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| {{IPL-Lognormal}}
| | #REDIRECT [[Inverse_Power_Law_(IPL)_Relationship]] |
| ==IPL-Weibull==
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| <br>
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| The IPL-Weibull model can be derived by setting <math>\eta =L(V)</math> in the Weibull <math>pdf</math>, yielding the following IPL-Weibull <math>pdf</math>:
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| <br>
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| ::<math>f(t,V)=\beta K{{V}^{n}}{{\left( K{{V}^{n}}t \right)}^{\beta -1}}{{e}^{-{{\left( K{{V}^{n}}t \right)}^{\beta }}}}</math>
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| <br>
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| This is a three parameter model. Therefore it is more flexible but it also requires more laborious techniques for parameter estimation. The IPL-Weibull model yields the IPL-exponential model for <math>\beta =1.</math>
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| <br>
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| ===IPL-Weibull Statistical Properties Summary===
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| <br>
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| ====Mean or MTTF====
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| <br>
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| The mean, <math>\overline{T}</math> (also called <math>MTTF</math> ), of the IPL-Weibull model is given by:
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| <br>
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| ::<math>\overline{T}=\frac{1}{K{{V}^{n}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)</math>
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| <br>
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| where <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)</math> is the gamma function evaluated at the value of <math>\left( \tfrac{1}{\beta }+1 \right)</math> .
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| <br>
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| ====Median====
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| <br>
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| The median, <math>\breve{T},</math> of the IPL-Weibull model is given by:
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| <br>
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| ::<math>\breve{T}=\frac{1}{K{{V}^{n}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}</math>
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| <br>
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| ====Mode====
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| <br>
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| The mode, <math>\tilde{T},</math> of the IPL-Weibull model is given by:
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| <br>
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| ::<math>\tilde{T}=\frac{1}{K{{V}^{n}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}</math>
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| ====Standard Deviation====
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| <br>
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| The standard deviation, <math>{{\sigma }_{T}},</math> of the IPL-Weibull model is given by:
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| <br>
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| ::<math>{{\sigma }_{T}}=\frac{1}{K{{V}^{n}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}</math>
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| <br>
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| ====IPL-Weibull Reliability Function====
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| <br>
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| The IPL-Weibull reliability function is given by:
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| <br>
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| ::<math>R(T,V)={{e}^{-{{\left( K{{V}^{n}}T \right)}^{\beta }}}}</math>
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| ====Conditional Reliability Function====
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| <br>
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| The IPL-Weibull conditional reliability function at a specified stress level is given by:
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| <br>
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| ::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left[ K{{V}^{n}}\left( T+t \right) \right]}^{\beta }}}}}{{{e}^{-{{\left( K{{V}^{n}}T \right)}^{\beta }}}}}</math>
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| <br>
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| :or:
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| <br>
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| ::<math>R(T,t,V)={{e}^{-\left[ {{\left( K{{V}^{n}}\left( T+t \right) \right)}^{\beta }}-{{\left( K{{V}^{n}}T \right)}^{\beta }} \right]}}</math>
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| ====Reliable Life====
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| <br>
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| For the IPL-Weibull model, the reliable life, <math>{T}_{R}</math>, of a unit for a specified reliability and starting the mission at age zero is given by:
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| <br>
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| ::<math>{{T}_{R}}=\frac{1}{K{{V}^{n}}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}</math>
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| <br>
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| ====IPL-Weibull Failure Rate Function====
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| <br>
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| The IPL-Weibull failure rate function, <math>\lambda (T)</math> , is given by:
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| <br>
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| ::<math>\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\beta K{{V}^{n}}{{\left( K{{V}^{n}}T \right)}^{\beta -1}}</math>
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| ===Parameter Estimation===
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| <br>
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| ====Maximum Likelihood Estimation Method====
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| <br>
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| Substituting the inverse power law relationship into the Weibull log-likelihood function yields:
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| <br>
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| ::<math>\begin{align}
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| \Lambda = \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta KV_{i}^{n}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta -1}}{{e}^{-{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
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| \end{align}</math>
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| <br>
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| where:
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| <br>
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| ::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( KV_{i}^{n}T_{Li}^{\prime \prime } \right)}^{\beta }}}}</math>
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| <br>
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| ::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( KV_{i}^{n}T_{Ri}^{\prime \prime } \right)}^{\beta }}}}</math>
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| <br>
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| and:
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| <br>
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| • <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
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| <br>
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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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| <br>
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| • <math>\beta </math> is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
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| <br>
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| • <math>{K}</math> is the IPL parameter (unknown, the second of three parameters to be estimated).
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| <br>
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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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| <br>
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| • <math>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.
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| <br>
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| • <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
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| <br>
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| • <math>S</math> is the number of groups of suspension data points.
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| <br>
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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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| <br>
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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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| <br>
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| • <math>FI</math> is the number of interval data groups.
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| <br>
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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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| <br>
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| • <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.
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| <br>
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| • <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.
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| <br>
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| <br>
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| The solution (parameter estimates) will be found by solving for <math>\beta ,</math> <math>K,</math> <math>n</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0</math>, <math>\tfrac{\partial \Lambda }{\partial K}=0</math> and <math>\tfrac{\partial \Lambda }{\partial n}=0</math>, where:
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| <br>
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| <br>
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| ::<math>\begin{align}\frac{\partial \Lambda }{\partial \beta }=\ & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \,KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }}\ln \left( KV_{i}^{n}T_{i}^{\prime } \right) \\ & \overset{FI}{\mathop{\underset{i=1} {\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( KV_{i}^{n} \right)}^{\beta }}\left[ R_{Li}^{\prime \prime }T_{Li}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Li}^{\prime \prime }) \right)-R_{Ri}^{\prime \prime }T_{Ri}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Ri}^{\prime \prime }) \right) \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\
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| \frac{\partial \Lambda }{\partial K}=\ & \frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\frac{\beta }{K}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \overset{{}}{\mathop{-\beta \underset{i=1}{\mathop{\overset{FI}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{K}^{\beta -1}}V_{i}^{n\beta }\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\
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| \frac{\partial\Lambda }{\partial n}=\ & \beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} -\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{n{{K}^{\beta }}V_{i}^{\beta (n-1)}\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
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| \end{align}</math>
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| <br>
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