Arrhenius-Weibull MLE - Revision history

Lisa Hacker: Redirected page to Arrhenius Relationship

2015-06-26T00:15:41Z

Redirected page to Arrhenius Relationship

← Older revision		Revision as of 00:15, 26 June 2015
Line 1:		Line 1:
	~~{{InProgress\| }}~~		#REDIRECT [[Arrhenius_Relationship]]


	~~==Parameter Estimation==~~
	~~===Maximum Likelihood Estimation Method===~~
	~~<br>~~
	~~The Arrhenius-Weibull log-likelihood function is as follows:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left~~[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}} \right] \\
	& & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \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>~~

	~~<br>~~
	~~where:~~

	~~<br>~~
	~~::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>~~


	~~<br>~~
	~~::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>~~

	~~<br>~~
	~~and~~
	~~<br>~~
	~~::• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.~~
	~~::• .. is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.~~
	~~::• <math>\beta </math> is the Weibull shape parameter (unknown, the first of three parameters to be estimated).~~
	~~::• <math>B</math> is the Arrhenius parameter (unknown, the second of three parameters to be estimated).~~
	~~::• <math>C</math> is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).~~
	~~::• <math>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.~~
	~~::• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.~~
	~~::• <math>S</math> is the number of groups 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.~~
	~~::• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.~~
	~~::• <math>FI</math> is the number of interval data groups.~~
	~~::• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.~~
	~~::• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.~~
	~~::• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.~~

	The solution (parameter estimates) will be found by solving for <math>\widehat{\beta },</math> <math>\widehat{B},</math> <math>\widehat{C}</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math> <math>\tfrac{\partial \Lambda }{\partial B}=0</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:

	~~<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( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
	& -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
	& -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
	& \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Li}^{\prime \prime }-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
	~~\end{align}</math>~~



	~~<br>~~
	~~::<math>\begin{align}~~
	\frac{\partial \Lambda }{\partial B}= & -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}+\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}{{\left( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }} \\
	& +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
	~~\end{align}</math>~~



	~~<br>~~
	~~::<math>\begin{align}~~
	\frac{\partial \Lambda }{\partial C}= & -\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} \\
	& +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
	~~\end{align}</math>~~

Pantelis at 15:21, 16 July 2011

2011-07-16T15:21:05Z

← Older revision		Revision as of 15:21, 16 July 2011
Line 1:		Line 1:
	====Maximum Likelihood Estimation Method====		{{InProgress\| }}


			==Parameter Estimation==
			===Maximum Likelihood Estimation Method===
	<br>		<br>
	The Arrhenius-Weibull log-likelihood function is as follows:		The Arrhenius-Weibull log-likelihood function is as follows:

Pantelis: Created page with '====Maximum Likelihood Estimation Method====
The Arrhenius-Weibull log-likelihood function is as follows:
:: $\begin{align} & \Lambda = & \underset{i=1}{\overset…'$

2011-07-16T14:53:52Z

Created page with '====Maximum Likelihood Estimation Method==== The Arrhenius-Weibull log-likelihood function is as follows: ::<math>\begin{align} & \Lambda = & \underset{i=1}{\overset…'

New page

====Maximum Likelihood Estimation Method====
 
The Arrhenius-Weibull log-likelihood function is as follows:

 
::<math>\begin{align}
& \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}} \right] \\
& & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \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>

 
where:

 
::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>

 
::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}</math>

 
and
 
::• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
::• .. is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.
::• <math>\beta </math> is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
::• <math>B</math> is the Arrhenius parameter (unknown, the second of three parameters to be estimated).
::• <math>C</math> is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).
::• <math>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.
::• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
::• <math>S</math> is the number of groups 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.
::• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
::• <math>FI</math> is the number of interval data groups.
::• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.
::• <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.
::• <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.

The solution (parameter estimates) will be found by solving for <math>\widehat{\beta },</math> <math>\widehat{B},</math> <math>\widehat{C}</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,</math> <math>\tfrac{\partial \Lambda }{\partial B}=0</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:

 
::<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( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
& -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
& -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
& \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Li}^{\prime \prime }-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
\end{align}</math>

 
::<math>\begin{align}
\frac{\partial \Lambda }{\partial B}= & -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}+\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}{{\left( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }} \\
& +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
\end{align}</math>

 
::<math>\begin{align}
\frac{\partial \Lambda }{\partial C}= & -\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} \\
& +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
\end{align}</math>

Arrhenius-Weibull MLE - Revision history

Lisa Hacker: Redirected page to Arrhenius Relationship

Pantelis at 15:21, 16 July 2011

Pantelis: Created page with '====Maximum Likelihood Estimation Method==== The Arrhenius-Weibull log-likelihood function is as follows: ::\begin{align} & \Lambda = & \underset{i=1}{\overset…'

Pantelis: Created page with '====Maximum Likelihood Estimation Method====
The Arrhenius-Weibull log-likelihood function is as follows:
:: $\begin{align} & \Lambda = & \underset{i=1}{\overset…'$