Lifetime Data Analysis, 11, 511–527, 2005 2005 Springer Science+Business Media, Inc. Manufactured in The Netherlands.
Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes CHANSEOK PARK [email protected] Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA W. J. PADGETT Department of Mathematical Sciences, Clemson University, Department of Statistics, University of South Carolina, Columbia, SC, 29208, USA Received May 8, 2005; Accepted June 15, 2005 Abstract. Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented. Keywords: inverse Gaussian (Wald) distribution, degradation process, accelerated life test, geometric Brownian motion process, gamma process, censoring
1.
Introduction
Many materials and systems degrade over time before they break down or fail. To model such degradation over time and make inference about the failure of such systems or materials, we need to develop a methodology considering both observed failures and degradation data. Recently, Lu (1995), Pettit and Young (1999), and Padgett and Tomlinson (2004) introduced some methodology assuming that degradation is a Gaussian process. In this paper, a somewhat different approach is proposed in that the assumed degradation process can be different from a Gaussian process. The pitfall of the Gaussian assumption is that the process is not increasing and can possibly be negative. That is, at some time point before the degradation is measured, the degradation value can be larger than the measurement or can be even a negative value. This presents a very difficult physical interpretation and motivates the need for more physically realistic models for describing degradation. With the development of engineering and science technology, many modern products have longer lifetimes and greater reliability than those in the past. Thus, lifetime measurements and degradation measurements take much more time than
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they used to. It is therefore difficult to observe failure times, or even degradation measurements, under normal use conditions. Since time-consuming tests under normal use conditions are costly, one needs to use accelerated tests. Accelerated life tests decrease the time to failure by exposing the products to higher levels of stress conditions (high-usage rates, increased levels of environmental variables) which cause earlier failures. Accelerated degradation tests expose the products to greater stress levels for degradation than the normal use stresses to obtain degradation measurements in a more timely fashion, and then time to failure is estimated by using the degradation measurements. Early work on degradation models is referenced by Nelson (1990), while more recent results are mentioned by Bagdonavicius and Nikulin (2002) and Meeker and Escobar (1998). In particular, degradation models based on Gaussian or other stochastic processes have been considered most recently by Doksum and Normand (1995), Lu (1995), Whitmore (1995), Whitmore and Schenkelberg (1997), Whitmore et al. (1998), Pettit and Young (1999), and Padgett and Tomlinson (2004). Bagdonavicius and Nikulin (2000) used a gamma degradation process that allowed covariates, but did not use exact failure times (first passage time to a damage threshold) for the likelihood, as we develop here. Lawless and Crowder (2004) also used the gamma process with covariates and random effects to model degradation under accelerated environments or degradation occurring at different rates in the same environment. Using regression-type methods, general degradation path models and special cases have been fitted and studied by several authors, including Lu and Meeker (1993), Boulanger and Escobar (1994), Hamada (1995), and Meeker et al. (1998). Also, degradation models applied to specific problems in engineering have been presented by several investigators including Carey and Koenig (1991), Yanagisawa (1997) and Meeker and Escobar (1998). In this paper, we develop several new models for degradation and failure data using a stochastic process, such as the geometric Brownian motion or gamma process, and incorporating an accelerated test variable. It is shown that in many cases, the models can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Our framework is quite general, allowing for different degradation processes, as illustrated in Section 2 for geometric Brownian motion and gamma processes. Even though our approach is different, the specific model using the gamma degradation process with acceleration in Section 2 is a special case of Lawless and Crowder (2004) model, but in addition, we show that our resulting gamma process model can often be closely approximated by an inverse Gaussian model. To develop the general framework, suppose that as the tensile load or stress level on a material specimen or a system is increased, the cumulative damage Xn+1 after n+1 increments of stress can be described by the cumulative damage model proposed by Durham and Padgett (1997), Xnþ1 ¼ Xn þ Dn hðXn Þ; where Dn denotes the damage incurred at the (n+1)st increment and h() is the damage model function. Recently, Park and Padgett (2005) proposed a new
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cumulative damage model using a damage accumulation function. This new model is given by cðXnþ1 Þ ¼ cðXn Þ þ Dn hðXn Þ; where c() is the damage accumulation function. In many cases, it is more appropriate to describe the damage process with a continuous process. A continuous version can be represented by dcðXu Þ ¼ hðXu ÞdDu : Then the cumulative damage or level of degradation of the system at the stress level (or time) t is given by Z t Z t 1 dcðXu Þ ¼ dDu ¼ Dt D0 : 0 hðXu Þ 0 Note that the above integral is a stochastic integral as defined by Jacod and Shiryaev (1987). By selecting various forms of the functions c() and h() with an appropriate stochastic process Du, several new models for degradation can be obtained. For example, assuming that Du is a Brownian motion process with h(u)=1 and c(u)=u, we obtain a degradation model based on a Gaussian process as studied by Lu (1995), Pettit and Young (1999), and Padgett and Tomlinson (2004). In this paper, we consider cðuÞ ¼ log u and h(u)=1 with a Brownian motion process Du, which results in a geometric Brownian motion process model for the degradation process Xt and c(u)=u and h(u)=1 with a gamma process Du. In some applications, the load or stress on a unit causes failure and is recorded as the ‘failure stress’ rather than the ‘failure time’. For example, a tensile load on a material specimen may be the cause of breakage of the specimen. Hence, when appropriate in the remainder of this paper, ‘failure time’ may be interpreted as ‘failure stress’.
2.
The Density Functions of the Failure Time and the Degradation Value
In this section, we derive the probability density function (pdf) of the failure time (or first passage time) assuming a geometric Brownian motion or a gamma process for degradation. We also present the pdf of the level of degradation at the time observation of the process is terminated, i.e., the terminal value of the process, conditional on the event that the terminal value does not exceed the damage threshold C where failure occurs. Such a terminated process is commonly referred to as a truncated stochastic process.
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Geometric Brownian Motion Process
In practice, it is often the case that a degradation process should be always positive. We first consider the geometric Brownian motion as a degradation process Xt which is always positive, while the Brownian motion process is not. Using the damage model function h(u)=1 and the damage accumulation function c(u)=log u, we have the following cumulative damage or level of degradation of the system at the time t, log Xt log X0 ¼ Dt : We assume that the stochastic process {Du: 0 £ u £ t} is a Brownian motion process with positive drift coefficient a and diffusion b2. This implies that the time at which log Xt reaches a critical value can be considered as the threshold for failure (or first passage time) for a Brownian motion with positive drift a and diffusion b2. Thus, the system degradation Xt is a geometric Brownian motion process. The damage threshold level for failure is assumed to be a known positive constant C and the initial value of the process Xt is given by X0=x0. In what follows, we will derive the pdf of the first passage time to the threshold C and the pdf of the terminal value Xt conditioning on the event that the terminal value does not exceed the threshold C during the process. For convenience, let Zt=log Xt. Using the result of Cox and Miller (1965 §5.3), we have the following probability that Zt £ z given that the initial value z0=log x0, z z0 at pffiffi ; P½Zt z ¼ U b t where F() denotes the standard normal cumulative distribution function (cdf) and 10 or passes the threshold. Analogous to the approach of Lu (1995), the former event is denoted here by A and the latter by Ac (complement of A). Therefore, the pdf f() consists of two parts representing the joint pdfs contributed by sample paths that do not exceed the threshold C, fA ðx; x0 ; tÞ, and sample paths passing the threshold, fAc ðx; x0 ; tÞ. Hence, we have
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fðx; x0 ; tÞ ¼ fA ðx; x0 ; tÞ þ fAc ðx; x0 ; tÞ: Notice that the support of fA ðx; x0 ; tÞ is (0, C) and that of fAc ðx; x0 ; tÞ is (C, ¥). The joint pdf fA() will be used for constructing the likelihood function for the degradation data. First, we consider the pdf of the first passage time. Let the random variable S be the first passage time of the geometric Brownian motion process Xt to the threshold C. The pdf of S will be used for constructing the likelihood function for the observed failure data. In order for the random variable S to be the first passage time, we require: (i) X0=x0, (ii) Xt n ffi P½Xn n ¼ P½Xn t ¼ P max Xu
Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes CHANSEOK PARK [email protected] Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA W. J. PADGETT Department of Mathematical Sciences, Clemson University, Department of Statistics, University of South Carolina, Columbia, SC, 29208, USA Received May 8, 2005; Accepted June 15, 2005 Abstract. Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented. Keywords: inverse Gaussian (Wald) distribution, degradation process, accelerated life test, geometric Brownian motion process, gamma process, censoring
1.
Introduction
Many materials and systems degrade over time before they break down or fail. To model such degradation over time and make inference about the failure of such systems or materials, we need to develop a methodology considering both observed failures and degradation data. Recently, Lu (1995), Pettit and Young (1999), and Padgett and Tomlinson (2004) introduced some methodology assuming that degradation is a Gaussian process. In this paper, a somewhat different approach is proposed in that the assumed degradation process can be different from a Gaussian process. The pitfall of the Gaussian assumption is that the process is not increasing and can possibly be negative. That is, at some time point before the degradation is measured, the degradation value can be larger than the measurement or can be even a negative value. This presents a very difficult physical interpretation and motivates the need for more physically realistic models for describing degradation. With the development of engineering and science technology, many modern products have longer lifetimes and greater reliability than those in the past. Thus, lifetime measurements and degradation measurements take much more time than
512
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they used to. It is therefore difficult to observe failure times, or even degradation measurements, under normal use conditions. Since time-consuming tests under normal use conditions are costly, one needs to use accelerated tests. Accelerated life tests decrease the time to failure by exposing the products to higher levels of stress conditions (high-usage rates, increased levels of environmental variables) which cause earlier failures. Accelerated degradation tests expose the products to greater stress levels for degradation than the normal use stresses to obtain degradation measurements in a more timely fashion, and then time to failure is estimated by using the degradation measurements. Early work on degradation models is referenced by Nelson (1990), while more recent results are mentioned by Bagdonavicius and Nikulin (2002) and Meeker and Escobar (1998). In particular, degradation models based on Gaussian or other stochastic processes have been considered most recently by Doksum and Normand (1995), Lu (1995), Whitmore (1995), Whitmore and Schenkelberg (1997), Whitmore et al. (1998), Pettit and Young (1999), and Padgett and Tomlinson (2004). Bagdonavicius and Nikulin (2000) used a gamma degradation process that allowed covariates, but did not use exact failure times (first passage time to a damage threshold) for the likelihood, as we develop here. Lawless and Crowder (2004) also used the gamma process with covariates and random effects to model degradation under accelerated environments or degradation occurring at different rates in the same environment. Using regression-type methods, general degradation path models and special cases have been fitted and studied by several authors, including Lu and Meeker (1993), Boulanger and Escobar (1994), Hamada (1995), and Meeker et al. (1998). Also, degradation models applied to specific problems in engineering have been presented by several investigators including Carey and Koenig (1991), Yanagisawa (1997) and Meeker and Escobar (1998). In this paper, we develop several new models for degradation and failure data using a stochastic process, such as the geometric Brownian motion or gamma process, and incorporating an accelerated test variable. It is shown that in many cases, the models can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Our framework is quite general, allowing for different degradation processes, as illustrated in Section 2 for geometric Brownian motion and gamma processes. Even though our approach is different, the specific model using the gamma degradation process with acceleration in Section 2 is a special case of Lawless and Crowder (2004) model, but in addition, we show that our resulting gamma process model can often be closely approximated by an inverse Gaussian model. To develop the general framework, suppose that as the tensile load or stress level on a material specimen or a system is increased, the cumulative damage Xn+1 after n+1 increments of stress can be described by the cumulative damage model proposed by Durham and Padgett (1997), Xnþ1 ¼ Xn þ Dn hðXn Þ; where Dn denotes the damage incurred at the (n+1)st increment and h() is the damage model function. Recently, Park and Padgett (2005) proposed a new
ACCELERATED DEGRADATION MODELS
513
cumulative damage model using a damage accumulation function. This new model is given by cðXnþ1 Þ ¼ cðXn Þ þ Dn hðXn Þ; where c() is the damage accumulation function. In many cases, it is more appropriate to describe the damage process with a continuous process. A continuous version can be represented by dcðXu Þ ¼ hðXu ÞdDu : Then the cumulative damage or level of degradation of the system at the stress level (or time) t is given by Z t Z t 1 dcðXu Þ ¼ dDu ¼ Dt D0 : 0 hðXu Þ 0 Note that the above integral is a stochastic integral as defined by Jacod and Shiryaev (1987). By selecting various forms of the functions c() and h() with an appropriate stochastic process Du, several new models for degradation can be obtained. For example, assuming that Du is a Brownian motion process with h(u)=1 and c(u)=u, we obtain a degradation model based on a Gaussian process as studied by Lu (1995), Pettit and Young (1999), and Padgett and Tomlinson (2004). In this paper, we consider cðuÞ ¼ log u and h(u)=1 with a Brownian motion process Du, which results in a geometric Brownian motion process model for the degradation process Xt and c(u)=u and h(u)=1 with a gamma process Du. In some applications, the load or stress on a unit causes failure and is recorded as the ‘failure stress’ rather than the ‘failure time’. For example, a tensile load on a material specimen may be the cause of breakage of the specimen. Hence, when appropriate in the remainder of this paper, ‘failure time’ may be interpreted as ‘failure stress’.
2.
The Density Functions of the Failure Time and the Degradation Value
In this section, we derive the probability density function (pdf) of the failure time (or first passage time) assuming a geometric Brownian motion or a gamma process for degradation. We also present the pdf of the level of degradation at the time observation of the process is terminated, i.e., the terminal value of the process, conditional on the event that the terminal value does not exceed the damage threshold C where failure occurs. Such a terminated process is commonly referred to as a truncated stochastic process.
514
2.1.
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Geometric Brownian Motion Process
In practice, it is often the case that a degradation process should be always positive. We first consider the geometric Brownian motion as a degradation process Xt which is always positive, while the Brownian motion process is not. Using the damage model function h(u)=1 and the damage accumulation function c(u)=log u, we have the following cumulative damage or level of degradation of the system at the time t, log Xt log X0 ¼ Dt : We assume that the stochastic process {Du: 0 £ u £ t} is a Brownian motion process with positive drift coefficient a and diffusion b2. This implies that the time at which log Xt reaches a critical value can be considered as the threshold for failure (or first passage time) for a Brownian motion with positive drift a and diffusion b2. Thus, the system degradation Xt is a geometric Brownian motion process. The damage threshold level for failure is assumed to be a known positive constant C and the initial value of the process Xt is given by X0=x0. In what follows, we will derive the pdf of the first passage time to the threshold C and the pdf of the terminal value Xt conditioning on the event that the terminal value does not exceed the threshold C during the process. For convenience, let Zt=log Xt. Using the result of Cox and Miller (1965 §5.3), we have the following probability that Zt £ z given that the initial value z0=log x0, z z0 at pffiffi ; P½Zt z ¼ U b t where F() denotes the standard normal cumulative distribution function (cdf) and 10 or passes the threshold. Analogous to the approach of Lu (1995), the former event is denoted here by A and the latter by Ac (complement of A). Therefore, the pdf f() consists of two parts representing the joint pdfs contributed by sample paths that do not exceed the threshold C, fA ðx; x0 ; tÞ, and sample paths passing the threshold, fAc ðx; x0 ; tÞ. Hence, we have
ACCELERATED DEGRADATION MODELS
515
fðx; x0 ; tÞ ¼ fA ðx; x0 ; tÞ þ fAc ðx; x0 ; tÞ: Notice that the support of fA ðx; x0 ; tÞ is (0, C) and that of fAc ðx; x0 ; tÞ is (C, ¥). The joint pdf fA() will be used for constructing the likelihood function for the degradation data. First, we consider the pdf of the first passage time. Let the random variable S be the first passage time of the geometric Brownian motion process Xt to the threshold C. The pdf of S will be used for constructing the likelihood function for the observed failure data. In order for the random variable S to be the first passage time, we require: (i) X0=x0, (ii) Xt n ffi P½Xn n ¼ P½Xn t ¼ P max Xu
