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:


 * $${{R}_{1,...,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,{{\rho }_{i}}\cdot {{e}^{-{{\left( \tfrac{T}{{{\eta }_{i}}} \right)}^}}}$$
 * where:


 * $$\underset{i=1}{\overset{S}{\mathop \sum }}\,{{\rho }_{i}}=1$$

to the parameters $$\widehat$$   $$\widehat,$$   $$\widehat,$$   $$\widehat\widehat,$$   $$\widehat,...,$$   $$\widehat{{{\rho }_{S,}}\text{ }}\widehat,$$   $$\widehat,$$  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. $$RRX$$  or  $$RRY,$$  is of no consequence since non-linear regression is utilized.