Template:Appr conf bounds for arr-weibull - Revision history

Richard House: Redirected page to Arrhenius Relationship#Approximate Confidence Bounds for the Arrhenius-Lognormal

2012-08-16T06:04:11Z

Redirected page to Arrhenius Relationship#Approximate Confidence Bounds for the Arrhenius-Lognormal

← Older revision		Revision as of 06:04, 16 August 2012
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	~~===Approximate Confidence Bounds for the Arrhenius~~-Lognormal~~===~~		#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Lognormal]]
	~~<br>~~

	~~{{acb-w on the parameters}}~~

	~~{{acb-w on reliability}}~~

	~~{{acb-w on time}}~~

Athanasios Gerokostopoulos: /* Approximate Confidence Bounds for the Arrhenius-Lognormal */

2012-02-14T00:03:12Z

Approximate Confidence Bounds for the Arrhenius-Lognormal

← Older revision		Revision as of 00:03, 14 February 2012
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	==Approximate Confidence Bounds for the Arrhenius-Lognormal==		===Approximate Confidence Bounds for the Arrhenius-Lognormal===
	<br>		<br>

Nicolette Young: /* Confidence Bounds on Time */

2012-01-12T22:21:34Z

Confidence Bounds on Time

← Older revision		Revision as of 22:21, 12 January 2012
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	{{acb-w on reliability}}		{{acb-w on reliability}}

	~~===Confidence Bounds on Time===~~		{{acb-w on time}}
	~~<br>~~
	~~The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:~~

	~~<br>~~
	~~::<math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}~~{~~V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>~~

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

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\~~
	~~& z= & {{\Phi }^~~{-~~1}}\left[ F({T}') \right]~~
	~~\end{align}</math>~~

	~~<br>~~
	~~and:~~

	~~<br>~~
	~~::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>~~

	~~<br>~~
	~~The next step is to calculate the variance of <math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}):</math>~~

	~~<br>~~
	~~::<math>\begin{align}~~
	& Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
	~~& & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\~~
	~~& & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\~~
	~~& & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)~~
	~~\end{align}</math>~~


	~~<br>~~
	~~or:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\~~
	~~& & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\~~
	~~& & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\~~
	~~& & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)~~
	~~\end{align}</math>~~


	~~<br>~~
	~~The upper and lower bounds are then found by:~~


	~~<br>~~
	~~::<math>\begin{align}~~
	~~& T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\~~
	~~& T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> yields:~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\~~
	~~& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}~~
	~~\end{align}</math>~~





	~~{{RS Copyright~~}}

Nicolette Young: /* Bounds on Reliability */

2012-01-12T22:20:55Z

Bounds on Reliability

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 {{acb-w on the parameters}}
-===Bounds on Reliability===
+{{acb-w on reliability}}
 ===Confidence Bounds on Time===

Nicolette Young: /* Bounds on the Parameters */

2012-01-12T22:20:17Z

Bounds on the Parameters

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 <br>
-===Bounds on the Parameters===
+{{acb-w on the parameters}}
 ===Bounds on Reliability===

Nicolette Young: Created page with '==Approximate Confidence Bounds for the Arrhenius-Lognormal==
===Bounds on the Parameters===
The lower and upper bounds on $B$ are estimated from:
…'

2012-01-12T22:19:29Z

Created page with '==Approximate Confidence Bounds for the Arrhenius-Lognormal== ===Bounds on the Parameters=== The lower and upper bounds on <math>B</math> are estimated from: …'

New page

==Approximate Confidence Bounds for the Arrhenius-Lognormal==
 

===Bounds on the Parameters===

 
The lower and upper bounds on <math>B</math> are estimated from:

 
::<math>\begin{align}
& {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\
& {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)}
\end{align}</math>

 
Since the standard deviation, <math>{{\widehat{\sigma }}_{{{T}'}}}</math> , and the parameter <math>C</math> are positive parameters, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math> and <math>\ln (C)</math> are treated as normally distributed. The bounds are estimated from:

 
::<math>\begin{align}
& {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\
& {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)}
\end{align}</math>

 
and:

 
::<math>\begin{align}
& {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
& {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}
\end{align}</math>

 
The variances and covariances of <math>B,</math> <math>C,</math> and <math>{{\sigma }_{{{T}'}}}</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B},</math> <math>\widehat{C}</math> , <math>{{\widehat{\sigma }}_{{{T}'}}}),</math> as follows:
 

::<math>\left[ \begin{matrix}
Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\
Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\
\end{matrix} \right]=</math>
 
::<math>={{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}</math>
 

===Bounds on Reliability===
 
The reliability of the lognormal distribution is:

 
::<math>R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\mathop{}_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt</math>

 
Let <math>\widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then ..
For <math>t={T}'</math> , <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}</math> , and for <math>t=\infty ,</math> <math>\widehat{z}=\infty .</math> The above equation then becomes:

 
::<math>R(\widehat{z})=\mathop{}_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

 
The bounds on <math>z</math> are estimated from:

 
::<math>\begin{align}
& {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\
& {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})}
\end{align}</math>

 
where:

 
::<math>\begin{align}
& Var(\widehat{z})= & \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) \\
& & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\
& & +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) \\
& & +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right)
\end{align}</math>

 
or:

 
::<math>\begin{align}
& Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
& & +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)]
\end{align}</math>

 
The upper and lower bounds on reliability are:

 
::<math>\begin{align}
& {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\
& {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)}
\end{align}</math>

===Confidence Bounds on Time===
 
The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

 
::<math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>

 
where:

 
::<math>\begin{align}
& {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\
& z= & {{\Phi }^{-1}}\left[ F({T}') \right]
\end{align}</math>

 
and:

 
::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

 
The next step is to calculate the variance of <math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}):</math>

 
::<math>\begin{align}
& Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
& & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\
& & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
& & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
\end{align}</math>

 
or:

 
::<math>\begin{align}
& Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
& & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\
& & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
& & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
\end{align}</math>

 
The upper and lower bounds are then found by:

 
::<math>\begin{align}
& T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\
& T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}
\end{align}</math>

 
Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> yields:

 
::<math>\begin{align}
& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\
& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
\end{align}</math>

{{RS Copyright}}