Duane Linear Regression Examples - Revision history

Lisa Hacker at 21:30, 18 September 2023

2023-09-18T21:30:54Z

← Older revision		Revision as of 21:30, 18 September 2023
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	<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->		<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->
	<noinclude>{{Banner RGA Examples}}{{Navigation box}}		<noinclude>{{Banner RGA Examples}}{{Navigation box}}
	''These examples appear in the [~~[Duane Model\|~~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]''.

	The following examples demonstrate how to estimate the parameters of the Duane model using a mathematical approach.		The following examples demonstrate how to estimate the parameters of the Duane model using a mathematical approach.

David J. Groebel at 21:54, 3 June 2014

2014-06-03T21:54:08Z

← Older revision		Revision as of 21:54, 3 June 2014
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	The following examples demonstrate how to estimate the parameters of the Duane model using a mathematical approach.		The following examples demonstrate how to estimate the parameters of the Duane model using a mathematical approach.
	==~~Least Squares~~ Example 1==		==Example 1==
	</noinclude>		</noinclude>
	<includeonly>====~~Least Squares~~ Example 1====		<includeonly>====Example 1====

	Using the same data set from the graphical approach example, estimate the parameters of the MTBF model using least squares.		Using the same data set from the graphical approach example, estimate the parameters of the MTBF model using least squares.
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	<noinclude>==~~Least Squares~~ Example 2==</noinclude>		<noinclude>==Example 2==</noinclude>
	<includeonly>====~~Least Squares~~ Example 2====</includeonly>		<includeonly>====Example 2====</includeonly>
	For the data given in columns 1 and 2 of the following table, estimate the Duane parameters using least squares.		For the data given in columns 1 and 2 of the following table, estimate the Duane parameters using least squares.

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	\end{align}\,\!</math>		\end{align}\,\!</math>

	[[Image:rga4.6.png\|center\|~~400px\|Duane plot for Example 3.~~]]		[[Image:rga4.6.png\|center\|450px]]

	<noinclude>==~~Least Squares~~ Example 3==</noinclude>		<noinclude>==Example 3==</noinclude>
	<includeonly>====~~Least Squares~~ Example 3====</includeonly>		<includeonly>====Example 3====</includeonly>

	For the data given in the following table, estimate the Duane parameters using least squares.		For the data given in the following table, estimate the Duane parameters using least squares.

David J. Groebel at 22:03, 2 June 2014

2014-06-02T22:03:37Z

← Older revision		Revision as of 22:03, 2 June 2014
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	<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->		<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->
	<noinclude>{{Banner RGA Examples}}{{Navigation box}}		<noinclude>{{Banner RGA Examples}}{{Navigation box}}
	''These examples appear in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''These examples appear in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference]]''.

	The following examples demonstrate how to estimate the parameters of the Duane model using a mathematical approach.		The following examples demonstrate how to estimate the parameters of the Duane model using a mathematical approach.
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	Do the following:		Do the following:

	1) Plot the cumulative MTBF growth curve.		#Plot the cumulative MTBF growth curve.
			#Write the equation of this growth curve.
	2) Write the equation of this growth curve.		#Write the equation of the instantaneous MTBF growth model.
			#Plot the instantaneous MTBF growth curve.
	3) Write the equation of the instantaneous MTBF growth model.

	4) Plot the instantaneous MTBF growth curve.

	</noinclude>'''Solution'''		</noinclude>'''Solution'''
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	\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left[ \ln ({{T}_{i}}) \right]}^{2}}&= & 168.99		\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left[ \ln ({{T}_{i}}) \right]}^{2}}&= & 168.99
	\end{align}\,\!</math>		\end{align}\,\!</math>


	Obtain the value of <math>\hat{\alpha}\,\!</math> from the least squares analysis, or:		Obtain the value of <math>\hat{\alpha}\,\!</math> from the least squares analysis, or:

Chris Kahn at 16:20, 14 May 2014

2014-05-14T16:20:44Z

← Older revision		Revision as of 16:20, 14 May 2014
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	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|+'''Failure ~~times data~~'''		\|+'''Failure Times Data'''
	\|-		\|-
	!(1)Failure Number		!(1)Failure Number
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	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|+'''Multiple ~~systems~~ (~~known operating times~~) ~~data~~}'''		\|+'''Multiple Systems (Known Operating Times) Data}'''
	!Run Number		!Run Number
	!Failed Unit		!Failed Unit

Kate Racaza at 16:48, 5 February 2014

2014-02-05T16:48:59Z

← Older revision		Revision as of 16:48, 5 February 2014
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	<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->		<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}{{Navigation box}}
	''These examples appear in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference book]]''.		''These examples appear in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference book]]''.

David J. Groebel at 22:54, 30 January 2014

2014-01-30T22:54:46Z

Show changes

Kate Racaza at 20:42, 29 January 2014

2014-01-29T20:42:04Z

← Older revision		Revision as of 20:42, 29 January 2014
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	<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->		<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''~~This example appears~~ in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference book]]''.		''These examples appear in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference book]]''.

	The following examples demonstrate how to estimate the parameters of the Duane model using a mathematical approach.		The following examples demonstrate how to estimate the parameters of the Duane model using a mathematical approach.

Kate Racaza at 21:19, 21 November 2013

2013-11-21T21:19:23Z

← Older revision		Revision as of 21:19, 21 November 2013
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	<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET IN THE ~~PAGE~~: Duane Graphical Method Example. -->		<!-- EVERYTHING ON THIS PAGE, EXCEPT FOR THE DATA SET IN THE FIRST EXAMPLE, IS TRANSCLUDED IN PAGE: Duane Model Examples. IF YOU EDIT THE DATA SET IN EXAMPLE 1, YOU MUST ALSO EDIT THE DATA SET AND RESULTS ON THE FOLLOWING PAGES: Duane Graphical Method Example and Duane Confidence Bounds Example. -->
	<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 book]]''.

Kate Racaza: /* Least Squares Example 1 */

2013-11-21T20:54:38Z

Least Squares Example 1

← Older revision		Revision as of 20:54, 21 November 2013
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	:3) Write the equation of the instantaneous MTBF growth model.		:3) Write the equation of the instantaneous MTBF growth model.
	:4) Plot the instantaneous MTBF growth curve.		:4) Plot the instantaneous MTBF growth curve.
	</noinclude>		</noinclude>'''Solution'''
	'''Solution'''

	From the data table:		From the data table:

Kate Racaza at 20:50, 21 November 2013

2013-11-21T20:50:21Z

← Older revision		Revision as of 20:50, 21 November 2013
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	</includeonly>		</includeonly>
	A complex system's reliability growth is being monitored and the data set is given in the table below.		<noinclude>A complex system's reliability growth is being monitored and the data set is given in the table below.

	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
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	:3) Write the equation of the instantaneous MTBF growth model.		:3) Write the equation of the instantaneous MTBF growth model.
	:4) Plot the instantaneous MTBF growth curve.		:4) Plot the instantaneous MTBF growth curve.
			</noinclude>
	'''Solution'''		'''Solution'''