Composite Experimental Design

Overview

Composite experimental design refers to the successive sampling of parameter space in such a way as to construct a first or second order polynomial function.

Explanation

Setting Up the Whole Design

1. Select 5 (or 3) levels for each variable. Code each level with a numerical value, typically between $ -1,1 $ (but can be, e.g., between $ -2,2 $, see Box and Draper 1987).

2. Create variable transforms to translate between the coded levels and the actual input parameter values (see below)

3. Create the full composite design matrix

4. Parse the full factorial matrix from above

5. Parse the fractional factorial matrix from above

6. Parse the one-factor-at-a-time matrix from above

7. Sample function in the following order:

• One factor at a time
• Fractional factorial
• Full factorial
• Full composite

Variable Transforms

For a variable $ x_i $ with range $ \alpha_i \leq x_i \leq \beta_i $,

• the transformed variable $ \hat{x}_i $ has the range $ -1 \leq \hat{x}_i \leq +1 $ for factorial design

• the transformed variable $ \hat{x}_i $ has the range $ -2 \leq \hat{x}_i \leq +2 $ for composite design

Linear Variables

To transform a linear variable $ x_i $ to the variable $ \hat{x}_i \in [-1, +1] $:

$ \hat{x}_i = \frac{ x_i - \left( \frac{\beta_i - \alpha_i}{2} + \alpha_i \right) }{ \frac{\beta_i - \alpha_i}{2} } $

To transform a linear variable $ x_i $ to the variable $ \hat{x}_i \in [-2, +2] $:

$ \hat{x}_i = \frac{ x_i - \left( \frac{\beta_i - \alpha_i}{2} + \alpha_i \right) }{ \frac{\beta_i - \alpha_i}{4} } $

Log Variables

To transform a log variable $ x_i $ to the variable $ \hat{x}_i \in [-1, +1] $:

$ \hat{x}_i = \frac{ \log{(x_i)} - \left( \frac{ \log{(\beta_i)} - \log{(\alpha_i)}}{2} + \log{(\alpha_i)} \right) }{ \frac{ \log{(\beta_i)} - \log{(\alpha_i)} }{2} } $

To transform a log variable $ x_i $ to the variable $ \hat{x}_i \in [-2, +2] $:

$ \hat{x}_i = \frac{ \log{(x_i)} - \left( \frac{ \log{(\beta_i)} - \log{(\alpha_i)}}{2} + \log{(\alpha_i)} \right) }{ \frac{ \log{(\beta_i)} - \log{(\alpha_i)} }{4} } $

Example

Mass flowrate	k(T)	Lmix
+1	+1	+1
+1	+1	-1
+1	-1	+1
+1	-1	-1

>> help ff2n
 FF2N   Two-level full-factorial design.
    X = FF2N(N) creates a two-level full-factorial design, X.
    N is the number of columns of X. The number of rows is 2^N.

    Reference page in Help browser
       doc ff2n