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.

A Note on Visualization

Response surfaces are difficult to visualize if they are more than 2 dimensions. For example, imagine reducing the dimension of a 1-D function (e.g. $ y = \log{(x)} $) by one dimension (a point).

Even worse is reducing by more than one dimension: for example, a plane described by a 2-D polynomial to a 0-D point.

For this reason, it is important to use more reliable metrics than visual inspection in order to judge how well a response surface represents the actual response.

Quadratic Surface, 6 Dimensions

A quadratic response surface for $ y_{p,exit} $, a quadratic function of 6 input parameters of the form:

$ \hat{y}(\boldsymbol{x}) = b_0 + \sum_{i=1}^{6} b_i x_i + \sum_{i < j} \sum_{j=1}^{6} b_{ij} x_i x_j + \sum_{i=1}^{6} b_i x_i^2 $

was computed using Matlab's regstats command [1].

Because the response surface is six dimensions, graphical representation is difficult (see preceding section). However, the surface was visualized using the mean values of each of the 4 non-visualized dimensions. The two dimensions visualized were $ L_{mix} $ and $ k(T) $.

The resulting polynomial coefficient vector $ \mathbf{b} $ is:

b(01) = 4.0870e+03 
b(02) = -2.0956e+03 
b(03) = -1.2574e+03 
b(04) = -4.1912e+02 
b(05) = -2.6527e-01 
b(06) = 8.2956e-02 
b(07) = -8.3864e+02 
b(08) = 4.1912e+02 
b(09) = 4.0102e-09 
b(10) = 4.1912e+02 
b(11) = 1.2271e-08 
b(12) = 1.0050e-08 
b(13) = 4.1912e+02 
b(14) = 1.2039e-10 
b(15) = 1.1920e-10 
b(16) = 1.1952e-10 
b(17) = 7.9500e-02 
b(18) = 1.2627e-11 
b(19) = 1.2676e-11 
b(20) = 1.2491e-11 
b(21) = 6.4480e-03 
b(22) = -9.1954e-04 
b(23) = 9.1895e-09 
b(24) = 7.8094e-09 
b(25) = 8.7553e-09 
b(26) = 1.4867e-02 
b(27) = 1.1544e-02 
b(28) = 4.1922e+02

The corresponding polynomial term for each coefficient (i.e. the order of polynomial terms) match the order described in Matlab's x2fx function documentation [2]. That is:

1. Constant term

2. Linear terms $ x_1, x_2, \dots x_n $

3. Interaction terms $ x_{1,2}, x_{1,3}, \dots x_{1,n}, x_{2,3}, \dots x_{n-1,n} $

4. Squared terms, in order $ x_1^2, x_2^2, \dots x_n^2 $

The resulting response surface, holding all other parameters constant at their mean value, looks like:

Some key statistics for the response surface are given here:

---------------------------------------------------
Response surface summary of information:
Number of variables in response surface is 6. 
Number of terms in polynomial is 28. 
Degree of response surface is 2.
MSE =			 0.03845480 
MSE DoF = 			 17 

L-inf norm resid = 	 0.34272386 

R^2 =			 0.86371957 
adjusted R^2 =		 0.64727417 
---------------------------------------------------

Quadratic Surface, 2 Dimensions

The response surface resulting from the regression of only the two dimensions visualized (of the same form, but lower in dimension) results in a polynomial coefficient vector of:

b(01) = 0.2019 
b(02) = -0.1065 
b(03) = 0.1115 
b(04) = 0.0269 
b(05) = -0.0145 
b(06) = -0.0009

It also results in the following response surface:

This surface has the following statistics:

---------------------------------------------------
Response surface summary of information:
Number of variables in response surface is 2. 
Number of terms in polynomial is 6. 
Degree of response surface is 2.
MSE =			 0.00690353 
MSE DoF = 			 39 

L-inf norm resid = 	 0.13735696 

R^2 =			 0.93490530 
adjusted R^2 =		 0.92655983 
---------------------------------------------------

It is obvious that removing the 4 non-visualized dimensions yields very significant differences in the response surface statistics.

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.