Appendix C: Benchmark Examples
Reference Appendix C: Benchmark 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{matrix}
{{\widehat{\sigma }}_{{{T}'}}}=0.59673 \\
\widehat{B}=9920.195 \\
\widehat{C}=9.69517\cdot {{10}^{-7}} \\
\end{matrix} }[/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{matrix}
{{\widehat{\sigma }}_{{{T}'}}}=0.59678 \\
\widehat{B}=9924.804 \\
\widehat{C}=9.58978\cdot {{10}^{-7}} \\
\end{matrix} }[/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: