Appendix C: Benchmark Examples: Difference between revisions

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==Computed Results for Example 3==
==Computed Results for Example 3==
Use the inverse power law model and Weibull as the underlying life distribution.
Use the inverse power law model and Weibull as the underlying life distribution.


• ALTA computed parameters are:
• ALTA computed parameters are:
<br>
<br>
::<math>\begin{matrix}
::<math>\begin{align}
   \widehat{\beta }=1.243375  \\
   \widehat{\beta }=\ & 1.243375  \\
   \widehat{K}=0.4350548  \\
   \widehat{K}=\ & 0.4350548  \\
   \widehat{n}=13.8529  \\
   \widehat{n}=\ & 13.8529  \\
\end{matrix}</math>
\end{align}</math>
 
   
   
• The 95% 2-sided confidence bounds on the parameters:
• The 95% 2-sided confidence bounds on the parameters:
::<math>\begin{align}
& \left\{ 0.9745811,1.586303 \right\}\text{ for }\widehat{\beta } \\
& \left\{ 0.330007,0.573542 \right\}\text{ for }\widehat{K } \\
& \left\{ 11.43510,16.27079 \right\}\text{ for }\widehat{n }
\end{align}</math>
   
   
• Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:
• Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:

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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{align} \widehat{\beta }=\ & 0.777 \\ \widehat{K}=\ & 6.8742\cdot {{10}^{-29}} \\ \widehat{n}=\ & 17.72958 \\ \end{align} }[/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{align} \widehat{\beta }=\ & 0.7765, \\ \widehat{K}=\ & 6.8741\cdot {{10}^{-29}} \\ \widehat{n}=\ & 17.7296 \\ \end{align} }[/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{align} \widehat{\beta }=\ & 1.243396 \\ \widehat{K}=\ & 0.4350735 \\ \widehat{n}=\ & 13.8528 \\ \end{align} }[/math]


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


[math]\displaystyle{ \begin{align} & \left\{ 0.9746493,1.586245 \right\}\text{ for }\widehat{\beta } \\ & \left\{ 0.332906,0.568596 \right\}\text{ for }\widehat{K } \\ & \left\{ 11.43569,16.26991 \right\}\text{ for }\widehat{n } \end{align} }[/math]


• 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{align} \widehat{\beta }=\ & 1.243375 \\ \widehat{K}=\ & 0.4350548 \\ \widehat{n}=\ & 13.8529 \\ \end{align} }[/math]


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

[math]\displaystyle{ \begin{align} & \left\{ 0.9745811,1.586303 \right\}\text{ for }\widehat{\beta } \\ & \left\{ 0.330007,0.573542 \right\}\text{ for }\widehat{K } \\ & \left\{ 11.43510,16.27079 \right\}\text{ for }\widehat{n } \end{align} }[/math]


• 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: