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

How Many Levels?

The question of whether to choose 3 or 5 levels depends entirely on the case.

Typically, 3-level designs are chosen for experiments where multiple levels create difficulty in experimental setup. In this case, the minimum number of levels is desirable.

However, in simulations, 5-level designs are best, because there is no significant effort on the part of the user when running with a large number of levels.

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

Problem Information

For details about the problem, including the input uncertainty map, see Example Problem for Experimental Design

Code

Main article: Composite Experimental Design Matlab Code

Computing Response Surface

See Response Surface Methodology for general information on response surface methodology.

See Composite Experimental Design Matlab Code for the actual Matlab code used to generate the results below.

(2d response surf)

Response surfaces are difficult to visualize if they are more than 2 dimensions. For this reason, it is important to rely on more than simple visual metrics for judging how well a response surface represents the actual response.

Box-Behnken Designs

The relationship between composite and Box Behnken designs is that, if you use a face-centered (i.e. a 3-level) composite design and combine it with a Box Behnken design, you will get a full $ 3^{k} $ factorial design. So composite and Box Behnken designs are both fractional $ 3^{k} $ factorial designs.