Template:Ipl weibull

IPL-Weibull
The IPL-Weibull model can be derived by setting $$\eta =L(V)$$ in the Weibull $$pdf$$, yielding the following IPL-Weibull $$pdf$$:


 * $$f(t,V)=\beta K{{V}^{n}}{{\left( K{{V}^{n}}t \right)}^{\beta -1}}{{e}^{-{{\left( K{{V}^{n}}t \right)}^{\beta }}}}$$

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 $$\beta =1.$$

Mean or MTTF
The mean, $$\overline{T}$$  (also called  $$MTTF$$ ), of the IPL-Weibull model is given by:


 * $$\overline{T}=\frac{1}{K{{V}^{n}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)$$

where $$\Gamma \left( \tfrac{1}{\beta }+1 \right)$$  is the gamma function evaluated at the value of  $$\left( \tfrac{1}{\beta }+1 \right)$$.

Median
The median, $$\breve{T},$$ of the IPL-Weibull model is given by:
 * $$\breve{T}=\frac{1}{K{{V}^{n}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}$$

Mode
The mode, $$\tilde{T},$$  of the IPL-Weibull model is given by:
 * $$\tilde{T}=\frac{1}{K{{V}^{n}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}$$

Standard Deviation
The standard deviation, $${{\sigma }_{T}},$$  of the IPL-Weibull model is given by:


 * $${{\sigma }_{T}}=\frac{1}{K{{V}^{n}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}$$

IPL-Weibull Reliability Function
The IPL-Weibull reliability function is given by:


 * $$R(T,V)={{e}^{-{{\left( K{{V}^{n}}T \right)}^{\beta }}}}$$

Conditional Reliability Function
The IPL-Weibull conditional reliability function at a specified stress level is given by:


 * $$R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac$$


 * or:


 * $$R(T,t,V)={{e}^{-\left[ {{\left( K{{V}^{n}}\left( T+t \right) \right)}^{\beta }}-{{\left( K{{V}^{n}}T \right)}^{\beta }} \right]}}$$

Reliable Life
For the IPL-Weibull model, the reliable life, $${T}_{R}$$, of a unit for a specified reliability and starting the mission at age zero is given by:


 * $${{T}_{R}}=\frac{1}{K{{V}^{n}}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}$$

IPL-Weibull Failure Rate Function
The IPL-Weibull failure rate function, $$\lambda (T)$$, is given by:


 * $$\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\beta K{{V}^{n}}{{\left( K{{V}^{n}}T \right)}^{\beta -1}}$$

Maximum Likelihood Estimation Method
Substituting the inverse power law relationship into the Weibull log-likelihood function yields:


 * $$\begin{align}

\Lambda = \underset{i=1}{\overset{\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 }] \end{align}$$

where:


 * $$R_{Li}^{\prime \prime }={{e}^{-{{\left( KV_{i}^{n}T_{Li}^{\prime \prime } \right)}^{\beta }}}}$$


 * $$R_{Ri}^{\prime \prime }={{e}^{-{{\left( KV_{i}^{n}T_{Ri}^{\prime \prime } \right)}^{\beta }}}}$$

and: •	 $${{F}_{e}}$$ is the number of groups of exact times-to-failure data points. •	 $${{N}_{i}}$$ is the number of times-to-failure data points in the  $${{i}^{th}}$$  time-to-failure data group. •	 $$\beta $$ is the Weibull shape parameter (unknown, the first of three parameters to be estimated). •	 $${K}$$ is the IPL parameter (unknown, the second of three parameters to be estimated). •	 $${n}$$ is the second IPL parameter (unknown, the third of three parameters to be estimated). •	 $${{V}_{i}}$$ is the stress level of the  $${{i}^{th}}$$  group. •	 $${{T}_{i}}$$ is the exact failure time of the  $${{i}^{th}}$$  group. •	 $$S$$ is the number of groups of suspension data points. •	 $$N_{i}^{\prime }$$ is the number of suspensions in the $${{i}^{th}}$$  group of suspension data points. •	 $$T_{i}^{\prime }$$ is the running time of the  $${{i}^{th}}$$  suspension data group. •	 $$FI$$ is the number of interval data groups. •	 $$N_{i}^{\prime \prime }$$ is the number of intervals in the  $${{i}^{th}}$$  group of data intervals. •	 $$T_{Li}^{\prime \prime }$$ is the beginning of the  $${{i}^{th}}$$  interval. •	 $$T_{Ri}^{\prime \prime }$$ is the ending of the  $${{i}^{th}}$$  interval. The solution (parameter estimates) will be found by solving for $$\beta ,$$   $$K,$$   $$n$$  so that  $$\tfrac{\partial \Lambda }{\partial \beta }=0$$,  $$\tfrac{\partial \Lambda }{\partial K}=0$$  and    $$\tfrac{\partial \Lambda }{\partial n}=0$$, where:
 * $$\begin{align}\frac{\partial \Lambda }{\partial \beta }=\ & \frac{1}{\beta }\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}\ln \left( KV_{i}^{n}{{T}_{i}} \right) -\underset{i=1}{\overset{\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 }} \\

\frac{\partial \Lambda }{\partial K}=\ & \frac{\beta }{K}\underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{\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 }} \\ \frac{\partial\Lambda }{\partial n}=\ & \beta \underset{i=1}{\overset{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) -\beta \underset{i=1}{\overset{\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 }} \end{align}$$