Template:Acb-w on the parameters - Revision history

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

2012-08-16T06:05:25Z

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

← Older revision		Revision as of 06:05, 16 August 2012
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	~~====Bounds on the Parameters====~~		#REDIRECT [[Arrhenius_Relationship#Approximate_Confidence_Bounds_for_the_Arrhenius-Lognormal]]

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

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

	~~<br>~~
	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:

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

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

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

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

	~~::<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]= {{\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>~~
	~~<br>~~

Athanasios Gerokostopoulos at 01:18, 14 February 2012

2012-02-14T01:18:36Z

← Older revision		Revision as of 01:18, 14 February 2012
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	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{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) \\		Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\
	\end{matrix} \right]~~=</math>~~		\end{matrix} \right]= {{\left[ \begin{matrix}
	~~<br>~~
	~~::<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 \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 B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

2012-02-14T01:15:15Z

Bounds on the Parameters

← Older revision		Revision as of 01:15, 14 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\		& {{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)}		& {{B}_{L}}= \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)}
	\end{align}</math>		\end{align}</math>

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

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

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

2012-02-14T00:03:39Z

Bounds on the Parameters

← Older revision		Revision as of 00:03, 14 February 2012
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	===Bounds on the Parameters===		====Bounds on the Parameters====

	<br>		<br>

Nicolette Young: Created page with '===Bounds on the Parameters===
The lower and upper bounds on $B$ are estimated from:
:: $\begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\…'$

2012-01-12T22:20:26Z

Created page with '===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 }}\…'

New page

===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>

Template:Acb-w on the parameters - Revision history

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

Athanasios Gerokostopoulos at 01:18, 14 February 2012

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

Athanasios Gerokostopoulos: /* Bounds on the Parameters */

Nicolette Young: Created page with '===Bounds on the Parameters=== The lower and upper bounds on B are estimated from: ::\begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\…'

Nicolette Young: Created page with '===Bounds on the Parameters===
The lower and upper bounds on $B$ are estimated from:
:: $\begin{align} & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\…'$