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==Arrhenius-Lognormal==
#REDIRECT [[Arrhenius_Relationship#Arrhenius-Lognormal]]
<br>
The  <math>pdf</math>  of the lognormal distribution is given by:
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<br>
::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\bar{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
 
<br>
where:
 
<br>
::<math>{T}'=\ln(T) </math>
 
<br>
and:
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• <math>T=</math>  times-to-failure.
 
• <math>{T}'=</math>  mean of the natural logarithms of the times-to-failure.
 
• <math>T=</math>  times-to-failure.
 
• <math>{{\sigma }_{{{T}'}}}=</math>  standard deviation of the natural logarithms of the times-to-failure.
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<br>
The median of the lognormal distribution is given by:
 
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::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>
 
<br>
 
The Arrhenius-lognormal model  <math>pdf</math>  can be obtained first by setting <math>\breve{T}=L(V)</math>  in Eqn. (arrhenius). Therefore:
 
::<math>\breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}}</math>
 
or:
 
::<math>{{e}^{{{\overline{T}}^{\prime }}}}=C{{e}^{\tfrac{B}{V}}}</math>
 
Thus:
 
::<math>{{\overline{T}}^{\prime }}=\ln (C)+\frac{B}{V}</math>
 
 
Substituting Eqn. (arrh-logn-mean) into Eqn. (arrh-logn-pdf) yields the Arrhenius-lognormal model  <math>pdf</math>  or:
 
::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
 
<br>
Note that in Eqn. (arrh-logn-pdf), it was assumed that the standard deviation of the natural logarithms of the times-to-failure,  <math>{{\sigma }_{{{T}'}}},</math>  is independent of stress. This assumption implies that the shape of the distribution does not change with stress ( <math>{{\sigma }_{{{T}'}}}</math>  is the shape parameter of the lognormal distribution).
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{{alta al stat prop sum}}
 
===Parameter Estimation===
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<br>
====Maximum Likelihood Estimation Method====
<br>
The lognormal log-likelihood function for the Arrhenius-lognormal model is as follows:
 
<br>
::<math>\begin{align}
  & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \\
&  & \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \\
&  & +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}</math>
 
<br>
where:
 
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::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-\ln C-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>
 
 
<br>
::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-\ln C-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>
 
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and:
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• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
 
• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
 
• ..  is the standard deviation of the natural logarithm of the times-to-failure (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{\sigma }}_{{{T}'}}},</math>  <math>\widehat{B},</math>  <math>\widehat{C}</math>  so that  <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=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 B}= & \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\frac{1}{{{V}_{i}}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}{{{V}_{i}}}) \\
&  & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }{{V}_{i}}(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} 
\end{align}</math>
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial C}= & \frac{1}{C\cdot \sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}{{{V}_{i}}}) \\
&  & +\frac{1}{C\cdot {{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }C(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
&  &  \\
& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-\ln (C)-\tfrac{B}{{{V}_{i}}} \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) \\
&  & +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} 
\end{align}</math>
 
<br>
and:
 
<br>
::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>
 
 
<br>
::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>

Latest revision as of 05:28, 16 August 2012

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