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| ==IPL-Lognormal==
| | #REDIRECT [[Inverse_Power_Law_(IPL)_Relationship#IPL-Lognormal]] |
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| The <math>pdf</math> for the Inverse Power Law relationship and the lognormal distribution is given next.
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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}'}</math>
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| The IPL-lognormal model <math>pdf</math> 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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| ::<math>\lambda (T,\,V)=\frac{f(T,\,V)}{R(T,\,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (K)+n\ln (V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{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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| ===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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| ::<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}}}\varphi \left( \frac{\ln \left( {{T}_{i}} \right)+\ln (K)+n\ln ({{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 (K)+n\ln ({{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 })]
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| \end{align}</math>
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| where:
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| ::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }+\ln K+n\ln {{V}_{i}}}{\sigma _{T}^{\prime }}</math>
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| ::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }+\ln K+n\ln {{V}_{i}}}{\sigma _{T}^{\prime }}</math>
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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 <math>{i}^{th}</math> 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 <math>{i}^{th}</math> group.
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| * <math>Ti</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</math> is the number of suspensions in the <math>{i}^{th}</math> group of suspension data points.
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| * <math>T^{'}_{i}</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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| * is the number of intervals in the <math>{i}^{th}</math> group of data intervals.
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| * is the beginning of the <math>{i}^{th}</math> interval.
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| * 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> {{\hat {\sigma}}_{{{T}'}}}</math>, <math>\hat {K}</math>, <math>\hat {n}</math> so that <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,</math>, <math>\tfrac{\partial \Lambda }{\partial K}=0</math> and <math>\tfrac{\partial \Lambda }{\partial n}=0\ \ :</math>:
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| ::<math>\begin{align}
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| \frac{\partial \Lambda }{\partial K}= & -\frac{1}{K\cdot \sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})+\ln (K)+n\ln ({{V}_{i}})) \ -\frac{1}{K\cdot {{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\varphi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{+\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\phi (z_{Ri}^{\prime \prime })-\phi (z_{Li}^{\prime \prime })}{K\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
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| \frac{\partial \Lambda }{\partial n}= & -\frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln ({{V}_{i}})\left[ \ln ({{T}_{i}})+\ln (K)+n\ln ({{V}_{i}}) \right] -\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln ({{V}_{i}})\frac{\varphi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\ln {{V}_{i}}\left( \phi (z_{Ri}^{\prime \prime })-\phi (z_{Li}^{\prime \prime }) \right)}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
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| \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})+\ln (K)+n\ln ({{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 (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)\,\varphi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{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 }\phi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\phi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
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| \end{align}</math>
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| and:
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| ::<math>\varphi \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 }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt</math>
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