Power Law Model Parameter Estimation Example - Revision history

Lisa Hacker at 21:15, 18 September 2023

2023-09-18T21:15:14Z

← Older revision		Revision as of 21:15, 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>

Chris Kahn at 17:45, 12 June 2014

2014-06-12T17:45:58Z

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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"
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	\|colspan="3" style="text-align:center"\|'''Repairable ~~system failure data~~'''		\|colspan="3" style="text-align:center"\|'''Repairable System Failure Data'''
	\|-		\|-
	!System 1 ( <math>{{X}_{i1}}\,\!</math> )		!System 1 ( <math>{{X}_{i1}}\,\!</math> )

David J. Groebel at 20:03, 2 June 2014

2014-06-02T20:03:51Z

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…'

2014-01-29T18:55:25Z

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>

For the data in the following table, the starting time for each system is equal to 0 and the ending time for each system is 2,000 hours. Calculate the maximum likelihood estimates <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|-
|colspan="3" style="text-align:center"|'''Repairable system failure data'''
|-
!System 1 ( <math>{{X}_{i1}}\,\!</math> )
!System 2 ( <math>{{X}_{i2}}\,\!</math> )
!System 3 ( <math>{{X}_{i3}}\,\!</math> )
|-
|1.2|| 1.4|| 0.3
|-
|55.6|| 35.0|| 32.6
|-
|72.7|| 46.8|| 33.4
|-
|111.9|| 65.9|| 241.7
|-
|121.9|| 181.1|| 396.2
|-
|303.6|| 712.6|| 444.4
|-
|326.9|| 1005.7|| 480.8
|-
|1568.4|| 1029.9 ||588.9
|-
|1913.5|| 1675.7|| 1043.9
|-
| ||1787.5|| 1136.1
|-
| ||1867.0|| 1288.1
|-
| || ||1408.1
|-
| || ||1439.4
|-
| || ||1604.8
|-
|<math>{{N}_{1}}=9\,\!</math> || <math>{{N}_{2}}=11\,\!</math> ||<math>{{N}_{3}}=14\,\!</math>
|}

'''Solution'''

Because the starting time for each system is equal to zero and each system has an equivalent ending time, the general equations for <math>\widehat{\beta }\,\!</math> and <math>\widehat{\lambda }\,\!</math> reduce to the closed form equations. The maximum likelihood estimates of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> are then calculated as follows:

::<math>\widehat{\beta }= \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{iq}}})} = 0.45300 </math>

::<math>\widehat{\lambda }= \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{K{{T}^{\beta }}} = 0.36224 \,\!</math>

The system failure intensity function is then estimated by:

::<math>\widehat{u}(t)=\widehat{\lambda }\widehat{\beta }{{t}^{\widehat{\beta }-1}},\text{ }t>0\,\!</math>

The next figure is a plot of <math>\widehat{u}(t)\,\!</math> over the period (0, 3000). Clearly, the estimated failure intensity function is most representative over the range of the data and any extrapolation should be viewed with the usual caution.

[[File:Instantaneous Failure Intensity vs. Time plot.png|450px|thumb|center|Instantaneous Failure Intensity vs. Time plot]]

← Older revision		Revision as of 20:03, 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>

	For the data in the following table, the starting time for each system is equal to 0 and the ending time for each system is 2,000 hours. Calculate the maximum likelihood estimates <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>.		For the data in the following table, the starting time for each system is equal to 0 and the ending time for each system is 2,000 hours. Calculate the maximum likelihood estimates <math>\widehat{\lambda }\,\!</math> and <math>\widehat{\beta }\,\!</math>.


	{\|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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	Because the starting time for each system is equal to zero and each system has an equivalent ending time, the general equations for <math>\widehat{\beta }\,\!</math> and <math>\widehat{\lambda }\,\!</math> reduce to the closed form equations. The maximum likelihood estimates of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> are then calculated as follows:		Because the starting time for each system is equal to zero and each system has an equivalent ending time, the general equations for <math>\widehat{\beta }\,\!</math> and <math>\widehat{\lambda }\,\!</math> reduce to the closed form equations. The maximum likelihood estimates of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> are then calculated as follows:

	::<math>\widehat{\beta }= \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{iq}}})} = 0.45300 ~~</math>~~		:<math>\widehat{\beta }= \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,\underset{i=1}{\overset{{{N}_{q}}}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{iq}}})} = 0.45300 </math>


	~~::<math>\widehat{\lambda }= \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{K{{T}^{\beta }}} = 0.36224 \,\!~~</math>

			:<math>\widehat{\lambda }= \frac{\underset{q=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{q}}}{K{{T}^{\beta }}} = 0.36224 \,\!</math>

	The system failure intensity function is then estimated by:		The system failure intensity function is then estimated by:

	::<math>\widehat{u}(t)=\widehat{\lambda }\widehat{\beta }{{t}^{\widehat{\beta }-1}},\text{ }t>0\,\!</math>		:<math>\widehat{u}(t)=\widehat{\lambda }\widehat{\beta }{{t}^{\widehat{\beta }-1}},\text{ }t>0\,\!</math>


	The next figure is a plot of <math>\widehat{u}(t)\,\!</math> over the period (0, 3000). Clearly, the estimated failure intensity function is most representative over the range of the data and any extrapolation should be viewed with the usual caution.		The next figure is a plot of <math>\widehat{u}(t)\,\!</math> over the period (0, 3000). Clearly, the estimated failure intensity function is most representative over the range of the data and any extrapolation should be viewed with the usual caution.

	[[File:Instantaneous Failure Intensity vs. Time plot.png\|450px~~\|thumb~~\|center~~\|Instantaneous Failure Intensity vs. Time plot~~]]		[[File:Instantaneous Failure Intensity vs. Time plot.png\|450px\|center]]