# Central composite design

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In statistics, a central composite design is an experimental design, useful in response surface methodology, for building a second order (quadratic) model for the response variable without needing to use a complete three level factorial experiment.

After the designed experiment is performed, linear regression is used, sometimes iteratively, to obtain results. Coded variables are often used when constructing this design.

## Design matrix Edit

The matrix, d, for an experiment involving k factors consists of the following three different parts:

1. The matrix obtained from the $2^k$ factorial experiment. This will be denoted by F.
2. The centre of the system of interest, denoted in coded variables as (0,0,0,...,0), where there are k zeros. This point is often repeated in order to improve the resolution of the method. This part will be denoted by C.
3. A matrix, with $2k$ row, where each factor is placed at $\pm\alpha$ and all other factors are at zero. The α value is determined by the designer, and it can have just about any value. This part is denoted by E and would look like this:
$\bold E =$$\begin{bmatrix} \alpha & 0 & 0 & \cdots & \cdots & \cdots & 0 \\ { - \alpha } & 0 & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & \alpha & 0 & \cdots & \cdots & \cdots & 0 \\ 0 & { - \alpha } & 0 & \cdots & \cdots & \cdots & 0 \\ \vdots & {} & {} & {} & {} & {} & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \cdots & \alpha \\ 0 & 0 & 0 & 0 & \cdots & \cdots & { - \alpha } \\ \end{bmatrix}$

Thus, the d matrix will look like this:

$\bold d=$$\begin{bmatrix} \bold F \\ \bold C \\ \bold E \end{bmatrix}$
.

The X matrix used in linear regression would be constructed as follows:

$\bold X=$$\begin{bmatrix} \bold 1 & \bold d & \bold d(1)*\bold d(2) & \bold d(1)* \bold d(3) & \cdots & \bold d(k-1)* \bold d(k) & \bold d(1)^2 &\bold d(2)^2 &\cdots & \bold d(k)^2 \end{bmatrix}$

where d(i) represents the $i^{th}$ column in d. The multiplication is to be done memberwise.

### Determining the value of $\alpha$Edit

There are many different methods to determine the value of α. Define $F = 2k$, the number of points due to the factorial design and $T = 2k + n$, the number of additional points, where $n$ is the number of central points in the design. Common values are as follows (Myers, 1971):

1. Orthogonal design:: $\alpha = (0.25QF)^{0.25}$, where $Q = ((\sqrt{F + T} -\sqrt{F})^2$;
2. Rotatable design: $\alpha = F^{0.25}$, which is the design implemented by MATLAB’s “ccdesign(k)” function.

## ReferenceEdit

Myers, Raymond H. Response Surface Methodology. Boston: Allyn and Bacon, Inc., 1971