Template:Cb for the eyring-log - Revision history

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

2012-08-17T01:12:26Z

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

← Older revision		Revision as of 01:12, 17 August 2012
Line 1:		Line 1:
	~~==Approximate Confidence Bounds for the Eyring~~-Lognormal==		#REDIRECT [[Eyring_Relationship#Approximate_Confidence_Bounds_for_the_Eyring-Lognormal]]
	~~<br>~~
	~~{{eyring-log cb on parameters}}~~

	~~{{eyring-log cb on reliability}}~~

	~~{{eyring-log cb on time}}~~

Nicolette Young: /* Confidence Bounds on Time */

2012-01-12T23:46:48Z

Confidence Bounds on Time

← Older revision		Revision as of 23:46, 12 January 2012
Line 5:		Line 5:
	{{eyring-log cb on reliability}}		{{eyring-log cb on reliability}}

	~~===Confidence Bounds on Time===~~		{{eyring-log cb 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{A},\widehat~~{~~B},~~{~~{\widehat{\sigma }}_{{{T}'}}})=~~-~~\ln (V)-\widehat{A}+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>~~

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

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& {T}'(V;\widehat{A},\widehat{B},{{\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{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}}):</math>~~

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

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

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

Nicolette Young: /* Bounds on Reliability */

2012-01-12T23:46:24Z

Bounds on Reliability

@@ Line 3: / Line 3: @@
 {{eyring-log cb on parameters}}
-===Bounds on Reliability===
+{{eyring-log cb on reliability}}
 ===Confidence Bounds on Time===

Nicolette Young: /* Bounds on the Parameters */

2012-01-12T23:45:35Z

Bounds on the Parameters

@@ Line 1: / Line 1: @@
 ==Approximate Confidence Bounds for the Eyring-Lognormal==
 <br>
-===Bounds on the Parameters===
+{{eyring-log cb on parameters}}
 ===Bounds on Reliability===

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

2012-01-12T23:40:58Z

Created page with '==Approximate Confidence Bounds for the Eyring-Lognormal== ===Bounds on the Parameters=== The lower and upper bounds on <math>A</math> and <math>B</math> are estima…'

New page

==Approximate Confidence Bounds for the Eyring-Lognormal==
 
===Bounds on the Parameters===
 
The lower and upper bounds on <math>A</math> and <math>B</math> are estimated from:

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

 
:and:

 
::<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> is a positive parameter, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math> is treated as normally distributed, and the bounds are estimated from:

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

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

 
:where:

 
::<math>F=\left( \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{A}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial A\partial B} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} \\
\end{matrix} \right)</math>

 

===Bounds on Reliability===

 
The reliability of the lognormal distribution is given by:

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

 
Let <math>\widehat{z}(t,V;A,B,{{\sigma }_{T}})=\tfrac{t+\ln (V)+\widehat{A}-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.</math>
 
For <math>t={T}'</math> , <math>\widehat{z}=\tfrac{{T}'+\ln (V)+\widehat{A}-\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}',V)}^{\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 A} \right)_{\widehat{A}}^{2}Var(\widehat{A})+\left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) \\
& & +2{{\left( \frac{\partial \widehat{z}}{\partial A} \right)}_{\widehat{A}}}{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}Cov\left( \widehat{A},\widehat{B} \right) \\
& & +2{{\left( \frac{\partial \widehat{z}}{\partial A} \right)}_{\widehat{A}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{A},{{\widehat{\sigma }}_{T}} \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)
\end{align}</math>

 
:or:

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

 
:where:

 
::<math>\begin{align}
& {T}'(V;\widehat{A},\widehat{B},{{\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{A},\widehat{B},{{\widehat{\sigma }}_{{{T}'}}}):</math>

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

 
:or:

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

Template:Cb for the eyring-log - Revision history

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

Nicolette Young: /* Confidence Bounds on Time */

Nicolette Young: /* Bounds on Reliability */

Nicolette Young: /* Bounds on the Parameters */

Nicolette Young: Created page with '==Approximate Confidence Bounds for the Eyring-Lognormal== ===Bounds on the Parameters=== The lower and upper bounds on A and B are estima…'

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