Arrhenius-Lognormal Model: Difference between revisions
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{{Reference Example| | {{Reference Example|{{Banner ALTA Reference Examples}}}} | ||
This example validates the | This example validates the results for the Arrhenius life stress relationship with a Lognormal distribution in ALTA standard folios. | ||
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{{Reference_Example_Heading2}} | {{Reference_Example_Heading2}} | ||
Device | Device A was tested under several different temperature settings. The following table shows the data. | ||
{| {{table}} | {| {{table}} | ||
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| 1||S||5000||353.15||4 | | 1||S||5000||353.15||4 | ||
|} | |} | ||
{{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 the 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} | |||
0.0176 & -0.195 & 68.4695\\ | 0.0176 & -0.195 & 68.4695\\ | ||
-0.195 & 8.336 & -2773.5950\\ | -0.195 & 8.336 & -2773.5950\\ | ||
68.4695 & -2773.5950 & 929264.5725 | 68.4695 & -2773.5950 & 929264.5725 | ||
\end{bmatrix}</math> | \end{bmatrix}</math> | ||
*The log-likelihood value is -321.7. | |||
{{Reference_Example_Heading4|ALTA}} | |||
*ML estimations for the model parameters are: | |||
[[image:Arrhenius Lognormal_Analysis Summary.png|center]] | |||
*The 95% confidence intervals are: | |||
[[image:Arrhenius Lognormal_Parameter Bounds.png|center]] | |||
The variance/covariance matrix for <math>\,\!\sigma</math> , <math>\,\!\alpha _{0}</math> and <math>\,\!\alpha _{1}</math> is: | |||
[[image:Arrhenius Lognormal_Var_Cov Results.png|center]] | |||
*The log-likelihood value is -321.7. | |||
It can be seen that all the results in ALTA are very close to the results in the book. | |||
Latest revision as of 18:20, 28 September 2015
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