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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====
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| • 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}
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| & \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}}}
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| \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}
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| & {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} \\
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| & = & \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)}
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| \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}
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| & \tilde{T}= & {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} \\
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| & = & {{e}^{\ln (C)-n\ln (U)+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}}
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| \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===
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| ====Maximum Likelihood Estimation Method====
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| The complete T-NT lognormal log-likelihood function is:
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| ::<math>\begin{align}
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| & \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] \\
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| & & \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] \\
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| & & +\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 })]
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| \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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