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 [[Repairable_Systems_Analysis|Reliability Growth and Repairable System Analysis Reference book]]''.
''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}
:<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}
:<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>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}
::<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}
:<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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\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|Conditional Reliability vs. Time plot with Fisher Matrix confidence bounds.]]
[[Image:rga13.3.png|center|450px]]
 


[[Image:rga13.4.png|center|450px|Conditional Reliability vs. Time plot with Crow confidence bounds.]]
[[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].

Rga13.3.png
Rga13.4.png