Make ALT data
This function is used to generate accelerated life testing (ALT) data. It is primarily used for testing the functions within ALT_fitters. The function Other_functions.make_ALT_data accepts the life distribution (Weibull, Lognormal, Normal, Exponential) and the life-stress model (Exponential, Eyring, Power, Dual_Exponential, Dual_Power, Power_Exponential), along with the parameters of the model and will create an object with the data in the correct format for the ALT models contained within reliability.ALT_fitters. The function contains many more inputs than are required and these inputs will only be used if they are part of the model. Please see the equations of the ALT model you are using to determine what parameters are required. The function is designed to automatically censor a fraction of the data using the input fraction_censored.
API Reference
For inputs and outputs see the API reference.
Example 1
In this first example we will create ALT data from a Weibull_Eyring model. To verify the accuracy of the fitter we can compare the fitted model’s parameters to the parameters we used to generate the data. Note that we only need to specify a, c, and beta since these are the three parameters of the Weibull_Exponential model.
from reliability.Other_functions import make_ALT_data
from reliability.ALT_fitters import Fit_Weibull_Eyring
ALT_data = make_ALT_data(distribution='Weibull',life_stress_model='Eyring',a=1500,c=-10,beta=2,stress_1=[500,400,350],number_of_samples=100,fraction_censored=0.2,seed=1)
Fit_Weibull_Eyring(failures=ALT_data.failures, failure_stress=ALT_data.failure_stresses, right_censored=ALT_data.right_censored, right_censored_stress=ALT_data.right_censored_stresses, use_level_stress=300, show_probability_plot=False, show_life_stress_plot=False)
'''
Results from Fit_Weibull_Eyring (95% CI):
Analysis method: Maximum Likelihood Estimation (MLE)
Optimizer: TNC
Failures / Right censored: 240/60 (20% right censored)
Parameter Point Estimate Standard Error Lower CI Upper CI
a 1458.14 92.7751 1276.31 1639.98
c -10.1231 0.230245 -10.5743 -9.6718
beta 1.92551 0.0921087 1.75318 2.11477
stress original alpha original beta new alpha common beta beta change acceleration factor
500 945.271 1.94553 920.348 1.92551 -1.03% 11.6466
400 2193.3 1.75376 2385.03 1.92551 +9.79% 4.49426
350 4822.96 2.11256 4588.29 1.92551 -8.85% 2.33615
Goodness of fit Value
Log-likelihood -2000.5
AICc 4007.09
BIC 4018.12
At the use level stress of 300, the mean life is 9507.77152
'''
Example 2
In this second example we will create ALT data from a Lognormal_Dual_Power model. To verify the accuracy of the fitter we can compare the fitted model’s parameters to the parameters we used to generate the data. Note that we only need to specify c, m, n, and sigma since these are the four parameters of the Lognormal_Dual_Power model.
from reliability.Other_functions import make_ALT_data
from reliability.ALT_fitters import Fit_Lognormal_Dual_Power
use_level_stress = [250, 7]
ALT_data = make_ALT_data(distribution='Lognormal', life_stress_model='Dual_Power', c=1e15, m=-4, n=-2, sigma=0.5, stress_1=[500, 400, 350, 420, 245], stress_2=[12, 8, 6, 9, 10], number_of_samples=100, fraction_censored=0.5, seed=1,use_level_stress=use_level_stress)
Fit_Lognormal_Dual_Power(failures=ALT_data.failures, failure_stress_1=ALT_data.failure_stresses_1, failure_stress_2=ALT_data.failure_stresses_2, right_censored=ALT_data.right_censored, right_censored_stress_1=ALT_data.right_censored_stresses_1, right_censored_stress_2=ALT_data.right_censored_stresses_2, use_level_stress=use_level_stress, show_probability_plot=False, show_life_stress_plot=False)
print('The mean life from the true model is',ALT_data.mean_life_at_use_stress)
'''
Results from Fit_Lognormal_Dual_Power (95% CI):
Analysis method: Maximum Likelihood Estimation (MLE)
Optimizer: TNC
Failures / Right censored: 250/250 (50% right censored)
Parameter Point Estimate Standard Error Lower CI Upper CI
c 9.48288e+14 6.67128e+14 2.38844e+14 3.76502e+15
m -3.9731 0.11982 -4.20795 -3.73826
n -1.99518 0.123271 -2.23678 -1.75357
sigma 0.491039 0.0212097 0.45118 0.534419
stress original mu original sigma new mu common sigma sigma change acceleration factor
500, 12 4.85616 0.496646 4.83656 0.491039 -1.13% 46.0321
420, 9 6.15963 0.525041 6.10326 0.491039 -6.48% 12.97
400, 8 6.39217 0.392671 6.53211 0.491039 +25.05% 8.44684
350, 6 7.69905 0.550747 7.63662 0.491039 -10.84% 2.79905
245, 10 8.02546 0.457947 8.03454 0.491039 +7.23% 1.88017
Goodness of fit Value
Log-likelihood -1859.62
AICc 3727.32
BIC 3744.1
At the use level stress of 250, 7, the mean life is 6545.04098
The mean life from the true model is 5920.122530308318
'''
Recommended values
Some parameters are more suitable than others for these models. The following parameters are recommended for use as a starting point if you are having difficulty in determining the rough order of magnitude of the values you should use:
Exponential: a=2000, b=10
Eyring: a=1500, c=-10
Power: a=5e15, n=-4
Dual_Exponential: a=50, b=0.1, c=500
Dual_Power: c=1e15, m=-4, n=-2
Power_Exponential: a=200, c=400, n=-0.5