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Modeling Accelerated Degradation Data Using Wiener Diffusion With A Time Scale Transformation
 

Modeling Accelerated Degradation Data Using Wiener Diffusion With A Time Scale Transformation

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My first technical paper and it's on accelerated degradation modeling. My entry to reliability engineering. ...

My first technical paper and it's on accelerated degradation modeling. My entry to reliability engineering.

Engineering degradation tests allow industry to assess the potential life span of long-life products
that do not fail readily under accelerated conditions in life tests. A general statistical model is presented here for
performance degradation of an item of equipment. The degradation process in the model is taken to be a Wiener
diffusion process with a time scale transformation. The model incorporates Arrhenius extrapolation for high stress
testing. The lifetime of an item is defined as the time until performance deteriorates to a specified failure threshold.
The model can be used to predict the lifetime of an item or the extent of degradation of an item at a specified future
time. Inference methods for the model parameters, based on accelerated degradation test data, are presented. The
model and inference methods are illustrated with a case application involving self-regulating heating cables. The
paper also discusses a number of practical issues encountered in applications.

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Modeling Accelerated Degradation Data Using Wiener Diffusion With A Time Scale Transformation Modeling Accelerated Degradation Data Using Wiener Diffusion With A Time Scale Transformation Document Transcript

  • Lifetime Data Analysis, 3, 27–45 (1997) c 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.Modelling Accelerated Degradation Data UsingWiener Diffusion With A Time Scale TransformationG. A. WHITMOREFaculty of Management, McGill University, Montreal, Quebec H3A 1G5, CanadaFRED SCHENKELBERGHewlett-Packard Company, Vancouver, Washington 98683, U.S.A.Received February 9, 1996; revised December 2, 1996; accepted December 20, 1996Abstract. Engineering degradation tests allow industry to assess the potential life span of long-life productsthat do not fail readily under accelerated conditions in life tests. A general statistical model is presented here forperformance degradation of an item of equipment. The degradation process in the model is taken to be a Wienerdiffusion process with a time scale transformation. The model incorporates Arrhenius extrapolation for high stresstesting. The lifetime of an item is defined as the time until performance deteriorates to a specified failure threshold.The model can be used to predict the lifetime of an item or the extent of degradation of an item at a specified futuretime. Inference methods for the model parameters, based on accelerated degradation test data, are presented. Themodel and inference methods are illustrated with a case application involving self-regulating heating cables. Thepaper also discusses a number of practical issues encountered in applications.Keywords: Acceleration, Arrhenius, Degradation, Likelihood methods, Prediction, Statistical inference, Wienerprocess1. IntroductionEngineering degradation tests allow industry to assess the potential life span of long-lifeproducts that do not fail readily under accelerated conditions in life tests. A general sta-tistical model is presented here for performance degradation of an item of equipment. Thedegradation process in the model is taken to be a Wiener diffusion process with a time scaletransformation. The model incorporates Arrhenius extrapolation for high stress testing.The lifetime of an item is defined as the time until performance deteriorates to a specifiedfailure threshold. The model can be used to predict the lifetime of an item or the extent ofdegradation of an item at a specified future time. Inference methods for the model param-eters, based on accelerated degradation test data, are presented. The model and inferencemethods are illustrated with a case application involving self-regulating heating cables. Thepaper also discusses a number of practical issues encountered in applications.2. ModelMost items deteriorate or degrade as they age. The items fail when their degradation reachesa specified failure threshold. We assume that one key degradation measure governs failureand take the statistical model for this measure to be a Wiener diffusion process {W (t)} with
  • 28 G. A. WHITMORE AND FRED SCHENKELBERGFigure 1. Representative sample path of a Wiener diffusion process. The path traces out the degradation of anitem over time.mean parameter δ and variance parameter ν > 0. A Wiener process has found applicationas a degradation model in other studies—see, for example, Doksum and H´ yland (1992) oand Lu (1995). This process represents an adequate model for the case study examinedin Section 4. Its appropriateness for general application is examined in Section 5. Basictheoretical properties of a Wiener process may be found in Cox and Miller (1965). Figure 1shows a representative sample path of a Wiener process. The path traces out the degradationof an item over time, with larger values representing greater deterioration. We assume thatdegradation is measured at n +1 time points ti , where t0 < t1 < · · · < tn . The measurementat time ti is denoted by Wi = W (ti ). The preceding Wiener model assumes that δ, the rate of degradation drift, is constant.Where it is not constant, however, it is often found that a monotonic transformation of thetime scale can make it constant. We assume that this kind of transformation is appropriatehere and denote this transformation by t = τ (r ), (2.1)where r denotes the clock or calendar time and t, the transformed time. We shall requirethe transformation to satisfy the initial condition τ (0) = 0. The transformed time t is oftenreferred to as operational time and, in physical terms, can be considered as measuring thephysical progress of degradation, such as oxidation, wearout, and so on. Because of thisphysical basis for t, the transformation t = τ (r ) will depend on the particular degradationmechanism that dominates in a given application. We shall let ti = τ (ri ) denote theoperational time corresponding to clock time ri , for i = 0, 1, . . . , n.
  • MODELLING ACCELERATED DEGRADATION DATA USING WIENER DIFFUSION 29 In the case study, we consider the following two forms of time transformation (2.1) t = 1 − exp(−λr γ ), (2.2a) t = r λ, (2.2b)where both λ and γ are positive parameters. The exponential time transformation in (2.2a)is suitable for many applications in which degradation approaches a saturation point orasymptote where deterioration ends (for example, where oxidation ceases). The transfor-mation has two parameters. Parameter λ usually is unknown and must be estimated in eachapplication. Parameter γ , on the other hand, is often a fundamental constant that is set tosome whole number or simple fractional value based on physical properties of the item. Anexample of (2.2a) is found in Carey and Koenig (1991). In their case study, the propagationdelay of a logical circuit degrades (increases) along an expected path of the form (2.2a)with γ = 1/2. Whitmore (1995) encountered a similar transformation of the time scalein connection with the degradation of transistor gain. In his case, γ = 1. The power timetransformation in (2.2b) is suggested by the prevalence of power relationships in physicalmodels generally. Transformation (2.2b) implies that degradation will continue to increasewithout bound. As with (2.2a), the parameter λ in (2.2b) usually must be estimated in eachapplication. In developing the model in this section and the inference methodology in the followingsection, we shall use time transformation (2.1) in its general form. As a final remark,we note that Doksum and H´ yland (1992) use a similar transformed time scale to model ovariable levels of stress in an accelerated life test. Their work prompted us to consider thesame approach in this research. When degradation test data are gathered under accelerated or high stress conditions (suchas elevated temperature), the relation between the level of stress and the model parametersmust be established so that the parameters can be extrapolated to the lower stress conditionsencountered in actual use of the item. In our model, we have the Wiener process parameters,δ and ν, and the parameters of the time transformation (2.1). For expository convenience,we shall assume that the time transformation t = τ (r ) has only one parameter that isunspecified and, hence, requires estimation. We denote this parameter by λ, as we havedone in (2.2a) and (2.2b). We then write the transformation as t = τ (r ; λ) to show thedependence on the parameter. Both parsimony and actual experience argue for using simplefunctional forms to describe the relations between the model parameters and the level ofstress. We shall let the stress be described by a single quantitative measure which wedenote by s. After applying continuous monotonic transformations to the parameters andto the stress measure s, we assume that the relations between the transformed stress and thetransformed parameters are linear as follows. A(δ) = a0 + a1 H (s) B(ν) = b0 + b1 H (s) (2.3) C(λ) = c0 + c1 H (s)Here A(δ), B(ν), C(λ) and H (s) denote the transformations. Note that A, B and C in (2.3)can be interpreted as a new set of parameters for the model while H can be considered as
  • 30 G. A. WHITMORE AND FRED SCHENKELBERGa surrogate stress measure. For expository convenience, we consider only nonparametrictransformations in (2.3), such as a logarithmic or reciprocal transformation. Nonparametrictransformations are sufficient for our case study, although general applications may requiretransformations with parameters that must be estimated. The linear relationship between the log-rate of a chemical reaction and the reciprocalabsolute temperature is called the Arrhenius equation. The physical equation is widelyencountered in stress tests. As the equations in (2.3) relate the statistical parameters ofthe degradation model to the level of stress, we shall refer to them here as the Arrheniusequations and the transformation H (s) as the Arrhenius transformation. In the traditionalArrhenius relation, H (s) = 1/s where s represents the absolute temperature in o K . If the linear coefficients of the Arrhenius equations in (2.3) were known then reliabilitypredictions would proceed as follows. Given the use-level of stress, say s0 , we wouldfirst calculate H0 = H (s0 ) and then calculate A0 = a0 + a1 H0 , B0 = b0 + b1 H0 andC0 = c0 + c1 H0 from (2.3). Next, we would solve for the process parameters δ0 andν0 and the time transformation parameter λ0 under use-level conditions by inverting thefollowing monotonic transformations: A0 = A(δ0 ), B0 = B(ν0 ), C0 = C(λ0 ). Two kindsof predictions could now be made.Prediction of a Future Degradation LevelSuppose that we wish to predict the level of degradation W f at some future (clock) time r ffor a new item. We assume that the degradation measure is calibrated so a new item hasno degradation and, hence, W (0) = 0. We then exploit the fact that W f ∼ N (δ0 t f , ν0 t f )where t f denotes the operational time corresponding to r f , i.e., t f = τ (r f ; λ0 ). This normaldistribution is the predictive distribution of W f at clock time r f under a use-level of stress. √Prediction limits for W f take the form δ0 t f ± z ν0 t f , where z denotes an appropriatestandard normal number.Prediction of a LifetimeSuppose that we wish to predict the lifetime of a new item. We must first specify a failurethreshold for the degradation process. We let ω > 0 denote this threshold level. We thenuse the fact that the first passage time T to this threshold will follow the inverse Gaussiandistribution I G(ω/δ0 , ω2 /ν0 ) under use conditions (see, for example, Chhikara and Folks,1989). The cumulative distribution function F(t) of this distribution can then be used tomake predictive statements about T . This function has the form 1 1 F(t) = (δ0 t − ω)(ν0 t)− 2 + exp(2δ0 /ν0 ) −(δ0 t + ω)(ν0 t)− 2 , (2.4)where denotes the standard normal distribution function. It must be kept in mind herethat T refers to the operational life-time of the item under use-level conditions. The inverseof the time transformation t = τ (r ; λ0 ) can be used to translate these predictive statementsabout T into equivalent statements about the item’s lifetime measured in clock time.
  • MODELLING ACCELERATED DEGRADATION DATA USING WIENER DIFFUSION 313. InferenceIn practice, the linear coefficients of the Arrhenius equations in (2.3) must be estimatedfrom accelerated degradation test data. We now give new notation to represent these testdata and specify the general characteristics of the degradation test. We assume that m k new items are placed on test at each of K stress levels, denoted by sk ,for k = 1, . . . , K . Let Wi jk and ri jk denote the observed degradation level and clock time,respectively, for the ith reading on the jth item at the kth stress level. Here j = 1, . . . , m kand i = 0, . . . , n jk . Thus, m k is the number of items on test at the kth stress level andn jk + 1 is the number of observations made on the jth item at that stress level. We assumethat n jk ≥ 2 for each item so that the data are sufficient for parameter estimation. We use the method of maximum likelihood to estimate the parameters δ, ν and λ for each ˆ ˆ ˆitem. Let these estimates be denoted by (δ jk , ν jk , λ jk ) for the jth item at the kth stress level.To describe the estimation method, we will suppress the subscripts j and k and focus onthe n + 1 observations (Wi , ri ), i = 0, 1, . . . , n, made on an individual item. Because non-overlapping increments in a Wiener process are independent, we consider first differencesof the observations. For i = 1, 2, . . . , n, define Wi = Wi − Wi−1 and ti = ti − ti−1 ,where ti = τ (ri ; λ). The dependence of the differences ti on the unknown parameter λ issuppressed in this notation. As we have Wi ∼ N (δ ti , ν ti ), (3.1)the sample likelihood function is n n 1 1 ( Wi − δ ti )2 L(δ, ν, λ) = (2πν ti )− 2 exp − . (3.2) i=1 2ν i=1 tiBy basing the sample likelihood function on first differences, the initial observation W0 =W (t0 ) does not appear explicitly in the function. For the case study in Section 4, the initialobservation for an item is independent of the degradation process parameters and, hence, noinformation is lost by omitting this initial reading from the likelihood. In some applications,there may be a need to model the initial reading and include it in the inference structure. The likelihood function (3.2) can be maximized directly by using a three-dimensional nu-merical optimization routine (the approach used in the case study). An alternative approachis to fix λ initially and then maximize the likelihood function (3.2) with respect to δ andν. This optimization yields the following conventional estimators, each being conditionalon λ. Wn − W 0 ˆ δ(λ) = (3.3a) tn − t 0 1 n ˆ [ Wi − δ(λ) ti ]2 ν (λ) = ˆ (3.3b) n i=1 ti
  • 32 G. A. WHITMORE AND FRED SCHENKELBERGSubstituting these conditional estimators back into (3.2) and simplifying, we obtain thefollowing partially maximized profile likelihood function. n n 1 L(λ) = [2π ν(λ)]− 2 ˆ ( ti )− 2 exp(−n/2) (3.4) i=1Recall that ti here is a function of λ. The function in (3.4) can be maximized using aone-dimensional search over λ, yielding the maximum likelihood estimate λ. Substitution ˆof this estimate into each of (3.3a) and (3.3b) yields the unconditional maximum likelihood ˆestimates, δ and ν . ˆ By one of the preceding numerical methods, therefore, we obtain the triplet of parameter ˆ ˆ ˆestimates (δ jk , ν jk , λ jk ) for the jth item at the kth stress level. Continuing to assume that thetransformations A, B, C and H are known for the moment, we can compute the transformed ˆ ˆstatistics A jk = A(δ jk ), Bjk = B(ˆ jk ), C jk = C(λ jk ), and Hk = H (sk ), for all j and k. ν The exact multivariate distribution of the triplets Y jk = (A jk , Bjk , C jk ) is unknown,although we know from likelihood theory that they will be asymptotically trivariate normal.Thus, if we let Xk = (1, Hk ) and a0 b0 c0 B= a1 b1 c1then we know that Y jk ∼ approx. N3 (Xk B, Σk ), (3.5)where Σk denotes the covariance matrix of the triplet for each item at stress level k. Using(3.5), the linear coefficients of the Arrhenius equations (2.3) can be estimated by multivariateregression of the triplets Y jk on the values of Hk (see Johnson and Wichern, 1992, page314). This regression must take account of the heteroscedastic error structure reflected inthe covariance matrix Σk . The precise form of the covariance matrix will vary from oneapplication to another. The regression analysis is illustrated in Section 4. Once the estimated Arrhenius equations are available, the predictive analysis described inthe preceding section can be applied, provided two remaining hurdles are overcome. First,account must be taken of the sampling errors in the estimated coefficients in (2.3) whenthey are used for predictive analysis. Second, the correct transformations A(δ), B(ν), C(λ)and H (s) must be identified. These two hurdles need to be dealt with on a case-by-casebasis. In some applications, the sampling errors in the estimated coefficients of (2.3) representeffects of secondary importance and, hence, can be ignored in predictive analysis. Wherethe sampling errors will have a material effect on predictive statements, a strategy is neededto take their effect into account. Development of exact analytical results appears to beinfeasible. A practical approach might employ sensitivity analysis, simulation or a Bayesianprocedure; these methods being in increasing order of sophistication. The estimator of thecoefficient matrix B in regression model (3.5) is the key input to predictions. The estimator, ˆwhich we denote by B, has an asymptotic multivariate normal distribution that can beestimated using conventional regression theory. The estimated distribution can then be
  • MODELLING ACCELERATED DEGRADATION DATA USING WIENER DIFFUSION 33used to set the parameter ranges for a simple sensitivity analysis. It can also be used toconstruct composite prediction intervals for a future degradation level W f or the lifetime Tof an item by generating simulated outcomes of B. Finally, the asymptotic distribution canbe used to calculate Bayesian prediction intervals. Identifying the correct transformations A(δ), B(ν), C(λ) and H (s) can be approached byusing a combination of subject matter knowledge and statistical procedures. The subjectmatter may suggest, for example, that H (s) has the Arrhenius form mentioned earlier. Thesubject matter may also suggest forms for A(δ), B(ν) and C(λ). Candidate transformationsmight also be identified by examining appropriate graphs, backed up by formal goodness-of-fit tests. To illustrate the fitting approach for A(δ), we note that there are m k observationsA jk , j = 1, . . . , m k , at each stress level sk . Hence, an ANOVA test of linearity may be used,based on sums of squares for pure error and lack of fit. This technique will be demonstratedin the case study presented in the next section.4. Case ApplicationThe Chemelex Division of the Raychem Corporation in Redwood City, California makesa self-regulating heating cable that finds extensive application in conditions that requirehigh reliability. These cables are subjected to extensive degradation testing, both to ensurethe dependability of current products and to support the development of new and improvedproduct designs. The data set presented in this case study is a disguised version of a typicaltest set. The disguise does not alter the essential statistical properties being studied but doesprotect the proprietary interests of the company. The test items in this case application are sample cable lengths cut from a productionlot. Degradation of the cable is indicated by a rise in its electrical resistance with age. Weshall use the natural logarithm of resistance as the degradation measure W . Degradation ofthe cable is accelerated by thermal stress so temperature (measured in ◦ K ) is used as thestress measure. The test data consist of readings on log-resistance at several time points foreach item. The readings of log-resistance are standardized to 0 at time r = 0 to adjust forsmall differences in the cable lengths of the items. This type of cable experiences a slightimprovement in performance (i.e., a drop in resistance) when it is first placed in operation.This initial improvement is attributed to curing of the cable polymer. For this reason, onlydegradation readings taken subsequent to this cure phase are considered for the analysis. Five test items were baked in an oven at each test temperature. Three test temperatureswere used, 200◦ C, 240◦ C, 260◦ C, giving a total of 15 test items. Ten readings were recordedafter the cure phase on each item tested at 200◦ C. Similarly, 11 readings were recorded oneach of four items tested at 240◦ C. Only 10 readings were available for the 5th item testedat this temperature. The last reading is missing because of a test fixture failure. Finally,seven readings were recorded on each item tested at 260◦ C. The test data appear in Table 1.The clock times in the table are in thousands of hours. The test design and protocol are based on an internal company specification. The selectedtemperature levels and number of items tested at each temperature were determined by thelimitations of available test equipment and the logistics of handling test fixtures. Thetemperature levels were chosen to span the range from the maximum level that would be
  • 34 G. A. WHITMORE AND FRED SCHENKELBERG Table 1. Heating cable test data. Degradation is measured as the natural logarithm of resistance. Time is measured in thousands of hours. A period denotes a missing value. Item Time 1 2 3 4 5 (a) Test temperature 200◦ C. 0.496 −0.120682 −0.118779 −0.123600 −0.126501 −0.124359 0.688 −0.112403 −0.109853 −0.115186 −0.118941 −0.111966 0.856 −0.103608 −0.101593 −0.105657 −0.110288 −0.107869 1.024 −0.096047 −0.094567 −0.098569 −0.103419 −0.100304 1.192 −0.085673 −0.084698 −0.088613 −0.095465 −0.085916 1.360 −0.077677 −0.076070 −0.079332 −0.084769 −0.077947 2.008 −0.045218 −0.040623 −0.045835 −0.052268 −0.045597 2.992 0.000526 0.004237 0.000533 −0.008265 0.000524 4.456 0.059261 0.063742 0.061032 0.051139 0.059544 5.608 0.093394 0.095117 0.093612 0.082414 0.084912 (b) Test temperature 240◦ C. 0.160 −0.005152 −0.019888 −0.045961 −0.023188 −0.044267 0.328 0.056930 0.046278 0.015198 0.040737 0.018173 0.496 0.112631 0.101628 0.067119 0.095504 0.072214 0.688 0.173202 0.162705 0.128670 0.156129 0.131555 0.856 0.214266 0.202604 0.168271 0.196349 0.171394 1.024 0.272668 0.257563 0.221611 0.250900 0.225281 1.192 0.311422 0.297875 0.260910 0.291937 0.266314 1.360 0.351988 0.338902 0.302126 0.332887 0.306105 2.008 0.489847 0.461855 0.440738 0.473130 0.443941 2.992 0.656780 0.629991 0.606275 0.638651 0.611724 4.456 0.851985 0.798431 0.834114 0.798457 . (c) Test temperature 260◦ C. 0.160 0.123360 0.127605 0.120759 0.105206 0.120115 0.328 0.251084 0.254944 0.247156 0.232389 0.247949 0.496 0.393107 0.394496 0.391516 0.375789 0.388406 0.688 0.517137 0.518485 0.513872 0.500556 0.511850 0.856 0.598797 0.599265 0.595704 0.583362 0.595220 1.024 0.693925 0.694445 0.688930 0.679117 0.690324 1.192 0.774347 0.774428 0.770313 0.758314 0.770782encountered in field installations to a level where the cable polymer undergoes a physicalphase change. The earliest test readings were made at roughly weekly intervals, withthe interval lengthening for later readings. The test was discontinued when the test itemswent through the failure threshold for the 240◦ C and 260◦ C test temperatures. According tocompany specifications, the cable is deemed to have failed when the resistance doubles (i.e.,
  • MODELLING ACCELERATED DEGRADATION DATA USING WIENER DIFFUSION 35the log-resistance reaches ln(2) = 0.693). The test continued at the lowest test temperature(200◦ C) until the test equipment was required for other projects. Both forms of the time transformation given in (2.2) were used in the study. Once theanalyses were completed and the final results were in hand, company scientists found reasonsto prefer one transformation over the other based on scientific considerations. Still, someuncertainty about the appropriate transformation remains which only additional empiricalevidence can eliminate. It is instructive to compare the two sets of results that the scientistshad to consider. We begin with the results for transformation (2.2a).Exponential Time TransformationThe NONLIN procedure of the statistical software package SYSTAT was used to obtain ˆ ˆ ˆthe estimates (δ jk , ν jk , λ jk ) for the jth item ( j = 1, . . . , 5) at the kth temperature level(k = 1, 2, 3) by maximizing the sample likelihood function (3.2) for each of the 15 items.The exponential time transformation (2.2a) was used in this application of the maximumlikelihood method with the parameter γ set to 1. Thus, in setting up the sample likelihoodfunction (3.2), each clock time was transformed as follows. ti jk = 1 − exp(−λri jk ) The parameter estimates for each of the items appear in Table 2. The results show that theparameter estimates are reasonably consistent across items at each temperature level. Twoexceptions might be item 5 at test temperature 200◦ C and item 3 at test temperature 240◦ C.The findings presented here are based on the whole data set because there was no externalreason to believe that either of these two outlying points is invalid or unrepresentative. Each set of parameter estimates was plotted against the reciprocal of the absolute temper-ature. Based on these three plots, the following parameter transformations were tentativelyidentified. A(δ) = δ, B(ν) = ln(ν), C(λ) = 1/λ (4.1)The plots for the transformed parameters appear in Figure 2. Consultations with companyscientists revealed that little was known of the statistical properties of the physical degra-dation mechanism. It was therefore difficult to judge whether these transformations werereasonable from a scientific point of view. Thus, the final judgment was to be based on theadequacy of the statistical fit, which we return to describe shortly. The plots in Figure 2 show roughly linear relationships but also exhibit patterns of het-eroscedasticity that need to be accommodated. It was especially noted in the plots for ln(ν) ˆ ˆand 1/λ in Figures 2b and 2c that the scatter of points shrinks as the reciprocal temperaturedecreases (i.e., as the temperature increases). In fact, it seems that the variability vanishesjust above the highest test temperature at 260◦ C or 533◦ K , where 1/s3 = 0.001876. Wehave already mentioned that the cable polymer experiences a phase change near this tem-perature level. As the resistance properties of the polymer are altered by the phase change,it was suspected that this factor might explain why the variability of resistance readings
  • 36 G. A. WHITMORE AND FRED SCHENKELBERG Table 2. Parameter estimates for all test items based on the exponential time transformation. (a) Test temperature 200◦ C Parameter Item Estimate 1 2 3 4 5 ˆ δ 0.495231 0.454562 0.474481 0.473190 0.378924 ˆ λ 0.120107 0.137024 0.131168 0.124015 0.181333 ˆ ν 0.000094 0.000129 0.000122 0.000170 0.000398 (b) Test temperature 240◦ C Parameter Item Estimate 1 2 3 4 5 ˆ δ 1.278160 1.145816 1.478103 1.130811 1.157966 ˆ λ 0.281411 0.324826 0.223447 0.339209 0.320289 ˆ ν 0.000898 0.000819 0.001357 0.000423 0.000470 (c) Test temperature 260◦ C Parameter Item Estimate 1 2 3 4 5 ˆ δ 1.491041 1.492564 1.482068 1.485528 1.506090 ˆ λ 0.640612 0.632789 0.644703 0.648073 0.629577 ˆ ν 0.001701 0.001618 0.001876 0.001740 0.001467converged to zero upon approaching this critical temperature. We shall denote this criti-cal temperature by sc and set it to 535◦ K . The patterns of heteroscedasticity in the threeplots therefore suggested that the following scaled version of the Arrhenius transformationshould be considered. 1 1 H (s) = − for s ≤ sc (4.2) s scObserve that H (s) in (4.2) is a decreasing function of s with H (sc ) = 0. The patterns ofheteroscedasticity suggest therefore that the appropriate form for the covariance matrix in(3.5) is Σk = Hk Σ, (4.3)where Hk = H (sk ) is based on the function in (4.2) and Σ denotes a baseline covariancematrix that is common to all items. A statistical software routine for weighted multivariateregression may now be used to estimate the coefficient matrix B in (3.5) and covariancematrix Σ in (4.3), using the reciprocals of the Hk values as weights.
  • MODELLING ACCELERATED DEGRADATION DATA USING WIENER DIFFUSION 37 (a) Estimates of A(δ) = δ. (b) Estimates of B(ν) = ln(ν). (c) Estimates of C(λ) = 1/λ.Figure 2. Plots of the transformed parameter estimates against the reciprocal of the test temperature (in ◦ K ) foreach test item, based on the exponential time transformation t = 1 − exp(−λr ). The result in (4.3) implies that the following transformations will yield a set of centeredquantities with a stable variance at each temperature level sk . A jk − a0 − a1 Hk A jk = √ (4.4a) Hk
  • 38 G. A. WHITMORE AND FRED SCHENKELBERG Bjk − b0 − b1 Hk B jk = √ (4.4b) Hk C jk − c0 − c1 Hk C jk = √ (4.4c) HkThe pairs of regression coefficients, (a0 , a1 ), (b0 , b1 ) and (c0 , c1 ), can also be estimateddirectly from the quantities in (4.4) by minimizing the total sum of squares for each of thethree sets of values. The resulting estimates of the Arrhenius equations are ˆ δ = 1.5237 − 4173.0H (s), (4.5a) ln(ˆ = −6.3255 − 10, 224H (s), ν) (4.5b) ˆ 1/λ = 1.3961 + 24, 617H (s), (4.5c)where H (s) is defined as in (4.2) with sc = 535. An ANOVA test for linearity was performed for each parametric transformation in (4.1)based on the three sets of quantities in (4.4). First, taking the A jk values in (4.4a), aHartley test for equal population variances at each of the three test temperature levelswas performed (Neter, Wasserman and Kutner, 1990, page 619), yielding a test statisticof H = 5.680. The 0.95 fractile of H under the null hypothesis is 15.5 which suggeststhat the hypothesis of equal variances cannot be rejected. Next, the centered total sum ofsquares of the quantities A jk was decomposed into component sums of squares relating towithin and between temperature-level variation, respectively. The P- value of the resultingANOVA test was 0.389, which is not significant. Similar tests were conducted for B jk andC jk . Their H statistics were 1.564 and 10.874, and the P-values for their ANOVA testswere 0.846 and 0.948, respectively. None of these test results is significant. Thus, thetests support the conclusion that each parameter transformation in (4.1) produces a linearArrhenius equation.Power Time TransformationThe same analysis as just described was repeated using the power time transformation (2.2b)in lieu of the exponential time transformation. In setting up the sample likelihood function(3.2), each clock time was transformed as follows. ti jk = riλjkTable 3 contains the maximum likelihood estimates of the parameters δ, ν and λ for eachitem. Although the parameter symbols are the same as for the exponential time transfor-mation, their interpretation is different. Parameter λ here denotes the power parameterin the time transformation. The drift parameter δ and variance parameter ν now relate to
  • MODELLING ACCELERATED DEGRADATION DATA USING WIENER DIFFUSION 39 Table 3. Parameter estimates for all test items based on the power time transformation. (a) Test temperature 200◦ C Parameter Item Estimate 1 2 3 4 5 ˆ δ 0.067033 0.070529 0.070460 0.064614 0.083440 ˆ λ 0.770625 0.746937 0.754399 0.776336 0.662939 ˆ ν 0.000026 0.000039 0.000035 0.000039 0.000120 (b) Test temperature 240◦ C Parameter Item Estimate 1 2 3 4 5 ˆ δ 0.402380 0.425770 0.367597 0.427560 0.369068 ˆ λ 0.6028l7 0.553013 0.662507 0.552947 0.665162 ˆ ν 0.000442 0.000523 0.000229 0.000753 0.000214 (c) Test temperature 260◦ C Parameter Item Estimate 1 2 3 4 5 ˆ δ 0.788621 0.781160 0.788423 0.791786 0.784428 ˆ λ 0.661233 0.664679 0.659068 0.660399 0.666644 ˆ ν 0.001172 0.001096 0.001283 0.001255 0.001017a different operational time scale. The results in Table 3 again show that the parameterestimates are reasonably consistent across items at each temperature level. Item 5 at testtemperature 200◦ C and item 3 at 240◦ C remain somewhat outlying, as does item 5 at 240◦ C.The analysis continues to be based on the whole data set, however, because there was noexternal reason to believe that any of these outlying points is invalid or unrepresentative. Each set of parameter estimates was plotted against the reciprocal of the absolute temper-ature. Based on two of these plots, the following parameter transformations were tentativelyidentified. A(δ) = ln(δ), B(ν) = ln(ν) (4.6)The plots for the transformed parameters are shown in Figures 3a and 3b. The two plotsappear to be nearly linear. An appropriate transformation for λ could not be identified. ˆ ˆFigure 3c shows a plot of λ against the reciprocal temperature. The value of λ appears to ◦be lowest at the intermediate reciprocal-temperature level (240 C) and distinctly elevatedat the highest reciprocal-temperature level (200◦ C).
  • 40 G. A. WHITMORE AND FRED SCHENKELBERG (a) Estimates of A(δ) = ln(δ). (b) Estimates of B(ν) = ln(ν). (c) Estimates of λ (untransformed).Figure 3. Plots of the transformed parameter estimates against the reciprocal of the test temperature (in ◦ K ) foreach test item, based on the power time transformation t = r λ . To explore the parameter λ further, we have computed the combined profile log-likelihoodfunction ln L k (λ) for all items tested at temperature level sk , as follows. mk ln L k (λ) = ln L jk (λ) for k = 1, 2, 3 (4.7) j=1
  • MODELLING ACCELERATED DEGRADATION DATA USING WIENER DIFFUSION 41Figure 4. Plot of the combined profile log-likelihood function ln L k (λ) for all items on test at each test temperaturesk . The vertical scale measures the decrement from the maximum log-likelihood level for each function.Here L jk (λ) denotes the profile likelihood function of form (3.4) for the jth item testedat temperature sk . Figure 4 shows a plot of this combined profile log-likelihood functionfor each temperature level. The vertical scale measures the decrement from the maximumlog-likelihood level. The functions are close to being quadratic in shape. From likelihoodtheory we know that the interval spanned by each function at a decrement of −1.92 from themaximum log-likelihood is an approximate 95% confidence interval for λ at that temperaturelevel. The intervals suggest that parameter λ is larger at 200◦ C than at 240◦ C, with theparameter taking an intermediate value at 260◦ C. Thus, both Figure 3c and Figure 4 suggestthat λ is not related to temperature in a monotonic fashion. The plots in Figure 3 show patterns of heteroscedasticity that are similar to those seenin Figure 2. We therefore employ the same Arrhenius transformation as in (4.2). Thecritical temperature for this transformation remains at sc = 535◦ K . The estimation of theArrhenius equations proceeds as with the exponential time transformation except that wewill omit the estimation of c0 and c1 for the equation involving λ because we cannot decide
  • 42 G. A. WHITMORE AND FRED SCHENKELBERGon the appropriate form for the transformation C(λ). The resulting estimated Arrheniusequations are as follows ˆ ln(δ) = −0.16728 − 9952.8H (s), (4.7a) ln(ˆ ) = −6.6652 − 14, 011H (s), ν (4.7b)where H (s) is defined as in (4.2) with sc = 535. ANOVA tests for linearity, together with Hartley tests for equal variances, indicate that theparameter transformations in (4.6) yield linear Arrhenius equations. The Hartley statisticsare H = 4.159 and H = 1.669 and the ANOVA P-values are 0.175 and 0.905, respectively. The statistical analyses based on the two forms of the time transformation have yieldedresults that are comparable in terms of the fit of the model to the data. We now look at howcompany scientists weighed the competing results. Oxidation of the cable polymer is considered to be the principal degradation mechanism.If oxidation were a self-limiting reaction then the exponential time transformation mightbe appropriate. Company scientists did not rule out this self-limiting feature. However,if oxidation is self-limiting then the asymptotic log-resistance of the cable would havean expected value of δ, corresponding to r = ∞ or t = 1. The Arrhenius equationA(δ) = a0 +a1 H (s) fitted under the exponential time transformation—see (4.5a)—indicatesthat δ varies with the test temperature, which implies that the asymptotic log-resistancevaries with the aging temperature. Company scientists felt that this feature was implausiblealthough there is no empirical evidence based on long aging studies to settle the question.Hence, the implications of the exponential time transformation were unacceptable. Thebehaviour of the model based on the power time transformation was more satisfactory tothe scientists. Interestingly, the transformations A(δ) and B(ν) were both of a logarithmicform which was a satisfying feature. The power transformation also implied that the log-resistance would rise without limit as the cable aged, which was a plausible behavior.The uncertainty about the form of the Arrhenius equation for the power parameter λ was adifficulty but the fact that the estimates of λ had only a small range of values was reassuring.Further testing is being planned that might settle the question of the appropriate form forC(λ). In summary, company scientists felt more comfortable with the implications of thepower time transformation. We now look at predictive inference for the power transformation model. Recall that thecable is deemed to have failed when the resistance doubles. Thus, ω = ln(2) = 0.69315.The normal use temperature of the cable is 175◦ C, which gives s0 = 448◦ K . At thistemperature, we have 1 1 1 1 H0 = − = − = 0.0003630. s0 sc 448 535The Arrhenius equations in (4.7a) and (4.7b) give the following results. ˆ ln(δ0 ) = −0.16728 − 9952.8(0.0003630) = −3.780 ln(ˆ 0 ) = −6.6652 − 14, 011(0.0003630) = −11.751 ν
  • MODELLING ACCELERATED DEGRADATION DATA USING WIENER DIFFUSION 43Thus, the corresponding use-level parameter estimates are ˆ δ0 = exp(−3.780) = 0.02282, ν0 = exp(−11.751) = 0.7883 × 10−5 . ˆ The use-level estimate of λ0 is still required and, as we saw in Figure 3c, it is unclear howto extrapolate the test results to obtain this estimate. Rather than attempt an estimate, wecarry out a sensitivity analysis. We wish to calculate the probability that the resistance of anew item will double within, say, its first 10 years of life. We must first convert this clocktime to operational time using the transformation t f = 87.6λ0 , where r f = 87.6 thousandhours corresponds to 10 years. Taking a range of plausible values for λ0 and ignoringthe sampling errors in the other parameter estimates, we use the cumulative distributionfunction in (2.4) with t = t f to calculate the required failure probability. The followingtable gives this failure probability for part of the range of λ0 values of interest. λ0 : 0.74 0.75 0.76 0.77 0.78 P(T ≤ t f ): 0.0000 0.0042 0.2629 0.9145 0.9996 The table shows that the failure probability is very sensitive to λ0 in this part of the range,varying from 0 to near 1 in a short interval. The sensitivity analysis suggests that the failureprobability is small only if λ0 is below 0.75. As 200◦ C is the test temperature that is closestto the use-level temperature, it is instructive to look at the λ estimates for the five items thatwere tested at that temperature (see Table 3a). Three of the five items have λ estimates thatare larger than the critical value 0.75 and two that are smaller—a mixed result. Thus, underthe power time transformation, this data set has not provided precise information aboutthe 10-year failure probability for this product. As already mentioned, further degradationtesting is being planned. This additional testing may yield a better assessment of the use-level parameter λ0 and settle the question of the appropriate form for transformation C(λ).As a concluding remark, we add that this kind of heating cable has proven to be very reliablein field installations to date, which suggests that λ0 may indeed be well below the criticalvalue.5. Discussion and ConclusionsThe model and case study presented here have left a number of practical issues untouched.Many of these can be dealt with by appropriate technical extensions. Measurement errors are produced by imperfect laboratory personnel, procedures andequipment. An extension of our model, similar to that described in Whitmore (1995), cantake measurement errors into account. In our experimental setting, however, measurementerrors are likely to be intercorrelated because readings on test items are made at the sametime under the same test conditions. For instance, in the case study, all test items in thesame oven are withdrawn together and cooled to room temperature before their resistancesare measured. They are then returned to the ovens and brought back up to test temperature.Any measurement imperfections in this setting will affect all items in the batch.
  • 44 G. A. WHITMORE AND FRED SCHENKELBERG A Wiener diffusion process may not describe the degradation process of interest in someapplications. For example, degradation may proceed in a strictly monotonic fashion orinvolve jump behavior. In these circumstances, the Wiener process might be replaced bya more appropriate process, such as a Hougaard process (see Lee and Whitmore, 1993,and Fook Chong, 1993). The Hougaard family includes the gamma and inverse Gaussianprocesses as special cases. Degradation may also be a multidimensional process in whichthe boundary conditions that define failure are more complicated than a simple failurethreshold. The model presented here assumes that stress s is a single quantitative measure, suchas temperature. This assumption is more restrictive than necessary. Stress s can be amultidimensional physical measure. The model would still apply, provided a functionH (s) exists that maps each measure s into a real number. The number H (s) then becomesa stress index for the multidimensional measure s. The challenge in this extension wouldbe to discover the appropriate form of the function H (s). The scaled version of the Arrhenius transformation in (4.2) assumes that sc is the samefor all parameters and is known. In the case study, however, there is some evidence thatthis critical temperature is not sharply delineated and may be a little different for differentparameters. Perhaps sc should be treated as another parameter that must be estimated foreach Arrhenius equation. This issue is one of many that demonstrate the general need for abetter understanding of the link between statistical parameters and the physical parametersof the scientific theories that underpin each application. The results of the case study demonstrate the need to examine the appropriate designfor degradation tests. The model provides a framework for studying design issues such asthe selection of stress levels, the number of items on test, and the number and spacings ofreadings on each item. The design should aim not only to optimize predictive accuracybut also to have a self-testing capability that would allow the model itself to be validatedthrough appropriate diagnostics.AcknowledgmentsThis research was completed while one of the authors (Schenkelberg) was employed by theRaychem Corporation. We thank the Corporation for permission to use the case study re-sults. We also are deeply indebted to scientists of the Corporation for providing backgroundscientific information and expert opinion about the materials and degradation processes in-volved. We thank two anonymous referees for helpful suggestions on an earlier version ofthis paper. Finally, we are grateful for the financial support provided for this research bythe Natural Sciences and Engineering Research Council of Canada.ReferencesM. Boulanger Carey and R. H. Koenig, “Reliability assessment based on accelerated degradation: A case study,” IEEE Transactions on Reliability vol. 40(5) pp. 499–506, 1991.R. S. Chhikara and J. L. Folks, The Inverse Gaussian Distribution: Theory, Methodology and Applications, Marcel Dekker: New York, 1989.
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