0(u[ti;β, xi(ti)]; θ0)
1−F0(u[ti;β, xi(ti)]; θ0)
where Xi(t1, t2) = {Xi(s) : t1< s ≤t2}. Since model (2) is assumed for Xi(ti, ti+s) and
Xi(ti) = xi(ti), the multivariate normal distribution theory can be used to obtain the condi-
tional distribution.
The Monte Carlo simulation is used to evaluate ρi(s;b
θ) since an analytical expression for
ρi(s;θ) is unavailable. The following procedure is used to compute ρi(s;b
θ).
1. Substitute θXwith the ML estimates b
θXin the distribution of Xi(ti, ti+s)|Xi(ti) =
xi(ti), and draw X∗
i(ti, ti+s) from the distribution.
2. Let X∗
i(ti+s) = {xi(ti),X∗
i(ti, ti+s)}be the simulated covariate process in the time
interval (ti, ti+s).
3. Compute the DRL given X∗
i(ti, ti+s) and the ML estimates b
θTof θT= (θ′
0, β) by
ρ∗
i(s;b
θ) =
F0u[ti+s;b
β, X∗
i(ti+s)]; b
θ0−F0u[ti;b
β, xi(ti)]; b
θ0
1−F0u[ti;b
β, xi(ti)]; b
θ0.
4. Repeat steps 1-3 Mtimes and obtain ρ∗m
i(s;b
θ), m = 1,··· , M.
5. The estimate is computed by ρi(s;b
θ) = M−1PM
m=1 ρ∗m
i(s;b
θ).
The confidence intervals (CIs) for ρi(s;b
θ) can be obtained through the following procedure:
1. Draw b
θ∗′
Tand b
θ∗′
X)′from N(b
θT,Σb
θT) and N(b
θX,Σb
θX), respectively.
2. Let b
θ∗= (b
θ∗′
T,b
θ∗′
X)′and obtain ρ∗∗
i(s;b
θ∗) following the above algorithm.
3. Repeat steps 1-2 Btimes to obtain ρ∗∗b
i(s) = ρ∗∗b
i(s;b
θ∗b), b = 1,··· , B.
4. The 100(1 −α)% CI is computed by hρ∗∗[αB]
i(s), ρ∗∗[(1−α)B]
i(s)i. Here ρ∗∗[b]
i(s) is the [b]th
ordered value of ρ∗∗b
i(s) and [ ·] is the function for rounding to the nearest integer.
Figure 2 shows the estimated DRL for two representative units. One unit has a higher
use rate which increases quickly over time ( bw0= 0.4061,bw1= 0.4184) and the other has a
lower use rate which increases slowly over time ( bw0= 0.1704,bw1= 0.0168). The trends in the
7

0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
Weeks after DFD
Estimated Distribution of Remaining Life (DRL)
DRL for high use rate
95% pointwise CI
DRL for low use rate
95% pointwise CI
Figure 2: The estimated DRLs and the 95% pointwise CIs for two representative at-risk units.
Figure reproduced with permission.
plot are consistent with our expectation that units with higher use rates tend to have higher
failure risk.
To assess the prediction variability, one may also want to calculate the prediction interval
(PI) of individual remaining life, denoted by hS
ei,e
Sii. A 100(1 −α)% PI of the remaining
lifetime can be obtained by using the method introduced by [46] as in
ρi(S
ei;b
θ) = vα/2and ρi(e
Si;b
θ) = v1−α/2.(8)
Here vαis the αquantile of the ρi(Si;b
θ) distribution, and Sirepresents the remaining life for
unit i. The vαcan be obtained through a Monte Carlo simulation.
In real applications, obtaining the predicted cumulative number of failures is also important
because this could help with the decisions for warranty cost control or the long-term production
plan. Suppose N(s) = Pi∈RS Ii(s) is the total failure counts at time safter DFD. The RS is
the risk set at DFD in this expression and Ii(s) is the binary indicator for the occurrence of a
failure at time swith Ii(s)∼Bernoulli[ρi(s;θ)].Let FN(nk;θ), nk= 0,1,...,n∗denote the cdf
of N(s), where n∗is the count of units in the risk set. Then FN(nk;θ) can be computed in its
explicit form using a discrete Fourier transformation [47]. The PI for N(s) can be calculated
similarly as the individual predictions.
For the Product D2 application, 1731 units are remained at risk after 69 out of 1800
installed units failed. The point predictions and the pointwise 90% PIs of the total number
8

0 20 40 60 80 100
0
50
100
150
200
Weeks after DFD
Cumulative Number of Failures
point prediction
90% pointwise PI
Figure 3: The point predictions and the pointwise 90% PIs for the cumulative number of
failures after the DFD out of the 1731 units at risk. Figure reproduced with permission.
of failures after the DFD are shown in Figure 3.
3 Degradation Data Analysis
In this section, we briefly introduce how to leverage the dynamic covariates for modeling
degradation data as described in Hong et al. [25].
3.1 Background and Data
Hong et al. [25] develop a general path model utilizing shape-restricted splines with random
effects to model the degradation process with dynamic covariates. This paper considers an
application of modeling the photodegradation process of organic coating materials due to
exposure to the time-varying ultraviolet (UV) radiation and the outdoor environmental con-
ditions. In this work, to study the service life of a coating material, scientists at NIST placed
36 specimens in outdoor setting with varied UV spectrum and intensity, temperature, and rel-
ative humidity (RH) recorded over an approximate 5-year period. The specimens started at
different time points to allow different degradation paths to be observed. For each specimen,
degradation measurements were taken periodically using Fourier transform infrared (FTIR)
spectroscopy. Since a particular compound or structure is tied with a peak at a certain wave-
9

length on the FTIR spectrum, the change in the height of the peak was used to measure the
decrease in the concentration of the compound. One of the compounds of interest for the
NIST data was C-O stretching of aryl ether, which was measured at the wavelength 1250 cm.
Figure 4(a) shows the degradation paths of nine representative specimens with varied starting
times in the study. We can observe very different trajectories with the degradation rate varies
over time and among different specimens as well. Figures 4(b)-(d) show the dynamic covariate
information on the daily UV dosage, RH, and temperature as well as the fitted smooth lines
for showing the mean process of one specimen over the study period. The vertical lines are
used to label time windows separated by every six months. We can observe both a seasonal
pattern and a random oscillation of the daily records for each individual covariate. There
are stronger seasonal patterns for the UV dosage and temperature than the RH. There also
appears to be a larger variation of the daily observations in the summer than in the winter,
which indicates a varied degree of variability of the covariates over different time periods.
3.2 Model for Degradation Paths and Parameter Estimation
The general additive model for degradation data with dynamic covariates is given in the form
yi(tij ) = D[tij;xi(tij )] + G(tij ;wi) + εi(tij ),(9)
where yi(tij ) for i= 1,··· , n, j = 1,··· , niis the degradation measurement at time tij for unit
i,εi(tij )∼N(0, σ2
ε) denotes the measurement error, and xi(tij) = [xi1(tij ),...,xip(tij )]′is the
vector containing the dynamic covariate information at the time tij . The actual degradation
level at tij is modeled by D[tij;xi(tij)] + G(tij ;wi) as the sum of a fixed component and
a random component. The fixed component is the population common degradation path,
modeled in a cumulative damage form given by
D[tij ;xi(tij)] = β0+
p
X
l=1 Ztij
0
fl[xil(u); βl]du. (10)
This model incorporates the dynamic covariates through the covariate-effect functions fl(·)
for l= 1,··· , p. Here, β0is the initial degradation, fl[xil(u); βl] is the lth covariate-effect of
xil(u) on the degradation process at time u, and Rtij
0fl[xil(u); βl]du is the cumulative effect of
xil up to time tij. The random component includes the random effect terms for modeling the
unit-to-unit variation, which is specified in G(tij ;wi) = w0i+w1itij . Here, wi= (w0i, w1i)′is
the vector of random effects for the initial degradation and the growth rate over time, and it
is assumed to follow a bivariate normal distribution N(0,Σw) with the covariance matrix
Σw=σ2
0ρσ0σ1
ρσ0σ1σ2
1.
10

0 50 100 150 200
−0.5
−0.4
−0.3
−0.2
−0.1
0.0
Days since first measurement
Damage
5 10 15 20 25 30 35
0
10
20
30
40
50
60
70
Months since 01JAN2002
UV Dosage
data points fitted
(a) Degradation paths (b) Daily UV dosage
5 10 15 20 25 30 35
−10
0
10
20
30
40
50
Months since 01JAN2002
Temperature (°C)
5 10 15 20 25 30 35
20
40
60
80
100
Months since 01JAN2002
Relative humidity (%)
(c) Daily temperature (d) Daily RH
Figure 4: Plots of (a) nine representative degradation paths and (b)-(d) dynamic covariate
information on the daily UV dosage, temperature and relative humidity for a single sample.
The black dots connected by green lines show the daily values. The vertical lines show the
time windows by every 6 months from January 2002. The red smooth curves are the estimated
mean process. Figure reproduced with permission.
11

Also we use σw= (σ0, σ1, ρ)′to denote all distinct parameters included in Σw.
The ML method is used for estimating the parameters. Since the degradation measure-
ments and the dynamic covariates are observed at discrete time points, the discrete version
of the degradation path model is used for computing the likelihood by replacing D[tij;xi(tij )]
in (10) by
D[tij ;xi(tij )] = β0+
p
X
l=1 X
uik≤tij
fl[xil(uik ); βl](uik −ui,k−1),(11)
where uik is the kth time point when the degradation and covariates are measured for unit i
and ui0= 0. Let θD={β,σw, σε}denote all the model parameters. Then the likelihood is
L(θD) =
n
Y
i=1 Zwi
Y
tij ≤tini
1
σε
φC[yi(tij); xi(tij),wi]
σεgwi(wi;σw)
dwi(12)
where C[yi(tij); xi(tij ),wi] = yi(tij )−D[tij ;xi(tij )] −G(tij ;wi), φ(·) and gwi(·) are the pdfs
of a standard normal distribution and a bivariate N(0,Σw) distribution, respectively.
Considering there was not sufficient knowledge on what might be a sensible form for the
covariate-effect functions, the paper chose to estimate the fl(·) using a linear combination
of spline bases. To leverage the physical understanding of the relationships between the
degradation process and the covariates, the shape-restricted splines [48] were used to ensure
monotonic decreasing bases (I-splines) for the UV dosage and temperature and concave bases
(C-splines) for the RH. Let Blq[xil (uik )] for q= 1,··· , aldenote the spline bases for the
covariate xl, then the covariate-effect function is modeled as
fl[xil(uik ); βl] =
al
X
q=1
Blq[xil(uik )]βlq ,
where βlq’s are the spline coefficients. Define Ulq(tij ) = Puik ≤tij Blq [xil(uik)](uik −ui,k−1).
Then the model in (9) with D[tij ;xi(tij)] given in (11) can be written as a linear mixed effects
model in the form of yi=Xiβ+Ziwi+εi,where
Xi=


1U11(ti1)··· U1a1(ti1)··· Up1(ti1)··· Upap(ti1)
.
.
..
.
.....
.
.....
.
.....
.
.
1U11(tini)··· U1a1(tini)··· Up1(tini)··· Upap(tini)


,Zi=


1ti1
.
.
..
.
.
1tini


,
and the coefficient vector β= (β′
u,β′
c)′, where βuand βcdenote the unconstrained and
constrained parameters, respectively.
The following algorithm was proposed [25] to obtain the ML estimate b
θDthat maxi-
mizes (12):
12

1. Initiallize σwand σεby fitting a linear mixed-effects model with no constraints.
2. Compute Vi=ZiΣwZ′
i+σ2
ǫIi.
3. The mixed primal-dual bases algorithm in [49] is used to estimate β. That is to minimize
Pn
i=1(yi−Xiβ)′V−1
i(yi−Xiβ) subject to βc≥0.
4. Fit a linear mixed-effects model ri=Ziwi+εiwith ri=yi−Xib
βto get the updated
estimates of σwand σε.
5. Repeat steps 2 to 4 until the estimated parameters converge.
With the shape-restricted splines, the ML estimates of some parameters might locate
on the boundary of the parameter space. In this case, the bootstrap method is useful for
assessing the variability and making inference about the parameters. An adjusted bootstrap
procedure by [50] was applied to resample the residuals and the estimated random effects for
constructing bootstrap resamples of the original data to avoid underestimating variability and
producing too narrow CIs. Then the bias-corrected bootstrap CIs were constructed based on
obtaining the ML estimates of model parameters using the above mentioned algorithm for a
large number of bootstrap samples.
3.3 Model for Covariates
To predict the degradation process and reliability, it is necessary to model the dynamic co-
variate process through a parametric model. Hong et al. [25] propose the following model
Xj(t) = Trj(t) + Sj(t) + ξj(t),
where Trj(t) models the long-term trend of the covariate process for the jth covariate, Sj(t)
captures the seasonal pattern, and ξj(t) depicts the random error which is assumed to be a
stationary process. For the NIST outdoor weathering data, there was no significant long-term
trend observed, and hence Trj(t) = µjfor j= 1,2,3. However, the seasonal pattern was quite
prominent and there were seasonal effects observed for both the mean and variance of the
process. So two sine functions were included in both the seasonal and error terms (except for
RH which shows no seasonal effect assumed for the variation of the process from Figure 4) in
the following form


S1(t)
S2(t)
S3(t)
=


κ1sin 2π
365 (t−η1)
κ2sin 2π
365 (t−η2)
κ3sin 2π
365 (t−η3)

,

ξ1(t)
ξ2(t)
ξ3(t)
=

1 + ν11 + sin 2π
365 (t−ς1)ε1(t)
1 + ν21 + sin 2π
365 (t−ς2)ε2(t)
ε3(t)

.
(13)
13

To capture the autocorrelation within and among the covariate processes, a lag-2 VAR model
[i.e. Var(2)] was used, where the error term was modeled by


ε1(t)
ε2(t)
ε3(t)
=Q1

ε1(t−1)
ε2(t−1)
ε3(t−1) 
+Q2

ε1(t−2)
ε2(t−2)
ε3(t−2) 
+

e1(t)
e2(t)
e3(t)
(14)
In the above equation, Q1and Q2are regression coefficients matrices, and [e1(t), e2(t), e3(t)]′∼
N(0,Σe) are multivariate normal random errors that do not change over time.
The parameters in models (13) and (14) are estimated in two steps. First, the ML estimates
of the seasonal effects in the process mean and variance structures are obtained by ignoring the
autocorrelation in the error terms. Then the VAR model is fitted to the residuals calculated
from the first step using the multivariate least squares approach [51]. The bootstrap method
is used for obtaining the CIs of the parameters in the dynamic covariate process.
3.4 Reliability Prediction
To predict the failure time and reliability, let Dfdenote the threshold for a soft failure. For any
X(∞) = x(∞) and w, the degradation path is fixed and the failure time can be determined
by
tD= min{t:D[t;x(∞)] + G(t;w) = Df}.(15)
However, for a random unit, the covariate process X(∞) and ware random. Hence, the cdf
of the failure time, T=T[Df,X(∞),w], can be defined as
F(t;θ) = EX(∞)EwPr {T[Df,X(∞),w]≤t}, t > 0,(16)
with θ={θD,θX}denoting all the unknown parameters. There is usually no closed form
expression of F(t;θ). Hence, the cdf at any estimated b
θis estimated through Monte Carlo
simulation outlined in the following steps [25].
1. One need to simulate the covariate process based on the estimated parameter b
θX.
2. Then one can simulate the random effects wfrom N(0,Σw) with the estimated parameter
b
θD.
3. Compute D[t;X(∞)] + G(t;w) based on the simulated covariate process and random
effects.
4. For the degradation path in step 3, determine the failure-time tDby Eqn. (15).
14

0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
Days after first measurement
Probability
point estimates
95% pointwise CIs
Figure 5: The estimated cdf and corresponding 95% pointwise CIs for a population of units
with random starting time between 161 and 190 days. Figure reproduced with permission.
5. Repeat steps 1 to 4 for Mtimes to obtain the simulated failure-times tm
D, m = 1,...,M.
Then F(t;b
θ) is estimated by the empirical cdf, F(t;b
θ) = M−1PM
m=1 1(tm
D≤t).
By using the bootstrap approach, the point estimates and the CIs of F(t;θ) can be calculated
using the sample mean and quantiles of the bootstrap version of the estimates of F(t;b
θ) based
on a large number of bootstrap estimates b
θ. By using Df=−0.4, M= 200 Monte Carlo
simulations, and 10000 bootstrap samples, Figure 5 shows the predicted F(t;θ) and its 95%
pointwise CIs for the NIST coating degradation data. We can see that for a population of
units with random starting time between 161 and 190 days, a majority of the population will
fail between 50 to 150 days in service.
4 Recurrent Event Data Analysis
In this section, we briefly introduce the multi-level trend renewal process (MTRP) model and
its application on the Vehicle B data as described in Xu et al. [36].
4.1 Background and Data
Xu et al. [36] consider the modeling and analysis of the Vehicle B data, which consist of
recurrent event data from a batch of industrial systems. Vehicle B is a two-level repairable
system. During its life span, Vehicle B may experience event at subsystem level (e.g., engine
15

Months
16
31
40
41
107
119
134
150
171
182
ID No.
0 20 40 60 80 100
current age
Component event Subsystem event
0 20 40 60 80 100
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Months
Response
(a) Recurrent event processes for (b) Cumulative usage processes for
a subset of system units a subset of system units
Figure 6: Plots of (a) the event processes and (b) the cumulative usage processes for ten
randomly selected units in the Vehicle B fleet. Figure reproduced with permission.
failures) and/or event at component level (e.g., oil pump failures). In the field data, we have
n= 203 units from a 110-month observation period. There are 219 component events and 44
subsystem events observed during the study period. Figure 6(a) shows the event plot for ten
randomly selected units. We also have the cumulative usage information available for each
unit, which is dynamic covariate. The cumulative usage information is shown in Figure 6(b).
The goal is to make a prediction for the cumulative number of component event occurrences
at a future time.
We need some notation to introduce the MTRP model. Suppose there are nunits under
observation from time 0 to τi. Let Xi(t) be the time-dependent covariate at time tfor system
i. Let Nis (t) be the number of subsystem events and and Nic(t) be the number of component
events up to time t. The total number of replacement events is Ni(t) = Nis(t) + Nic (t). The
subsystem event time is sorted as 0 < ts
i1<···< ts
i,Nis(τi)< τi. The component event time is
sorted as 0 < tc
i1<··· < tc
i,Nic(τi)< τi. Let 0 < ti1<··· < ti,Ni(τi)< τibe the replacement
event times, regardless of the types.
16

4.2 The MTRP Model and Parameter Estimation
For a two-level repairable system, Xu et al. [36] propose the following MTRP model to describe
events occurred at component level. In particular, the intensity function is
λc
i(t|Fi,t−;θc) = hcΛs
i(t|F s
i,t−)−Λs
iti,Ni(t−)| F s
i,t−
i,Ni(t−);θcλs
i(t|F s
i,t−;θc).(17)
Here Fs
i,t−denotes the historical information. In this multi-level model framework, the effect
of subsystem events on the component event process is modeled by λs
i(t|F s
i,t−;θc), which takes
the form
λs
i(t|F s
i,t−;θc) = hs{Λi(t)−Λi[ts
i,Nis(t−)]; θc}λi(t;θc),(18)
Here, θcdenotes the unknown parameters. The cumulative event intensity functions can be
obtained as Λi(t) = Rt
0λi(u;θc)du, and Λs
i(t|F s
i,t−) = Rt
0λs
i(u|F s
i,u−;θc)du. The baseline
function λi(t;θc) models the intensity of the component process when there is no event ad-
justment, and the function hs(·) is used to model the adjustment for effect of events from
the subsystem. The renewal distribution function Fc(·) is used to describe the distribution
of gap times under the transformed scale. The model in (18) can be extended to incorporate
dynamic covariates and random effects.
To model the dynamic covariates, the intensity function can be extended as
λi(t;θc) = λb(t) exp{γg[Xi(t)]},(19)
where λb(t) denotes intensity trend function under the baseline and γis the regression coef-
ficient. In the Vehicle B application, we use g[Xi(t)] = log[Xi(t)]. To incorporate random
effects, the intensity function can be further extended as
λi(t;θc) = λb(t) exp{γlog[Xi(t)] + wi}.(20)
Here wi’s are independent and identically distributed with N(0, σ2
r). The MTRP with random
effects is referred to as HMTRP(Fc, F s, λi), in which the HMTRP stands for heterogenous
MTRP.
To estimate the model parameters, one need to construct the likelihood function. The
component events data can be denoted as {tij , δc
ij}with tij be the event time and δc
ij be the
component-event indicator. The event history is denoted as F={Nic (u), Nis(u), Xi(u) : 0 <
u≤τi, i = 1,··· , n}. The likelihood function is
L(θc) ==
n
Y
i=1
Ni(τi)+1
Y
j=1 nfc[Λs
i(tij |Fs
i,t−
ij )−Λs
i(ti,j−1|F s
i,t−
i,j−1)]λs
i(tij|Fs
i,t−
ij ;θc)oδc
ij
×nSc[Λs
i(tij |Fs
i,t−
ij )−Λs
i(ti,j−1|F s
i,t−
i,j−1)]o1−δc
ij .(21)
17

Xu et al. [36] use Bayesian methods with diffuse priors to estimate the model parameters.
The Metropolis-within-Gibbs algorithm is used to obtained the posterior distributions and
then the inference can be carried out using the Markov Chain Monte Carlo (MCMC) samples
from the posterior distributions.
4.3 Prediction for Component Events
To make predictions for component events, let θdenote the vector of all the parameters and
Xi(t1, t2) = {Xi(t); t1< t ≤t2}is the covariate information between the time t1and t2. The
prediction for the counts of component events at time t∗is
Nc(t∗;θ) =
n
X
i=1
Nic(t∗;θ) =
n
X
i=1
EXi(τi,τi+t∗)|X(τi)EwiNic[t∗,Xi(τi, τi+t∗), wi;θ].(22)
Here Nic(t∗;θ) denotes the prediction for unit i. Because there is no closed form expression
for (22), the Monte Carlo simulation is used.
By fitting the MTRP model to the Vehicle B data using Bayesian estimation, one needs
to specify the prior distributions for the unknown parameters. The Weibull distribution was
used as renewal functions for Fcand Fs. To check the performance of prediction, first the last
15 months of the Vehicle B data were held back and only the first 95 months data were used to
estimate the MTRP model and then generate predictions for the last 15 months. Figure 7(a)
shows the prediction of the cumulative component events for the last 15 months based on the
earlier data. One can see that the actual observed cumulative numbers of component events
are closely located around the predicted values and also well bounded within the pointwise
prediction intervals. Figure 7(b) shows the predicted future events given all the observed
data for the next 30 months, which indicates that the total number of component events are
expected to range between 62 and 90 with a 95% confidence level.
5 Sequential Test Planning of Accelerated Life Tests
In this section, we briefly introduce the sequential Bayesian design (SBD) for fatigue test
experiments described in Lee et al. [43].
5.1 Background and Historical Data
A sequential Bayesian test planning strategy for the accelerated life tests was proposed by Lee
et al. [43]. The fatigue test for glass fiber, a composite material is considered to illustrate the
sequential design strategy. In the test, a tensile/compressive stress σ(positive/negative value)
is applied to the test unit and the material life is observed under that stress. In this work,
18

0 5 10 15
0
10
20
30
40
50
Months after DFD
Cumulative Number of Events
Actual number
Predicted number
95% PI
0 5 10 15 20 25 30
0
20
40
60
80
Months after DFD
Cumulative Number of Events
Predicted mean
95% PI
(a) Back test based on an early subset of the data (b) Prediction of future events
Figure 7: Plots of the predicted cumulative number of component events for Vehicle b for (a)
the last 15 months based on the earlier 95 months data and (b) the future 30 months based
on all observed data. Figure reproduced with permission.
14 observations of E-glass are made including 11 failed and 3 right-censored units. Historical
data of the observations are show in Figure 8. Several other important factors in the test
are set as follow. Let R=σm/σMdenote the stress ratio, where σmis the minimum stress
and σMis the maximum stress. The range of Rcan reveal different test type and it is set at
R= 0.1 for a tension-tension loading test in this application. The ultimate stress σult, where
the material breaks at the first cycle is set to be 1339.67 MPa. The frequency of the cyclic
stress testing (f) is set at 2 Hz, and the angle (α) between the testing direction and material
is set at 0.
5.2 Lifetime Model
Consider using a log-location-scale distribution to model the cycles-to-failure, T. The cdf and
pdf are given as
F(t;θ) = Φ log (t)−µ
νand f(t;θ) = 1
νt φlog (t)−µ
ν,
where Φ(·) and φ(·) are the standard cdf and pdf, respectively. The lognormal and Weibull
distributions are the common choices. In the ALT modeling, we assume a constant scale
parameter νand the location parameter is µ=µβ(x), where xis the stress level and βis the
unknown parameter. The following nonlinear model for composite materials proposed in [52]
19

0 500000 1000000 1500000 2000000
0 200 400 600 800 1000 1200
Exact observations
Censored observations
Cycles
Stress
0 200 400 600 800 1000 1200
6 8 10 12 14
Exact observations
Censored observations
Fitted S−N curve by θ0
log(Cycles)
Stress
(a) Stress-life relationship (b) Log of cycles vs. Stress leves
Figure 8: The plots show the historical data from a fatigue testing of the glass fiber with the
fitted stress-life relationship. Figure reproduced with permission.
is used to describe µ=µβ(x) in the form of
µβ(x) = 1
Blog B
AhBσult
x−1σult
xγ(α)−1[1 −ψ(R)]−γ(α)+ 1,(23)
In the above model, Aand Bare effects from environment and material, and β= (A, B)′. The
function ψ(R) = 1/R if R≥1 and ψ(R) = Rif −∞ < R < 1, and γ(α) = 1.6−ψ|sin (α)|.
Then θ= (β′, ν)′denotes the unknown parameter in the ALT modeling.
The lower quantile of the cycles-to-failure distribution is of interest as it contains material
life information. The log of the pth quantile is
log (ξp,u) = µβ(u) + zpν, (24)
where ξp,u is the pth quantile at the use condition uand zpis the pth quantile of the standard
distribution. Our goal is to propose test planning under multiple use conditions to approximate
the real scenarios. The use stress profile consists of a set of use levels, {u1,··· , uK}, with
weights {w1,··· , wK}and PK
k=1 wk= 1.
Let (xi, ti, δi) denote the data for the ith testing unit, where xiis the stress level of the
accelerating factor and tiis the observed cycles to failure (or censoring). Let δibe a censoring
indicator where δi= 1 if the observation is censored and δi= 0 if the observation fails. Then,
the log-likelihood function is given by
l(θ|xn,tn,δn) =
n
X
i=1
(1 −δi) [log φ(zi)−log(ti)−log(ν)] + δilog [1 −Φ (zi)] ,(25)
20

where zi= [log(ti)−µβ(xi)] /ν. Let b
θbe the ML estimates of θand log(b
ξp,u) be the ML
estimate of the logarithm of the pth quantile at the use level u, obtained by substituting β
and νby b
βand bνin (24). Given the use level u, the asymptotic variance of log(b
ξp,u) is
Avar hlog b
ξp,ui=c′Σθ(xn)c,
where c= [∂µβ(u)/∂A, ∂µβ(u)/∂B, zp]′, Σθ(xn) = I−1
n(θ), and In(θ) is the Fisher informa-
tion matrix based on nobserved data. The details for calculating In(θ) can be found in [43].
A weighted version of asymptotic variance can be expressed as
K
X
k=1
wkAvar hlog b
ξp,uki.(26)
Given {(uk, wk)}K
k=1, the weighted asymptotic variance only depends on the observed test-
ing levels xi, where i= 1,...,n. Therefore, the optimum design points should determine
x1,...,xnto minimize the weighted asymptotic variance in (26).
5.3 Test Plan Development
To obtain an accurate prediction from an efficient ALT, the optimum test planning can be
determined by minimizing the asymptotic variance in (26). In the literature, when determining
an optimal test plan, it often requires pre-specifying the values of parameters (known as the
planning values). The optimal design based on some chosen planning values of parameters
is known as the local c-optimality design. However, the planning values are not precisely
known a priori for many experiments in practice. Hence, the SBD is useful for improving
our understanding of the unknown parameters as more data becoming available during the
experiment, when there is little knowledge or historical data available.
Before the test planning, the stress levels are often standardized to be between 0 and 1,
denoted by qi=xi/σult. In practice, a range of testing levels, [qL, qU], is often determined
at the very early stage of the test planing, where qLis the lower bound and qUis the upper
bound. To design an efficient ALT via the sequential planning, the objective function based
on (26) is chosen as
ϕ(qnew) = ZΘ"K
X
k=1
wkck′Σθ(qnew)ck#π(θ|qn,tn,δn)dθ,(27)
where Σθ(qnew) = [In(θ,qn) + I1(θ, qnew )]−1,qnew = (q′
n, qnew)′,qn= (q1,...,qn)′, and
π(θ|qn,tn,δn) is the posterior distribution of θ. Specifically,
π(θ|qn,tn,δn)∝f(tn|θ,xn,δn)π(θ),
21

where f(tn|θ,xn,δn) is the joint pdf of the historical data and π(θ) is the prior distribution
of θ. Then, the optimum (n+ 1)th design point is determined by
q∗
n+1 = arg min
qnew∈[qL,qU]ϕ(qnew).(28)
The procedure of the sequential Bayesian design is summarized as follows.
1. Specify prior distributions of model parameters. Specify prior distributions of Aand B
as A∼N(µA, σ2
A) and B∼N(µB, σ2
B), where µA,σ2
A,µB, and σ2
Bare the parameters of
the normal distribution and set to be known constants. Let ν2∼Inverse Gamma(κ, γ),
where κand γcan be known from the historical data or experience.
2. Evaluate the asymptotic variance. Use the technique of MCMC to approximate (27).
The details of the related algorithms can be found in [43].
3. Determine the optimum next testing point q∗
n+1.Given a candidate set of design points,
their corresponding values of the objective function in (27) can be evaluated in Step 2.
Then, determine the optimum next design point, which has the smallest value of the
asymptotic variance.
4. Obtain the failure data at the level q∗
n+1.Under the stress level q∗
n+1, conduct the exper-
iment and obtain the failure information (tn+1, δn+1).
5. Repeat Steps 2 to 4 till the desired number of testing units are obtained. Add the new
lifetime data, (q∗
n+1, tn+1 , δn+1), to the historical dataset and repeat Steps 2 to 4 till the
desired number of new design points are obtained.
5.4 Illustration of Test Plans
For the original data, we can fit the lognormal distribution and the corresponding ML esti-
mates are θ0=ˆ
θ= (0.0157,0.3188,0.7259)′. Before the testing planning, the setup for the
sequential Bayesian design is as follows.
1. Prior information: Let Aand Bbe from the normal distributions, where A∼N(0.08,
0.0008) and B∼N(1,0.0833). The prior distribution for ν2is Inverse Gamma(4.5,3).
2. Historical data: In practical implementation, the sample size at the beginning of testing
is limited. Choose the three failed observations at stress levels x3= (621,690,965) from
Figure 8 as the historical dataset.
3. Total size of design points: Let the sample size of the new design points be 12.
22

1 5 7 9 11 14
0.3 0.5 0.7 0.9
Number of units in current data
Stress level
historical stress level
sequential design
Simulation 1
1 5 7 9 11 14
0.3 0.5 0.7 0.9
Number of units in current data
Stress level
historical stress level
sequential design
Simulation 2
1 5 7 9 11 14
0.3 0.5 0.7 0.9
Number of units in current data
Stress level
historical stress level
sequential design
Simulation 3
1 5 7 9 11 14
0.3 0.5 0.7 0.9
Number of units in current data
Stress level
historical stress level
sequential design
Simulation 4
0 2 4 6 8 10
3 4 5 6 7 8 9 11 13 15
Avar
Number of units in the current data
Simulation 1
Simulation 2
Simulation 3
Simulation 4
(a) Design points for SBD (b) Asymptotic variance
Figure 9: Plots show the results of the 4 simulation trials including the sequential design points
and their corresponding values of asymptotic variance. Figure reproduced with permission.
4. Design candidate: The standardized levels of historical data are 0.46, 0.52, and 0.72,
and the candidate points are from qL= 0.35 to qU= 0.75 with a 5% increase.
For the illustrative purpose, assume that the true values of parameters are θ0. When an
optimum design point is determined, the new observation is generated from the lognormal
distribution with parameter θ0and the censoring time at 2 ×106cycles. Repeat Steps 2 to
4 in Section 5.3 till 12 testing locations are obtained. Then, the results of 4 simulation trials
are shown in Figure 9. It consistently shows that only two stress levels at 0.35 and 0.75 are
selected, and 8 and 4 units are allocated to the levels 0.35 and 0.75, respectively. And the
resulting asymptotic variances decrease as the size of sequential runs increases.
Using the same historical data, the developed SBD is also compared with the local c-
optimality design. For the locally c-optimal design, the estimated values of parameters from
historical data are usually used as the planning values of the parameters. With only 3 obser-
vations available from the historical data, the ML estimates are ˆ
θ1= (0.0005,0.7429,0.1658)′.
Hence, the local c-optimality design chooses 11 and 1 unit at the testing levels at 0.65 and
0.75, respectively. Now, we compare the performance on the value of the asymptotic variance
based on the ML estimates of the final dataset including the 12 new testing observations and
3 historical observations. With 100 simulations, the averages of asymptotic variances for the
SBD and the local c-optimality designs are 0.6048 and 4.0337, respectively. It shows that the
SBD is more efficient than the traditional local c-optimality design when there is too little
historical data available to provide accurate estimates of the model parameters. The proposed
23

SBD can be also applied when there is no historical data but only prior information based on
subject matter expertise.
6 Concluding Remarks
In this chapter, we review recent developments on statistical reliability analysis utilizing dy-
namic covariates and sequential test planning. For time to event data, we introduce a cumu-
lative damage model to account for the effect of dynamic covariates and illustrate the method
with the Product D2 application. For degradation data, we present the general path model for
incorporating dynamic covariates and illustrate the method with the NIST coating degrada-
tion data. We also introduce the MTRP model for recurrent events using dynamic covariates
and illustrate it with the Vehicle B data. Regarding to test planning for ALT, we focus on
the SBD and illustrate it with the ALT design for polymer composites fatigue testing.
Looking forward, more versatile data become available due to the rapid advance of modern
technology, and new statistical methods need to be developed to make use of those new data
for improving reliability modeling and prediction. As described in [3], many data types such
as spatial data, functional data, image data, and text data, all have great potential to be
used for reliability modeling and analysis. New methods that are available in statistics and
machine learning can also be transformed and integrated with reliability domain knowledge
for reliability analysis, which provides tremendous opportunity in reliability research.
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