Inverse Power Law (IPL)-Lognormal Model: Difference between revisions

Revision as of 18:49, 11 June 2014

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ALTA_Reference_Examples

This example compares the results for the IPL life stress relationship with a lognormal distribution.

Reference Case

The data set is from Example 19.10 on page 504 in book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.

Data

A Mylar-Polyurethane Insulating structure was tested under several different voltage settings. The test data is given in the table shown next.

Time Failed (Hr)	style="width:50%"
15	style="width:50%"
16	219
36	219
50	219
55	219
95	219
122	219
129	219
625	219
700	219
49	157.1
99	157.1
154.5	157.1
180	157.1
291	157.1
447	157.1
510	157.1
600	157.1
1656	157.1
1721	157.1
188	122.4
297	122.4
405	122.4
744	122.4
1218	122.4
1340	122.4
1715	122.4
3382	122.4
606	100.3
1012	100.3
2520	100.3
2610	100.3
3988	100.3
4100	100.3
5025	100.3
6842	100.3

Result

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

[math]\displaystyle{ \,\!\mu {}'=\beta _{0}+\beta _{1}\times log\left ( V \right ) }[/math]

where V is the voltage and its natural log transform is used in the above life stress relation.

This function can be written in the following way:

[math]\displaystyle{ \,\!e^{\mu {}'}=e^{\alpha _{0}+\alpha _{1}log\left ( V \right )} }[/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 IPL (inverse power law) model:

[math]\displaystyle{ \,\!e^{\mu {}'}=\frac{1}{K\cdot V^{n}} }[/math]

where [math]\displaystyle{ \,\!K=e^{-\alpha _{0}} }[/math] and [math]\displaystyle{ \,\!n=-\alpha _{1} }[/math].

In the book, the following results are provided:

ML estimations for the model parameters are: [math]\displaystyle{ \,\!\sigma =1.05 }[/math] , [math]\displaystyle{ \,\!\beta _{0}=27.5 }[/math] and [math]\displaystyle{ \,\!\beta _{1}=-4.29 }[/math].

The standard deviation of each parameter are: [math]\displaystyle{ \,\!std\left ( \sigma \right )=0.12 }[/math] , [math]\displaystyle{ \,\!std\left ( \beta _{0} \right )=3.0 }[/math] and [math]\displaystyle{ \,\!std\left ( \beta _{1} \right )=0.6 }[/math].

Therefore, their variances are: [math]\displaystyle{ \,\!Var\left ( \sigma \right )=0.0144 }[/math] , [math]\displaystyle{ \,\!Var\left ( \beta _{0} \right )=9 }[/math] and [math]\displaystyle{ \,\!Var\left ( \beta _{1} \right )=0.36 }[/math].

The log-likelihood value is -271.4.

The 95% two-sided confidence intervals are: for [math]\displaystyle{ \,\!\sigma }[/math] , it is [0.83, 1.32]; for [math]\displaystyle{ \,\!\beta _{0} }[/math] , it is [21.6, 33.4]; and for [math]\displaystyle{ \,\!\beta _{1} }[/math] , it is [-5.46, -3.11].

Results in ALTA