Template:Ipl lognormal

IPL-Lognormal
The pdf for the Inverse Power Law relationship and the lognormal distribution is given next.

The pdf of the lognormal distribution is given by:


 * $$f(T)=\frac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline}{{{\sigma }_}} \right)}^{2}}}}$$

where:


 * $$T'=ln(T)$$.

and:


 * $$T$$ = times-to-failure.


 * $$\overline{T}'$$ = mean of the natural logarithms of the times-to-failure.


 * $$\sigma_{T'}$$ = standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:


 * $$\breve{T}=e^{\overline{T}'}$$

The IPL-lognormal model pdf can be obtained first by setting $$\breve{T}=L(V)$$ in the lognormal $$pdf$$. Therefore:


 * $$ \breve{T}=L(V)=\frac{1}{K \cdot V^n}$$

or:


 * $$e^{\overline{T'}}=\frac{1}{K \cdot V^n}$$

Thus:


 * $$\overline{T}'=-ln(K)-n ln(V) $$

So the IPL-lognormal model $$pdf$$ is:


 * $$f(T,V)=\frac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+ln(K)+n ln(V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}$$

The Mean
The mean life of the IPL-lognormal model (mean of the times-to-failure), $$\bar{T}$$, is given by:


 * $$\bar{T}=\ {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _^{2}}}= {{e}^{{-ln(K)-nln(V)}+\tfrac{1}{2}\sigma _^{2}}}$$

The mean of the natural logarithms of the times-to-failure, $${{\bar{T}}^{^{\prime }}}$$, in terms of $$\bar{T}$$ and $${{\sigma }_{T}}$$ is given by:


 * $${{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}+1 \right)$$

The Standard Deviation
The standard deviation of the IPL-lognormal model (standard deviation of the times-to-failure), $${{\sigma }_{T}}$$, is given by:


 * $$\begin{align}

{{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _^{2}}} \right)\,\left( {{e}^{\sigma _^{2}}}-1 \right)} = \sqrt{\left( {{e}^{2\left( -\ln (K)-n\ln (V) \right)+\sigma _^{2}}} \right)\,\left( {{e}^{\sigma _^{2}}}-1 \right)} \end{align}$$

The standard deviation of the natural logarithms of the times-to-failure, $${{\sigma }_}$$, in terms of $$\bar{T}$$ and $${{\sigma }_{T}}$$ is given by:


 * $${{\sigma }_}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}+1 \right)}$$

The Mode
The mode of the IPL-lognormal model is given by:


 * $$\tilde{T}={{e}^{{\bar{T}}'-\sigma _^{2}}}={{e}^{-\ln (K)-n\ln (V)-\sigma _^{2}}}$$

IPL-Lognormal Reliability
The reliability for a mission of time T, starting at age 0, for the IPL-lognormal model is determined by:


 * $$R(T,\,V)=\int_{T}^{\infty }f(t,\,V)dt$$

or:


 * $$R(T,\,V)=\int_^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t+\ln (K)+n\ln (V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt$$

Reliable Life
The reliable life, or the mission duration for a desired reliability goal, $${{t}_{R}},$$ is estimated by first solving the reliability equation with respect to time, as follows:


 * $$T_{R}^{\prime }=-\ln (K)-n\ln (V)+z\cdot {{\sigma }_}$$

where:


 * $$z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },\,V \right) \right]$$

and:


 * $$\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',\,V)}{{e}^{-\tfrac{2}}}dt$$

Since $${T}'=\ln (T)$$ the reliable life, $${{t}_{R}},$$, is given by:


 * $${{t}_{R}}={{e}^{T_{R}^{\prime }}}$$

Lognormal Failure Rate
The lognormal failure rate is given by:


 * $$\lambda (T,\,V)=\frac{f(T,\,V)}{R(T,\,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (K)+n\ln (V)}{{{\sigma }_}} \right)}^{2}}}}}{\int_^{\infty }\tfrac{1}{{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'+\ln (K)+n\ln (V)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}$$

Maximum Likelihood Estimation Method
The complete IPL-lognormal log-likelihood function is:


 * $$\begin{align}

\ln (L)= & \Lambda =\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}\varphi \left( \frac{\ln \left( {{T}_{i}} \right)+\ln (K)+n\ln ({{V}_{i}})} \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}})} \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 })] \end{align}$$

where:


 * $$z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }+\ln K+n\ln {{V}_{i}}}{\sigma _{T}^{\prime }}$$


 * $$z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }+\ln K+n\ln {{V}_{i}}}{\sigma _{T}^{\prime }}$$

and:
 * Fe is the number of groups of exact times-to-failure data points.


 * Ni is the number of times-to-failure data points in the ith time-to-failure data group.


 * $$s_{T'}$$ is the standard deviation of the natural logarithm of the times-to-failure (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).


 * $$Vi$$ is the stress level of the ith group.


 * $$Ti$$ is the exact failure time of the ith group.


 * $$S$$ is the number of groups of suspension data points.


 * $$N'_i$$ is the number of suspensions in the ith group of suspension data points.


 * $$T^{'}_{i}$$ is the running time of the ith suspension data group.


 * $$FI$$ is the number of interval data groups.


 * is the number of intervals in the ith group of data intervals.


 * is the beginning of the ith interval.


 * is the ending of the ith interval.

The solution (parameter estimates) will be found by solving for $$ {{\hat {\sigma}}_}$$, $$\hat {K}$$, $$\hat {n}$$ so that $$\tfrac{\partial \Lambda }{\partial {{\sigma }_}}=0,$$, $$\tfrac{\partial \Lambda }{\partial K}=0$$ and  $$\tfrac{\partial \Lambda }{\partial n}=0\ \ :$$:


 * $$\begin{align}

\frac{\partial \Lambda }{\partial K}= & -\frac{1}{K\cdot \sigma _^{2}}\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})+\ln (K)+n\ln ({{V}_{i}})) \ -\frac{1}{K\cdot {{\sigma }_}}\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}})} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})} \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 }))} \\ \frac{\partial \Lambda }{\partial n}= & -\frac{1}{\sigma _^{2}}\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln ({{V}_{i}})\left[ \ln ({{T}_{i}})+\ln (K)+n\ln ({{V}_{i}}) \right] -\frac{1}\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}})} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})} \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 }))} \\ \frac{\partial \Lambda }{\partial {{\sigma }_}}= & \underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\left( \frac{\sigma _^{3}}-\frac{1} \right) \ +\frac{1}\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}})} \right)\,\varphi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)+\ln (K)+n\ln ({{V}_{i}})} \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 }))} \end{align}$$

and:


 * $$\varphi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}$$


 * $$\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{2}}}dt$$