|
|
| (38 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| 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\sigma_{T'}\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{T'-\overline{T'}}{\sigma_{T'}})^2}</math>(6)
| |
| | |
| | |
| where:
| |
| | |
| :<math>T'=ln(T)</math> = ln(T).
| |
| | |
| :<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>(7)
| |
| | |
| | |
| The IPL-lognormal model pdf can be obtained first by setting = L(V) in Eqn. ( 30). Therefore:
| |
| | |
| | |
| <math> \breve{T}=L(V)=\frac{1}{K*V^n}</math>
| |
| | |
| | |
| or:
| |
| | |
| <math>e^{\overline{T'}}=\frac{1}{K*V^n}</math>
| |
| | |
| Thus:
| |
| | |
| <math>\overline{T}'=-ln(K)-n ln(V) </math>(8)
| |
| | |
| Substituting Eqn. (8) into Eqn. (6) yields the IPL- lognormal model pdf or:
| |
| | |
| <math>f(T,V)=\frac{1}{T\sigma_{T'} \sqrt{2 \pi} e^{\frac{1}{2}(\frac{t'+ln(K)+ln(V)}{\sigma_{T'}})^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:
| |
| | |
| | |
| | |
| 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.
| |
| 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.
| |
| is the number of suspensions in the ith group of suspension data points.
| |
| is the running time of the ith suspension data group.
| |
| FIis 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:
| |
| | |
| | |
| | |
| | |
| | |
| See Also:
| |