Template:Mixed weibull regression solution: Difference between revisions
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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: | 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}}( | ::<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> | ::<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. | 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. | ||
Revision as of 18:12, 14 February 2012
Regression Solution
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]\displaystyle{ {{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]\displaystyle{ \underset{i=1}{\overset{S}{\mathop \sum }}\,{{\rho }_{i}}=1 }[/math]
to the parameters [math]\displaystyle{ \widehat{{{\rho }_{1,\text{ }}}} }[/math] [math]\displaystyle{ \widehat{{{\beta }_{1}}}, }[/math] [math]\displaystyle{ \widehat{{{\eta }_{1}}}, }[/math] [math]\displaystyle{ \widehat{{{\rho }_{2,\text{ }}}}\widehat{{{\beta }_{2}}}, }[/math] [math]\displaystyle{ \widehat{{{\eta }_{2}}},..., }[/math] [math]\displaystyle{ \widehat{{{\rho }_{S,}}\text{ }}\widehat{{{\beta }_{S}}}, }[/math] [math]\displaystyle{ \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]\displaystyle{ RRX }[/math] or [math]\displaystyle{ RRY, }[/math] is of no consequence since non-linear regression is utilized.