Template:Ipl weibull - Revision history

Richard House: Redirected page to Inverse Power Law (IPL) Relationship

2012-08-15T23:27:52Z

Redirected page to Inverse Power Law (IPL) Relationship

Chris Kahn: /* Maximum Likelihood Estimation Method */

2012-03-09T18:25:08Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 18:25, 9 March 2012
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	• <math>S</math> is the number of groups of suspension data points.		• <math>S</math> is the number of groups of suspension data points.
	<br>		<br>
	• <math>N_{i}^{\prime }</math> is the number of suspensions in the .. group of suspension data points.		• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
	<br>		<br>
	• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.		• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
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	<br>		<br>
	<br>		<br>
	The solution (parameter estimates) will be found by solving for <math>\beta ,</math> <math>K,</math> <math>n</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math> <math>\tfrac{\partial \Lambda }{\partial K}=0</math> and .. , where:		The solution (parameter estimates) will be found by solving for <math>\beta ,</math> <math>K,</math> <math>n</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0</math>, <math>\tfrac{\partial \Lambda }{\partial K}=0</math> and <math>\tfrac{\partial \Lambda }{\partial n}=0</math>, where:
	<br>		<br>
	<br>		<br>

Chris Kahn: /* Maximum Likelihood Estimation Method */

2012-03-09T18:10:42Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 18:10, 9 March 2012
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	• <math>{K}</math> is the IPL parameter (unknown, the second of three parameters to be estimated).		• <math>{K}</math> is the IPL parameter (unknown, the second of three parameters to be estimated).
	<br>		<br>
	• <math>n</math> is the second IPL parameter (unknown, the third of three parameters to be estimated).		• <math>{n}</math> is the second IPL parameter (unknown, the third of three parameters to be estimated).
	<br>		<br>
	• <math>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.		• <math>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.

Chris Kahn: /* Maximum Likelihood Estimation Method */

2012-03-09T18:09:37Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 18:09, 9 March 2012
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	• <math>\beta </math> is the Weibull shape parameter (unknown, the first of three parameters to be estimated).		• <math>\beta </math> is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
	<br>		<br>
	• .. is the IPL parameter (unknown, the second of three parameters to be estimated).		• <math>{K}</math> is the IPL parameter (unknown, the second of three parameters to be estimated).
	<br>		<br>
	• <math>n</math> is the second IPL parameter (unknown, the third of three parameters to be estimated).		• <math>n</math> is the second IPL parameter (unknown, the third of three parameters to be estimated).

Chris Kahn: /* Reliable Life */

2012-03-09T18:07:30Z

Reliable Life

← Older revision		Revision as of 18:07, 9 March 2012
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	====Reliable Life====		====Reliable Life====
	<br>		<br>
	For the IPL-Weibull model, the reliable life, .. , of a unit for a specified reliability and starting the mission at age zero is given by:		For the IPL-Weibull model, the reliable life, <math>{T}_{R}</math>, of a unit for a specified reliability and starting the mission at age zero is given by:

	<br>		<br>
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	<br>		<br>

	====IPL-Weibull Failure Rate Function====		====IPL-Weibull Failure Rate Function====
	<br>		<br>

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-15T23:18:12Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 23:18, 15 February 2012
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	<br>		<br>
	<br>		<br>
	<math>\begin{align}\frac{\partial \Lambda }{\partial \beta }=\ & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \,KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }}\ln \left( KV_{i}^{n}T_{i}^{\prime } \right) \\ & \overset{FI}{\mathop{\underset{i=1} {\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( KV_{i}^{n} \right)}^{\beta }}\left[ R_{Li}^{\prime \prime }T_{Li}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Li}^{\prime \prime }) \right)-R_{Ri}^{\prime \prime }T_{Ri}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Ri}^{\prime \prime }) \right) \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\		::<math>\begin{align}\frac{\partial \Lambda }{\partial \beta }=\ & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \,KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }}\ln \left( KV_{i}^{n}T_{i}^{\prime } \right) \\ & \overset{FI}{\mathop{\underset{i=1} {\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( KV_{i}^{n} \right)}^{\beta }}\left[ R_{Li}^{\prime \prime }T_{Li}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Li}^{\prime \prime }) \right)-R_{Ri}^{\prime \prime }T_{Ri}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Ri}^{\prime \prime }) \right) \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\
	\frac{\partial \Lambda }{\partial K}=\ & \frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\frac{\beta }{K}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \overset{{}}{\mathop{-\beta \underset{i=1}{\mathop{\overset{FI}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{K}^{\beta -1}}V_{i}^{n\beta }\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\		\frac{\partial \Lambda }{\partial K}=\ & \frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\frac{\beta }{K}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \overset{{}}{\mathop{-\beta \underset{i=1}{\mathop{\overset{FI}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{K}^{\beta -1}}V_{i}^{n\beta }\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\
	\frac{\partial\Lambda }{\partial n}=\ & \beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{n{{K}^{\beta }}V_{i}^{\beta (n-1)}\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}		\frac{\partial\Lambda }{\partial n}=\ & \beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{n{{K}^{\beta }}V_{i}^{\beta (n-1)}\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}

Athanasios Gerokostopoulos at 18:24, 15 February 2012

2012-02-15T18:24:30Z

← Older revision		Revision as of 18:24, 15 February 2012
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	<br>		<br>
	<br>		<br>
	<math>\begin{align}\frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \,KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }}\ln \left( KV_{i}^{n}T_{i}^{\prime } \right) \\ & \overset{FI}{\mathop{\underset{i=1} {\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( KV_{i}^{n} \right)}^{\beta }}\left[ R_{Li}^{\prime \prime }T_{Li}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Li}^{\prime \prime }) \right)-R_{Ri}^{\prime \prime }T_{Ri}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Ri}^{\prime \prime }) \right) \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\		<math>\begin{align}\frac{\partial \Lambda }{\partial \beta }=\ & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \,KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }}\ln \left( KV_{i}^{n}T_{i}^{\prime } \right) \\ & \overset{FI}{\mathop{\underset{i=1} {\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( KV_{i}^{n} \right)}^{\beta }}\left[ R_{Li}^{\prime \prime }T_{Li}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Li}^{\prime \prime }) \right)-R_{Ri}^{\prime \prime }T_{Ri}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Ri}^{\prime \prime }) \right) \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\
	\frac{\partial \Lambda }{\partial K}= & \frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\frac{\beta }{K}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \overset{{}}{\mathop{-\beta \underset{i=1}{\mathop{\overset{FI}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{K}^{\beta -1}}V_{i}^{n\beta }\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}		\frac{\partial \Lambda }{\partial K}=\ & \frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\frac{\beta }{K}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \overset{{}}{\mathop{-\beta \underset{i=1}{\mathop{\overset{FI}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{K}^{\beta -1}}V_{i}^{n\beta }\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\
	\~~end{align}</math>~~		\frac{\partial\Lambda }{\partial n}=\ & \beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{n{{K}^{\beta }}V_{i}^{\beta (n-1)}\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
	~~<math>\begin{align}~~\frac{\partial\Lambda }{\partial n}=\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) \\
	-\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \\
	-\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \\
	\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{n{{K}^{\beta }}V_{i}^{\beta (n-1)}\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
	\end{align}</math>		\end{align}</math>

	<br>		<br>

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-15T17:59:26Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 17:59, 15 February 2012
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	<br>		<br>
	<br>		<br>
	<math>\begin{align}\frac{\partial \Lambda }{\partial \beta }=\frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( KV_{i}^{n}{{T}_{i}} \right) \\		<math>\begin{align}\frac{\partial \Lambda }{\partial \beta }= & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \,KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }}\ln \left( KV_{i}^{n}T_{i}^{\prime } \right) \\ & \overset{FI}{\mathop{\underset{i=1} {\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( KV_{i}^{n} \right)}^{\beta }}\left[ R_{Li}^{\prime \prime }T_{Li}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Li}^{\prime \prime }) \right)-R_{Ri}^{\prime \prime }T_{Ri}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Ri}^{\prime \prime }) \right) \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\
	-\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}\ln \left( KV_{i}^{n}{{T}_{i}} \right) \\		\frac{\partial \Lambda }{\partial K}= & \frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\frac{\beta }{K}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \overset{{}}{\mathop{-\beta \underset{i=1}{\mathop{\overset{FI}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{K}^{\beta -1}}V_{i}^{n\beta }\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
	-\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \,KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }}\ln \left( KV_{i}^{n}T_{i}^{\prime } \right) \\
	\overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( KV_{i}^{n} \right)}^{\beta }}\left[ R_{Li}^{\prime \prime }T_{Li}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Li}^{\prime \prime }) \right)-R_{Ri}^{\prime \prime }T_{Ri}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Ri}^{\prime \prime }) \right) \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\
	\frac{\partial \Lambda }{\partial K}= & \frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \\
	-\frac{\beta }{K}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \\
	\overset{{}}{\mathop{-\beta \underset{i=1}{\mathop{\overset{FI}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{K}^{\beta -1}}V_{i}^{n\beta }\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
	\end{align}</math>		\end{align}</math>
	<math>\begin{align}\frac{\partial\Lambda }{\partial n}=\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) \\		<math>\begin{align}\frac{\partial\Lambda }{\partial n}=\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) \\

Athanasios Gerokostopoulos: /* Maximum Likelihood Estimation Method */

2012-02-15T17:41:32Z

Maximum Likelihood Estimation Method

← Older revision		Revision as of 17:41, 15 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	\Lambda = \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta KV_{i}^{n}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta -1}}{{e}^{-{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}}} \right] \\		\Lambda = \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta KV_{i}^{n}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta -1}}{{e}^{-{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \\
	+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	:where:		where:

	<br>		<br>
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	<br>		<br>
	:and:		and:
	<br>		<br>
	• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.		• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.

Athanasios Gerokostopoulos: /* Median */

2012-02-15T17:40:23Z

Median

← Older revision		Revision as of 17:40, 15 February 2012
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	<br>		<br>
	The median, <math>\breve{T},</math> of the IPL-Weibull model is given by:		The median, <math>\breve{T},</math> of the IPL-Weibull model is given by:

	<br>		<br>
	::<math>\breve{T}=\frac{1}{K{{V}^{n}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>		::<math>\breve{T}=\frac{1}{K{{V}^{n}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>

	<br>		<br>