Duane Confidence Bounds Example - Revision history

Lisa Hacker at 21:23, 18 September 2023

2023-09-18T21:23:31Z

← Older revision		Revision as of 21:23, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Duane Model\|~~Reliability ~~Growth and Repairable System Analysis Reference]~~]''.		''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.

	</noinclude>		</noinclude>

David J. Groebel at 19:55, 8 August 2018

2018-08-08T19:55:13Z

← Older revision		Revision as of 19:55, 8 August 2018
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	:<math>\begin{align}		:<math>\begin{align}
	{{m}_{c}}{{(t)}_{l}}= & ~~803~~.~~6695~~ \\		{{m}_{c}}{{(t)}_{l}}= & 855.9815 \\
	{{m}_{c}}{{(t)}_{u}}= & ~~997~~.~~4658~~		{{m}_{c}}{{(t)}_{u}}= & 936.5071
	\end{align}\,\!</math>		\end{align}\,\!</math>

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	:<math>\begin{align}		:<math>\begin{align}
	{{m}_{i}}{{(t)}_{l}}= & ~~2077~~.~~9204~~ \\		{{m}_{i}}{{(t)}_{l}}= & 2213.1753 \\
	{{m}_{i}}{{(t)}_{u}}= & ~~2578~~.~~9886~~		{{m}_{i}}{{(t)}_{u}}= & 2421.3776
	\end{align}\,\!</math>		\end{align}\,\!</math>

David J. Groebel at 19:12, 8 August 2018

2018-08-08T19:12:32Z

← Older revision		Revision as of 19:12, 8 August 2018
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	:<math>\begin{align}		:<math>\begin{align}
	{{[{{\lambda }_{c}}(t)]}_{L}}= & 0.~~00100254~~ \\		{{[{{\lambda }_{c}}(t)]}_{L}}= & 0.00106780 \\
	{{[{{\lambda }_{c}}(t)]}_{U}}= & 0.~~00124429~~		{{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00116825
	\end{align}\,\!</math>		\end{align}\,\!</math>

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	:<math>\begin{align}		:<math>\begin{align}
	{{[{{\lambda }_{i}}(t)]}_{L}}= & 0.~~00038775~~ \\		{{[{{\lambda }_{i}}(t)]}_{L}}= & 0.00041299 \\
	{{[{{\lambda }_{c}}(t)]}_{U}}= & 0.~~00048125~~		{{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00045184
	\end{align}\,\!</math>		\end{align}\,\!</math>

David J. Groebel at 22:12, 2 June 2014

2014-06-02T22:12:43Z

← Older revision		Revision as of 22:12, 2 June 2014
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	:<math>\hat{\alpha}=0.6133\,\!</math>		:<math>\hat{\alpha}=0.6133\,\!</math>

	Calculate the 90% confidence bounds for:		Calculate the 90% confidence bounds for the following:

	#The parameters <math>\alpha\,\!</math> and <math>b\,\!</math>.		#The parameters <math>\alpha\,\!</math> and <math>b\,\!</math>.

David J. Groebel at 22:06, 2 June 2014

2014-06-02T22:06:16Z

← Older revision		Revision as of 22:06, 2 June 2014
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference]]''.

	</noinclude>		</noinclude>
Line 9:		Line 9:
	:<math>\hat{\alpha}=0.6133\,\!</math>		:<math>\hat{\alpha}=0.6133\,\!</math>

	~~calculate~~ the 90% confidence bounds for:		Calculate the 90% confidence bounds for:

	#The parameters <math>\alpha\,\!</math> and <math>b\,\!</math>.		#The parameters <math>\alpha\,\!</math> and <math>b\,\!</math>.
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	'''Solution'''		'''Solution'''

	1. Use the values of <math>\hat{b}\,\!</math> and <math>\hat{\alpha }\,\!</math> estimated from the least squares analysis. Then:		<ol>
			<li>Use the values of <math>\hat{b}\,\!</math> and <math>\hat{\alpha }\,\!</math> estimated from the least squares analysis. Then:

	:<math>\begin{align}		:<math>\begin{align}
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	{{b}_{U}}= & 2.1231		{{b}_{U}}= & 2.1231
	\end{align}\,\!</math>		\end{align}\,\!</math>
			</li>
	2. The cumulative failure intensity is:		<li>The cumulative failure intensity is:

	:<math>\begin{align}		:<math>\begin{align}
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	The following figures show the graphs of the cumulative and instantaneous failure intensity. Both are plotted with confidence bounds.		The following figures show the graphs of the cumulative and instantaneous failure intensity. Both are plotted with confidence bounds.

	[[Image:rga4.7.png\|center\|~~400px\|Cumulative Failure Intensity plot with 2-sided 90% confidence bounds.~~]]		[[Image:rga4.7.png\|center\|450px]]

	~~[[Image:rga4.8.png\|center\|400px\|Instantaneous Failure Intensity plot with 2-sided 90% confidence bounds.]]~~


	3. The cumulative MTBF is:		[[Image:rga4.8.png\|center\|450px]]
			</li>
			<li>The cumulative MTBF is:

	:<math>\begin{align}		:<math>\begin{align}
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	The figure below displays the cumulative MTBF.		The figure below displays the cumulative MTBF.

	[[Image:rga4.9.png\|center\|~~400px\|Cumulative MTBF plot with 2-sided 90% condfidence bounds.~~]]		[[Image:rga4.9.png\|center\|450px]]

	The next figure displays the instantaneous MTBF. Both are plotted with confidence bounds.		The next figure displays the instantaneous MTBF. Both are plotted with confidence bounds.

	[[Image:rga4.10.png\|center\|~~400px\|Instantaneous MTBF plot with 2-sided 90% confidence bounds.~~]]		[[Image:rga4.10.png\|center\|450px]]
			</li>
			</ol>

David J. Groebel at 22:51, 30 January 2014

2014-01-30T22:51:09Z

← Older revision		Revision as of 22:51, 30 January 2014
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	</noinclude>		</noinclude>

	Using the values of <math>\~~widehat~~{b}\,\!</math> and <math>\~~widehat~~{\alpha }\,\!</math> estimated from the least squares analysis in <noinclude>[[Duane Linear Regression Examples\|		Using the values of <math>\hat{b}\,\!</math> and <math>\hat{\alpha }\,\!</math> estimated from the least squares analysis in <noinclude>[[Duane Linear Regression Examples\|
	Least Squares Example 2]]</noinclude><includeonly>Least Squares Example 2</includeonly>:		Least Squares Example 2]]</noinclude><includeonly>Least Squares Example 2</includeonly>:
	:<math>\~~widehat~~{b}=1.9453\,\!</math>		:<math>\hat{b}=1.9453\,\!</math>
	:<math>\~~widehat~~{\alpha}=0.6133\,\!</math>		:<math>\hat{\alpha}=0.6133\,\!</math>

	calculate the 90% confidence bounds for:		calculate the 90% confidence bounds for:
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	'''Solution'''		'''Solution'''

	1. Use the values of <math>\~~widehat~~{b}\,\!</math> and <math>\~~widehat~~{\alpha }\,\!</math> estimated from the least squares analysis. Then:		1. Use the values of <math>\hat{b}\,\!</math> and <math>\hat{\alpha }\,\!</math> estimated from the least squares analysis. Then:

	:<math>\begin{align}		:<math>\begin{align}

David J. Groebel at 22:50, 30 January 2014

2014-01-30T22:50:40Z

Show changes

Kate Racaza at 20:41, 29 January 2014

2014-01-29T20:41:15Z

← Older revision		Revision as of 20:41, 29 January 2014
Line 5:		Line 5:

	Using the values of <math>\widehat{b}\,\!</math> and <math>\widehat{\alpha }\,\!</math> estimated from the least squares analysis in <noinclude>[[Duane Linear Regression Examples\|		Using the values of <math>\widehat{b}\,\!</math> and <math>\widehat{\alpha }\,\!</math> estimated from the least squares analysis in <noinclude>[[Duane Linear Regression Examples\|
	Least ~~Square~~ Example 2]]</noinclude><includeonly>Least ~~Square~~ Example 2</includeonly>:		Least Squares Example 2]]</noinclude><includeonly>Least Squares Example 2</includeonly>:
	::<math>\widehat{b}=1.9453\,\!</math>		::<math>\widehat{b}=1.9453\,\!</math>
	::<math>\widehat{\alpha}=0.6133\,\!</math>		::<math>\widehat{\alpha}=0.6133\,\!</math>

Kate Racaza at 21:13, 21 November 2013

2013-11-21T21:13:49Z

← Older revision		Revision as of 21:13, 21 November 2013
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	#The cumulative and instantaneous failure intensity.		#The cumulative and instantaneous failure intensity.
	#The cumulative and instantaneous MTBF.		#The cumulative and instantaneous MTBF.


	'''Solution'''		'''Solution'''

Kate Racaza: Created page with '{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. Using the valu…'

2013-11-21T21:12:30Z

Created page with '<noinclude>{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. </noinclude> Using the valu…'

New page

<noinclude>{{Banner RGA Examples}}
''This example appears in the [[Duane Model|Reliability Growth and Repairable System Analysis Reference book]]''.

</noinclude>

Using the values of <math>\widehat{b}\,\!</math> and <math>\widehat{\alpha }\,\!</math> estimated from the least squares analysis in <noinclude>[[Duane Linear Regression Examples|
Least Square Example 2]]</noinclude><includeonly>Least Square Example 2</includeonly>:
::<math>\widehat{b}=1.9453\,\!</math>
::<math>\widehat{\alpha}=0.6133\,\!</math>

calculate the 90% confidence bounds for:

#The parameters <math>\alpha\,\!</math> and <math>b\,\!</math>.
#The cumulative and instantaneous failure intensity.
#The cumulative and instantaneous MTBF.

'''Solution'''

1. Use the values of <math>\widehat{b}\,\!</math> and <math>\widehat{\alpha }\,\!</math> estimated from the least squares analysis. Then:

::<math>\begin{align}
{{S}_{xx}}&=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{(\ln {{t}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{t}_{i}}) \right)}^{2}} \\
& = 1400.9084-1301.4545 \\
& = 99.4539
\end{align}\,\!</math>

::<math>\begin{align}
SE(\hat{\alpha })= & \frac{\sigma }{\sqrt{{{S}_{xx}}}} \\
= & \frac{0.08428}{9.9727} \\
= & 0.008452
\end{align}\,\!</math>

::<math>\begin{align}
SE(\ln \hat{b})= & \sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{(\ln {{T}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}} \\
= & 0.065960
\end{align}\,\!</math>

Thus, the 90% confidence bounds on parameter <math>\alpha \,\!</math> are:

::<math>C{{B}_{\alpha }}=\hat{\alpha }\pm {{t}_{n-2,\alpha /2}}SE(\hat{\alpha })\,\!</math>

::<math>\begin{align}
{{\alpha }_{L}}= & 0.602050 \\
{{\alpha }_{U}}= & 0.624417
\end{align}\,\!</math>

And 90% confidence bounds on parameter <math>b\,\!</math> are:

::<math>C{{B}_{b}}=\hat{b}{{e}^{\pm {{t}_{n-2,\alpha /2}}SE\left[ \ln (\hat{b}) \right]}}\,\!</math>

::<math>\begin{align}
{{b}_{L}}= & 1.7831 \\
{{b}_{U}}= & 2.1231
\end{align}\,\!</math>

2. The cumulative failure intensity is:

::<math>\begin{align}
{{\lambda }_{c}}= & \frac{1}{1.9453}\cdot {{22000}^{-0.6133}} \\
= & 0.00111689
\end{align}\,\!</math>

And the instantaneous failure intensity is equal to:

::<math>\begin{align}
{{\lambda }_{i}}= & \frac{1}{1.9453}\cdot (1-0.6133)\cdot {{22000}^{-0.6133}} \\
= & 0.00043198
\end{align}\,\!</math>

So, at the 90% confidence level and for <math>T=22,000\,\!</math> hours, the confidence bounds on cumulative failure intensity are:

::<math>\begin{align}
{{[{{\lambda }_{c}}(t)]}_{L}}= & 0.00100254 \\
{{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00124429
\end{align}\,\!</math>

For the instantaneous failure intensity:

::<math>\begin{align}
{{[{{\lambda }_{i}}(t)]}_{L}}= & 0.00038775 \\
{{[{{\lambda }_{c}}(t)]}_{U}}= & 0.00048125
\end{align}\,\!</math>

The following figures show the graphs of the cumulative and instantaneous failure intensity. Both are plotted with confidence bounds.

[[Image:rga4.7.png|center|400px|Cumulative Failure Intensity plot with 2-sided 90% confidence bounds.]]

[[Image:rga4.8.png|center|400px|Instantaneous Failure Intensity plot with 2-sided 90% confidence bounds.]]

3. The cumulative MTBF is:

::<math>\begin{align}
{{m}_{c}}(T)= & 1.9453\cdot {{22000}^{0.6133}} \\
= & 895.3395
\end{align}\,\!</math>

And the instantaneous MTBF is:

::<math>\begin{align}
{{m}_{i}}(T)= & \frac{1.9453}{1-0.6133}\cdot {{22000}^{0.6133}} \\
= & 2314.9369
\end{align}\,\!</math>

So, at 90% confidence level and for <math>T=22,000\,\!</math> hours, the confidence bounds on the cumulative MTBF are:

::<math>\begin{align}
{{m}_{c}}{{(t)}_{l}}= & 803.6695 \\
{{m}_{c}}{{(t)}_{u}}= & 997.4658
\end{align}\,\!</math>

The confidence bounds for the instantaneous MTBF are:

::<math>\begin{align}
{{m}_{i}}{{(t)}_{l}}= & 2077.9204 \\
{{m}_{i}}{{(t)}_{u}}= & 2578.9886
\end{align}\,\!</math>

The figure below displays the cumulative MTBF.

[[Image:rga4.9.png|center|400px|Cumulative MTBF plot with 2-sided 90% condfidence bounds.]]

The next figure displays the instantaneous MTBF. Both are plotted with confidence bounds.

[[Image:rga4.10.png|center|400px|Instantaneous MTBF plot with 2-sided 90% confidence bounds.]]