General Log-Linear (GLL)-Weibull Model: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
No edit summary
 
(14 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{Reference Example|ALTA_Reference_Examples_Banner.png|ALTA_Reference_Examples}}
{{Reference Example|{{Banner ALTA Reference Examples}}}}
This example validates the calculation of GLL relationship for Weibull distribution.
 
This example validates the results for the GLL life-stress relationship with a Weibull distribution in ALTA standard folios.




Line 10: Line 11:
{{Reference_Example_Heading2}}
{{Reference_Example_Heading2}}


The data is given below.
The following table shows the data.
{| {{table}}
{| {{table}}
!State F/S
!State F/S
Line 85: Line 86:


::<math>\,\!ln\left ( \eta  \right )=\alpha _{0}+\alpha _{1}\frac{1}{T}</math>  
::<math>\,\!ln\left ( \eta  \right )=\alpha _{0}+\alpha _{1}\frac{1}{T}</math>  


The book has the following results:
The book has the following results:
*The model parameters are: <math>\,\!\alpha _{0}=-3.156</math> , <math>\,\!\alpha _{1}=4390</math> and <math>\,\!\beta =2.27</math>.
*The model parameters are: <math>\,\!\alpha _{0}=-3.156</math> , <math>\,\!\alpha _{1}=4390</math> and <math>\,\!\beta =2.27</math>.


*The variance of each parameter is: <math>\,\!Var\left ( \alpha _{0} \right )=3.08</math> , <math>\,\!Var\left ( \alpha _{1} \right )=484,819.5</math> and <math>\,\!Var\left ( \beta\right )=0.1396</math> .


*The two-sided 90% confidence intervals for the model parameters are:  <math>\,\!\left [ \alpha _{0,L},\alpha _{0,U}  \right ]=\left [ -6.044,-0.269 \right ]</math>  , <math>\,\!\left [ \alpha _{1,L},\alpha _{1,U}  \right ]=\left [ 3244.8,5535.3 \right ]</math> and <math>\,\!\left [ \beta _{1,L},\beta _{1,U}  \right ]=\left [ 1.73,2.97 \right ]</math> .  
*The variance of each parameter is: <math>\,\!Var\left ( \alpha _{0} \right )=3.08</math> , <math>\,\!Var\left ( \alpha _{1} \right )=484819.5</math> and <math>\,\!Var\left ( \beta\right )=0.1396</math> .
 
 
*The two-sided 90% confidence intervals for the model parameters are:  <math>\,\!\left [ \alpha _{0,L},\alpha _{0,U}  \right ]=\left [ -6.044, -0.269 \right ]</math>  , <math>\,\!\left [ \alpha _{1,L},\alpha _{1,U}  \right ]=\left [ 3244.8, 5535.3 \right ]</math> and <math>\,\!\left [ \beta _{1,L},\beta _{1,U}  \right ]=\left [ 1.73, 2.97 \right ]</math> .
 
 
*The estimated B10 life at temperature of 35°C is 24,286 hours. The two-sided 90% confidence interval is [10371, 56867].


*The estimated B10 life at temperature of 35°C is 24,286 hours. The two-sided 90% confidence interval is [10,371, 56,867].
    
    
*The estimated reliability at 35°C and 10,000 hours is <math>\,\!R\left ( 10,000 \right )=0.9860</math> . The two-sided 90% confidence interval is [0.892, 0.998].
*The estimated reliability at 35°C and 10,000 hours is <math>\,\!R\left ( 10000 \right )=0.9860</math> . The two-sided 90% confidence interval is [0.892, 0.998].




Latest revision as of 18:21, 28 September 2015

ALTA Reference Examples Banner.png


New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images and more targeted search.

As of January 2024, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest references at ALTA examples and ALTA reference examples.



General Log-Linear (GLL)-Weibull Model

This example validates the results for the GLL life-stress relationship with a Weibull distribution in ALTA standard folios.


Reference Case

The data set is from Example 7.14 on page 297 in book Life Cycle Reliability Engineering by Dr. Guangbin Yang, John Wiley & Sons, 2007.


Data

The following table shows the data.

State F/S Time to State (Hr) Temperature (°C) Group ID
F 1138 100 1
F 1944 100 1
F 2764 100 1
F 2846 100 1
F 3246 100 1
F 3803 100 1
F 5046 100 1
F 5139 100 1
S 5500 100 1
S 5500 100 1
S 5500 100 1
S 5500 100 1
F 1121 120 2
F 1572 120 2
F 2329 120 2
F 2573 120 2
F 2702 120 2
F 3702 120 2
F 4277 120 2
S 4500 120 2
F 420 150 3
F 650 150 3
F 703 150 3
F 838 150 3
F 1086 150 3
F 1125 150 3
F 1387 150 3
F 1673 150 3
F 1896 150 3
F 2037 150 3


Result

The model used in the book is:

[math]\displaystyle{ \,\!ln\left ( \eta \right )=\alpha _{0}+\alpha _{1}\frac{1}{T} }[/math]


The book has the following results:

  • The model parameters are: [math]\displaystyle{ \,\!\alpha _{0}=-3.156 }[/math] , [math]\displaystyle{ \,\!\alpha _{1}=4390 }[/math] and [math]\displaystyle{ \,\!\beta =2.27 }[/math].


  • The variance of each parameter is: [math]\displaystyle{ \,\!Var\left ( \alpha _{0} \right )=3.08 }[/math] , [math]\displaystyle{ \,\!Var\left ( \alpha _{1} \right )=484819.5 }[/math] and [math]\displaystyle{ \,\!Var\left ( \beta\right )=0.1396 }[/math] .


  • The two-sided 90% confidence intervals for the model parameters are: [math]\displaystyle{ \,\!\left [ \alpha _{0,L},\alpha _{0,U} \right ]=\left [ -6.044, -0.269 \right ] }[/math] , [math]\displaystyle{ \,\!\left [ \alpha _{1,L},\alpha _{1,U} \right ]=\left [ 3244.8, 5535.3 \right ] }[/math] and [math]\displaystyle{ \,\!\left [ \beta _{1,L},\beta _{1,U} \right ]=\left [ 1.73, 2.97 \right ] }[/math] .


  • The estimated B10 life at temperature of 35°C is 24,286 hours. The two-sided 90% confidence interval is [10371, 56867].


  • The estimated reliability at 35°C and 10,000 hours is [math]\displaystyle{ \,\!R\left ( 10000 \right )=0.9860 }[/math] . The two-sided 90% confidence interval is [0.892, 0.998].


Results in ALTA

In ALTA, the GLL model with Weibull distribution is used. Since temperature is the stress, the reciprocal transform is used. The results are:

  • The model parameters are:
Temperature GLL Weibull Analysis Summary.png


  • The variances of the parameters are:
Temperature GLL Weibull Var Cov Results.png


  • The two-sided 90% confidence intervals for the model parameters are:
Temperature GLL Weibull Parameter Bounds.png


  • The estimated B10 life and its two-sided 90% confidence intervals are:
Temperature GLL Weibull QPC B10 Life.png


  • The estimated reliability with its two-sided 90% confidence interval at 35°C and 10,000 hours are:
Temperature GLL Weibull QPC Reliability.png
Retrieved from "https://www.reliawiki.com/index.php?title=General_Log-Linear_(GLL)-Weibull_Model&oldid=60835"