Crow-AMSAA (NHPP) - Revision history

Lisa Hacker at 00:12, 13 July 2023

2023-07-13T00:12:13Z

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	{{template:RGA BOOK\|3.2\|Crow-AMSAA}}		{{template:RGA BOOK\|3.2\|Crow-AMSAA}}
	Dr. Larry H. Crow [[RGA_References\|[17]]] noted that the [[Duane Model]] could be stochastically represented as a Weibull process, allowing for statistical procedures to be used in the application of this model in reliability growth. This statistical extension became what is known as the Crow-AMSAA (NHPP) model. This method was first developed at the U.S. Army Materiel Systems Analysis Activity (AMSAA). It is frequently used on systems when usage is measured on a continuous scale. It can also be applied for the analysis of one shot items when there is high reliability and a large number of trials.		Dr. Larry H. Crow [[RGA_References\|[17]]] noted that the [[Duane Model]] could be stochastically represented as a Weibull process, allowing for statistical procedures to be used in the application of this model in reliability growth. This statistical extension became what is known as the Crow-AMSAA (NHPP) model. This method was first developed at the U.S. Army Materiel Systems Analysis Activity (AMSAA). It is frequently used on systems when usage is measured on a continuous scale. It can also be applied for the analysis of one shot items when there is high reliability and a large number of trials

	Test programs are generally conducted on a phase by phase basis. The Crow-AMSAA model is designed for tracking the reliability within a test phase and not across test phases. A development testing program may consist of several separate test phases. If corrective actions are introduced during a particular test phase, then this type of testing and the associated data are appropriate for analysis by the Crow-AMSAA model. The model analyzes the reliability growth progress within each test phase and can aid in determining the following:		Test programs are generally conducted on a phase by phase basis. The Crow-AMSAA model is designed for tracking the reliability within a test phase and not across test phases. A development testing program may consist of several separate test phases. If corrective actions are introduced during a particular test phase, then this type of testing and the associated data are appropriate for analysis by the Crow-AMSAA model. The model analyzes the reliability growth progress within each test phase and can aid in determining the following:

Sam Eisenberg: Updated failure terminated denominator per R-DE4648.

2020-04-13T16:14:38Z

Updated failure terminated denominator per R-DE4648.

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	For failure terminated data (the test ends after a specified number of failures):		For failure terminated data (the test ends after a specified number of failures):

	:<math>\bar{\beta }=\frac{N-2}{N}\hat{\beta }\,\!</math>		:<math>\bar{\beta }=\frac{N-2}{N-1}\hat{\beta }\,\!</math>

	By default <math>\hat{\beta }\,\!</math> is returned. <math>\bar{\beta }\,\!</math> can be returned by selecting the '''Calculate unbiased beta''' option on the Calculations tab of the Application Setup.		By default <math>\hat{\beta }\,\!</math> is returned. <math>\bar{\beta }\,\!</math> can be returned by selecting the '''Calculate unbiased beta''' option on the Calculations tab of the Application Setup.

Kate Racaza: /* Example - Helicopter System */

2015-12-01T18:24:38Z

Example - Helicopter System

← Older revision		Revision as of 18:24, 1 December 2015
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	{{:Crow-AMSAA_Model_-_Helicopter_System_Example}}		{{:Crow-AMSAA_Model_-_Helicopter_System_Example}}

			<div class="noprint">
	{{Examples Box\|RGA Examples\|<p>More grouped data examples are available! See also:</p>		{{Examples Box\|RGA Examples\|<p>More grouped data examples are available! See also:</p>
	{{Examples Link External\|http://www.reliasoft.com/rga/examples/rgex1/index.htm\|Simple MTBF Determination}}<nowiki/>		{{Examples Link External\|http://www.reliasoft.com/rga/examples/rgex1/index.htm\|Simple MTBF Determination}}<nowiki/>
	}}		}}
			</div>

	<!-- ==Goodness-of-Fit Tests== This section is no longer necessary-->		<!-- ==Goodness-of-Fit Tests== This section is no longer necessary-->

Kate Racaza at 23:06, 23 November 2015

2015-11-23T23:06:52Z

← Older revision		Revision as of 23:06, 23 November 2015
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	~~{{template:InProgress}}~~
	{{template:RGA BOOK\|3.2\|Crow-AMSAA}}		{{template:RGA BOOK\|3.2\|Crow-AMSAA}}
	Dr. Larry H. Crow [[RGA_References\|[17]]] noted that the [[Duane Model]] could be stochastically represented as a Weibull process, allowing for statistical procedures to be used in the application of this model in reliability growth. This statistical extension became what is known as the Crow-AMSAA (NHPP) model. This method was first developed at the U.S. Army Materiel Systems Analysis Activity (AMSAA). It is frequently used on systems when usage is measured on a continuous scale. It can also be applied for the analysis of one shot items when there is high reliability and a large number of trials.		Dr. Larry H. Crow [[RGA_References\|[17]]] noted that the [[Duane Model]] could be stochastically represented as a Weibull process, allowing for statistical procedures to be used in the application of this model in reliability growth. This statistical extension became what is known as the Crow-AMSAA (NHPP) model. This method was first developed at the U.S. Army Materiel Systems Analysis Activity (AMSAA). It is frequently used on systems when usage is measured on a continuous scale. It can also be applied for the analysis of one shot items when there is high reliability and a large number of trials.

David J. Groebel: /* Cramér-von Mises Test */

2015-09-25T15:40:30Z

Cramér-von Mises Test

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	If the individual failure times are known, a Cramér-von Mises statistic is used to test the null hypothesis that a non-homogeneous Poisson process with the failure intensity function <math>\rho \left( t \right)=\lambda \,\beta \,{{t}^{\beta -1}}\left( \lambda >0,\beta >0,t>0 \right)\,\!</math> properly describes the reliability growth of a system. The Cramér-von Mises goodness-of-fit statistic is then given by the following expression:		If the individual failure times are known, a Cramér-von Mises statistic is used to test the null hypothesis that a non-homogeneous Poisson process with the failure intensity function <math>\rho \left( t \right)=\lambda \,\beta \,{{t}^{\beta -1}}\left( \lambda >0,\beta >0,t>0 \right)\,\!</math> properly describes the reliability growth of a system. The Cramér-von Mises goodness-of-fit statistic is then given by the following expression:

	:<math>C_{M}^{2}=\frac{1}{12M}+\underset{i=1}{\overset{M}{\mathop \sum }}\,{{\left[ {{\left( \frac{{{T}_{i}}}{T} \right)}^{{\~~hat~~{\beta }}}}-\frac{2i-1}{2M} \right]}^{2}}\,\!</math>		:<math>C_{M}^{2}=\frac{1}{12M}+\underset{i=1}{\overset{M}{\mathop \sum }}\,{{\left[ {{\left( \frac{{{T}_{i}}}{T} \right)}^{{\bar{\beta }}}}-\frac{2i-1}{2M} \right]}^{2}}\,\!</math>

	where:		where:
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	N-1\text{ if the test is failure terminated} \\		N-1\text{ if the test is failure terminated} \\
	\end{matrix} \right\}\,\!</math>		\end{matrix} \right\}\,\!</math>
			:<math>{\bar{\beta }}\,\!</math> is the unbiased value of Beta.

	The failure times, <math>{{T}_{i}}\,\!</math>, must be ordered so that <math>{{T}_{1}}<{{T}_{2}}<\ldots <{{T}_{M}}\,\!</math>.		The failure times, <math>{{T}_{i}}\,\!</math>, must be ordered so that <math>{{T}_{1}}<{{T}_{2}}<\ldots <{{T}_{M}}\,\!</math>.

Harry Guo: /* Parameter Estimation for Failure Times Data */

2015-08-21T14:40:49Z

Parameter Estimation for Failure Times Data

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	:<math>f({{T}_{i}}\|{{T}_{i-1}})=\frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}\cdot {{e}^{-\tfrac{1}{{{\eta }^{\beta }}}\left( T_{i}^{\beta }-T_{i-1}^{\beta } \right)}}\,\!</math>		:<math>f({{T}_{i}}\|{{T}_{i-1}})=\frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}\cdot {{e}^{-\tfrac{1}{{{\eta }^{\beta }}}\left( T_{i}^{\beta }-T_{i-1}^{\beta } \right)}}\,\!</math>

	~~The~~ likelihood function is:		Let <math>\lambda =\tfrac{1}{{{\eta }^{\beta }}},\,\!</math>, the likelihood function is:

	:<math>L={{\lambda }^{n}}{{\beta }^{n}}{{e}^{-\lambda {{T}^{*\beta }}}}\underset{i=1}{\overset{n}{\mathop \prod }}\,T_{i}^{\beta -1}\,\!</math>		:<math>L={{\lambda }^{n}}{{\beta }^{n}}{{e}^{-\lambda {{T}^{*\beta }}}}\underset{i=1}{\overset{n}{\mathop \prod }}\,T_{i}^{\beta -1}\,\!</math>

Chris Kahn: /* More Examples */

2014-05-07T18:48:31Z

More Examples

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	==More Examples==		==More Examples==
	===Determining Whether a Design Meets the MTBF Goal===		===Determining Whether a Design Meets the MTBF Goal===
	{{:~~Failure_Times_Crow_AMSAA_Example~~}}		{{:Failure_Times_Crow-AMSAA_Example}}

	===Analyzing Mixed Data for a One-Shot System===		===Analyzing Mixed Data for a One-Shot System===
	{{:Mixed_Data_-~~_Crow_AMSAA_Example~~}}		{{:Mixed_Data_-_Crow-AMSAA_Example}}

Ricardo J Ortiz: /* Biasing and Unbiasing of Beta */

2014-05-07T15:25:02Z

Biasing and Unbiasing of Beta

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	====Biasing and Unbiasing of Beta==== <!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended - Continuous Evaluation. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		====Biasing and Unbiasing of Beta==== <!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended - Continuous Evaluation. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	The equation above returns the biased estimate, <math>\hat{\beta }\,\!</math>. The unbiased estimate, <math>\bar{\beta }\,\!</math>, can be calculated by using the following relationships. For time terminated data (the test ends after a specified ~~number of failures~~):		The equation above returns the biased estimate, <math>\hat{\beta }\,\!</math>. The unbiased estimate, <math>\bar{\beta }\,\!</math>, can be calculated by using the following relationships. For time terminated data (the test ends after a specified test time):

	:<math>\bar{\beta }=\frac{N-1}{N}\hat{\beta }\,\!</math>		:<math>\bar{\beta }=\frac{N-1}{N}\hat{\beta }\,\!</math>

	For failure terminated data (the test ends after a specified ~~test time~~):		For failure terminated data (the test ends after a specified number of failures):

	:<math>\bar{\beta }=\frac{N-2}{N}\hat{\beta }\,\!</math>		:<math>\bar{\beta }=\frac{N-2}{N}\hat{\beta }\,\!</math>

Chris Kahn: /* Mixed Data Confidence Bounds */

2014-03-14T20:25:37Z

Mixed Data Confidence Bounds

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	====Mixed Data Confidence Bounds====		====Mixed Data Confidence Bounds====
	'''Bounds on Average Failure Probability'''<br>		'''Bounds on Average Failure Probability'''<br>
	The process to calculate the average unreliability confidence bounds for ~~mixed~~ data is as follows:		The process to calculate the average unreliability confidence bounds for Mixed data is as follows:

	#Calculate the average failure probability <math>({{t}_{1}},{{t}_{2}})\,\!</math>.		#Calculate the average failure probability <math>({{t}_{1}},{{t}_{2}})\,\!</math>.
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	'''Bounds on Average Reliability'''<br>		'''Bounds on Average Reliability'''<br>
	The process to calculate the average reliability confidence bounds for ~~mixed~~ data is as follows:		The process to calculate the average reliability confidence bounds for Mixed data is as follows:

	#Calculate confidence bounds for average unreliability <math>({{t}_{1}},{{t}_{2}})\,\!</math> as described above.		#Calculate confidence bounds for average unreliability <math>({{t}_{1}},{{t}_{2}})\,\!</math> as described above.

Chris Kahn: /* Grouped per Configuration */

2014-03-14T18:49:00Z

Grouped per Configuration

← Older revision		Revision as of 18:49, 14 March 2014
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	:<math>{{f}_{i}}=\frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}}\,\!</math>		:<math>{{f}_{i}}=\frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}}\,\!</math>

	In this equation, <math>i\,\!</math> represents the trial number. Thus an equation for the reliability (probability of success) for the <math>{{i}^{th}}\,\!</math> configuration is obtained:		In this equation, <math>i\,\!</math> represents the trial number. Thus, an equation for the reliability (probability of success) for the <math>{{i}^{th}}\,\!</math> configuration is obtained:

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