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Chapter 23 Accelerated Destructive Degradation Test Data, Models, and Analysis William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University Copyright 2002-2003 W. Q. Meeker and L. A. Escobar. Complements to the authors’ text Statistical Methods for Reliability Data, John Wiley & Sons Inc., 1998. Work done jointly with Danny L. Kugler and Laura L. Kramer (Imaging & Printing Group, Hewlett-Packard) January 13, 2014 3h 42min 23 - 1 Accelerated Destructive Degradation Tests Data, Models, and Data Analysis Chapter 23 Objectives • Describe useful accelerated destructive degradation test (ADDT) reliability models. • Show the connection between degradation reliability models and failure-time reliability models. • Present methods of data analysis and reliability inference for ADDT. • Discuss the use of ADDT data to estimate acceleration factors. • Illustrate methods with examples. 23 - 2 AdhesiveBondB ADDT Data • Objective: Assess the strength of an AdhesiveBond over time; estimate the proportion of devices with a strength below 40 Newtons after 5 years of operation (approximately 260 weeks) at 25◦C. • The test is destructive; each unit can be measured only once. There were 6 right censored observations. • Test plan: 8 units with no ageing were measured at the start of the experiment. A total of 80 additional units were aged and measured according to the following temperatures and time schedule. Temp ◦C — 50 60 70 Totals 0 Weeks Aged 2 4 6 12 Totals 16 8 8 8 6 6 20 0 0 6 6 8 6 4 18 8 6 9 23 7 6 0 13 8 31 24 25 88 23 - 3 DegreesC AdhesiveBondB ADDT Plan Number of Units Measured at Each (Time, Temperature) Combination 70 6 60 50 25 6 4 9 6 6 6 6 8 8 8 7 6 12 16 8 0 2 4 Weeks 23 - 4 AdhesiveBondB ADDT Data Scatter Plot Linear–Linear Axes AdhesiveBondB Data Destructive Degradation Scatter Plot Resp:Linear,Time:Linear 100 Newtons 80 60 40 50DegreesC 60DegreesC 70DegreesC 20 0 0 5 10 15 20 Weeks 23 - 5 AdhesiveBondB ADDT Data Scatter Plot at Individual Conditions of Temperature Linear–Linear Axes AdhesiveBondB Data Resp:Linear,Time:Linear, Dist:Normal 50DegreesC 60DegreesC 100 100 80 80 60 60 40 40 20 20 0 0 5 10 15 20 15 20 0 5 10 15 20 Newtons 0 70DegreesC 100 80 60 40 20 0 0 5 10 Weeks 23 - 6 AdhesiveBondB ADDT Data Scatter Plot at Individual Conditions of Temperature Square Root–Log Axes AdhesiveBondB Data Resp:Log,Time:Square root, Dist:Normal 50DegreesC 60DegreesC 80 40 20 10 80 40 20 10 5 10 15 0 5 10 15 Newtons 0 70DegreesC 80 40 20 10 0 5 10 15 Weeks 23 - 7 Model for Accelerated Degradation Path • Actual degradation path model: Actual path of a unit at ti and accelerating variable(s) level (combinations) AccVarj (e.g., temperature or temperature and relative humidity) is Dij = D(τi, xj , β ) where τi = ht(ti) is a known monotone increasing transformations of ti and xj = ha(AccVarj ) are known transformations of the accelerating variables at the jth level (or combination of levels) AccVarj . When there is no possibility of confusion, τi and xj are called the time and the AccVar, respectively. • Rates in the model are with respect to τ = ht(t). • Path parameters: the elements of β are fixed but unknown. • Though no commonly used in accelerated testing it is possible to have models with interactions among the accelerating variables. 23 - 8 Statistical Model for ADDT Data • Sample path model: For observation k at time τi and accelerating variable(s) level xj the model is yijk = hd(Dij ) + ǫijk = µij + ǫijk , k = 1, . . . , nij where µij = hd(Dij ) and yijk are, respectively, monotone increasing transformations of Dij and the observed degradation. nij is the number of observations at time ti and AccVarj vector level xj . ǫijk is a residual deviation which describes unit-to-unit variability with (ǫijk /σ) ∼ Φ(z). • The transformations for the observed degradation, Dij , ti,, and the AccVarj might be suggested by physical/chemical theory, past experience, or the data. 23 - 9 Degradation Path Model: Single Temperature Level D(τ, x, β ) = exp[β0 + β1 exp(β2x)τ ] √ τ = Weeks and x is Arrhenius-Transformed Temp Linear–Linear Axes 100 70 DegreesC Newtons 80 60 40 20 0 0 10 20 30 40 50 60 Weeks 23 - 10 Degradation Path Model: Multiple Temperature Levels D(τ, x, β ) = exp[β0 + β1 exp(β2x)τ ] √ τ = Weeks, x is Arrhenius-Transformed Temp Linear–Linear Axes 100 70 DegreesC 60 DegreesC 50 DegreesC Newtons 80 60 40 20 0 0 10 20 30 40 50 60 Weeks 23 - 11 Using Transformations to Linearize a Degradation Model log[D(τ, x, β )] = β0 + β1 exp(β2x)τ √ τ = Weeks, x is Arrhenius-Transformed Temp Square Root–Log Axes 100 Newtons 50 20 70 DegreesC 60 DegreesC 50 DegreesC 10 0 10 20 30 40 50 60 Weeks 23 - 12 A Class of Linear Degradation Models • These models are of the form yijk = µij + ǫijk = β0 + β1 exp(β ′2xj )τi + ǫijk , k = 1, . . . , nij where yijk , τi, and xj may be transformations of the measured degradation, ti, and the accelerating variable(s) AccVarj , respectively. • The model is linear in the sense that for specified AccVar vector xj , the degradation is linear in τi. ◮ For a single (scalar) AccVar xj , β ′2xj = β2xj . ◮ For multiple AccVar, xj , xj = (x2, . . . , xp)′, β ′2 = (β2, β3, . . . ), and β ′2xj is a linear combination of the AccVar, i.e., β ′2xj = p X i=2 βixji = β2xj2 + β3xj3 + · · · + βpxjp. 23 - 13 Linear Degradation Model Interpretation of the Parameters For the linear degradation model yijk = β0 + β1 exp(β ′2xj )τi + ǫijk • β0 is path intercept parameter when τ = 0. For example, ◮ If τi = √ ti then β0 is the intercept at time t = 0. ◮ If τi = log(ti) then β0 is intercept at time t = 1. • The degradation rate at xj is υ(xj )(AccVarj ) = β1 exp(β ′2xj ). • The sign (±) of β1 determines if the degradation is increasing or decreasing in time. • For a power transformation of time τ = tκ, the components of the vector parameter β 2 are related to the amount of acceleration obtained by increasing the accelerating variables AccVar. 23 - 14 Linear Degradation Model for the AdhesiveBondB Data For the AdhesiveBondB data, the strength of the adhesive as a function of time and temperature is modeled by yijk = β0 + β1 exp(β2xj )τi + ǫijk where yijk = log(Newtonsijk ) p √ ti = Weeksi τi = xj = −11605/(◦Cj + 273.15) (ǫijk /σ) ∼ Φnor (z). The ǫijk term contains model and measurement errors. 23 - 15 ADDT Individual Analysis Likelihood for Fixed AccVar level xj • For the data at fixed xj of the AccVar with exact failure times and right-censored observations, the likelihood is nij Y Y 1 yijk − µij 1−δijk yijk − µij δijk × 1−Φ φ Lj (θ |DATA) = σ σ i k=1 σ where µij ≡ µ(τi, xj , β ) = β0 + β1 exp(β ′2xj )τi, δijk indicates whether observation yijk is a failure (δijk = 1) or a right censored observation (δijk = 0), θ = (β0, β1, β ′2, σ), and nij is the number of observations at (τi, xj ). • For fixed xj , the model parameters are: the spread σ, the intercept β0, and the slope of the line υj = β1 exp(β ′2xj ). The parameter υj can be interpreted as the degradation rate of µ(τ, xj , β ) with respect to the transformed time τ. 23 - 16 AdhesiveBondB ADDT Data Normal Model ML Individual Line Fits • For each temperature level three individual ML estimates [j] b [j]. are obtained: βb0 , υb [j], and σ • A summary of the Normal model individual ML fits for the AdhesiveBondB data is ML Estimates AccVarj [j] βb0 50◦C 4.490 60◦C 4.489 70◦C 4.400 υb [j] −0.1088 −0.2089 −0.3626 sceυb[j] 0.01494 0.02214 0.01944 95% Approximate Confidence Interval for υ [j] Lower Upper −0.1424 −0.08309 −0.2571 −0.4028 −0.16969 −0.32643 23 - 17 AdhesiveBondB ADDT Data Individual Normal Model ML Fits Square Root–Log Axes [j] b [j]τ b [j] = βb0 + υ µ AdhesiveBondB Data Resp:Log,Time:Square root, Dist:Normal 50DegreesC 60DegreesC 80 40 20 10 80 40 20 10 5 10 15 0 5 10 15 Newtons 0 70DegreesC 80 40 20 10 0 5 10 15 Weeks 23 - 18 AdhesiveBondB ADDT Data Overlay of Individual Normal Model Fits Square Root–Log Axes [j] b [j]τ b [j] = βb0 + υ µ AdhesiveBondB Data Destructive Degradation Individual Regression Analyses Resp:Log,Time:Square root, Dist:Normal 100 80 Newtons 60 50 40 30 20 50DegreesC 60DegreesC 70DegreesC 10 0 5 10 15 20 Weeks 23 - 19 Individual Degradation Rate Estimates • The ML estimates υb [j] (slopes of the individual lines) can be used to identify the relationship between the degradation rate and the AccVar. • When the degradation is decreasing, use absolute values of the degradation rate. • Because log(| υj |) = log(| β1 |) + β ′2xj the surface log(| υb [j] |) versus the AccVar xj should be approximately linear in the xj if the model relating degradation rate and the AccVar is adequate. Then ◮ For a single AccVar xj , the plot of log(| υb [j] |) versus xj should be approximately linear. ◮ For an AccVar vector xj , the plot of log(| υb [j] |) versus any of the accelerating variables, conditional on fixed values of the remaining accelerating variables, should be approximately linear. 23 - 20 Arrhenius Plot of |υb [j]| Individual Degradation Rates Normal Model ML Estimates Degradation rate versus DegreesC on Arrhenius Scale for AdhesiveBondB Data Resp:Log,Time:Square root,x:arrhenius, Dist:Normal 0.40 0.30 0.25 Slope 0.20 0.15 0.10 0.05 45 50 55 60 65 70 75 DegreesC on Arrhenius Scale 23 - 21 Likelihood for All Data With Right Censored Data • For a sample of n units consisting of exact failure times and right-censored observations, the likelihood can be expressed as L(θ |DATA) = L(θ |DATA) = Y j Lj (θ |DATA) Y 1 ijk σ φ yijk − µij δijk σ × 1−Φ yijk − µij 1−δijk σ where θ = (β0, β1, β ′2, σ)′, and δijk indicates whether observation ijk is a failure (δijk = 1) or a right censored observation (δijk = 0). • Note that µij is a nonlinear function of the accelerating variable xj . • For the AdhesiveBondB data where xj is a scalar, µij = β0 + β1 exp(β2xj )τi 11605 xj = − ◦ . Cj + 273.15 23 - 22 AdhesiveBondB Data ML Estimates for the Acceleration Model Fit • Parameter estimates Parameter β0 β1 β2 σ 95% Approximate Confidence Interval Lower Upper 4.396 4.547 ML Estimate 4.471 Standard Error 0.03864 −8.641 × 108 1.595 × 109 −3.989 × 109 2.261 × 109 0.1580 0.01233 0.1356 0.1841 0.6364 0.05488 0.5375 0.7536 • Normal model ML for the slopes (degradation rates) at b ◦Cj ) = βb1 exp(βb2xj ), each temperature are obtained from υ( where xj = −11605/(◦Cj + 273.15). In this case for the four temperatures of interest the ML estimates are b υ(25) = −0.0151, b υ(60) = −0.2035, b υ(50) = −0.1025 b υ(70) = −0.3883 23 - 23 AdhesiveBondB ADDT Data Normal Model Arrhenius ML Fit b ij = βb0 + βb1 exp(βb2xj )τi µ AdhesiveBondB data Destructive Degradation Regression Analyses Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal 100 80 Newtons 60 50 40 30 20 50DegreesC 60DegreesC 70DegreesC 25DegreesC 10 0 5 10 15 20 Weeks 23 - 24 Model Checking Residual Plots • Residuals versus fitted values. • Residuals versus accelerating variables. • Residuals versus time of exposure. • Residuals versus observation order. observations are taken sequentially. This is useful when • Residual probability plot. 23 - 25 AdhesiveBondB ADDT Data Residuals Versus Fitted Values AdhesiveBondB Data Destructive Degradation Residuals versus Fitted Values Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal 4 Standardized Residuals 2 .95 0 .50 .05 -2 -4 -6 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 Fitted Values 23 - 26 AdhesiveBondB ADDT Data Residuals Versus Temperature Conditions AdhesiveBondB Data Destructive Degradation Residuals versus DegreesC Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal 4 Standardized Residuals 2 .95 0 .50 .05 -2 -4 -6 50 55 60 65 70 DegreesC 23 - 27 AdhesiveBondB ADDT Data Sev Residual Probability Plot AdhesiveBondB Data Destructive Degradation Residual Probability Plot with 95% Simultaneous Confidence Bands Resp:Log,Time:Square root,DegreesC:Arrhenius, Dist:Normal Normal Probability Plot .9999 .999 Probability .995 .98 .95 .9 .8 .6 .4 .2 .1 .05 .01 .002 .0005 .0001 -3 -2 -1 0 1 2 3 Standardized Residuals 23 - 28 Some Comments on the AdhesiveBondB Residuals • The standardized residuals look approximately like a random sample from a NOR(0, 1). • The vertical line at 0 is the median life of the standardized distribution. Then approximately 50% of the residuals should be below that line. • There appears to be some evidence of nonconstant variance, but it is not systematic with temperature or times. 23 - 29 Distribution of Degradation at (Possibly Transformed) Time and AccVar (t, AccVar) • The degradation distribution, for given time t and AccVar vector x is FY (y; τ, x) = P (Y ≤ y; τ, x) = Φ " y − µ(τ, x, β ) σ # where y = hd(degradation), µ(τ, x, β ) = β0 + β1 exp(β ′2x)τ. • The p quantiles of the distribution is yp = µ(t, x, β )+σΦ−1(p). • The ML estimate of the degradation distribution for given (t, x) is b y−µ b FY (y; τ, x) = Φ b σ b x)τ , τ = h (t), x = h (AccVar), b = βb0 + βb1 exp(β where µ a t 2 b ,σ and βb0, βb1, β 2 b are ML estimates of the corresponding parameters. When AccVar is a vector, ha is generally a different function for each of its elements. 23 - 30 AdhesiveBondB ADDT Data Distribution of Degradation at Given Time and Temperature (t, Temp) • For the AdhesiveBondB data, the Normal model ML estimate of FY (y; τ, x) at time t and temperature ◦C is b y−µ FbY (y; τ, x) = Φnor b σ √ b b b b = β0+β1 exp(β2x)τ , τ = Weeks, x = −11605/(◦C+ where µ 273.15), βb0 = 4.471, βb1 = −864064160, βb2 = 0.6364, b = 0.1580 σ • The ML estimate of the p quantile (log Newtons) is b+σ b Φ−1 ybp = µ nor (p). 23 - 31 Induced Failure Time Distribution at Fixed Values of (AccVar, Df ) for Decreasing Linear Degradation • Observe that T ≤ t [i.e., ht(T ) ≤ τ ] is equivalent to observed degradation less than Df (i.e., Y ≤ µf ,) where µf = hd(Df ). Then FT (t; x, β ) = FY (µf ; x, β ) = Φ τ −ν = Φ ς " µf − µ(τ, x, β ) σ # for t ≥ 0 where τ = ht(t), (β0 − µf ) exp(−β ′2x) ν= | β1 | and σ exp(−β ′2x) ς= . | β1 | • The failure time distribution is a mixed distribution with a spike of Pr(T = 0) = Φ [(β0 − µf )/σ ] at t = 0. For t > 0 the cdf is continuous and it agrees with the cdf of a log-location-scale variable with standardized cdf Φ(z), location ν and scale ς. 23 - 32 50 Newtons 100 ML Estimate Showing Proportion Failing as a Function of Time at 25◦C b Φ−1 ybp = βb0 + βb1 exp(βb2x)τ + σ nor (p) 1% 0.1% 0 20 100 200 500 1000 Weeks 23 - 33 Quantiles for the Induced Failure Time Distribution at AccVar vector x and Critical Condition Df Decreasing Linear Degradation • For p ≥ Φ [(β0 − µf )/σ ] ht(tp) = τp = ν + ςΦ−1(p) where (β0 − µf ) exp(−β ′2x) ν= | β1 | Thus σ exp(−β ′2x) ς= . | β1 | and h i −1 −1 t p = ht ν + ςΦ (p) . • Substituting the expressions for ν and ς and after simplifications, shows that log[ht(tp)] = log(τp) = −β ′2x + log " (β0 − µf ) + σΦ−1(p) | β1 | # . Thus, the log of the transformed failure-time distribution quantiles are linear in the transformed AccVar vector x. 23 - 34 ML Estimates for Quantiles of the Induced Failure Time Distribution at AccVar x and Critical Degradation Df Decreasing Linear Degradation • For p ≥ Φ [(β0 − µf )/σ ] h i −1 −1 tbp = ht νb + ςbΦ (p) where h−1 t (·) is the inverse of the ht (·) function. • An approximate 100(1−α)% confidence interval for tp based on the large-sample approximate NOR(0, 1) distribution of Zlog(bt ) = [log(tbp) − log(tp)]/scelog(bt ) is p p [tp, e t̃p] = [tbp/w, where w = exp[z(1−α/2)scebt /tbp]. tbp × w] p 23 - 35 AdhesiveBondB ADDT Data Failure Time Distribution Quantile Estimates • For the AdhesiveBondB accelerated test data, ht(t) = Then log[ q √ t. # −1 b b Φ (p) (β 0 − µ f ) + σ . tbp] = −βb2x + log b | β1 | " This implies that # −1 b b (β0 − µf ) + σ Φ (p) . log(tbp) = −2βb2x + 2 log b | β1 | " Thus the logs of the quantiles of the failure-time distribution are linear in the AccVar vector x. 23 - 36 AdhesiveBondB Data Model Plot ML Estimate of Failure Time Distribution as Functionn of Temperature o −1 b b b b b Φ (p)]/ | β1 | log(tp) = −2β2x + 2 log [(β0 − µf ) + σ Model plot for AdhesiveBondB data Resp:log,Time:Square root,DegreesC:arrhenius, Dist:normal Failure−time distribution for degradation failure level of 40 Newtons 10000 Weeks 1000 100 10 90% 50% 10% 1 Probability spike at time zero = 3.6473e−007 20 30 40 50 60 70 80 DegreesC on Arrhenius Scale 23 - 37 Acceleration Factors • Here we consider time power transformations, i.e., τ = ht(t) = tκ, where κ 6= 0. • To obtain the accelerating effect of the AccVar x, let τ (x) and τ (xU ) be the (transformed) times to reach the critical degradation Df when the (transformed) accelerating variable take values x and xU , respectively. • Solving for τ (x) and τxU the equation Df = D[τ (x), x, β ] = D[τ (xU ), xU , β ] gives τ (x U ) ht[t(xU )] = = exp[β ′2(x − xU )]. τ (x ) ht[t(x)] Using τ (x) = ht[t(x)] = [t(x)]κ and solving for t(xU )/t(x) yields 1 t(xU ) = exp β ′2(x − xU ) . AF (x) = t(x) κ 23 - 38 AdhesiveBondB ADDT Data ML Estimates of Acceleration Factors √ • For the AdhesiveBondB, ht(t) = t, then κ = 1/2 and the acceleration factor for temperature is " AF (temp) = exp 2 β2 11605 11605 − tempU K temp K !# where temp K = temp ◦C + 273.15 is temperature in the absolute Kelvin scale. • The Normal model ML estimate are " d (temp) = exp 2 βb AF 2 11605 11605 − tempU K temp K !# where βb2 = 0.6364. • The ML estimate of the Arrhenius activation energy is 2 × βb2 = 2 × 0.6364 = 1.2728 d (temp) is done easily using SPLIDA. • The computation of AF d (60) = 182.12 For example, AF 23 - 39 AdhesiveBondB Data Normal Model Acceleration Factors as a Function of Temperature Acceleration Factor relative to 25 Degrees C Arrhenius relationship with activation energy in units of eV, 200 1.2728 100 50 20 10 5 2 1 25 30 35 40 45 50 55 60 Degrees C 23 - 40 A Two-Variables Linear Degradation Model With Temperature and Relative Humidity as AccVar • AdhesiveBondD Model yijk = β0 + β1 exp(β ′2xj )τi + ǫijk = β0 + β1 exp(β2xj2 + β3xj3)τi + ǫijk where yijk = log(Newtonsijk ) p √ τi = ti = Weeksi (ǫijk /σ) ∼ Φnor (z). • The AccVar transformations are Arrhenius for Temp (Temperature) and logit for RH (Relative Humidity), i.e., 11605 xj2 = − ◦ Cj + 273.15 RH xj3 = log . 1 − RH 23 - 41 AdhesiveBondD ADDT Data • Objective: Assess the strength of an AdhesiveBond over time. There is interest in estimating the proportion of devices with a strength below 25 Newtons after 5 years of operation at room temperature of 25◦C and relative humidity of 50%. • The test is destructive; each unit can be measured only once. • Experimental factors ◮ AccVar: Temp (Temperature): RH (Relative Humidity): 50◦C, 60◦C, 70◦C 20%, 80% ◮ Weeks : Measurements after 0, 1, 2, 4, 6, 8, 12, 16 Weeks • Sample size: 6 units allocated at each (Weeks, RH, Temp) combination. This gives a sample size equal to: 6 × 8 × 2 × 3 = 288 units. 23 - 42 AdhesiveBondD ADDT Data Structure of the Data The test plan is a completely balanced 8 × 3 × 2 full factorial arrangement in Weeks, Temp, and RH, with 6 units allocated at each combination of the experimental factors. RH % 20 Temp ◦C 50 60 70 80 50 60 70 Totals 0 1 6 6 6 6 6 6 36 6 6 6 6 6 6 36 Weeks Aged 2 4 6 8 6 6 6 6 6 6 36 6 6 6 6 6 6 36 6 6 6 6 6 6 36 6 6 6 6 6 6 36 Totals 12 16 6 6 6 6 6 6 36 6 6 6 6 6 6 36 48 48 48 48 48 48 288 23 - 43 AdhesiveBondD ADDT Data Acceleration Factors √ • For the AdhesiveBondD, ht(t) = t, then acceleration factor for temperature and relative humidity is t(xU ) 1 AF (x) = = exp β ′2(x − xU ) t(x) κ where κ = 1/2 because the square root transformation on time. Then ( ! 11605 11605 AF (temp, RH) = exp 2 β2 − + tempU K temp K !#) " RH RHU − log 2 β3 log 1 − RH 1 − RHU where tempU K = 25◦C + 273.15 and RHU = 50% are the AccVar at use conditions. • The estimation of AF (temp, RH) will be programmed in SPLIDA. 23 - 44 Another Linear Degradation Model for an Insulation Breakdown Nelson 1990, Chapter 11, discusses the following linear degradation model for time to an insulation Breakdown. yijk = β0 + β1 exp(β2xj )τi + ǫijk where yijk = log[(Breakdown Voltage)ijk ] τi = ti = Weeksi xj = −11605/(◦Cj + 273.15) (ǫijk /σ) ∼ Φnor (z). Notice that in this instance the variable time is used in the linear scale. 23 - 45 ADT Models with Interactions • Most accelerated test models used in practice do not contain interaction terms. • Interaction terms imply curvature in the degradation rate versus the AccVar surface, i.e., µij = β0 + β1 exp(β2xj2 + β3xj3 + β4xj2xj3) τi. • Extrapolation is hazardous, especially with surfaces involving curvature. • A physical model could suggest interactions. • Current version of the SPLIDA software does not allow for interactions. 23 - 46 Future Research in ADDT Data, Models, and Analysis • Nonlinear degradation models. • Coarse data. • Stochastic variability in the degradation response. • Prediction in non-constant environments. • Use of prior information. • Random initiation times. 23 - 47 Technical Details The following slides give technical details used in SPLIDA to implement the methodology. 23 - 48 A Reparameterization of the Linear ADDT Model for Numerical Stability • The model is as before yij = β0 + β1 exp(β ′2xj )τi + ǫij . • Suppose that x̄ is the centroid of the stress variables [i.e., x̄ = (x̄1, . . . , x̄k )′] and τ̄ is an average transformed time. Then the model can be reparameterized as yij = γ0 + γ1 exp ′ xj − x̄ γ 2 τi − τ̄ + ǫij . where γ0 is the intercept for the average stress line (i.e., degradation line for x̄) at τ̄ ; γ1 is the slope of the average stress line; and γ 2 = β 2 are the regression coefficients corresponding to the x variables. • It can be shown that γ1 is the geometric mean of the slopes β1 exp(β ′2xj ), for the values of xj stress variables in the data set. 23 - 49 Relationship Between Stable and Original Parameters • Then γ0 = β0 + β1 exp(β ′2x̄)τ̄ γ1 = β1 exp(β ′2x̄) γ 2 = β 2. • Solving for the βs β2 = γ 2 β1 = γ1 exp(−x̄′γ 2) β0 = γ0 − γ1τ̄ . 23 - 50 Induced Failure Time Distribution at Use Conditions x and Critical Level Df Decreasing Linear Degradation • In this case τ −ν FT (t, x, β ) = Φ ς for t ≥ 0 where (β0 − µf ) exp(−β ′2x) ν= | β1 | and σ exp(−β ′2x) ς= . | β1 | • The failure time distribution is a mixed distribution with a spike of Φ [(µf − β0)/σ ] = Φ (− ν/ς ) at t = 0. For t > 0 the cdf is continuous and it agrees with the cdf of a location-scale variable with standardized cdf Φ(z), location ν and scale ς. 23 - 51 Induced Failure Time Distribution at Use Conditions x and Critical Level Df Decreasing Linear Degradation (Continued) • In particular, the following follows: Y = hd(degradation) ∼ Φ and τ −ν ht(T ) ∼ Φ , ς " y − µ(τ, x, β ) σ # t > 0. • This implies that ht(T ) is location-scale distributed with parameters (ν, ς) and standardized distribution Φ(·). 23 - 52 Density to Plot Induced Failure Time Distribution at Use Conditions x and Critical Level Df Decreasing Linear Degradation Let W = log(T ), one needs to plot fW (w) which is giving by exp(w) τ −ν fW (w, β ) = ×φ ς ς where dτ × dt t=exp(w) for − ∞ < w < ∞ τ = ht[exp(w)] 1 if not transformation on time dτ = 1 dt t=exp(w) if square-root transformation on time 2τ and (β0 − µf ) exp(−β ′2x) ν= | β1 | and σ exp(−β ′2x) ς= . | β1 | 23 - 53 Induced Failure Time Distribution at Use Conditions x and Critical Level Df Increasing Linear Degradation • In this case −τ − ν FT (t, x, β ) = 1 − Φ ς where as before (β0 − µf ) exp(−β ′2x) ν= | β1 | and for t ≥ 0 σ exp(−β ′2x) ς= . | β1 | • The failure time distribution is a mixed distribution with a spike of 1 − Φ [(µf − β0)/σ ] = 1 − Φ (− ν/ς ) at t = 0. For t > 0 the cdf is continuous and it agrees with the cdf of continuous random variable. 23 - 54 Induced Failure Time Distribution at Use Conditions x and Critical Level Df Increasing Linear Degradation • In particular the following follows: " y − µ(τ, x, β ) Y = hd(degradation) ∼ Φ σ and −τ − ν −ht(T ) ∼ Φ , ς # t > 0. • This implies that that −ht(T ) is location-scale distributed with parameters (ν, ς) and standardized distribution Φ(·). 23 - 55 Density to Plot Induced Failure Time Distribution at Use Conditions x and Critical Level Df Increasing Linear Degradation Let W = log(T ), you need to plot fW (w) which is giving by exp(w) −τ − ν fW (w, β ) = ×φ ς ς where dτ × dt t=exp(w) for − ∞ < w < ∞ τ = ( ht[exp(w)] dτ 1 if not transformation on time = 1 if square-root transformation on time dt t=exp(w) 2τ and (β0 − µf ) exp(−β ′2x) ν= | β1 | and σ exp(−β ′2x) ς= . | β1 | 23 - 56 Quantiles of the Failure Time Distribution • For decreasing degradation, the quantiles of the failure time distribution are ν 0 if p ≤ Φ − ς tp = h i h−1 ν + ςΦ−1(p) otherwise t • For increasing degradation, the quantiles of the failure time distribution are ν 0 if p ≤ 1 − Φ − ς tp = n h io h−1 − ν + ςΦ−1(1 − p) otherwise t 23 - 57