The Inverse Gaussian Process as a Degradation Model

Zhi-Sheng Ye† and Nan Chen‡

†Department of Applied Mathematics, The Hong Kong Polytechnic University

‡Department of Industrial and Systems Engineering, National University of Singapore

June 8, 2013

Abstract

This paper systematically investigates the inverse Gaussian (IG) process as an

effective degradation model. The IG process is shown to be a limiting compound

Poisson process, which gives it a meaningful physical interpretation for modeling

degradation of products deteriorating in random environments. Treated as the

first passage process of a Wiener process, the IG process is flexible in incorpo-

rating random effects and covariates that account for heterogeneities commonly

observed in degradation problems. This flexibility makes the class of IG pro-

cess models much more attractive compared with the Gamma process, which

has been thoroughly investigated in the literature of degradation modeling. The

paper also discusses statistical inference for three random effects models and the

model selection. It concludes with a real world example to demonstrate the appli-

cability of the IG process in degradation analysis. This paper has supplementary

materials available online.

Keywords: Random effects, Covariate, Compound Poisson approximation, Wiener process,

Monotone degradation paths

1

1 Introduction

Degradation is an irreversible accumulation of damage over time that ultimately leads to a

system failure when the degradation hits a certain threshold, either fixed or random. Com-

pared with traditional failure time data analysis, degradation analysis aims to characterize

the underlying failure process and requires far fewer testing samples to achieve the same es-

timation accuracy (Meeker and Escobar 1998). In addition, correct specification and proper

monitoring of the degradation process are helpful for predicting remaining useful life and

scheduling necessary maintenance. In practice, the degradation over time is often modeled

by a stochastic process {Y (t); t ≥ 0} to account for inherent randomness. Based on the

assumption of additive accumulation of degradation, two classes of degradation processes

have been well exploited, i.e., the Wiener processes and the Gamma processes.

The Wiener process as a degradation model is based on the consideration that the degra-

dation increment in an infinitesimal time interval might be viewed as an additive superposi-

tion of a large number of small external effects. In this regard, the degradation increment is

independent and normally distributed due to the law of large numbers. For most products,

many observable environmental factors such as the usage rate, temperature, humidity, etc.,

influence the degradation behavior. Their effects on the degradation process can be taken

into account as covariates. Different ways of incorporating covariates in the Wiener process

models can be found in Doksum and Hyland (1992), Doksum and Normand (1995) and

Padgett and Tomlinson (2004). On the other hand, when unobservable factors, e.g., size of

an internal defect and the unobservable field use conditions, influence product degradation,

their effects are often represented by incorporating a random effect, or frailty term, into the

degradation model. Random effects variants of the Wiener process can be found in Peng

and Tseng (2009) and Wang (2010), and will be discussed in Section 3 when random effects

models for the inverse Gaussian (IG) process are developed.

A distinct feature of the Wiener process is that its sample path is not necessarily mono-

tone, which might not be meaningful in many degradation applications. As an alternative,

2

the Gamma process is often adopted when monotonicity is required. The Gamma process is

the limit of a compound Poisson process with the jump size conforming to a certain distribu-

tion (Lawless and Crowder 2004). This interpretation underpins the Gamma process as an

appropriate degradation model because many reliability engineers believe that degradation

is often caused by a series of external shocks, each with random and tiny damage (Esary and

Marshall 1973). Generally speaking, there are two ways to incorporate covariates (Bagdon-

avicius and Nikulin 2000; Lawless and Crowder 2004) and one way to incorporate random

effects (Lawless and Crowder 2004) for the Gamma processes.

Notwithstanding wide applications of the Wiener and Gamma processes in degradation

modeling, two classes of models are insufficient to fit all degradation data (Wang and Xu

2010). Another attractive degradation model with monotone paths is the IG process pro-

posed by Wasan (1968). Although the IG distribution has been well studied (c.f. Chhikara

and Folks 1989) because of its close relation with the Wiener process with drift, the IG

process is not well received in degradation modeling except for an excellent recent study

by Wang and Xu (2010). The scarce application of IG processes in degradation modeling

might be attributed to its unclear physical meanings to reliability engineers, in contrast to

the well-known Wiener and Gamma processes.

The main objective of this study is to investigate the application of IG processes in

degradation modeling. In particular, we first justify its physical meaning by exploring the

inherent relations between the IG process and the compound Poisson process. Wang and

Xu (2010) have proposed a method to incorporate random effects in the IG process. By

treating the IG process as the first passage process of a Wiener process, we further propose

two different approaches to incorporate random effects. These random effects models are

analytically tractable and can be efficiently estimated from degradation data. In addition,

we show that covariates, if available, can be flexibly incorporated into the IG processes. With

these nice properties, the class of IG processes greatly complements the family of stochastic

degradation models, especially when the degradation is monotone.

3

The remainder of the paper is organized as follows. Section 2 introduces the IG process,

discusses its first passage time distribution and gives the physical interpretation of the IG

process by showing that it is a limiting compound Poisson process. By linking the IG process

to the Wiener process, Section 3 discusses three approaches to incorporate random effects

in the IG process to account for unobservable heterogeneity across units. Section 4 studies

parameter estimation and model selection in the random effects models. Section 5 proposes

a number of methods to incorporate covariates in the IG process with or without random

effects. Section 6 uses a real life example to illustrate the effectiveness of the IG process.

Section 7 concludes the paper with a comparison between the IG process and the well-known

Gamma process, and discusses possible topics for future research.

2 The IG Process and its Physical Interpretation

2.1 Introduction to the IG process

The IG process {Y (t), t ≥ 0} is defined as the stochastic process satisfying:

• Y (t) has independent increments, i.e., Y (t2)−Y (t1) and Y (s2)−Y (s1) are independent

∀t2 > t1 ≥ s2 > s1;

• Y (t)−Y (s) follows an IG distribution IG (Λ(t)− Λ(s), η[Λ(t)− Λ(s)]2) , ∀t > s ≥ 0,

where Λ(t) is a monotone increasing function and IG(a, b), a, b > 0, denotes the IG distribu-

tion with probability density function (PDF)

fIG(y; a, b) =

√b

2πy3· exp

[−b(y − a)2

2a2y

], y > 0,

and cumulative distribution function (CDF)

FIG(y; a, b) = Φ

[√b

y(y

a− 1)

]+ e2b/aΦ

[−

√b

y(y

a+ 1)

], y > 0,

4

where Φ(·) is the standard normal CDF. As with the convention, we let Λ(0) = 0 and Y (0) =

0, and thus Y (t) follows IG(Λ(t), ηΛ(t)2) with mean Λ(t) and variance Λ(t)/η. The property

of independent increments for the IG process makes the prediction of future degradation

straightforward, i.e., [Y (t)|Y (t1), Y (t2), · · · , Y (tn)] ∼ IG (Λ(t)− Λ(tn), η[Λ(t)− Λ(tn)]2) for

t > tn > · · · > t1 ≥ 0.

In many engineering applications, the failure time TD for an item is defined as the

time upon which the degradation path first reaches a pre-determined threshold D. The

distribution of the first passage time TD plays an important role in predicting remaining

useful life and in determining the optimal maintenance strategies (see the review by van

Noortwijk 2009). Consider the IG process {Y (t); t ≥ 0} with Y (t) ∼ IG(Λ(t), η · Λ2(t)).

Because of the monotonicity property of the IG process, the CDF of TD can be readily

obtained from the relation P(TD < t) = P(Y (t) > D) = 1−FIG (D; Λ(t), η · Λ2(t)) as (Wang

and Xu 2010)

P(TD < t) = Φ

[√η

D(Λ(t)−D)

]− e2ηΛ(t)Φ

[−√η

D(Λ(t) +D)

]. (1)

When η ·Λ(t) is large, which is often true when t is large, Y (t) is approximately normal with

mean Λ(t) and variance Λ(t)/η (Chhikara and Folks 1989). Accordingly, the CDF of TD can

be approximated as

P(TD < t) ≈ 1− Φ

[D − Λ(t)√

Λ(t)/η

]= Φ

[√η · Λ(t)−

D√η√

Λ(t)

], (2)

which is a Birnbaum-Saunders type distribution (Tang and Chang 1995). To some extent,

this indicates the appropriateness of IG processes as a degradation model, as the Birnbaum-

Saunders distribution is a famous model for fatigue failure times caused by crack growth

passing a critical value (Desmond 1985). In particular, when Λ(t) = µt, (2) reduces to

a Birnbaum-Saunders distribution with shape parameter µ(Dη2)−1/4 and scale parameter√D/µ. In addition, it should be noted that both the derivatives of (1) and (2) with respect

5

to t have closed form expressions (see Section 1.1 of the supplementary materials for details).

This is an important property during parameter inference, as some data collection procedures

are designed to collect the times 0 < t1 < t2 < ... < tm at which the degradation Y (t) reaches

specified levels a1, a2, ..., am (e.g., Yang and Yang 2002; Wu and Ni 2003; Tang and Su 2008).

In contrast, the first passage time of the Gamma process does not bear an explicit PDF,

making exact statistical inference analytically intractable.

2.2 Physical interpretation of the IG process

To promote the IG process as a degradation model, it is important to present its physical

interpretation in a manner that is meaningful to practitioners. Many degradation phenomena

such as corrosion, fatigue crack growth and physical wear can be considered as accumulations

of additive and irreversible damage caused by a sequence of external random shocks. The

shock arrival process may be approximated by a Poisson process, while each shock causes

a small and random amount of damage to the system (Esary and Marshall 1973; Desmond

1985; Singpurwalla 1995; van Noortwijk 2009; Ye et al. 2013). For example, the capacity of a

lithium-ion battery fades with the charge/discharge cycling. Each cycle decreases the battery

capacity by a small and random amount; while the usage, i.e., the number of cycles, may be

approximated by a Poisson process (Singpurwalla and Wilson 1998; Lawless and Crowder

2010). As another example, wear of the air bearing slider in a hard disk drive is caused by

intermittent contacts between the trailing edge of the slider and the glide avalanche of the

disk. The number of contacts over time may be approximated as a Poisson process, while the

wear caused by each contact is random and tiny. The degradation physics in these examples

suggest that degradation can be modeled by a compound Poisson process or its variants.

A compound Poisson process is defined as C(t) =∑N(t)

i=1 Di, where N(t) is a homogeneous

Poisson process with arrival rate ν, and Di are i.i.d. positive random variables representing

the size of each shock, or the so-called jump size. When the rate ν becomes large and the

jump size becomes small, the compound Poisson process may be approximated by other

6

computationally more efficient processes. For example, the Gamma process can be viewed

as the limit of a compound Poisson process whose rate goes to infinity while the jump size

tends to zero in proportion (Lawless and Crowder 2004). This approximation provides a

sound physical underpinning for the Gamma process, and to a large extent makes it widely

received in degradation modeling.

Similar to the Gamma process, it can be shown that the IG process is also a limiting

compound Poisson process, but with different jump size distributions. This nice interpreta-

tion justifies the IG process as a degradation model. More precisely, Theorem 1 states that

when the arrival rate goes to infinity and the jump size goes to zero in a certain way, the

limiting compound Poisson process is indeed an IG process.

Theorem 1 Consider the compound Poisson process {Cn(t), t > 0} with arrival rate νn and

jump size distribution Gn(dx) = Πn(dx)/νn, where

Πn(dx) =

√η

2πx3exp

(− ηx

2µ2

)I[1/n,∞)(x)dx, νn =

∫ ∞1/n

Πn(dx),

where IA(x) is the indicator function which equals one when x ∈ A and zero otherwise.

When n → ∞, {Cn(t), t > 0} converges to the stationary IG process {Y (t), t ≥ 0} with

marginal distribution Y (t) ∼ IG(µt, ηt2).

The proof of this theorem is placed in the Appendix for reference. The IG process, as one

type of the processes with independent and positive increments, is also a pure jump Levy

process (Ferguson and Klass 1972) with Levy measure Π(dx) = limn→∞Πn(dx). As a pure

jump process, the jump behavior of the stationary IG process can be studied in detail. A

brief discussion on the jump behaviors and their applications has been included in Section

1.2 of the supplementary materials for reference.

The close relation with the compound Poisson process provides a solid physical ground

for the IG process as a degradation model. As we shall see in the next section, the flexibility

in handling heterogeneity makes the IG process even more attractive in practice.

7

3 IG Processes with Random Effects

The IG process investigated in Section 2 is called a simple IG process throughout the pa-

per. Random effects models are often needed to account for unexplained heterogeneous

degradation rates within a product population. By linking to the Wiener process, this sec-

tion investigates different options to incorporate random effects in the IG process model.

Consider the Wiener process W (x) = vx + σ · B(x), where v > 0 is the drift parameter,

σ > 0 is the volatility parameter and B(x) is the standard Brownian motion. Given a fixed

threshold Λ > 0, it is well known that the first passage time TΛ = inf{x > 0|W (x) ≥ Λ}

follows IG(Λ/v,Λ2/σ2). Going one step further, we consider a series of thresholds Λ(t) in-

dexed by t with Λ(0) = 0 and Λ(t) increasing in t, and define the first passage time process

Y (t) ≡ TΛ(t). It is easily verified that the induced {Y (t); t > 0} is an IG process with mean

function Λ(t)/v and variance function Λ(t)/σ2 by virtue of the stationary and independent

increment property of the Wiener process W (x) (detail derivations are included in Section

1.3 of the supplementary materials). In keeping with Tweedie (1945), we shall call this

relationship the inverse relation.

The inverse relation between the IG and Wiener processes motivates investigation of

the IG process from a new perspective. Existing results on the Wiener processes can lend

support to the development of IG process models with random effects. For ease of exposition,

the format of the Wiener process as well as the induced IG process considered in this section

will slightly vary from line to line. In real applications, proper constraints can be imposed

on the parameters (µ, η and Λ(t) below) to make the models identifiable.

3.1 Random drifts model

Consider a Wiener process W (x) = µ−1x + η−1/2B(x) with the induced IG process Y (t) ∼

IG(µΛ(t), ηΛ2(t)). A common practice to incorporate random effects in the Wiener process

is to let the drift parameter µ−1 vary randomly across units (Crowder and Lawless 2007;

8

Peng and Tseng 2009). As such, a plausible way to incorporate random effects in the IG

process is to let µ be a random variable. To avoid negative values of µ (Whitmore 1986)

and ensure mathematical tractability, we assume µ−1 follows a truncated normal distribution

T N (ω, κ−2), κ > 0 with PDF

g(µ−1;ω, κ−2) =κ · φ[κ(µ−1 − ω)]

1− Φ(−κω), µ > 0

where φ(·) is the standard normal PDF.

In a degradation test, if the degradation of the ith testing unit is observed at times

0 = ti0 < ti1 < · · · < tiniwith observations Yi(tij), j = 0, 1, 2, · · · , ni, the joint PDF of

Yi ≡ [Yi(ti1), Yi(ti2), · · · , Yi(tini)] can be computed by first conditioning on the random drift

parameter µi and then marginalizing it, which yields

fIG(Yi) =1− Φ(−κiωi)1− Φ(−κω)

· κκi·ni∏j=1

√ηλ2

ij

2πy3ij

· exp

[κ2i ω

2i − κ2ω2

2− η

ni∑j=1

λ2ij

2yij

], (3)

where yij = Yi(tij) − Yi(ti,j−1) is the observed increment, λij = Λ(tij) − Λ(ti,j−1), κi =√ηYi(ti,ni

) + κ2 and ωi = (ηΛ(ti,ni) + ωκ2)/κ2

i . When the degradation paths of N units are

observed, the likelihood function is simply∏N

i=1 fIG(Yi).

After observing Yi for unit i, the future degradation can be better predicted using the

conditional distribution of µ−1i given Yi, which can be computed as

g(µ−1i |Yi) ∝ exp

[−κ2(µ−1

i − ω)2]· I[0,∞)(µi) ·

ni∏j=1

exp

[−η(yij − µiλij)2

2µ2i yij

].

A routine calculation shows that [µ−1i |Yi] follows T N

(ωi, κ

−2i

)with updated parameters

depending on Yi(tini) and Λ(tini

) only. When Yi(tini) is large, which is often true when t is

large, κ−2i becomes small and thus [µ−1

i |Yi] tends to degenerate to the true value. By making

use of the observed degradation Yi, the conditional distribution of the future degradation

Yi(t), t > tini, can be obtained by first conditioning on µ−1

i and then marginalizing over

9

[µ−1i |Yi], which yields

fYi(t)|Yi(y) =

1− Φ [−ωiκi]1− Φ[−ωiκi]

· κiκi

√η[Λ(t)− Λ(tini

)]2

2π[y − Yi(tini)]3

· exp

[ω2i κ

2i − ω2

i κ2i

2− η[Λ(t)− Λ(tini

)]2

2[y − Yi(tini)]

], (4)

where κi =√ηYi(t) + κ2 and ωi = (ηΛ(t) + ωκ2)/κ2

i . It is interesting to observe that the

conditional distribution of [Yi(t)|Yi] is similar to (3) with ni = 1, with a difference being that

(4) uses the updated distribution of [µ−1i |Yi]. Furthermore, since the updated distribution

of µ−1i is only dependent on Yi(tini

), the resulting distribution of [Yi(t)|Yi] only depends on

the last observation Yi(tini), indicating that the Markovian property of the process is still

preserved.

Remarks: The truncated normal is adopted for µ−1 in this subsection. However, the normal

distribution is recommended when the possibility of a negative µ−1 is ignorable or when the

negativity is not of critical concern, which happens when the principal objective is to use

existing degradation information to predict degradation of a future unit. When the normal

distribution is used, Equations (3)-(4) can be simplified by dropping the terms of the form

1 − Φ(−κω). In addition, as will be discussed in Section 4.1, under the truncated normal

distribution, direct maximization of the likelihood function often yields a solution far away

from the MLE. Nevertheless, this problem is greatly alleviated when a normal distribution

is adopted.

3.2 Random volatility model

Again, consider the Wiener process W (x) = µ−1x + η−1/2B(x) with the induced IG process

Y (t) ∼ IG(µΛ(t), ηΛ2(t)). Another way of introducing unit-specific random effects to the

Wiener process is to assume that each unit has a distinct realization of the volatility param-

eter η. Accordingly, η in the IG process is random. With a random η in the IG process,

all units have the same mean degradation path µΛ(t) yet different variance functions. A

10

tractable model results when η follows a Gamma distribution with PDF

g(η; δ, γ) =γδηδ−1

Γ(δ)exp(−γη), η > 0,

where δ > 0, γ > 0 are the distribution parameters, and Γ(δ) is the Gamma function. As in

Section 3.1 (see also Wang and Xu 2010), the unconditional distribution of Yi for unit i can

be derived as

fIG(Yi) =Γ(δi)

Γ(δ)

γδ

γ δii

ni∏j=1

√λ2ij

2πy3ij

, (5)

where δi = ni/2 + δ and γi = γ +∑ni

j=1(yij − µλij)2/(2µ2yij). The conditional distribution

of ηi given observations Yi is

g(ηi|Yi) ∝ ηni/2+δ−1i exp

[−ηi

(γ +

ni∑j=1

(yij − µλij)2

2µ2yij

)]= ηδii exp(−γiηi), (6)

which is still a Gamma distribution with updated parameter δi, γi. Given Yi, the future

degradation Yi(t), t > tini, follows a distribution similar to (5) with ni = 1 and with δ, γ

replaced by the updated parameters. It is noted that (6) depends on all historical observa-

tions. Consequently, the conditional distribution of the future degradation depends on all

elements in Yi, and thus the Markovian property no longer holds.

The IG process with random volatility model was originally proposed by Wang and Xu

(2010), but we show that it arises in a natural way through the link to the Wiener process.

On the other hand, it is uncommon to use the volatility parameter to control heterogeneity in

the Wiener process. Therefore, application of this random volatility model may be limited.

3.3 Random drift-volatility model

In the forgoing two subsections, the random effects either diversify the mean degradation

rates or the degradation variances in the Wiener process. In practice, we may expect that a

unit with a higher degradation rate should also have higher volatility in the degradation path.

11

This motivates another format of the Wiener process as W (x) = µ−1x+ η−1/2µ−1B(x) with

the induced IG process Y (t) ∼ IG(µΛ(t), ηµ2Λ2(t)). If µ is a random effect, it is readily

seen that a Wiener process with a high degradation rate will also have a large variance

function. Analogously, we let µ in the IG process be random across the product population.

Assuming µ ∼ T N (ω, κ−2) leads to a tractable IG degradation model with the unconditional

distribution of Yi for unit i given by

fIG(Yi) = exp

[−ηYi(tini

) + ω2κ2 − ω2i κ

2i

2

]·κ ·∑ni

j=0 Cjni

(ωiκi)ni−jMj(−ωiκi)

κni+1i [1− Φ(−ωκ)]

·ni∏j=1

√ηλ2

ij

2πy3ij

, (7)

where κi =√κ2 + η

∑ni

i=1 λ2ij/yij, ωi = (ηΛ(tini

) +ωκ2)/κ2i , Cj

niis the combination number,

and Mj(u) is defined as the integral∫∞uxjφ(x)dx which can be computed recursively as

shown in Section 1.4 of the supplementary materials. Given Yi, the conditional distribution

of µi can be expressed as

g(µi|Yi) =1

C· µni · exp

[−

ni∑j=1

η(yij − θiλij)2

2yij− κ2(θi − ω)2

2

]· I[0,∞)(θi), (8)

with the normalizing constant C =√

2π ·∑ni

j=0 Cjni

(ωiκi)ni−jMj(−ωiκi)/κni+1

i . Although

(8) looks complex, the moments E(µki |Yi), k ≥ 1, can be computed efficiently in a recursive

way, which facilitates adoption of the EM algorithm for statistical inference. Given the

historical observations Yi, the distribution for the future degradation can be obtained by

first conditioning on and then numerically integrating out µ by utilizing (8). Similar to the

random volatility model, the random drift-volatility model also does not have the Markovian

property. In other words, p(Yi(t)|Yi) 6= p(Yi(t)|Yi(tini)) for t > tini

. Therefore, all the

historical observations should be kept in order to update the distributions of µ and the

future degradation, making the on-line prediction inefficient in the case of massive historical

data, say, continuous inspections. Alternatively, when ni is large, [µ|Yi] tends to degenerate

12

and thus g(µi|Yi) may be approximated by a point mass, e.g., E(µi|Yi) or the mode of the

distribution.

Remarks: Our model enforces the drift and volatility change according to the same random

effect. However, it is more general to allow both µ and η to be random and correlated.

Although the resulting random effects model is more computational intensive to estimate

the parameters and predict the future degradations, it is expected to include more realistic

scenarios. Efficient algorithms will be investigated in the future for such models.

4 Model Estimation and Selection

When the IG process is chosen to describe the degradation of a product, the model parame-

ters should be estimated from testing or historical data. In addition, after the development

in Section 3, there are four models in the class of IG processes. Therefore, this section is

aimed at statistical inference as well as model selection for the IG process. Parallel to the

notation in Section 3, we consider N testing samples, with ni observations Yi for unit i,

i = 1, 2, · · · , N .

4.1 Parameter estimation

Point estimation of the model parameters can be obtained through direct maximization of

the log-likelihood function l(θ) =∑N

i=1 ln fIG(Yi), where fIG(Yi) is the likelihood function

following the form (3), (5), or (7) depending on the random effects model adopted, and

θ is the vector of all unknown parameters in each corresponding model. In this paper,

we only consider the parametric shape function Λ(t). Nonparametric Λ(t) can be handled

analogously as in Wang and Xu (2010). For the simple IG process and the random volatility

model, the estimator from direct maximization is satisfactory and can be used as the MLE θ.

For the random drifts and random drift-volatility models with a truncated normal random

effect, however, estimators so obtained have large biases and mean squared errors (MSEs)

13

for moderate sample sizes according to our simulation experience in both Matlabr (using

fminsearch) and R (using optim). This indicates that such estimators are far away from

the true MLE. Failures of the brute-force maximization approach might be attributed to

rounding errors in the computation of the normalization terms 1 − Φ(·) in (3) and (7), in

conjunction with the fact that the log-likelihood function is quite flat in some directions

around θ. Under this circumstance, even a small rounding error would trap the resulting

solution somewhere far from θ. To support our speculation, we use a normally distributed

random effect so that the normalizing terms are casted away, and the simulation results (not

shown) reveal that θ from direct maximization is satisfactory.

When the truncated normal is adopted in the random drifts and random drift-volatility

models, the EM algorithm is found to be efficient in finding θ as it is free of the rounding

error problem. Detailed procedures of the EM algorithms for these two models can be found

in Section 1.5.1 and 1.5.3 of the supplementary materials. As for the random volatility

model, Wang and Xu (2010) have developed the EM algorithm under the semi-parametric

setting where Λ(t) is estimated nonparametrically. Under the parametric setting where a

parametric form of Λ(t) is available, direct maximization of l(θ) is recommended; in case

Λ(t) has a complicated form and direct maximization does not work well, the EM algorithm

can be invoked again. This procedure is also given in the supplementary materials (Section

1.5.2). From our experience, the EM algorithms usually converge to (locally) optimal points

in a few iterations. They can employ multiple starting points to improve the solution. It

should also be noted that the parametric form of Λ(t) should ensure the identifiability of the

model to make the algorithms work.

A comprehensive simulation study has been carried out. We let ni be a uniform random

number between 1 to T and tij are randomly selected in the sampling ranges [0, T ]. Under

each parameter setting, the simulation is repeated 1000 times, based on which the bias and

standard deviation of the MLE are computed. We only display the results with Λ(t) = tq in

Table 1. More simulation results are included in Section 2.1 of the supplementary materials

14

Table 1: Estimation errors of three random effects models. The numbers outside and insidethe parenthesis represent the bias and standard deviation respectively from 1000 simulationreplications.

RDmodel

True values η = 3 ω = 2 κ = 4 q = 0.3(N, T ) = (20, 20) 0.092(0.59) 0.0043(0.14) 3.13(3.94) -0.00019(0.017)(N, T ) = (40, 40) 0.033(0.35) -0.0059(0.087) 1.81(2.84) -0.00031(0.0089)

RVmodel

True Value µ = 2 δ = 10 γ = 3 q = 0.3(N, T ) = (20, 20) -0.024(0.24) 5.31(8.97) 1.87(2.85) 0.0040(0.014)(N, T ) = (40, 40) -0.011(0.15) 1.75(4.51) 0.68(1.50) 0.0026(0.0071)

RDVmodel

True Value η = 3 ω = 1.5 κ = 4 q = 0.3(N, T ) = (20, 20) 0.13(0.53) -0.044(0.092) 0.99(2.63) 0.0036(0.016)(N, T ) = (40, 40) 0.070(0.32) -0.029(0.061) 0.27(0.69) 0.0019(0.0089)

for reference. These results reveal that the parameters of the three random effects models

can be accurately estimated with reasonable sample sizes.

4.2 Goodness-of-fit and model selection

Two simple graphical methods can be used to visualize goodness-of-fit of the IG process

models. The first method is proposed by Wang and Xu (2010) by noting that if X ∼ IG(a, b),

b(X − a)2/(a2X) ∼ χ21. In the simple IG process Yi(t) ∼ IG (Λ(t), ηΛ2(t)), the increment

yij = Yi(tij)− Yi(ti,j−1) are independent and thus ζij ≡ η(yij − λij)2/yij, i = 1, 2, · · · , N, j =

1, 2, · · · , ni are i.i.d. χ21 variables. When the parameters θ are estimated from data, the

simulation results in Section 4.1 suggest that θ is consistent, so ζij ∼ χ21 asymptotically.

Therefore, a χ21 quantile-quantile (Q-Q) plot for {ζij; i = 1, 2, · · · , N, j = 1, 2, · · · , ni} can

be used to assess the goodness of fit. Analogously, Q-Q plots for the three random effects

models can be constructed with additional effort. For example, in the random drifts model

Yi(t) ∼ IG (µiΛ(t), ηΛ2(t)) with µi random, ζij = η(µ−1i yij − λij)

2/yij are approximately

i.i.d. χ21, where µ−1

i = E(µ−1i |θ,Yi). The second graphical method is based on the concept

of pseudo failure times (Meeker and Escobar 1998, Section 13.9) when there are multiple

observations for each unit. Given a fixed threshold D, the pseudo failure time T(D)i of the

ith unit is obtained through fitting Yi against the observation times by means of nonlinear

15

least squares and then interpolating or extrapolating. On the other hand, given a specific

IG process model, the first passage time distribution FTD(t) can be easily obtained, e.g., (1)

or (2) for the simple IG process. In the presence of random effects, FTD(t) is obtained by

integrating the random effects out of (1) or (2). If the degradation follows the specific IG

process model of interest, FTD(T(D)i ), i = 1, · · · , N , are approximately i.i.d. standard uniform

random variables. This approximation enables us to construct standard uniform Q-Q plots

for a goodness-of-fit check. For the purpose of goodness-of-fit test, D could be the failure-

defining threshold or other smaller convenient thresholds. We also want to point out that

when the sample size is small, especially when the number of observations from each unit is

small, the Q-Q plots are not effective in checking the goodness-of-fit. This is because given

the limited samples, the realizations of the random effects in each unit cannot be accurately

estimated. Consequently the ζij might be far from the χ21 distribution in such cases.

Given a degradation dataset with reasonably large sample size, the Q-Q plots discussed

above serve as intuitive tools for selecting the best-fit IG process model. Simulation examples

of model selection based on the Q-Q plots are included in Section 2.2 of the supplementary

materials. However, model selection based on the Q-Q plots is qualitative. To quantitatively

select the model, the most straightforward way is to use the Akaike Information Criterion

(AIC) defined as AIC = 2dim(θ)− 2l(θ), where dim(θ) is the number of parameters. Based

on this criterion, the model with the minimum AIC value is selected. Alternatively, we

can use the root mean squared prediction error (RMSPE) as the criterion, and the error

is estimated by means of leave-one-out cross-validation. More specifically, we routinely set

aside unit i, i = 1, · · · , N , use the remaining N − 1 units to estimate the model parameters

θ−i, and then compute the prediction error of the degradation of unit i as

eij = Yi(tij)− E[Yi (tij)

∣∣θ−i, Yi(tik); k < j], j = 1, 2, · · · , ni,

where E[Yi(tij)

∣∣θ−i, Yi(tik); k < j]

is the predicted degradation of unit i at time tij condi-

16

tional on all observations before tij as well as on θ−i. After obtaining eij for i = 1, 2, · · · , N

and j = 1, 2, · · · , ni, we can compute RMSPE =√∑N

i=1

∑ni

j=1 e2ij/[∑N

i=1 ni − dim(θ)]. The

model that best fits the data is expected to yield the minimum RMSPE. In the supplemen-

tary materials (Section 2.2), we include some numerical results on how these criteria perform

in model selection.

Remarks: It is also interesting to note that some features of the degradation paths can

help identify the appropriate model. For example, if the degradation paths of all the units

have different rates, they suggest that the random drifts model and random drift-volatility

model might fit. On the other hand, if the degradation rates are similar, but the increments

have larger variability, the random volatility model can be considered. To illustrate, sample

degradation paths from each model are plotted in the supplementary material (Section 2.3)

for reference. However, these features might not be distinctive enough for model selection.

In such cases, the model selection techniques discussed above shall be used complementarily.

5 IG Processes with Covariates

Covariates or explanatory variables, such as temperature, humidity, usage rate, etc., affect

the degradation process and should be incorporated into the degradation model when the

covariate information is available. This section considers IG processes with a single covariate

s. The case of multiple covariates can be handled in a similar vein. Incorporation of covariates

can be done by allowing some parameters in the IG process to depend on s through a link

function h(s). Some commonly used link functions include:

• Power law function: h(s) = ξ0 · sα;

• Arrhenius function: h(s) = ξ0 · e−α/s;• Inverse-logit function: h(s) = ξ0 + αes/(1 + es);

• Exponential function: h(s) = ξ0eαs;

where ξ0, and α are model parameters. The Arrhenius function is widely used when the

covariate is temperature and the power law is often used to characterize the effect of voltage

17

(Meeker and Escobar 1998). In real applications, appropriate link functions can be specified

based on the degradation physics, expert knowledge or data analysis.

We denote si, i = 1, · · · , N as the covariate values associated with the ith degradation

path. Consider the simple IG process Y (t) ∼ IG(µΛ(t), ηΛ2(t)). The first covariate model

has µi = h(si) whilst η independent of s. With the observed data Yi and si, i = 1, · · · , N ,

the log-likelihood function, up to a constant, is

l(θ) =ln η

2

N∑i=1

ni +N∑i=1

ni∑j=1

[lnλij −

η(yij/h(si)− λij)2

2yij

]. (9)

Direct numerical maximization of (9) yields the MLE θ. The second covariate model has µ

independent of s and ηi = h(si), leading to the likelihood function

l(θ) =N∑i=1

ni2

lnh(si) +N∑i=1

ni∑j=1

[lnλij − h(si)

(yij − µλij)2

2µ2yij

].

We can also consider Y (t) ∼ IG(µΛ(t), ηµ2Λ2(t)) and the third covariate model has µi =

h(si) and η independent of s. The associated likelihood function is

l(θ) =N∑i=1

ni2

ln(η · h2(si)) +N∑i=1

ni∑j=1

[lnλij −

η(yij − h(si)λij)2

2yij

].

The development of the three covariate models above follows a similar line of reasoning

with the random effects models in Section 3. From the viewpoint of equivalent cumulative

operating time (Bagdonavicius and Nikulin 2000), the fourth covariate model can be devel-

oped. The basic idea is that for a unit exposed at stress s with duration t, its degradation

is the same as if it is exposed at stress s0 for duration h(s) · t, where s0 is the nominal stress

with h(s0) = 1. Consider the IG process Y (t) ∼ IG(µΛ(t), ηΛ2(t)). The notion of equivalent

cumulative operating time amounts to replacing Λ(t) by Λ(h(si) · t). The resulting covariate

18

model has a likelihood function as

l(θ) =N∑i=1

ni2

ln η +N∑i=1

ni∑j=1

[lnλij(si)−

η(yij/µ− λij(si))2

2yij

], (10)

where λij(si) = Λ(h(si)tij) − Λ(h(si)ti,j−1) to emphasize its dependence on si. A figure

of merit of this covariate model is that it allows the covariate to be time-dependent. In

particular, when the covariate si(t) varies over time t, the process {Yi(t; si(t)); t > 0} at time

t has an equivalent cumulative operating time of∫ t

0h[si(u)]du under the nominal covariate

condition s0. The resulting likelihood function remains the same as (10), with λij(si) replaced

by Λ(∫ tij

0h[si(u)]du)− Λ(

∫ ti,j−1

0h[si(u)]du).

Random effects are not considered in the above covariate models. In the presence of

random effects, covariates can likewise be incorporated through the model parameters, or

through the method of equivalent cumulative operation time. For example, in the random

drifts model Y (t) ∼ IG(µΛ(t), ηΛ2(t)) with µ−1 ∼ T N (ω, κ−2), covariate models can be

obtained by letting ω = h(s) and other parameters independent of s, or by letting η = h(s).

Consequently, the unconditional distribution of Yi with covariate si resembles (3) with ω

replaced by h(si), or η replaced by h(si) in the later case. Alternatively, we can use the

method of equivalent cumulative operation time and replace Λ(t) by Λ(h(s) · t). In like

manner, covariates can be incorporated into the other two random effects models, which

lead to slightly different likelihood functions from those in Section 3.

Model estimation and selection in Section 4 apply analogously when both random effects

and covariates exist. Therefore, the detailed discussions are omitted here.

6 An Illustrative Example

Meeker and Escobar (1998) presented a degradation dataset of GaAs laser devices. Degra-

dation of the laser device in terms of the operation current increases over time and fails the

device when it is higher than a fixed threshold. Degradation data from 15 testing samples

19

10^3 Hours

Perc

enta

ge incre

ases in c

urr

ent

0

2

4

6

8

10

12

0 1 2 3 4

Figure 1: Degradation paths of the GaAs laser current.

are collected, with observations made at 250, 500, · · · , 4000 hr. The degradation data are

plotted in Figure 1. It shows that the degradation rates are considerably heterogeneous,

while the variability of the degradation increments is not very significant. This observation

also motivates the development of the random drifts model.

Wang and Xu (2010) thoroughly analyzed this dataset and concluded that neither the

Wiener nor the Gamma process provides a good fit. They found the IG process with random

volatility model fits the data reasonably well. In this section, we further apply the other two

random effects models proposed in Sections 3.1 and 3.3 to the data.

Performance of the three random effects models as well as the simple IG process are

compared. From the semi-parametric fitting in Wang and Xu (2010), we adopt the power

law Λ(t) = tq. Table 2 compares their estimated parameters as well as the 95% confi-

dence intervals using the parametric percentile bootstrap (Efron and Tibshirani 1993) with

B = 5000 bootstrap replications. It is noted from Table 2 that δ and γ in the random volatil-

ity model have wide confidence intervals. In addition, by means of the semi-parametric

method (Wang and Xu 2010), the estimated δ and γ in the random volatility model are

20

Table 2: MLEs and the 95% confidence intervals for the IG processes.

Simple Model RD ModelEstimation CI.lower CI.upper Estimation CI.lower CI.upper

µ 2.0416 1.8897 2.208 ω 0.5075 0.4517 0.5593η 54.7318 45.0245 69.723 κ 11.1932 8.2769 23.5635q 0.9984 0.9534 1.043 η 72.2011 59.5408 90.6289

q 0.9984 0.9594 1.0319

RV Model RDV ModelEstimation CI.lower CI.upper Estimation CI.lower CI.upper

δ 54.8830 42.4222 664.555 ω 2.009 1.7708 2.196γ 0.9991 0.8016 10.000 κ 2.570 1.9111 4.647µ 2.0366 1.8859 2.194 η 18.645 15.5330 23.261q 1.0000 0.9585 1.043 q 1.010 0.9881 1.049

δ = 12.3798 and γ = 0.1996, which are quite different from those estimated via the para-

metric results in this section. The discrepancy is mainly because the profile likelihood

l(δ, γ) = maxµ,p∏N

i=1 fIG(Yi|γ) is very flat in a large region of (δ, γ), as shown in Section 2.4

of the supplementary materials. Therefore, slight changes (e.g., nonparametric Λ(t) versus

parametric Λ(t)) may lead to significant differences in γ. These results tend to suggest that

the random volatility model may not be the best choice for the data.

To select the best-fit IG process model, the χ21 Q-Q plot based on the transformed

degradation increments and the uniform Q-Q plot based on the pseudo failure times with

D = 0.06, as discussed in Section 4.2, are shown in Figure 2. This figure suggests that the

random drifts model provides the best fit to the data, insofar as the points scatter around

the straight line. The random drift-volatility model also looks reasonable, though inferior

to the random drifts model. We then compare these four IG process models in terms of the

quantitative criteria discussed in Section 4.2. The AIC and RMSPE values of each model

are summarized in Table 3. The AIC criterion favors the random drift-volatility model while

the random drifts model yields the minimum RMSPE. In consideration of both Figure 2 and

Table 3, the random drift model is recommended for the laser degradation data, but the

random drift-volatility model is also a possible choice. Other Q-Q plots using the pseudo

failure times generated by different failure thresholds (see Section 2.4 of the supplementary

21

Theoretical Quantile

Sam

ple

Qua

ntile

0

2

4

6

8

0 2 4 6 8

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●

●●●● ●

●

●

Simple

0 2 4 6 8

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●

●●●●●●●●●

●●●

●●

●●

●

●

RD

0 2 4 6 8

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●

●●●●●●●●●●●

●●●●●●●

● ●●

●

RV

0 2 4 6 8

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●

●●●●

●●●●●

●●

●

RDV

(a) GOF using degradation increments

Theoretical Quantile

Sam

ple

Qua

ntile

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Simple

0.0 0.2 0.4 0.6 0.8 1.0

RD

0.0 0.2 0.4 0.6 0.8 1.0

RV

0.0 0.2 0.4 0.6 0.8 1.0

RDV

(b) GOF using failure times

Figure 2: Q-Q plots of different IG process models using GaAs laser degradation data.

materials) also agree with the conclusion. We note in passing that the variation in the

degradation increments is small (large η), which is consistent with the degradation paths. In

addition, ωκ is large enough that a normal distribution may be used for the random effects

instead of the truncated normal.

7 Conclusion

This study has systematically investigated the IG processes as a meaningful class of degra-

dation models. This class of models has a number of promising features. The IG process

has monotone and conditionally independent increments, while the increments have a simple

22

Table 3: Quantitative comparisons between the four IG process models.

Simple model RD RV RDVAIC -48.114 -86.702 -46.236 -90.508

RMSPE 0.201 0.184 0.202 0.187

closed form PDF and CDF. The first hitting time of the process to a fixed threshold has a

closed form PDF and CDF, and can be linked to the Birnbaum-Saunders distribution. Next,

we showed that the IG process is a limiting compound Poisson process. This justification

underpins it as an appropriate degradation model for products whose degradation is caused

by random environments. In addition to the random effects model proposed by Wang and

Xu (2010), we further developed two random effects models by linking the IG process to the

Wiener process. These three random effects models make the IG process more flexible than

the Gamma process, which has only one way to incorporate random effects. We also showed

that covariates can be easily incorporated into both the simple IG process and the random

effects models by allowing the model parameters to be a function of the covariates. This

leads to a wide variety of IG process models with covariates. In summary, the class of IG

processes has similar properties to the Gamma process, but is much more flexible.

Beyond our current scope, there are some open issues worth further investigation. The

random effects models are mathematically tractable only when the random effects follow

certain distributions. Therefore, it is of interest to further investigate how robust such

models are when the distribution of the random effects is not as assumed. In addition, it is

of interest to develop more powerful goodness-of-fit tests and more comprehensive selection

criteria to select the best fitting IG process model. Furthermore, applications of the IG

process in burn-in tests, accelerated degradation tests, preventive maintenance scheduling

and remaining useful life prediction are broad topics that require extensive investigations.

23

Appendix

Proof of Theorem 1: The log characteristic function of the IG process with Y (t) ∼

IG(µt, ηt2) can be expressed as

ΨY (t)(u) = ln E(eiuY (t)) =ηt

µ

[1−

√1− 2iµ2u

η

]

= t ·∫ ∞

0

(eiux − 1) ·√

η

2πx3· exp

(− ηx

2µ2

)I(0,∞)(x)dx

= t ·∫ ∞

0

(eiux − 1) · Π(dx),

where Π(dx) =√η/(2πx3) · exp [−ηx/(2µ2)] I(0,∞)(x)dx. We can verify that

∫R\{0}

min(x2, 1)Π(dx) <

∫ ∞0

x2

√η

2πx3· exp

(− ηx

2µ2

)dx =

2µ3

η√π· Γ(

3

2),

where Γ(x) is the gamma function. Consequently, according to Levy-Khintchine representa-

tion (Ferguson and Klass 1972), the log-characteristic function of the stationary IG process

Y (t) can be expressed as

ΨY (t)(u) = itςu+ t

∫R\{0}

(eiux − 1− iuxI|x|<1)Π(dx)

which is a pure jump Levy process without the diffusion component. Its Levy measure is

Π(dx), and the drift rate can be expressed as ς = µ · Γ (1/2, η/(2µ2)) /√π, where Γ(a, b) =∫ b

0ua−1e−udu is the lower incomplete Gamma function.

To prove that the IG process is the limiting process of a compound Poisson process, it

suffices to show that the log characteristic function of the corresponding compound Poisson

process converges to ΨY (t)(u). According to the construction of the arrival rate and jump

size distributions of the compound Poisson process in Theorem 1, their log characteristic

24

function can be expressed as

Ψn(u) = νnt ·[∫ ∞

0

eiuxGn(dx)− 1

]= t ·

∫ ∞0

(eiux − 1)Πn(dx).

It can be easily shown that limn→∞Πn(dx) = Π(dx) almost everywhere. As a result of the

dominant convergence theorem (Rudin 1986), limn→∞Ψn(u) = ΨY (t)(u). Since there is a

one-to-one correspondence between a characteristic function and a distribution function, we

can see that the constructed compound Poisson process converges to the IG process.

Supplementary Materials:

Technical details: The PDF file provides the technical details as referred in the paper. It

also includes additional figures and tables from simulation studies (PDF file).

Source code: The zipped package contains R codes to perform the EM estimation, diag-

nostic analysis, and case studies of the IG process (ZIP file).

Acknowledgements: We would like to thank the editor, associate editor, and two anony-

mous referees for their constructive comments and suggestions that have considerably im-

proved the paper.

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