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| ==IPL-Lognormal==
| | #REDIRECT [[Inverse_Power_Law_(IPL)_Relationship#IPL-Lognormal]] |
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| The pdf for the Inverse Power Law relationship and the lognormal distribution is given next.
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| The pdf 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}'}</math>
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| The IPL-lognormal model pdf can be obtained first by setting <math>\breve{T}=L(V)</math> in the lognormal <math>pdf</math>. Therefore:
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| ::<math> \breve{T}=L(V)=\frac{1}{K \cdot V^n}</math>
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| or:
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| ::<math>e^{\overline{T'}}=\frac{1}{K \cdot V^n}</math>
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| Thus:
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| ::<math>\overline{T}'=-ln(K)-n ln(V) </math>
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| So the IPL-lognormal model <math>pdf</math> is:
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| ::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+ln(K)+n ln(V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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| ====IPL-Lognormal Statistical Properties Summary====
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| ====The Mean====
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| The mean life of the IPL-lognormal model (mean of the times-to-failure), <math>\bar{T}</math> , is given by:
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| ::<math>\bar{T}=\ {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}= {{e}^{{-ln(K)-nln(V)}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}</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 IPL-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)} = \sqrt{\left( {{e}^{2\left( -\ln (K)-n\ln (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 IPL-lognormal model is given by:
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| ::<math>\tilde{T}={{e}^{{\bar{T}}'-\sigma _{{{T}'}}^{2}}}={{e}^{-\ln (K)-n\ln (V)-\sigma _{{{T}'}}^{2}}}</math>
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| ====IPL-Lognormal Reliability====
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| The reliability for a mission of time T, starting at age 0, for the IPL-lognormal model is determined by:
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| <math>R(T,\,V)=\int_{T}^{\infty }f(t,\,V)dt</math>
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| or:
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| <math>R(T,\,V)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (K)+n\ln (V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>
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| ====Reliable Life====
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| 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 (K)-n\ln (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 },\,V \right) \right]</math>
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| and:
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| ::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',\,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 lognormal failure rate is given by:
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| ===Parameter Estimation===
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| ====Maximum Likelihood Estimation Method====
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| The complete IPL-lognormal log-likelihood function is:
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| [[Image:chapter8_171.gif|center]]
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| where:
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| [[Image:chapter8_172.gif|center]]
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| [[Image:chapter8_173.gif|center]]
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| and:
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| * Fe is the number of groups of exact times-to-failure data points.
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| * Ni is the number of times-to-failure data points in the ith time-to-failure data group.
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| * <math>s_{T'}</math> is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).
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| * <math>K</math> is the IPL parameter (unknown, the second of three parameters to be estimated).
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| * <math>n</math> is the second IPL parameter (unknown, the third of three parameters to be estimated).
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| * <math>Vi</math> is the stress level of the ith group.
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| * <math>Ti</math> is the exact failure time of the ith group.
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| * <math>S</math> is the number of groups of suspension data points.
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| * <math>N'_i</math> is the number of suspensions in the ith group of suspension data points.
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| * <math>T^{'}_{i}</math> is the running time of the ith suspension data group.
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| * <math>FI</math> is the number of interval data groups.
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| * is the number of intervals in the ith group of data intervals.
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| * is the beginning of the ith interval.
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| * is the ending of the ith interval.
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| The solution (parameter estimates) will be found by solving for , , so that = 0, = 0 and = 0:
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| [[Image:chapter8_202.gif|center]]
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| and:
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| [[Image:chapter8_203.gif|center]]
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| [[Image:chapter8_204.gif|center]]
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