Template:TNT Lognormal: Difference between revisions

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==T-NT Lognormal==
#REDIRECT [[Temperature-NonThermal_Relationship#T-NT_Lognormal]]
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The  <math>pdf</math>  of the lognormal distribution is given by:
 
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::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
 
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where:
 
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::<math>{T}'=\ln (T)</math>
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and:
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• <math>T=</math>  times-to-failure.
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• <math>\overline{{{T}'}}=</math>  mean of the natural logarithms of the times-to-failure.
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• <math>{{\sigma }_{{{T}'}}}=</math> standard deviation of the natural logarithms of the times-to-failure.
 
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The median of the lognormal distribution is given by:
 
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::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>
 
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The T-NT lognormal model <math>pdf</math> can be obtained by setting <math>\breve{T}=L(V)</math>. Therefore:
 
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::<math>\breve{T}=L(V)=\frac{C}{{{U}^{n}}}{{e}^{\tfrac{B}{V}}}</math>
 
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or:
 
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::<math>{{e}^{{{\overline{T}}^{\prime }}}}=\frac{C}{{{U}^{n}}}{{e}^{\tfrac{B}{V}}}</math>
 
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Thus:
 
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::<math>{{\overline{T}}^{\prime }}=\ln (C)-n\ln (U)+\frac{B}{V}</math>
 
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Substituting the above equation into the lognormal <math>pdf</math> yields the T-NT lognormal model  <math>pdf</math> or:
 
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::<math>f(T,U,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
 
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===T-N-T Lognormal Statistical Properties Summary===
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====The Mean====
• The mean life of the T-NT lognormal model (mean of the times-to-failure),  <math>\bar{T}</math> , is given by:
 
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::<math>\begin{align}
  & \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} = & {{e}^{\ln (C)-n\ln (U)+\tfrac{B}{V}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} 
\end{align}</math>
 
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• The mean of the natural logarithms of the times-to-failure,  <math>{{\bar{T}}^{^{\prime }}}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is given by:
 
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::<math>{{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)</math>
 
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====The Standard Deviation====
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• The standard deviation of the T-NT lognormal model (standard deviation of the times-to-failure),  <math>{{\sigma }_{T}}</math> , is given by:
 
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::<math>\begin{align}
  & {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} = & \sqrt{\left( {{e}^{2\left( \ln (C)-n\ln (U)+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} 
\end{align}</math>
 
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• The standard deviation of the natural logarithms of the times-to-failure,  <math>{{\sigma }_{{{T}'}}}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is given by:
 
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::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>
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====The Mode====
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• The mode of the T-NT lognormal model is given by:
 
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::<math>\begin{align}
  & \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} = & {{e}^{\ln (C)-n\ln (U)+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}} 
\end{align}</math>
 
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====T-NT Lognormal Reliability====
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For the T-NT lognormal model, the reliability for a mission of time  <math>T</math> , starting at age 0, for the T-NT lognormal model is determined by:
 
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::<math>R(T,U,V)=\mathop{}_{T}^{\infty }f(t,U,V)dt</math>
 
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:or:
 
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::<math>R(T,U,V)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>
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====Reliable Life====
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For the T-NT lognormal model, the reliable life, or the mission duration for a desired reliability goal,  <math>{{t}_{R}},</math>  is estimated by first solving the reliability equation with respect to time, as follows:
 
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::<math>T_{R}^{\prime }=\ln (C)-n\ln (U)+\frac{B}{V}+z\cdot {{\sigma }_{{{T}'}}}</math>
 
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:where:
 
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::<math>z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },U,V \right) \right]</math>
 
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:and:
 
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::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{z({T}',U,V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
 
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Since  <math>{T}'=\ln (T)</math>  the reliable life,  <math>{{t}_{R}}</math> , is given by:
 
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::<math>{{t}_{R}}={{e}^{T_{R}^{\prime }}}</math>
 
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====Lognormal Failure Rate====
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The T-NT lognormal failure rate is given by:
 
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::<math>\lambda (T,U,V)=\frac{f(T,U,V)}{R(T,U,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}</math>
 
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===Parameter Estimation===
====Maximum Likelihood Estimation Method====
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The complete T-NT lognormal log-likelihood function is:
 
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::<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 }_{pdf}}\left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)+n\ln ({{U}_{i}})-\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)+n\ln ({{U}_{i}})-\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>
 
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:where:
 
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::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-\ln C+n\ln U_{i}^{\prime \prime }-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>
 
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::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-\ln C+n\ln U_{i}^{\prime \prime }-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}</math>
 
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::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}</math>
 
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::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\mathop{}_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</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.
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• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
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• <math>{{\sigma }_{{{T}'}}}</math>  is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of four parameters to be estimated).
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• <math>B</math>  is the first T-NT parameter (unknown, the second of four parameters to be estimated).
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• <math>C</math>  is the second T-NT parameter (unknown, the third of four parameters to be estimated).
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• <math>n</math>  is the third T-NT parameter (unknown, the fourth of four parameters to be estimated).
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• <math>{{V}_{i}}</math>  is the stress level for the first stress type (i.e. temperature) of the  <math>{{i}^{th}}</math>  group.
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• <math>{{U}_{i}}</math>  is the stress level for the second stress type (i.e. non-thermal) of the  <math>{{i}^{th}}</math>  group.
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• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
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• <math>S</math>  is the number of groups of suspension data points.
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• <math>N_{i}^{\prime }</math>  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
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• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
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• <math>FI</math>  is the number of interval data groups.
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• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the  <math>{{i}^{th}}</math>  group of data intervals.
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• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the  <math>{{i}^{th}}</math>  interval.
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• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the  <math>{{i}^{th}}</math>  interval.
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The solution (parameter estimates) will be found by solving for  <math>{{\widehat{\sigma }}_{{{T}'}}},</math>  <math>\widehat{B},</math>  <math>\widehat{C},</math>  <math>\widehat{n}</math>  so that  <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial B}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial C}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial n}=0</math> .
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Latest revision as of 06:16, 15 August 2012

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