|
|
| Line 1: |
Line 1: |
| '''Regression Solution'''
| | #REDIRECT [[The_Mixed_Weibull_Distribution#Mixed_Weibull_Parameter_Estimation]] |
| | |
| Weibull++ utilizes a modified Levenberg-Marquardt algorithm (non-linear regression) when performing regression analysis on a mixed Weibull distribution. The procedure is rather involved and is beyond the scope of this reference. It is sufficient to say that the algorithm fits a curved line of the form:
| |
| | |
| ::<math>{{R}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,{{\rho }_{i}}\cdot {{e}^{-{{\left( \tfrac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}}}}}</math>
| |
| where:
| |
| | |
| ::<math>\underset{i=1}{\overset{S}{\mathop \sum }}\,{{\rho }_{i}}=1</math>
| |
| | |
| to the parameters <math>\widehat{{{\rho }_{1,\text{ }}}}</math> <math>\widehat{{{\beta }_{1}}},</math> <math>\widehat{{{\eta }_{1}}},</math> <math>\widehat{{{\rho }_{2,\text{ }}}}\widehat{{{\beta }_{2}}},</math> <math>\widehat{{{\eta }_{2}}},...,</math> <math>\widehat{{{\rho }_{S,}}\text{ }}\widehat{{{\beta }_{S}}},</math> <math>\widehat{{{\eta }_{S}}},</math> utilizing the times-to-failure and their respective plotting positions. It is important to note that in the case of regression analysis, using a mixed Weibull model, the choice of regression axis, i.e. <math>RRX</math> or <math>RRY,</math> is of no consequence since non-linear regression is utilized.
| |