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:

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

or:

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

where:

and:

Since = ln(T) the reliable life, tR, is given by:

Lognormal Failure Rate
The lognormal failure rate is given by:

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



where:





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, ,  so that  = 0,  = 0 and  = 0:



and: