'''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:

- The matrix obtained from the $2^k$ factorial experiment This will be denoted by
**F**
- 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**
- 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 $\{cal\; E\}$ will have the following form:

$[\{cal\; E\}\; =\; left[\; \{begin\{array\}\{*\{20\}c\}\; alpha\; \&\; 0\; \&\; 0\; \&\; cdots\; \&\; cdots\; \&\; cdots\; \&\; 0\; \{\; -\; alpha\; \}\; \&\; 0\; \&\; 0\; \&\; cdots\; \&\; cdots\; \&\; cdots\; \&\; 0\; 0\; \&\; alpha\; \&\; 0\; \&\; cdots\; \&\; cdots\; \&\; cdots\; \&\; 0\; 0\; \&\; \{\; -\; alpha\; \}\; \&\; 0\; \&\; cdots\; \&\; cdots\; \&\; cdots\; \&\; 0\; vdots\; \&\; \{\}\; \&\; \{\}\; \&\; \{\}\; \&\; \{\}\; \&\; \{\}\; \&\; \{\}\; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \&\; cdots\; \&\; cdots\; \&\; alpha\; 0\; \&\; 0\; \&\; 0\; \&\; 0\; \&\; cdots\; \&\; cdots\; \&\; \{\; -\; alpha\; \}\; end\{array\}\}\; right$### 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):

**Orthogonal design:** $[$

alpha = (025QF)^{025}] where

$[Q\; =\; left(\; \{sqrt\; \{F\; +\; T\}\; -\; sqrt\; F\; \}\; right)^2]$;

- Rotatable : $[$

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