Appendix C: Benchmark Examples: Difference between revisions

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==Computed Results for Example 1==
==Computed Results for Example 1==
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This same data set can be entered into ALTA by selecting the data sheet for grouped times-to-failure data with suspensions and using the Arrhenius model, the lognormal distribution, and MLE.
This same data set can be entered into ALTA by selecting the data sheet for grouped times-to-failure data with suspensions and using the Arrhenius model, the lognormal distribution, and MLE. ALTA  computed parameters for maximum likelihood are:
ALTA  computed parameters for maximum likelihood are:




::<math>\begin{matrix}
::<math>\begin{align}
   {{\widehat{\sigma }}_{{{T}'}}}=0.59678  \\
   {{\widehat{\sigma }}_{{{T}'}}}=\ & 0.59678  \\
   \widehat{B}=9924.804  \\
   \widehat{B}=\ & 9924.804  \\
   \widehat{C}=9.58978\cdot {{10}^{-7}}  \\
   \widehat{C}=\ & 9.58978\cdot {{10}^{-7}}  \\
\end{matrix}</math>
\end{align}</math>


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Revision as of 21:14, 23 February 2012

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Chapter C: Appendix C: Benchmark Examples


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Chapter C  
Appendix C: Benchmark Examples  

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Available Software:
ALTA

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More Resources:
ALTA Examples

In this section, five published examples are presented for comparison purposes. ReliaSoft's R&D validated the ALTA software with hundreds of data sets and methods. ALTA also cross-validates each provided solution by independently re-evaluating the second partial derivatives based on the estimated parameters each time a calculation is performed. These partials will be equal to zero when a solution is reached. Double precision is used throughout ALTA.

Example 1


From Wayne Nelson [28, p. 135].

Published Results for Example 1

• Published Results:


[math]\displaystyle{ \begin{align} {{\widehat{\sigma }}_{{{T}'}}}=\ & 0.59673 \\ \widehat{B}=\ & 9920.195 \\ \widehat{C}=\ & 9.69517\cdot {{10}^{-7}} \\ \end{align} }[/math]


Computed Results for Example 1


This same data set can be entered into ALTA by selecting the data sheet for grouped times-to-failure data with suspensions and using the Arrhenius model, the lognormal distribution, and MLE. ALTA computed parameters for maximum likelihood are:


[math]\displaystyle{ \begin{align} {{\widehat{\sigma }}_{{{T}'}}}=\ & 0.59678 \\ \widehat{B}=\ & 9924.804 \\ \widehat{C}=\ & 9.58978\cdot {{10}^{-7}} \\ \end{align} }[/math]


Example 2


From Wayne Nelson [28, p. 453], time to breakdown of a transformer oil, tested at 26kV, 28kV, 30kV, 32kV, 34kV, 36kV and 38kV.

Published Results for Example 2


• Published Results:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=0.777 \\ \widehat{K}=6.8742\cdot {{10}^{-29}} \\ \widehat{n}=17.72958 \\ \end{matrix} }[/math]

• Published 95% confidence limits on [math]\displaystyle{ \beta }[/math] :

[math]\displaystyle{ \begin{matrix} \left\{ 0.653,0.923 \right\} \\ \end{matrix} }[/math]


Computed Results for Example 2


Use the inverse power law model and Weibull as the underlying life distribution.
• ALTA computed parameters are:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=0.7765, \\ \widehat{K}=6.8741\cdot {{10}^{-29}} \\ \widehat{n}=17.7296 \\ \end{matrix} }[/math]

• ALTA computed 95% confidence limits on the parameters:

[math]\displaystyle{ \left\{ 0.6535,0.9228 \right\}\text{ for }\widehat{\beta } }[/math]


Example 3


From Wayne Nelson [28, p. 157], forty bearings were tested to failure at four different test loads. The data were analyzed using the inverse power law Weibull model.

Published Results for Example 3

Nelson's [28, p. 306] IPL-Weibull parameter estimates:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=1.243396 \\ \widehat{K}=0.4350735 \\ \widehat{n}=13.8528 \\ \end{matrix} }[/math]


• The 95% 2-sided confidence bounds on the parameters:

• Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:

Percentile Life Estimate 95% Lower 95% Upper
1% 0.3913096 0.1251383 1.223632
10% 2.589731 1.230454 5.450596
90% 30.94404 19.41020 49.33149
99% 54.03563 33.02691 88.40821

Computed Results for Example 3

Use the inverse power law model and Weibull as the underlying life distribution.

• ALTA computed parameters are:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=1.243375 \\ \widehat{K}=0.4350548 \\ \widehat{n}=13.8529 \\ \end{matrix} }[/math]

• The 95% 2-sided confidence bounds on the parameters:

• Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:

Percentile Life Estimate 95% Lower 95% Upper
1% 0.3913095 0.1251097 1.223911
10% 2.589814 1.230384 5.451588
90% 30.94632 19.40876 49.34240
99% 54.04012 33.02411 88.43039


Example 4


From Meeker and Escobar [26, p. 504], Mylar-Polyurethane Insulating Structure data using the inverse power law lognormal model.

Published Results for Example 4


• Published Results:

[math]\displaystyle{ \begin{matrix} {{\widehat{\sigma }}_{{{T}'}}}=1.05, \\ \widehat{K}=1.14\cdot {{10}^{-12}}, \\ \widehat{n}=4.28. \\ \end{matrix} }[/math]

• The 95% 2-sided confidence bounds on the parameters:

Computed Results for Example 4

Use the inverse power law lognormal.

• ALTA computed parameters are:

[math]\displaystyle{ \begin{matrix} {{\widehat{\sigma }}_{{{T}'}}}=1.04979 \\ \widehat{K}=1.15\cdot {{10}^{-12}} \\ \widehat{n}=4.289 \\ \end{matrix} }[/math]

• ALTA computed 95% confidence limits on the parameters:

Example 5

From Meeker and Escobar [26, p. 515], Tantalum Capacitor data using the combination (Temperature-NonThermal) Weibull model.

Published Results for Example 5

• Published Results:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=0.4292 \\ \widehat{B}=3829.468 \\ \widehat{C}=4.513\cdot {{10}^{36}} \\ \widehat{n}=20.1 \\ \end{matrix} }[/math]

• The 95% 2-sided confidence bounds on the parameters:

Computed Results for Example 5

Use the Temperature-NonThermal model and Weibull as the underlying life distribution.
• ALTA computed parameters are:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=0.4287 \\ \widehat{B}=3780.298 \\ \widehat{C}=4.772\cdot {{10}^{36}} \\ \widehat{n}=20.09 \\ \end{matrix} }[/math]

• ALTA computed 95% confidence limits on the parameters: