Power Law Model Confidence Bounds Example - Revision history

Lisa Hacker at 21:23, 18 September 2023

2023-09-18T21:23:06Z

← Older revision		Revision as of 21:23, 18 September 2023
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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]~~]''.		''These examples appear in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

David J. Groebel at 20:02, 2 June 2014

2014-06-02T20:02:57Z

← Older revision		Revision as of 20:02, 2 June 2014
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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 [[Repairable_Systems_Analysis\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

David J. Groebel at 20:02, 2 June 2014

2014-06-02T20:02:38Z

← Older revision		Revision as of 20:02, 2 June 2014
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	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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	= & 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~~\|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]]

Chris Kahn at 21:36, 23 April 2014

2014-04-23T21:36:27Z

← Older revision		Revision as of 21:36, 23 April 2014
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	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>
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	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\|Conditional Reliability vs. Time plot with Fisher matrix confidence bounds.]]


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

Kate Racaza: Created page with '{{Banner RGA Examples}}{{Navigation box}} ''These examples appear in the [[Repairable_Systems_Analysis|Reliability Growth and Repairable System Analysis Reference book…'

2014-01-29T18:59:56Z

Created page with '<noinclude>{{Banner RGA Examples}}{{Navigation box}} ''These examples appear in the [[Repairable_Systems_Analysis|Reliability Growth and Repairable System Analysis Reference book…'

New page

<noinclude>{{Banner RGA Examples}}{{Navigation box}}
''These examples appear in the [[Repairable_Systems_Analysis|Reliability Growth and Repairable System Analysis Reference book]]''.
</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.

'''Solution'''

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{\lambda }= & 0.36224
\end{align}\,\!</math>

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]}} \\
= & 0.90292
\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:

::<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)]}_{U}}= & 0.94392
\end{align}\,\!</math>

The Crow confidence bounds for the mission reliability are:

::<math>\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>d=40\,\!</math>.

[[Image:rga13.3.png|center|450px|Conditional Reliability vs. Time plot with Fisher Matrix confidence bounds.]]

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