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==IPL-Lognormal==
#REDIRECT [[Inverse_Power_Law_(IPL)_Relationship#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:
 
::<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>
 
 
where:
 
::<math>T'=ln(T)</math>.
 
and:
 
:<math>T</math> = times-to-failure.
 
:<math>\overline{T}'</math> = mean of the natural logarithms of the times-to-failure.
 
:<math>\sigma_{T'}</math> = standard deviation of the natural logarithms of the times-to-failure.
 
The median of the lognormal distribution is given by:
 
::<math>\breve{T}=e^{\overline{T}'}</math>
 
 
The IPL-lognormal model pdf can be obtained first by setting  <math>\breve{T}=L(V)</math> in the lognormal <math>pdf</math>. Therefore:
 
 
::<math> \breve{T}=L(V)=\frac{1}{K \cdot V^n}</math>
 
 
or:
 
::<math>e^{\overline{T'}}=\frac{1}{K \cdot V^n}</math>
 
Thus:
 
::<math>\overline{T}'=-ln(K)-n ln(V) </math>
 
So the IPL-lognormal model <math>pdf</math> is:
 
::<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>
 
 
====IPL-Lognormal Statistical Properties Summary====
 
====The Mean====
 
The mean life of the IPL-lognormal model (mean of the times-to-failure), , is given by:
 
(9)
 
The mean of the natural logarithms of the times-to-failure,  , in terms of  and  is given by:
 
 
 
====The Standard Deviation====
 
The standard deviation of the IPL-lognormal model (standard deviation of the times-to-failure), , is given by:
 
(10)
 
The standard deviation of the natural logarithms of the times-to-failure, , in terms of  and  is given by:
 
 
 
====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:
 
 
 
===Parameter Estimation===
 
====Maximum Likelihood Estimation Method====
 
The complete IPL-lognormal log-likelihood function is:
 
[[Image:chapter8_171.gif|center]]
 
where:
 
[[Image:chapter8_172.gif|center]]
 
[[Image:chapter8_173.gif|center]]
 
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.
 
* <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).
 
* <math>K</math> is the IPL parameter (unknown, the second of three parameters to be estimated).
 
* <math>n</math> is the second IPL parameter (unknown, the third of three parameters to be estimated).
 
* <math>Vi</math> is the stress level of the ith group.
 
* <math>Ti</math> is the exact failure time of the ith group.
 
* <math>S</math> is the number of groups of suspension data points.
 
* <math>N'_i</math> is the number of suspensions in the ith group of suspension data points.
 
* <math>T^{'}_{i}</math> is the running time of the ith suspension data group.
 
* <math>FI</math> 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:
 
 
[[Image:chapter8_202.gif|center]]
 
and:
 
[[Image:chapter8_203.gif|center]]
 
[[Image:chapter8_204.gif|center]]

Latest revision as of 23:18, 15 August 2012

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