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# Central composite design

'''Central Composite Design''' is a [response surface methology] for fitting a second order model to a data set without needing to use a complete $3^k$ [factorial experiment] After the necessary experiment is created (multiple) linear regression is performed

## Method

The matrix A 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 (0000) where there are k zeros This point is often repeated so 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 $+-\alpha$ and all other factors are at zero The α value is determined by the designer and it can have just about any value Thus this part denoted by $\left\{cal E\right\}$ will have the following form:
$\left[\left\{cal E\right\} = left\left[ \left\{begin\left\{array\right\}\left\{*\left\{20\right\}c\right\} alpha & 0 & 0 & cdots & cdots & cdots & 0 \left\{ - alpha \right\} & 0 & 0 & cdots & cdots & cdots & 0 0 & alpha & 0 & cdots & cdots & cdots & 0 0 & \left\{ - alpha \right\} & 0 & cdots & cdots & cdots & 0 vdots & \left\{\right\} & \left\{\right\} & \left\{\right\} & \left\{\right\} & \left\{\right\} & \left\{\right\} 0 & 0 & 0 & 0 & cdots & cdots & alpha 0 & 0 & 0 & 0 & cdots & cdots & \left\{ - alpha \right\} end\left\{array\right\}\right\} right$

### 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: $\left[$
alpha = (025QF)^{025}] where $\left[Q = left\left( \left\{sqrt \left\{F + T\right\} - sqrt F \right\} right\right)^2\right]$;
1. Rotatable : $\left[$
alpha = F^{025}] which is the design implemented by MATLAB’s “ccdesign(k)” function

## Reference

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