Cumulative Damage Model for Progress Stress Profiles: Difference between revisions
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'''Validate the Likelihood Value Calculation in ALTA''' | '''Validate the Likelihood Value Calculation in ALTA''' | ||
For the ML solution given in the book, the log likelihood value is -1035.4269. | For the ML solution given in the book, the log likelihood value is -1035.4269. To validate what the LK value would be if the parameter values used in the book were calculated in ALTA, we use the Alter Parameters tool, as shown next. | ||
[[Image: Cum Damage_Stress Profile_alter parameters.png|center]] | |||
The resulting LK Value is -1489.6904, as shown next. | |||
[[Image: Cum Damage_Stress Profile_alter parameters results.png|center]] | |||
We need to validate the calculation for the log likelihood value in ALTA to make sure it is correct. | |||
First, we define the life characteristic of the Weibull distribution as a function of the two stresses: | |||
::<math>C = \int^{t}_{0} \frac{1}{\eta(V,A)} dV = \int^{t}_{0} e^{-(\alpha_{0}+\alpha_{1} \cdot ln(R \cdot V)+\alpha_{2} \cdot ln A)} dV\,\!</math> | |||
We get the equation for C as: | |||
::<math>C = e^{-\alpha_{0}} \cdot A^{-\alpha_{2}} \cdot R^{-\alpha_{1}} \cdot ]frac{1}{1-\alpha_{1}} \cdot t^{1-\alpha_{1}}\,\!</math> | |||
where ''R'' is the ramping rate for the voltage. | |||
Therefore, the probability density function at failure time ''t'' is: | |||
::<math>f(t) = \beta \cdot e^{-C^{\beta}} \cdot C^{\beta} \cdot (1-\alpha_{1}) \cdot \frac{1}{t}\,\!</math> | |||
The following table shows the value for the pdf and its log transformation for the first several failure times. The log likelihood value for all other failure times can be calculated using the same method. The total log likelihood value is the sum of all the individual log likelihood values. | |||
{| {{table}} | |||
| Time to Failure||Voltage Ramp Rate||Area||f(t)||log-f(t) | |||
|- | |||
| 41||10||1||0.069249||-2.67004 | |||
|- | |||
| 43||10||1||0.094154||-2.36283 | |||
|- | |||
| 42||10||1||0.08244||-2.49568 | |||
|- | |||
| 43||10||1||0.094154||-2.36283 | |||
|- | |||
| 44||10||1||0.102165||-2.28117 | |||
|- | |||
| 40||10||1||0.05623||-2.8783 | |||
|- | |||
| 38||10||1||0.034243||-3.37427 | |||
|- | |||
| 47||10||1||0.083568||-2.4821 | |||
|- | |||
| 43||10||1||0.094154||-2.36283 | |||
|- | |||
| 45||10||1||0.104043||-2.26295 | |||
|- | |||
| …||…||…||…||… | |||
|- | |||
|colspan="4" style="text-align: right;"|'''Total'''||-1035.42696 | |||
|- | |||
|} | |||
The total likelihood value is LKV = -2.67004 -2.36283--2.49568…= -1035.42696. | |||
We also can use the optimization tool in Excel to find the ML solutions for this data set. By using the GRG-nonlinear optimization method provided by the Solver in Excel, the result in Excel is the same as the result in ALTA. |
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