Template:Mixed weibull regression solution

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.