Power Law Model Confidence Bounds Example: Difference between revisions
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<noinclude>{{Banner RGA Examples}}{{Navigation box}} | <noinclude>{{Banner RGA Examples}}{{Navigation box}} | ||
''These examples appear in the [ | ''These examples appear in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''. | ||
</noinclude> | </noinclude> | ||
Using the data from the <noinclude>[[Power Law Model Parameter Estimation Example]]</noinclude><includeonly>power law model example given above</includeonly>, calculate the mission reliability at <math>t=2000\,\!</math> hours and mission time <math>d=40\,\!</math> hours along with the confidence bounds at the 90% confidence level. | Using the data from the <noinclude>[[Power Law Model Parameter Estimation Example]]</noinclude><includeonly>power law model example given above</includeonly>, calculate the mission reliability at <math>t=2000\,\!</math> hours and mission time <math>d=40\,\!</math> hours along with the confidence bounds at the 90% confidence level. | ||
'''Solution''' | '''Solution''' | ||
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The maximum likelihood estimates of <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math> from the example are: | The maximum likelihood estimates of <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math> from the example are: | ||
:<math>\begin{align} | |||
\widehat{\beta }= & 0.45300 \\ | \widehat{\beta }= & 0.45300 \\ | ||
\widehat{\lambda }= & 0.36224 | \widehat{\lambda }= & 0.36224 | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
The mission reliability at <math>t=2000\,\!</math> for mission time <math>d=40\,\!</math> is: | The mission reliability at <math>t=2000\,\!</math> for mission time <math>d=40\,\!</math> is: | ||
:<math>\begin{align} | |||
\widehat{R}(t)= & {{e}^{-\left[ \lambda {{\left( t+d \right)}^{\beta }}-\lambda {{t}^{\beta }} \right]}} \\ | \widehat{R}(t)= & {{e}^{-\left[ \lambda {{\left( t+d \right)}^{\beta }}-\lambda {{t}^{\beta }} \right]}} \\ | ||
= & 0.90292 | = & 0.90292 | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
At the 90% confidence level and <math>T=2000\,\!</math> hours, the Fisher matrix confidence bounds for the mission reliability for mission time <math>d=40\,\!</math> are given by: | At the 90% confidence level and <math>T=2000\,\!</math> hours, the Fisher matrix confidence bounds for the mission reliability for mission time <math>d=40\,\!</math> are given by: | ||
:<math>CB=\frac{\widehat{R}(t)}{\widehat{R}(t)+(1-\widehat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{R}(t))}/\left[ \widehat{R}(t)(1-\widehat{R}(t)) \right]}}}\,\!</math> | |||
:<math>\begin{align} | |||
{{[\widehat{R}(t)]}_{L}}= & 0.83711 \\ | {{[\widehat{R}(t)]}_{L}}= & 0.83711 \\ | ||
{{[\widehat{R}(t)]}_{U}}= & 0.94392 | {{[\widehat{R}(t)]}_{U}}= & 0.94392 | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
The Crow confidence bounds for the mission reliability are: | The Crow confidence bounds for the mission reliability are: | ||
:<math>\begin{align} | |||
{{[\widehat{R}(t)]}_{L}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}{{{\Pi }_{1}}}}} \\ | {{[\widehat{R}(t)]}_{L}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}{{{\Pi }_{1}}}}} \\ | ||
= & {{[0.90292]}^{\tfrac{1}{0.71440}}} \\ | = & {{[0.90292]}^{\tfrac{1}{0.71440}}} \\ | ||
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= & 0.93836 | = & 0.93836 | ||
\end{align}\,\!</math> | \end{align}\,\!</math> | ||
The next two figures show the Fisher matrix and Crow confidence bounds on mission reliability for mission time <math>d=40\,\!</math>. | The next two figures show the Fisher matrix and Crow confidence bounds on mission reliability for mission time <math>d=40\,\!</math>. | ||
[[Image:rga13.3.png|center|450px | [[Image:rga13.3.png|center|450px]] | ||
[[Image:rga13.4.png|center|450px | [[Image:rga13.4.png|center|450px]] | ||
Latest revision as of 21:23, 18 September 2023
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These examples appear in the Reliability growth reference.
Using the data from the Power Law Model Parameter Estimation Example, calculate the mission reliability at [math]\displaystyle{ t=2000\,\! }[/math] hours and mission time [math]\displaystyle{ d=40\,\! }[/math] hours along with the confidence bounds at the 90% confidence level.
Solution
The maximum likelihood estimates of [math]\displaystyle{ \widehat{\lambda }\,\! }[/math] and [math]\displaystyle{ \widehat{\beta }\,\! }[/math] from the example are:
- [math]\displaystyle{ \begin{align} \widehat{\beta }= & 0.45300 \\ \widehat{\lambda }= & 0.36224 \end{align}\,\! }[/math]
The mission reliability at [math]\displaystyle{ t=2000\,\! }[/math] for mission time [math]\displaystyle{ d=40\,\! }[/math] is:
- [math]\displaystyle{ \begin{align} \widehat{R}(t)= & {{e}^{-\left[ \lambda {{\left( t+d \right)}^{\beta }}-\lambda {{t}^{\beta }} \right]}} \\ = & 0.90292 \end{align}\,\! }[/math]
At the 90% confidence level and [math]\displaystyle{ T=2000\,\! }[/math] hours, the Fisher matrix confidence bounds for the mission reliability for mission time [math]\displaystyle{ d=40\,\! }[/math] are given by:
- [math]\displaystyle{ CB=\frac{\widehat{R}(t)}{\widehat{R}(t)+(1-\widehat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{R}(t))}/\left[ \widehat{R}(t)(1-\widehat{R}(t)) \right]}}}\,\! }[/math]
- [math]\displaystyle{ \begin{align} {{[\widehat{R}(t)]}_{L}}= & 0.83711 \\ {{[\widehat{R}(t)]}_{U}}= & 0.94392 \end{align}\,\! }[/math]
The Crow confidence bounds for the mission reliability are:
- [math]\displaystyle{ \begin{align} {{[\widehat{R}(t)]}_{L}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}{{{\Pi }_{1}}}}} \\ = & {{[0.90292]}^{\tfrac{1}{0.71440}}} \\ = & 0.86680 \\ {{[\widehat{R}(t)]}_{U}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}{{{\Pi }_{2}}}}} \\ = & {{[0.90292]}^{\tfrac{1}{1.6051}}} \\ = & 0.93836 \end{align}\,\! }[/math]
The next two figures show the Fisher matrix and Crow confidence bounds on mission reliability for mission time [math]\displaystyle{ d=40\,\! }[/math].
