Arrhenius-Lognormal Model: Difference between revisions

Revision as of 18:52, 10 June 2014

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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 & -.239& 0.0069 \end{bmatrix} }[/math]

In terms of [math]\displaystyle{ \,\! }[/math] , [math]\displaystyle{ \,\! }[/math] and [math]\displaystyle{ \,\! }[/math] , the variance/covariance matrix is

[math]\displaystyle{ \,\! }[/math]

Results in ALTA

@@ Line 114: / Line 114: @@
 *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  ,  , and   is
+*The variance/covariance matrix for <math>\,\!\sigma</math> , <math>\,\!\beta _{0}</math> and <math>\,\!\beta _{1}</math> is
 ::<math>\,\!\begin{bmatrix}

Arrhenius-Lognormal Model: Difference between revisions

Revision as of 18:52, 10 June 2014

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