Arrhenius-Lognormal Model: Difference between revisions

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{{Reference_Example_Heading3}}
{{Reference_Example_Heading3}}


The following function is used for the Ln-Mean <math>\,\!\mu {}'</math> :
The following function is used for the Ln-Mean <math>\,\!\mu {}'</math>:
 


::<math>\,\!\mu {}'=\beta _{0}+\beta _{1}\times \frac{11605}{T}</math>  
::<math>\,\!\mu {}'=\beta _{0}+\beta _{1}\times \frac{11605}{T}</math>  


where ''T'' is the temperature; <math>\,\!\beta _{1}</math> is the activation energy; <math>\,\!11605</math> is from reciprocal of the Boltzmann constant . This function can be written in the following way:
where ''T'' is the temperature; <math>\,\!\beta _{1}</math> is the activation energy; <math>\,\!11605</math> is from reciprocal of the Boltzmann constant . This function can be written in the following way:


::<math>\,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}}</math>  
::<math>\,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}}</math>  


The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by <math>\,\!\alpha _{i}</math> .
 
The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by <math>\,\!\alpha _{i}</math>.


In fact, the above model also can be expressed using the traditional Arrhenius model:
In fact, the above model also can be expressed using the traditional Arrhenius model:


::<math>\,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}}=C\times e^{\frac{B}{T}}</math>  
::<math>\,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}}=C\times e^{\frac{B}{T}}</math>  


In the book, the following results are provided:
In the book, the following results are provided:
*ML estimations for the model parameters are: <math>\,\!\sigma =0.98</math> ,<math>\,\!\beta _{0}=-13.469</math> , <math>\,\!\beta _{1}=0.6279</math> (or <math>\,\!\alpha _{1}=7286.78</math>).  
*ML estimations for the model parameters are: <math>\,\!\sigma =0.98</math> ,<math>\,\!\beta _{0}=-13.469</math> , <math>\,\!\beta _{1}=0.6279</math> (or <math>\,\!\alpha _{1}=7286.78</math>).  


*The 95% confidence interval for <math>\,\!\sigma</math> is [0.75, 1.28], for <math>\,\!\beta _{0}</math> is [-19.1, -7.8] and for <math>\,\!\beta _{1}</math> is [0.47, 0.79].  
*The 95% confidence interval for <math>\,\!\sigma</math> is [0.75, 1.28], for <math>\,\!\beta _{0}</math> is [-19.1, -7.8] and for <math>\,\!\beta _{1}</math> is [0.47, 0.79].  


*The variance/covariance matrix for <math>\,\!\sigma</math> , <math>\,\!\beta _{0}</math> and <math>\,\!\beta _{1}</math> is
*The variance/covariance matrix for <math>\,\!\sigma</math> , <math>\,\!\beta _{0}</math> and <math>\,\!\beta _{1}</math> is


::<math>\,\!\begin{bmatrix}
::<math>\,\!\begin{bmatrix}
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In terms of <math>\,\!\sigma</math> , <math>\,\!\alpha _{0}</math>  and <math>\,\!\alpha _{1}</math>, the variance/covariance matrix is:
In terms of <math>\,\!\sigma</math> , <math>\,\!\alpha _{0}</math>  and <math>\,\!\alpha _{1}</math>, the variance/covariance matrix is:


:<math>\,\!\begin{bmatrix}
:<math>\,\!\begin{bmatrix}
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{{Reference_Example_Heading4|ALTA}}
{{Reference_Example_Heading4|ALTA}}

Revision as of 21:20, 10 June 2014

ALTA_Reference_Examples_Banner.png

ALTA_Reference_Examples

This example validates the calculation for the Arrhenius life stress relationship with a Lognormal distribution.


Reference Case

The data set is from Example 19.5 on page 498 in book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.


Data

Device-A was tested under several different temperature settings.

Number in Group State F/S Time to State Temperature (°F) Subset ID
30 S 5000 283.15 1
1 F 1298 313.15 2
1 F 1390 313.15 2
1 F 3187 313.15 2
1 F 3241 313.15 2
1 F 3261 313.15 2
1 F 3313 313.15 2
1 F 4501 313.15 2
1 F 4568 313.15 2
1 F 4841 313.15 2
1 F 4982 313.15 2
90 S 5000 313.15 2
1 F 581 333.15 3
1 F 925 333.15 3
1 F 1432 333.15 3
1 F 1586 333.15 3
1 F 2452 333.15 3
1 F 2734 333.15 3
1 F 2772 333.15 3
1 F 4106 333.15 3
1 F 4674 333.15 3
11 S 5000 333.15 3
1 F 283 353.15 4
1 F 361 353.15 4
1 F 515 353.15 4
1 F 638 353.15 4
1 F 854 353.15 4
1 F 1024 353.15 4
1 F 1030 353.15 4
1 F 1045 353.15 4
1 F 1767 353.15 4
1 F 1777 353.15 4
1 F 1856 353.15 4
1 F 1951 353.15 4
1 F 1964 353.15 4
1 F 2884 353.15 4
1 S 5000 353.15 4


Result

The following function is used for the Ln-Mean [math]\displaystyle{ \,\!\mu {}' }[/math]:


[math]\displaystyle{ \,\!\mu {}'=\beta _{0}+\beta _{1}\times \frac{11605}{T} }[/math]


where T is the temperature; [math]\displaystyle{ \,\!\beta _{1} }[/math] is the activation energy; [math]\displaystyle{ \,\!11605 }[/math] is from reciprocal of the Boltzmann constant . This function can be written in the following way:


[math]\displaystyle{ \,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}} }[/math]


The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by [math]\displaystyle{ \,\!\alpha _{i} }[/math].

In fact, the above model also can be expressed using the traditional Arrhenius model:


[math]\displaystyle{ \,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}}=C\times e^{\frac{B}{T}} }[/math]


In the book, the following results are provided:

  • ML estimations for the model parameters are: [math]\displaystyle{ \,\!\sigma =0.98 }[/math] ,[math]\displaystyle{ \,\!\beta _{0}=-13.469 }[/math] , [math]\displaystyle{ \,\!\beta _{1}=0.6279 }[/math] (or [math]\displaystyle{ \,\!\alpha _{1}=7286.78 }[/math]).


  • The 95% confidence interval for [math]\displaystyle{ \,\!\sigma }[/math] is [0.75, 1.28], for [math]\displaystyle{ \,\!\beta _{0} }[/math] is [-19.1, -7.8] and for [math]\displaystyle{ \,\!\beta _{1} }[/math] is [0.47, 0.79].


  • The variance/covariance matrix for [math]\displaystyle{ \,\!\sigma }[/math] , [math]\displaystyle{ \,\!\beta _{0} }[/math] and [math]\displaystyle{ \,\!\beta _{1} }[/math] is


[math]\displaystyle{ \,\!\begin{bmatrix} 0.0176 & -0.195 & 0.0059\\ -0.195 & 8.336 & -0.239\\ 0.0059 & -0.239 & 0.0069 \end{bmatrix} }[/math]


In terms of [math]\displaystyle{ \,\!\sigma }[/math] , [math]\displaystyle{ \,\!\alpha _{0} }[/math] and [math]\displaystyle{ \,\!\alpha _{1} }[/math], the variance/covariance matrix is:


[math]\displaystyle{ \,\!\begin{bmatrix} 0.0176 & -0.195 & 68.4695\\ -0.195 & 8.336 & -2773.5950\\ 68.4695 & -2773.5950 & 929264.5725 \end{bmatrix} }[/math]


Results in ALTA

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