Template:Mixed weibull regression solution: Difference between revisions

Latest revision as of 00:38, 15 August 2012

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-====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.