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 (see below)

3. Create the full composite design matrix (this will include a full factorial matrix)

4. Reduce the full factorial matrix to create the fractional factorial and one-factor-at-a-time designs

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