AcceleratedLifeModel

Accelerated life model for lifetime analysis

Since R2026a

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Description

An AcceleratedLifeModel object contains the results of fitting an accelerated life model to stressor and failure time data.

Use the properties of an AcceleratedLifeModel object to investigate a fitted accelerated life model. The object properties include the stressor and failure time data, and information about coefficient estimates, statistics, and censoring. Use the object functions to predict mean failure times at stressor levels, compute confidence intervals for coefficient estimates, and visualize the accelerated life model. For more information, see What Is Accelerated Life Analysis?

Creation

Create an AcceleratedLifeModel object using fitacclife.

Properties

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This property is read-only.

Stressor levels, represented as a table with n variables, where n is the number of elements in StressorNames. Each column (variable) in StressorLevels contains a stressor in the accelerated life model, and each row contains a stressor level.

This property is read-only.

Failure times, represented as a numeric column vector or a two-column numeric matrix. Each element in FailureTimes corresponds to the row with the same index in StressorLevels. When FailureTimes is a two-column matrix, the first column contains the lower bounds for the interval-censored failure times, and the second column contains the upper bounds.

Data Types: double

This property is read-only.

Indicator of censored failure times, represented as a vector consisting of 0, –1, and 1, which indicate fully observed, left-censored, and right-censored failure times, respectively. Each element of Censoring indicates the censorship status of the corresponding failure time in FailureTimes. The default is a vector of 0s, indicating all failure times are fully observed.

Data Types: logical | single | double

This property is read-only.

Life distribution, represented as a string scalar. The following table shows the supported distributions.

Distribution NameDistribution Object
"exponential"ExponentialDistribution
"logistic"LogisticDistribution
"loglogistic"LoglogisticDistribution
"lognormal"LognormalDistribution
"normal"NormalDistribution
"weibull" (default)WeibullDistribution

For more information about life distributions, see What Is Accelerated Life Analysis?

This property is read-only.

Life stress model, represented as a string scalar or a function handle (@stressName). The following table shows the supported life stress models, where X1 is the value of the first stressor, X2 is the value of the second stressor (if applicable), and bx are the model coefficients.

ValueLife Stress ModelStressor Data Conditions
"linear"b0 + b1*X1 + b2*X2None
"log"b0 + b1*log(X1) + b2*log(X2)X1 and X2 must be positive.
"exponential"b0*exp(b1*X1 + b2*X2)None
"arrhenius" (default)b0*exp(b1/X1 + b2/X2)X1 and X2 must be nonzero.
"power"b0*X1b1*X2b2X1 and X2 must be nonnegative.
"reciprocal"b0 + b1/X1 + b2/X2X1 and X2 must be nonzero.
"sqrt"b0 + b1*sqrt(X1) + b2*sqrt(X2)X1 and X2 must be nonnegative.
"eyring"(1/X1)*e–(b0 – b1/X1)X1 and X2 must be nonzero.
"temp-nonthermal"b0*e-b1/X1*X2b2X1 must be nonzero, and X2 must be nonnegative.

For more information about life stress models, see What Is Accelerated Life Analysis?

Data Types: string | function handle

This property is read-only.

Coefficient values, represented as a table that contains one row for each coefficient and these columns:

  • Source — Source of the coefficient value, where "StressModel" indicates a life stress model coefficient, and "Distribution" indicates a life distribution coefficient

  • Estimate — Estimated coefficient value

  • SE — Standard error of the estimate

Use the coefci function to find the confidence intervals of the coefficient estimates.

To obtain any of the columns as a vector, index into the property using dot notation. For example, obtain the estimated coefficient vector in the model mdl:

beta = mdl.Coefficients.Estimate

This property is read-only.

Covariance matrix for the coefficient estimates, represented as a square matrix with the number of rows equal to the number of coefficients.

Data Types: double

This property is read-only.

Coefficient names, represented as a cell array of character vectors, each containing the name of the corresponding term.

Data Types: cell

This property is read-only.

Loglikelihood of the fitted model, represented as a numeric scalar.

You can use loglikelihood values to compare different models and assess the significance of the effects of terms in the model.

Data Types: double

This property is read-only.

Criterion for model comparison, represented as a structure with these fields:

  • AIC — Akaike information criterion. AIC = –2*logL + 2*m, where logL is the loglikelihood and m is the number of estimated parameters.

  • AICc — Akaike information criterion corrected for the sample size. AICc = AIC + (2*m*(m + 1))/(n – m – 1), where n is the number of observations.

  • BIC — Bayesian information criterion. BIC = –2*logL + m*log(n).

  • CAIC — Consistent Akaike information criterion. CAIC = –2*logL + m*(log(n) + 1).

Information criteria are model selection tools that you can use to compare multiple models fit to the same data. These criteria are likelihood-based measures of model fit that include a penalty for complexity (specifically, the number of parameters). Different information criteria are distinguished by the form of the penalty.

When you compare multiple models, the model with the lowest information criterion value is the best-fitting model. The best-fitting model can vary depending on the criterion used for model comparison.

To obtain any of the criterion values as a scalar, index into the property using dot notation. For example, obtain the AIC value aic in the model mdl:

aic = mdl.ModelCriterion.AIC

Data Types: struct

This property is read-only.

Baseline stressor level, represented as a numeric vector with the same number of elements as StressorNames.

Data Types: double

This property is read-only.

Stressor names, represented as a string array with n elements, where n is the number of stressors in the model. Each element of StressorNames contains the name of the corresponding stressor in StressorLevels.

Data Types: string

Object Functions

accelfactorAcceleration factors of accelerated life model
coefciConfidence intervals for accelerated life model coefficients
distfcnDistribution functions of accelerated life model
distplotPlot distribution functions of accelerated life model
icdfInverse cumulative distribution function of accelerated life model
meanfailplotPlot failure times of accelerated life model
meanfailtimeMean failure times and life distribution coefficients of accelerated life model
probplotPlot failure probabilities of accelerated life model

Examples

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Open Live Script

Perform an accelerated life analysis on a machine part that is subject to different stress levels.

Load and Visualize Data

Load the machineFailure data set, which contains 100 simulated failure time measurements of the machine part at five evenly spaced temperatures (T) between 300 K and 400 K. Approximately 5% of the failure time measurements are right-censored. These measurements are indicated in the censored variable of the table.

load machineFailure

Create a histogram of the failure times at the lowest stressor level in the data set (T = 300).

histogram(failureTable.failureTime(failureTable.T==300),BinWidth=2)
xlabel("Failure Time");
title("Failure Times at T = 300 K")

Figure contains an axes object. The axes object with title Failure Times at T = 300 K, xlabel Failure Time contains an object of type histogram.

The histogram shows that the failure times at the first stressor level are approximately normally distributed.

Create a plot of mean failure time versus stressor level.

s = groupsummary(failureTable,"T","mean","failureTime");
plot(s.T,s.mean_failureTime,"-o")
xlabel("T")
ylabel("Failure Time");

Figure contains an axes object. The axes object with xlabel T, ylabel Failure Time contains an object of type line.

The plot indicates that the mean failure time has a nonlinear dependence on temperature. The failure times in this data set follow an Eyring relationship, which has the functional form f(T) = 1/T*exp(–(b0–b1/T)), where b0 and b1 are model coefficients.

Fit Accelerated Life Model

Fit an accelerated life model to the data using the fitacclife function. Specify a normal life distribution and an Eyring life stress model. Use 230 K as the baseline stressor level.

mdl = fitacclife(failureTable,"failureTime", ...
    Censoring=failureTable.censored,Distribution="normal", ...
    StressModel="eyring",StressorName="T",BaselineStressorLevel=230)
mdl = 
AcceleratedLifeModel

Life distribution: normal
Stress model: eyring
Baseline stressor level: 230

     T     NormalMu    MeanFailureTime    AccelerationFactor
    ___    ________    _______________    __________________

    400     90.742         90.742               1.7712      
    375     96.951         96.951               1.6577      
    350     104.07         104.07               1.5443      
    325     112.32         112.32               1.4309      
    300     121.99         121.99               1.3175      
    230     160.72         160.72                    1      


Log-likelihood: -195.5267

mdl is an AcceleratedLifeModel object, which you can use to compute mean failure times, calculate failure probabilities at specific stressor levels, and create plots. The first column of the output contains the unique stressor levels in the data and the baseline stressor level. The second and third columns list the fitted life distribution parameter values and mean failure times, respectively. The fourth column lists the acceleration factor, which is the ratio of the mean failure time at the stressor level to the mean failure time at the baseline stressor level.

Display information about the fitted model coefficients.

mdl.Coefficients
ans=3×3 table
                       Source        Estimate       SE   
                   ______________    ________    ________

    b0             "StressModel"     -10.475     0.016919
    b1             "StressModel"      9.8762        5.709
    NormalSigma    "Distribution"     1.7779      0.13054

The table lists the estimated value and the standard error of each coefficient in the life stress model, and the estimated value and the standard error of the life distribution parameter. The life distribution parameter (NormalSigma) and the first coefficient (b0) of the life stress model are well constrained. The second coefficient of the life stress model (b1) is not well constrained.

List the 95% confidence intervals for each fitted model coefficient and parameter.

coefci(mdl)
ans = 3×2

  -10.5084  -10.4412
   -1.4545   21.2070
    1.5188    2.0370

Display the mean failure times according to the model.

meanfailtime(mdl)
ans=6×2 table
     T     MeanFailureTime
    ___    _______________

    400        90.742     
    375        96.951     
    350        104.07     
    325        112.32     
    300        121.99     
    230        160.72

The last row in the table indicates that the mean failure time at the baseline stressor level (T = 230) is 160.7.

Create a plot of the mean failure times.

meanfailplot(mdl)

Figure contains an axes object. The axes object with title Failure Time Plot, xlabel T, ylabel Failure Time contains 7 objects of type line. One or more of the lines displays its values using only markers These objects represent Mean Failure Time, 300, 325, 350, 375, 400, 230 (Baseline).

The plot indicates that the Eyring model provides a good fit to the mean failure times at different stressor levels. The mean predicted failure time at the baseline stressor level is indicated on the plot with an asterisk marker.

Create a probability plot of the model and the data.

probplot(mdl)

Figure contains an axes object. The axes object with title Probability Plot for Normal Distribution, xlabel Failure Time, ylabel Failure Probability contains 10 objects of type functionline, line. One or more of the lines displays its values using only markers These objects represent 400, 375, 350, 325, 300.

The log-linear plot shows each observation in the data, grouped by stressor level. Each dashed reference line connects the first and third quartiles of the failure time data for that stressor level and extends to the ends of the data. The failure times are shorter at higher temperatures. Also, at a fixed stressor level, the failure probability increases nonlinearly with time.

Open Live Script

Load the partFailure data set, which contains simulated observations of failure times for an assembly line part at specific humidity and temperature levels.

load partFailure.mat

Fit an accelerated life model to the data in the partFailure table using the fitacclife function. Use the FailureTime table variable as the failure times, and the other table variables as the stressors. 

mdl = fitacclife(partFailure,"FailureTime")
mdl = 
AcceleratedLifeModel

Life distribution: weibull
Stress model: arrhenius

    Humidity    Temperature    WeibullA    MeanFailureTime
    ________    ___________    ________    _______________

       90           35          1.4502         1.3928     
       90           30          1.5114         1.4516     
       90           25          1.6015         1.5382     
       90           20          1.7468         1.6777     
       90           15          2.0189          1.939     
       90           12          2.3333          2.241     
       90            8          3.3507         3.2181     
       90            5          6.4271         6.1729     
       80           35          1.4458         1.3886     
       80           30          1.5069         1.4473     
       80           25          1.5967         1.5335     
       80           20          1.7415         1.6727     
       80           15          2.0128         1.9332     
       80           12          2.3263         2.2343     
       80            8          3.3406         3.2084     
       80            5          6.4077         6.1543     
       70           35          1.4402         1.3832     
       70           30           1.501         1.4417     
       70           25          1.5905         1.5276     
       70           20          1.7348         1.6662     
       70           15           2.005         1.9257     
       70           12          2.3173         2.2256     
       70            8          3.3276          3.196     
       70            5          6.3829         6.1304     
       60           35          1.4328         1.3761     
       60           30          1.4933         1.4342     
       60           25          1.5823         1.5197     
       60           20          1.7259         1.6576     
       60           15          1.9946         1.9158     
       60           12          2.3053         2.2141     
       60            8          3.3104         3.1795     
       60            5          6.3499         6.0988     
       50           35          1.4224         1.3662     
       50           30          1.4825         1.4239     
       50           25          1.5709         1.5087     
       50           20          1.7134         1.6456     
       50           15          1.9803         1.9019     
       50           12          2.2887         2.1981     
       50            8          3.2866         3.1566     
       50            5          6.3041         6.0548     


Log-likelihood: 47.0683

mdl is an AcceleratedLifeModel object, which contains information about the fitted model coefficient estimates. By default, the fitacclife function fits an Arrhenius life stress model to the data, and uses a Weibull life distribution. The first and second columns of the displayed output list the unique stressor levels in partFailure. The third and fourth columns list the fitted life distribution parameter values and mean failure times, respectively.

Display information about the fitted model coefficients.

mdl.Coefficients
ans=4×3 table
                    Source        Estimate       SE   
                ______________    ________    ________

    b0          "StressModel"      1.1592     0.039112
    b1          "StressModel"     -2.1731       2.0553
    b2          "StressModel"      8.6848      0.11616
    WeibullB    "Distribution"     12.764      0.72011

The table lists the estimated value and the standard error of each coefficient in the life stress model (b0, b1, and b2), and lists the estimated value and the standard error of the life distribution parameter (WeibullB).

List the 95% confidence intervals for each fitted model coefficient and parameter.

ci = coefci(mdl)
ci = 4×2

    1.0819    1.2364
   -6.2330    1.8868
    8.4554    8.9143
   11.3419   14.1867

All the coefficients are well constrained except the b1 coefficient of the Arrhenius life stress model.

Open Live Script

Load the partFailure data set, which contains simulated observations of failure times for an assembly line part at specific humidity and temperature levels.

load partFailure.mat

Fit an accelerated life model to the data in the partFailure table using the fitacclife function. Use the FailureTime table variable as the failure times, and the other table variables as the stressors. Set the baseline stressor level to a humidity of 50% and a temperature of 20 degrees Celsius.

mdl = fitacclife(partFailure,"FailureTime",BaselineStressorLevel=[50 20]);

Display the mean failure times of the model at each stressor level in the data.

meanfailtime(mdl)
ans=41×3 table
    Humidity    Temperature    MeanFailureTime
    ________    ___________    _______________

       90           35             1.3928     
       90           30             1.4516     
       90           25             1.5382     
       90           20             1.6777     
       90           15              1.939     
       90           12              2.241     
       90            8             3.2181     
       90            5             6.1729     
       80           35             1.3886     
       80           30             1.4473     
       80           25             1.5335     
       80           20             1.6727     
       80           15             1.9332     
       80           12             2.2343     
       80            8             3.2084     
       80            5             6.1543     
      ⋮

Create a table of acceleration factors for the unique stressor levels in mdl.

tbl = accelfactor(mdl)
tbl=41×4 table
    Humidity    Temperature    MeanFailureTime    AccelerationFactor
    ________    ___________    _______________    __________________

       90           35             1.3928               1.1815      
       90           30             1.4516               1.1336      
       90           25             1.5382               1.0699      
       90           20             1.6777              0.98087      
       90           15              1.939              0.84869      
       90           12              2.241              0.73432      
       90            8             3.2181              0.51136      
       90            5             6.1729              0.26659      
       80           35             1.3886               1.1851      
       80           30             1.4473               1.1371      
       80           25             1.5335               1.0731      
       80           20             1.6727              0.98383      
       80           15             1.9332              0.85125      
       80           12             2.2343              0.73654      
       80            8             3.2084              0.51291      
       80            5             6.1543               0.2674      
      ⋮

Each acceleration factor is the ratio of the mean failure time at the baseline stressor level to the mean failure time at the stressor level.

Create a 3-D scatter plot showing the acceleration factors of different stressor levels.

scatter3(tbl,"Humidity","Temperature","AccelerationFactor")

Figure contains an axes object. The axes object with xlabel Humidity, ylabel Temperature contains an object of type scatter.

Open Live Script

Load the partFailure data set, which contains simulated observations of failure times for an assembly line part at specific humidity and temperature levels.

load partFailure.mat

Fit an accelerated life model to the data in the partFailure table using the fitacclife function. Use the FailureTime table variable as the failure times, and the other table variables as the stressors.

mdl = fitacclife(partFailure,"FailureTime");

Plot Survivor Function

Return the survivor function values of the model, evaluated at 50 equally spaced time values between 0.8 and 10. By default, the software calculates the survivor function at the lowest unique stressor level in mdl.StressorLevels. In this example, the lowest unique stressor level corresponds to a humidity value of 50% and a temperature of 5 degrees Celsius.

points = linspace(0.8,10,50)';
x = distfcn(mdl,EvaluationTimes=points)
x = 50×1

    0.8000
    0.9878
    1.1755
    1.3633
    1.5510
    1.7388
    1.9265
    2.1143
    2.3020
    2.4898
    2.6776
    2.8653
    3.0531
    3.2408
    3.4286
      ⋮

Plot the survivor function at the evaluation points.

distplot(mdl,EvaluationTimes=points)

Figure contains an axes object. The axes object with title Survivor Function for Humidity=50, Temperature=5, xlabel Time, ylabel Survivor Function contains an object of type line. This object represents Humidity=50, Temperature=5.

The plot shows that, at this stressor level, the assembly line part has a survival probability close to 100% for time values smaller than 4. The survival probability drops to approximately zero for time values greater than 7.5.

Plot Probability Density Function

Plot the probability density function (pdf). 

distplot(mdl,Type="pdf",EvaluationTimes=points)

Figure contains an axes object. The axes object with title Probability Density Function for Humidity=50, Temperature=5, xlabel Time, ylabel Probability Density Function contains an object of type line. This object represents Humidity=50, Temperature=5.

The plot shows that the most likely failure time at this stressor level is approximately 6.2.

Plot Cumulative Distribution Function

Plot the cumulative distribution function (cdf) at a stressor level that corresponds to 90% humidity and a temperature of 20 degrees. 

distplot(mdl,[90 20],Type="cdf",EvaluationTimes=points)

Figure contains an axes object. The axes object with title Cumulative Distribution Function for Humidity=90, Temperature=20, xlabel Time, ylabel Cumulative Distribution Function contains an object of type line. This object represents Humidity=90, Temperature=20.

The plot shows that, at this stressor level, approximately half of all assembly line parts have failure times smaller than 1.7.

Plot Inverse Cumulative Distribution Function

Calculate the inverse cumulative distribution function (icdf) at 90% humidity and 35 degrees, and 90% humidity and 20 degrees. Evaluate the icdf at 10 equally spaced probability values between 0.1 and 0.9.

pts = linspace(0.05,0.9,10);
icdfval = [icdf(mdl,pts); icdf(mdl,pts,[90,20])]
icdfval = 2×10

    4.9953    5.4502    5.6944    5.8737    6.0227    6.1562    6.2834    6.4119    6.5525    6.7298
    1.3842    1.5102    1.5779    1.6275    1.6688    1.7058    1.7411    1.7767    1.8156    1.8648

Create a plot of the icdf values for the two stressor levels.

plot(pts,icdfval,"-o")
xlabel("Probability")
ylabel("ICDF Value")
legend(["Humidity = 90, Temperature = 35", ...
    "Humidity = 90, Temperature = 20"],Location="east")

Figure contains an axes object. The axes object with xlabel Probability, ylabel ICDF Value contains 2 objects of type line. These objects represent Humidity = 90, Temperature = 35, Humidity = 90, Temperature = 20.

The icdf values are consistently lower at the lower temperature stressor level.

Open Live Script

Load the diodeFailure data set, which contains simulated observations of failure times for a diode at different current levels.

load diodeFailure.mat

Fit an accelerated life model to the data in the diodeFailure table using the fitacclife function. Use the FailureTime table variable as the failure times.

mdl = fitacclife(diodeFailure,"FailureTime")
mdl = 
AcceleratedLifeModel

Life distribution: weibull
Stress model: arrhenius

    Current    WeibullA    MeanFailureTime
    _______    ________    _______________

      10        1.5323         1.4721     
       5        2.4956         2.3975     
       3        4.7821         4.5941     


Log-likelihood: 0.4676

mdl is an AcceleratedLifeModel object, which contains information about the fitted model coefficient estimates. By default, the fitacclife function fits an Arrhenius life stress model to the data, using a Weibull life distribution. The first column of the displayed output contains the unique stressor levels in diodeFailure. The second and third columns contain the fitted life distribution coefficient values and mean failure times, respectively.

Create a probability plot.

probplot(mdl)

Figure contains an axes object. The axes object with title Probability Plot for Weibull Distribution, xlabel Failure Time, ylabel Failure Probability contains 6 objects of type functionline, line. One or more of the lines displays its values using only markers These objects represent 10, 5, 3.

The log-log scatter plot of failure probability versus failure time shows each observation in the data, grouped by stressor level. The dashed lines are linear regression fits to the failure times belonging to each group.

The plot indicates that failure times are shorter at higher current levels, and that the failure probability increases nonlinearly with time at each stressor level.

Open Live Script

Load the diodeFailure data set, which contains simulated observations of failure times for a diode at different current levels.

load diodeFailure.mat

Fit an accelerated life model to the data in the diodeFailure table using the fitacclife function. Specify a power law life stress model and use the FailureTime table variable as the failure times.

mdl = fitacclife(diodeFailure,"FailureTime",StressModel="power")
mdl = 
AcceleratedLifeModel

Life distribution: weibull
Stress model: power

    Current    WeibullA    MeanFailureTime
    _______    ________    _______________

      10        1.4869           1.41     
       5        2.8445         2.6973     
       3         4.588         4.3506     


Log-likelihood: -3.6697

mdl is an AcceleratedLifeModel object, which contains information about the fitted model coefficient estimates. By default, the fitacclife function uses a Weibull life distribution. The first column of the displayed output lists the unique stressor levels in diodeFailure. The second and third columns list the fitted life distribution values and mean failure times, respectively.

Use the meanfailtime function to compute the predicted mean failure time and the Weibull life distribution parameter at current levels 12 and 15.

[meanFailTimes,lifeCoeffs] = meanfailtime(mdl,[12; 15])
meanFailTimes=2×2 table
    Current    MeanFailureTime
    _______    _______________

      12            1.1888    
      15           0.96474    


lifeCoeffs=2×2 table
    Current    WeibullA
    _______    ________

      12        1.2537 
      15        1.0174

Create a plot of failure time versus current level.

meanfailplot(mdl)

Figure contains an axes object. The axes object with title Failure Time Plot, xlabel Current, ylabel Failure Time contains 4 objects of type line. One or more of the lines displays its values using only markers These objects represent Mean Failure Time, 3, 5, 10.

The plot shows that the mean failure time decreases nonlinearly with increasing current level.

More About

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Version History

Introduced in R2026a

See Also

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