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. Each level will be coded with $ +2, +1, 0, -1, -2 $.

2. Create variable transforms (see below)

3. Create a one-factor-at-a-time design

4. Fill that in to create a fractional factorial design

5. Fill that in to create a full factorial design

6. Fill that in to create a full composite design

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

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Mass flowrate	k(T)	Lmix