Change of Slope Analysis - Revision history

Lisa Hacker at 20:37, 18 September 2023

2023-09-18T20:37:03Z

← Older revision		Revision as of 20:37, 18 September 2023
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	<noinclude>{{Banner RGA Articles}}{{Navigation box}}		<noinclude>{{Banner RGA Articles}}{{Navigation box}}
	''This article also appears in ~~ReliaSoft's~~ Reliability ~~Growth and Repairable System Analysis Reference~~.'' </noinclude>		''This article also appears in [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference].'' </noinclude>

	The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline.		The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline.

Lisa Hacker at 22:52, 15 September 2023

2023-09-15T22:52:20Z

← Older revision		Revision as of 22:52, 15 September 2023
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	<noinclude>{{Banner RGA Articles}}{{Navigation box}}		<noinclude>{{Banner RGA Articles}}{{Navigation box}}
	''This article also appears in ~~the [[Crow-AMSAA (NHPP)\|~~Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>		''This article also appears in ReliaSoft's Reliability Growth and Repairable System Analysis Reference.'' </noinclude>

	The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline.		The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline.

Lisa Hacker at 21:01, 24 June 2015

2015-06-24T21:01:52Z

← Older revision		Revision as of 21:01, 24 June 2015
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	<noinclude>{{Banner RGA Articles}}{{Navigation box}		<noinclude>{{Banner RGA Articles}}{{Navigation box}}
	''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>		''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>

Lisa Hacker at 21:01, 24 June 2015

2015-06-24T21:01:14Z

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	<noinclude>{{Banner RGA Articles}}{{Navigation box}~~}[[Category: Shared Articles]]~~		<noinclude>{{Banner RGA Articles}}{{Navigation box}
	''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>		''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>

David J. Groebel at 00:41, 27 May 2015

2015-05-27T00:41:56Z

← Older revision		Revision as of 00:41, 27 May 2015
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	The Crow-AMSAA (NHPP) model will be used again to analyze the data after <math>{{T}_{1}}\,\!</math>. However, the information collected during the first segment will be used when creating the model for the second segment. Given that, the ML estimators of the model parameters in the second segment are:		The Crow-AMSAA (NHPP) model will be used again to analyze the data after <math>{{T}_{1}}\,\!</math>. However, the information collected during the first segment will be used when creating the model for the second segment. Given that, the ML estimators of the model parameters in the second segment are:

	:<math>\widehat{{{\lambda }_{2}}}=\frac{{{n~~}_{2~~}}}{T_{2}^{{{\beta }_{2}}}}\,\!</math>		:<math>\widehat{{{\lambda }_{2}}}=\frac{{{n}}}{T_{2}^{{{\beta }_{2}}}}\,\!</math>

	and:		and:

David J. Groebel at 18:25, 30 May 2014

2014-05-30T18:25:30Z

← Older revision		Revision as of 18:25, 30 May 2014
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	<noinclude>{{Banner RGA Articles}}{{Navigation box}}[[Category: Shared Articles]]		<noinclude>{{Banner RGA Articles}}{{Navigation box}}[[Category: Shared Articles]]
	''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]] ~~book~~.'' </noinclude>		''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]].'' </noinclude>

	The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline.		The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline.
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	Consider the data in the following plot from a reliability growth test.		Consider the data in the following plot from a reliability growth test.

	[[Image:rga5.19.png\|center\|~~400px\|Data from a reliability growth test.~~]]		[[Image:rga5.19.png\|center\|450px]]

	As discussed above, the cumulative number of failures vs. the cumulative time should be linear on logarithmic scales. The next figure shows the data plotted on logarithmic scales.		As discussed above, the cumulative number of failures vs. the cumulative time should be linear on logarithmic scales. The next figure shows the data plotted on logarithmic scales.

	[[Image:rga5.20.png\|center\|~~400px\|Cumulative number of failures plotted on logarithmic axes.~~]]		[[Image:rga5.20.png\|center\|450px]]

	One can easily recognize that the failure behavior is not constant throughout the duration of the test. Just by observing the data, it can be asserted that a major change occurred at around 140 hours that resulted in a change in the rate of failures. Therefore, using a single model to analyze this data set likely will not be appropriate.		One can easily recognize that the failure behavior is not constant throughout the duration of the test. Just by observing the data, it can be asserted that a major change occurred at around 140 hours that resulted in a change in the rate of failures. Therefore, using a single model to analyze this data set likely will not be appropriate.

David J. Groebel at 00:05, 23 April 2014

2014-04-23T00:05:31Z

← Older revision		Revision as of 00:05, 23 April 2014
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	where:		where:

	:*<math>{{T}_{1}}\,\!</math> is the time when the change occurs		*<math>{{T}_{1}}\,\!</math> is the time when the change occurs
	:*<math>{{n}_{1}}\,\!</math> is the number of failures observed up to time <math>{{T}_{1}}\,\!</math>		*<math>{{n}_{1}}\,\!</math> is the number of failures observed up to time <math>{{T}_{1}}\,\!</math>
	:*<math>{{t}_{i}}\,\!</math> is the time at which each corresponding failure was observed		*<math>{{t}_{i}}\,\!</math> is the time at which each corresponding failure was observed

	The equation for <math>\widehat{\beta_{1}}\,\!</math> can be rewritten as follows:		The equation for <math>\widehat{\beta_{1}}\,\!</math> can be rewritten as follows:
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	where:		where:

	:*<math>{{n}_{2}}\,\!</math> is the number of failures that were observed after <math>{{T}_{1}}\,\!</math>		*<math>{{n}_{2}}\,\!</math> is the number of failures that were observed after <math>{{T}_{1}}\,\!</math>
	:*<math>n={{n}_{1}}+{{n}_{2}}\,\!</math> is the total number of failures observed throughout the test		*<math>n={{n}_{1}}+{{n}_{2}}\,\!</math> is the total number of failures observed throughout the test
	:*<math>{{T}_{2}}\,\!</math> is the end time of the test. The test can either be failure terminated or time terminated		*<math>{{T}_{2}}\,\!</math> is the end time of the test. The test can either be failure terminated or time terminated

	===Example - Multiple MLE===		===Example - Multiple MLE===
	{{:Change of Slope Analysis Example}}		{{:Change of Slope Analysis Example}}

David J. Groebel: /* Change of Slope Example */

2014-02-04T15:56:48Z

Change of Slope Example

← Older revision		Revision as of 15:56, 4 February 2014
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	:*<math>{{T}_{2}}\,\!</math> is the end time of the test. The test can either be failure terminated or time terminated		:*<math>{{T}_{2}}\,\!</math> is the end time of the test. The test can either be failure terminated or time terminated

	===~~Change of Slope~~ Example===		===Example - Multiple MLE===
	{{:Change of Slope Analysis Example}}		{{:Change of Slope Analysis Example}}

David J. Groebel at 17:56, 27 January 2014

2014-01-27T17:56:15Z

← Older revision		Revision as of 17:56, 27 January 2014
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	<noinclude>{{Banner RGA Articles}}{{Navigation box}}[[Category: Shared Articles]]		<noinclude>{{Banner RGA Articles}}{{Navigation box}}[[Category: Shared Articles]]
	''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]] book.'' </noinclude>		''This article also appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]] book.'' </noinclude>


	The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline.		The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline.
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	[[Image:rga5.19.png\|center\|400px\|Data from a reliability growth test.]]		[[Image:rga5.19.png\|center\|400px\|Data from a reliability growth test.]]


	As discussed above, the cumulative number of failures vs. the cumulative time should be linear on logarithmic scales. The next figure shows the data plotted on logarithmic scales.		As discussed above, the cumulative number of failures vs. the cumulative time should be linear on logarithmic scales. The next figure shows the data plotted on logarithmic scales.
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	The Change of Slope methodology proposes to split the data into two segments and apply a Crow-AMSAA (NHPP) model to each segment. The time of change that will be used to split the data into the two segments (it will be referred to as <math>{{T}_{1}}\,\!</math> ) could be estimated just by observing the data, but will most likely be dictated by engineering knowledge of the specific change to the system design or operating conditions. It is important to note that although two separate models will be applied to each segment, the information collected in the first segment (i.e., data up to <math>{{T}_{1}}\,\!</math> ) will be considered when creating the model for the second segment (i.e., data after <math>{{T}_{1}}\,\!</math> ). The models presented next can be applied to the reliability growth analysis of a single system or multiple systems.		The Change of Slope methodology proposes to split the data into two segments and apply a Crow-AMSAA (NHPP) model to each segment. The time of change that will be used to split the data into the two segments (it will be referred to as <math>{{T}_{1}}\,\!</math> ) could be estimated just by observing the data, but will most likely be dictated by engineering knowledge of the specific change to the system design or operating conditions. It is important to note that although two separate models will be applied to each segment, the information collected in the first segment (i.e., data up to <math>{{T}_{1}}\,\!</math> ) will be considered when creating the model for the second segment (i.e., data after <math>{{T}_{1}}\,\!</math> ). The models presented next can be applied to the reliability growth analysis of a single system or multiple systems.


	===Model for First Segment (Data up to <i>T<sub>1</sub></i>)===		===Model for First Segment (Data up to <i>T<sub>1</sub></i>)===
	The data up to the point of the change that occurs at <math>{{T}_{1}}\,\!</math> will be analyzed using the Crow-AMSAA (NHPP) model. Based on the ML equations for <math>\lambda \,\!</math> and <math>\beta \,\!</math> (in the section [[Crow-AMSAA (NHPP)#Maximum_Likelihood_Estimators\|Maximum Likelihood Estimators]]), the ML estimators of the model are:		The data up to the point of the change that occurs at <math>{{T}_{1}}\,\!</math> will be analyzed using the Crow-AMSAA (NHPP) model. Based on the ML equations for <math>\lambda \,\!</math> and <math>\beta \,\!</math> (in the section [[Crow-AMSAA (NHPP)#Maximum_Likelihood_Estimators\|Maximum Likelihood Estimators]]), the ML estimators of the model are:

			:<math>\widehat{{{\lambda }_{1}}}=\frac{{{n}_{1}}}{T_{1}^{{{\beta }_{1}}}}\,\!</math>

	~~::<math>\widehat{{{\lambda }_{1}}}=\frac{{{n}_{1}}}{T_{1}^{{{\beta }_{1}}}}\,\!</math>~~		and

	:~~and~~		:<math>{{\widehat{\beta }}_{1}}=\frac{{{n}_{1}}}{{{n}_{1}}\ln {{T}_{1}}-\underset{i=1}{\overset{{{n}_{1}}}{\mathop{\sum }}}\,\ln {{t}_{i}}}\,\!</math>

	~~::<math>{{\widehat{\beta }}_{1}}=\frac{{{n}_{1}}}{{{n}_{1}}\ln {{T}_{1}}-\underset{i=1}{\overset{{{n}_{1}}}{\mathop{\sum }}}\,\ln {{t}_{i}}}\,\!</math>~~		where:

	:where:

	~~::*<math>{{T}_{1}}\,\!</math> is the time when the change occurs~~
	~~::*<math>{{n}_{1}}\,\!</math> is the number of failures observed up to time <math>{{T}_{1}}\,\!</math>~~
	~~::*<math>{{t}_{i}}\,\!</math> is the time at which each corresponding failure was observed~~

			:*<math>{{T}_{1}}\,\!</math> is the time when the change occurs
			:*<math>{{n}_{1}}\,\!</math> is the number of failures observed up to time <math>{{T}_{1}}\,\!</math>
			:*<math>{{t}_{i}}\,\!</math> is the time at which each corresponding failure was observed

	The equation for <math>\widehat{\beta_{1}}\,\!</math> can be rewritten as follows:		The equation for <math>\widehat{\beta_{1}}\,\!</math> can be rewritten as follows:

	::<math>\begin{align}		:<math>\begin{align}
	{{\widehat{\beta }}_{1}}= & \frac{{{n}_{1}}}{{{n}_{1}}\ln {{T}_{1}}-\left( \ln {{t}_{1}}+\ln {{t}_{2}}+...+\ln {{t}_{{{n}_{1}}}} \right)} \\		{{\widehat{\beta }}_{1}}= & \frac{{{n}_{1}}}{{{n}_{1}}\ln {{T}_{1}}-\left( \ln {{t}_{1}}+\ln {{t}_{2}}+...+\ln {{t}_{{{n}_{1}}}} \right)} \\
	= & \frac{{{n}_{1}}}{(\ln {{T}_{1}}-\ln {{t}_{1}})+(\ln {{T}_{1}}-\ln {{t}_{2}})+(...)+(\ln {{T}_{1}}-\ln {{t}_{{{n}_{1}}}})} \\		= & \frac{{{n}_{1}}}{(\ln {{T}_{1}}-\ln {{t}_{1}})+(\ln {{T}_{1}}-\ln {{t}_{2}})+(...)+(\ln {{T}_{1}}-\ln {{t}_{{{n}_{1}}}})} \\
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	:or		or

	~~::<math>{{\widehat{\beta }}_{1}}=\frac{{{n}_{1}}}{\underset{i=1}{\overset{{{n}_{1}}}{\mathop{\sum }}}\,\ln \tfrac{{{T}_{1}}}{{{t}_{i}}}}\,\!</math>~~

			:<math>{{\widehat{\beta }}_{1}}=\frac{{{n}_{1}}}{\underset{i=1}{\overset{{{n}_{1}}}{\mathop{\sum }}}\,\ln \tfrac{{{T}_{1}}}{{{t}_{i}}}}\,\!</math>

	===Model for Second Segment (Data after <i>T<sub>1</sub></i>)===		===Model for Second Segment (Data after <i>T<sub>1</sub></i>)===
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	The Crow-AMSAA (NHPP) model will be used again to analyze the data after <math>{{T}_{1}}\,\!</math>. However, the information collected during the first segment will be used when creating the model for the second segment. Given that, the ML estimators of the model parameters in the second segment are:		The Crow-AMSAA (NHPP) model will be used again to analyze the data after <math>{{T}_{1}}\,\!</math>. However, the information collected during the first segment will be used when creating the model for the second segment. Given that, the ML estimators of the model parameters in the second segment are:

	::<math>\widehat{{{\lambda }_{2}}}=\frac{{{n}_{2}}}{T_{2}^{{{\beta }_{2}}}}\,\!</math>		:<math>\widehat{{{\lambda }_{2}}}=\frac{{{n}_{2}}}{T_{2}^{{{\beta }_{2}}}}\,\!</math>

	and:		and:

	::<math>{{\widehat{\beta }}_{2}}=\frac{{{n}_{2}}}{{{n}_{1}}\ln \tfrac{{{T}_{2}}}{{{T}_{1}}}+\underset{i={{n}_{1}}+1}{\overset{n}{\mathop{\sum }}}\,\ln \tfrac{{{T}_{2}}}{{{t}_{i}}}}\,\!</math>		:<math>{{\widehat{\beta }}_{2}}=\frac{{{n}_{2}}}{{{n}_{1}}\ln \tfrac{{{T}_{2}}}{{{T}_{1}}}+\underset{i={{n}_{1}}+1}{\overset{n}{\mathop{\sum }}}\,\ln \tfrac{{{T}_{2}}}{{{t}_{i}}}}\,\!</math>

	~~::where:~~

	:~~:*<math>{{n}_{2}}\,\!</math> is the number of failures that were observed after <math>{{T}_{1}}\,\!</math>~~		where:
	~~::*<math>n={{n}_{1}}+{{n}_{2}}\,\!</math> is the total number of failures observed throughout the test~~
	~~::*<math>{{T}_{2}}\,\!</math> is the end time of the test. The test can either be failure terminated or time terminated~~

			:*<math>{{n}_{2}}\,\!</math> is the number of failures that were observed after <math>{{T}_{1}}\,\!</math>
			:*<math>n={{n}_{1}}+{{n}_{2}}\,\!</math> is the total number of failures observed throughout the test
			:*<math>{{T}_{2}}\,\!</math> is the end time of the test. The test can either be failure terminated or time terminated

	===Change of Slope Example===		===Change of Slope Example===
	{{:Change of Slope Analysis Example}}		{{:Change of Slope Analysis Example}}

Chris Kahn at 21:57, 22 January 2014

2014-01-22T21:57:58Z

← Older revision		Revision as of 21:57, 22 January 2014
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	===Model for First Segment (Data up to <i>T<sub>1</sub></i>)===		===Model for First Segment (Data up to <i>T<sub>1</sub></i>)===
	The data up to the point of the change that occurs at <math>{{T}_{1}}\,\!</math> will be analyzed using the Crow-AMSAA (NHPP) model. Based on the ML equations for <math>\lambda \,\!</math> and <math>\beta \,\!</math> (in the section [[Crow-~~AMSAA_-_NHPP~~#Maximum_Likelihood_Estimators\|Maximum Likelihood Estimators]]), the ML estimators of the model are:		The data up to the point of the change that occurs at <math>{{T}_{1}}\,\!</math> will be analyzed using the Crow-AMSAA (NHPP) model. Based on the ML equations for <math>\lambda \,\!</math> and <math>\beta \,\!</math> (in the section [[Crow-AMSAA (NHPP)#Maximum_Likelihood_Estimators\|Maximum Likelihood Estimators]]), the ML estimators of the model are: