Convert a quadratic L_Pseudo mixture model to Real
The following information is required for conversion from pseudo to real.
Convert Pseudo limits (0 = minimum, 1 = maximum) to proportion (real) limits.
Low |
High |
|
A |
AL |
AH |
B |
BL |
BH |
C |
CL |
CH |
Compute the sum of the low real settings (\(\sum_L\)). It is used for L_pseudo conversion.
The Pseudo model example:
\(Quadratic\, Pseudo = \beta_1A\, +\, \beta_2B\, +\, \beta_3C\, +\, \beta_{12}AB\, +\, \beta_{13}AC\, +\, \beta_{23}BC\)
Insignificant terms with a near zero coefficient should be included in this model.
\(L = \displaystyle\sum_{i=1}^q L_i\)
Rewrite the model substituting…
\(\frac{X_i\, -\, L_i\,}{1\, -\, L}\) for each component, while replacing the X with the component ID being replaced.
The rewrite of the Pseudo Example is:
Expand all terms and combine like terms, starting with higher order terms first.
Quadratic term expansion
Showing the BC quadratic term as an example. Use this procedure for each quadratic term.
\(\beta_{23}\frac{B\, -\, B_L}{1\, -\, L}\frac{C\, -\, C_L}{1\, -\, L}\)
\(\beta_{23}\frac{BC\, -\, B_LC\, -\, C_LB\, +\, B_LC_L}{(1\, -\, L)^2}\)
\(\beta_{23}\begin{bmatrix}\frac{BC}{(1\, -\, L)^2}\, +\, \frac{-B_LC}{(1\, -\, L)^2}\, +\, \frac{-C_LB}{(1\, -\, L)^2}\, +\, \frac{B_LC_L}{(1\, -\, L)^2} \end{bmatrix}\)
The BC coefficient is changed to \(\beta_{23}\,/\,(1\, -\, L)^2\) in the real model.
\(-\beta_{23}B_L\,/\,(1\, -\, L)^2\) is a correction that will be applied to the C coefficient.
\(-\beta_{23}C_L\,/\,(1\, -\, L)^2\) is a correction that will be applied to the B coefficient.
\(B_LC_L\,/\,(1\, -\, L)^2\) is a constant which requires special handling.
From the mixture design property of a constant total, we know that \(A\, +\, B\, +\, C\, =\, 1\) in terms of the reals. Rewrite \(B_LC_L\) as \(B_LC_L\, \cdot\, 1\) and substitute \([A\, +\, B\, +\, C]\) for 1, yielding \(B_LC_L\, \cdot\, [A\, +\, B\, +\, C]\). When expanded, the result is an adjustment to all the linear coefficients of \(\beta_{23}B_LC_L\ /\, (1\, -\, L)^2\).
Linear term expansion
Showing the C term as the example. Use this procedure for all the linear terms.
\(\beta_3\frac{C\, -\, C_L}{1\, -\, L}\, =\, \beta_3\begin{bmatrix}\frac{C}{1\, -\, L}\, +\, \frac{-C_L}{1\, -\, L}\end{bmatrix}\)
\(\beta_3\,/\,(1\, -\, L)\) is the base coefficient for the C linear effect. This will be adjusted by quadratic and other linear effect adjustments.
\(-C_L\,/\,(1\, -\, L)\) is a constant which is treated the same as the quadratic term’s constant becoming, \(-\beta_3C_L\,/\,(1\, -\, L)\, \cdot\, [A\, +\, B\, +\, C]\). Each linear term creates an adjustment to all linear terms.
Combine Like Terms
After working through each term in the model, combine like terms into new coefficients for the real model.
\(\beta_A = \frac{\beta_1}{1\, -\, L}\, -\, \frac{\beta_1A_L\, +\, \beta_2B_L\, +\, \beta_3C_L}{1\, -\, L}\, +\, \frac{\beta_{12}(A_LB_L\, -\, B_L)\, +\, \beta_{13}(A_LC_L\, -\, C_L)\, +\, \beta_{23}(B_LC_L)}{(1\, -\, L)^2}\)
\(\beta_B = \frac{\beta_2}{1\, -\, L}\, -\, \frac{\beta_1A_L\, +\, \beta_2B_L\, +\, \beta_3C_L}{1\, -\, L}\, +\, \frac{\beta_{12}(A_LB_L\, -\, A_L)\, +\, \beta_{23}(B_LC_L\, -\, C_L)\, +\, \beta_{13}(A_LC_L)}{(1\, -\, L)^2}\)
\(\beta_C = \frac{\beta_3}{1\, -\, L}\, -\, \frac{\beta_1A_L\, +\, \beta_2B_L\, +\, \beta_3C_L}{1\, -\, L}\, +\, \frac{\beta_{13}(A_LC_L\, -\, A_L)\, +\, \beta_{23}(B_LC_L\, -\, B_L)\, +\, \beta_{23}(A_LB_L)}{(1\, -\, L)^2}\)
\(\beta_{AB} = \frac{\beta_{12}}{(1\, -\, L)^2}\)
\(\beta_{AC} = \frac{\beta_{13}}{(1\, -\, L)^2}\)
\(\beta_{BC} = \frac{\beta_{23}}{(1\, -\, L)^2}\)
References
J. Cornell. Experiments with Mixtures. Wiley, 3rd edition, 2002.