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| =Appendix 6.A: Arrhenius Confidence Bounds=
| | #REDIRECT [[Arrhenius_Relationship#Arrhenius_Confidence_Bounds]] |
| <br>
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| {{appr conf bounds for arr-exp}}
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| | |
| ==Approximate Confidence Bounds for the Arrhenius-Weibull:==
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| <br>
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| ===Bounds on the Parameters===
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| <br>
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| From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },</math> and <math>\widehat{C}</math> are positive parameters, <math>\ln (\widehat{\beta }),</math> and <math>\ln (\widehat{C})</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{\beta }_{U}}= & \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
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| & {{\beta }_{L}}= & \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}
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| \end{align}</math>
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| <br>
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| also:
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| <br>
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| ::<math>\begin{align}
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| & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
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| & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
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| \end{align}</math>
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| <br>
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| and:
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| <br>
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| ::<math>\begin{align}
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| & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\
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| & {{C}_{L}}= & \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}
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| \end{align}</math>
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| <br>
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| The variances and covariances of <math>\beta ,</math> <math>B,</math> and <math>C</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{\beta },</math> <math>\widehat{B},</math> <math>\widehat{C})</math> , as follows:
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| <br>
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| ::<math>\left[ \begin{matrix}
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| Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\
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| Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
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| Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
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| \end{matrix} \right]={{\left[ \begin{matrix}
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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|
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| ===Confidence Bounds on Reliability===
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| <br>
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| The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}}</math>
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| <br>
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| or:
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| <br>
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| ::<math>\widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}}</math>
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| <br>
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| Setting:
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| <br>
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| ::<math>\widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]</math>
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| <br>
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| or:
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| <br>
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| ::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right]</math>
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| <br>
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| The reliability function now becomes:
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| <br>
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| ::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}}</math>
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| <br>
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| The next step is to find the upper and lower bounds on <math>\widehat{u}\ \ :</math>
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| <br>
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| ::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| ::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| where:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
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| \end{align}</math>
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| <br>
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| or:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) \\
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| & & -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
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| \end{align}</math>
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| <br>
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| The upper and lower bounds on reliability are:
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| <br>
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| ::<math>\begin{align}
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| & {{R}_{U}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
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| & {{R}_{L}}(T,V)= & {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}
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| \end{align}</math>
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| where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated from Eqns. (ArreibRupper) and (ArreibRlower).
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| <br>
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| ===Confidence Bounds on Time===
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| <br>
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| The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:
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| <br>
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| ::<math>\begin{align}
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| & \ln (R)= & -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\
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| & \ln (-\ln (R))= & \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right)
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| \end{align}</math>
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| <br>
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| or:
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| <br>
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| ::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V}</math>
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| <br>
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| where <math>\widehat{u}=\ln \widehat{T}</math> .
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| <br>
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| The upper and lower bounds on <math>u</math> are estimated from:
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| ::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| ::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>
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| <br>
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| where:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) \\
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| & & +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
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| \end{align}</math>
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| <br>
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| or:
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| <br>
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| ::<math>\begin{align}
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| & Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C}) \\
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| & & -\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) \\
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| & & +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
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| \end{align}</math>
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| <br>
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| The upper and lower bounds on time can then found by:
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| <br>
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
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| & {{T}_{L}}= & {{e}^{{{u}_{L}}}}
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| \end{align}</math>
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| <br>
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| where <math>{{u}_{U}}</math> and <math>{{u}_{L}}</math> are estimated using Eqns. (ArreibTupper) and (ArreibTlower).
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| | |
| ==Approximate Confidence Bounds for the Arrhenius-Lognormal==
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| <br>
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| ===Bounds on the Parameters===
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| <br>
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| The lower and upper bounds on <math>B</math> are estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{B}_{U}}= & \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\
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| & {{B}_{L}}= & \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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| 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:
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| <br>
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| ::<math>\begin{align}
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| & {{C}_{U}}= & \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\
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| & {{C}_{L}}= & \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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| and:
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| <br>
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| ::<math>\begin{align}
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| & {{\sigma }_{U}}= & {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
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| & {{\sigma }_{L}}= & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}
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| \end{align}</math>
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| <br>
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| 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:
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| <br>
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| ::<math>\left[ \begin{matrix}
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| Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\
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| Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\
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| \end{matrix} \right]=</math>
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| <br>
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| ::<math>={{\left[ \begin{matrix}
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| -\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} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
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| -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
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| \end{matrix} \right]}^{-1}}</math>
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| <br>
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| ===Bounds on Reliability===
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| <br>
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| The reliability of the lognormal distribution is:
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| <br>
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| ::<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>
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| <br>
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| Let <math>\widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then ..
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| 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:
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| <br>
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| ::<math>R(\widehat{z})=\mathop{}_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>
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| <br>
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| The bounds on <math>z</math> are estimated from:
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| <br>
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| ::<math>\begin{align}
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| & {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\
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| & {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})}
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| \end{align}</math>
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| <br>
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| where:
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| <br>
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| ::<math>\begin{align}
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| & 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}}) \\
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| & & +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) \\
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| & & +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) \\
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| & & +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)
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| \end{align}</math>
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| <br>
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| or:
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| <br>
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| ::<math>\begin{align}
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| & 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}'}}}) \\
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| & & +\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)]
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| \end{align}</math>
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| <br>
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| The upper and lower bounds on reliability are:
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| <br>
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| ::<math>\begin{align}
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| & {{R}_{U}}= & \mathop{}_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\
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| & {{R}_{L}}= & \mathop{}_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)}
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| \end{align}</math>
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| ===Confidence Bounds on Time===
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| <br>
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| The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:
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| <br>
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| ::<math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}</math>
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| <br>
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| where:
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| <br>
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| ::<math>\begin{align}
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| & {T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})= & \ln (T) \\
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| & z= & {{\Phi }^{-1}}\left[ F({T}') \right]
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| \end{align}</math>
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| <br>
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| and:
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| <br>
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| ::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>
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| <br>
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| The next step is to calculate the variance of <math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}):</math>
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| <br>
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| ::<math>\begin{align}
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| & 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}'}}}) \\
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| & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\
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| & & +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| & & +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
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| \end{align}</math>
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|
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| <br>
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| or:
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| <br>
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| ::<math>\begin{align}
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| & Var({T}')= & \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) \\
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| & & +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) \\
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| & & +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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| & & +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
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| \end{align}</math>
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| | |
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| <br>
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| The upper and lower bounds are then found by:
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| | |
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| <br>
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| ::<math>\begin{align}
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| & T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\
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| & T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}
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| \end{align}</math>
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| <br>
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| Solving for <math>{{T}_{U}}</math> and <math>{{T}_{L}}</math> yields:
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| <br>
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| ::<math>\begin{align}
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| & {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\
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| & {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
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| \end{align}</math>
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| {{RS Copyright}}
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