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

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

2. 2. 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.
3. 3. 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
5. 5. 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
6. 6. 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