Central composite design

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
The matrix, d, for an experiment involving k factors consists of the following three different parts: $$ \bold E =$$$$ \begin{bmatrix} \alpha & 0 & 0 &  \cdots  &  \cdots  &  \cdots  & 0  \\ { - \alpha } & 0 & 0 & \cdots  &  \cdots  &  \cdots  & 0  \\ 0 & \alpha & 0 &  \cdots  &  \cdots  &  \cdots  & 0  \\ 0 & { - \alpha } & 0 & \cdots  &  \cdots  &  \cdots  & 0  \\ \vdots & {} & {} & {} & {} & {} & \vdots  \\ 0 & 0 & 0 & 0 & \cdots  &  \cdots  & \alpha   \\ 0 & 0 & 0 & 0 & \cdots  &  \cdots  & { - \alpha }  \\ \end{bmatrix}$$
 * 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:

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 $$\alpha$$
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.

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