1.2 Starting with an example

FIG. 1.1 below is extracted from Delplanque and Louvet (2005) and summarizes an experiment for the design of an electrical outlet. The objective is to determine which combination leads to the lowest operating temperature.

**FIG. 1.1:** Experimental campaign and results.
$\begin{figure}\begin{centering} \begin{tabular}{ccccccc} \hline \char93 & Seal... ...\tabularnewline \hline \end{tabular}\par \end{centering}\par\par \end{figure}$

Definition 1.2.1 According to Stevens (1946), there are measurable quantities (additives), the length among them. There are ordered quantities, the temperature among them. And there are labelled quantities (where some identifier is recorded).

Remark 1.2.2 A labelled quantity is ever obtained with infinite precision (plastic or brass is either full plastic or full brass). Other quantities are prone to measure uncertainties that must be qualified, quantified and dealt with. In what follows, input quantities are ever assumed to be quantified with infinite precision.

Example 1.2.3 In FIG. 1.1, the inputs are five nominal factors, while the target is an ordered quantity.

Definition 1.2.4 The universe $\Omega$ is the set of all the achievable combinations of the factors. It is a subset of the product set of the factors.

Definition 1.2.5 A DOE (design of experiments) is an efficient choice of a subset $\omega$ of the universe $\Omega$ .

Definition 1.2.7 When factor

ranges over $\left(n+1\right)$ levels $\xi_{0}, \xi_{1}, \cdots, \xi_{n}$ , the complete code of a given value

is the $\left(n+1\right)-$ tuple $\left(x_{0}, x_{1}, \cdots, x_{n}\right)$ where $x_{i}=1$ when $X=\xi_{i}$ and $x_{i}=0$ otherwise.

Proposition 1.2.8 The following binding relationship ever holds :

$\displaystyle \forall\xi, \sum_{0}^{n}x_{i}=1.$

Definition 1.2.9 The reduced code of a $\left(n+1\right)-$ levels factor is a

tuple, that depends of the choice of a "reference level" $\xi_{0}$ . This code is obtained by :

$\displaystyle \xi_{0}$	$% latex2html id marker 7766 $\displaystyle \mapsto$$	$\displaystyle \left(-1, -1, \cdots, -1\right)$
$\displaystyle \xi_{i}$	$% latex2html id marker 7772 $\displaystyle \mapsto$$	$\displaystyle \left(\; x_{1},\; x_{2}, \cdots,\; x_{n}\right)\qquad\mathrm{when} i\neq0$

Definition 1.2.10 A balanced (fair) DOE is a design where all the levels of a given factor are visited the same number of times.

Example 1.2.11 Codes and number of visits relative to FIG. 1.1 are given in FIG. 1.2.

**FIG. 1.2:** Codes and numbers of visits
$\begin{figure}\begin{tabular}{\vert c\vert c\vert c\vert} \hline \multicolumn{3... ...& \char93 4\tabularnewline \hline \end{tabular}\hfill . \par\par \end{figure}$

Proposition 1.2.12 The weighted mean of the reduced codes for a given factor is null if and only if the DOE is balanced relative to this factor.

Definition 1.2.13 The coding of an experiment is "

" followed by the reduced codes relative to the factors.

Example 1.2.14 Here, the length of the code is

and the code of the experiment $\char93 12$ :
brass ; welded ; CuNiSn ; 4.0_mm2_flexible ; HYPO4 is :

$\displaystyle 1,\quad-1,\quad0,1,\quad0,1,\quad0,0,1,\quad1,0,0$

Exercise 1.2.15 Data relative to FIG. 1.1, obtained from Delplanque and Louvet (2005), are available at http://www.douillet.info/~douillet/cours/planx/datas/us-experimentique.txt. Import this file, convert it and obtain the $16\times12$ matrix :

$\begin{displaymath}ma, mb= \left[ \begin{array}{rrrrrrrrrrrr} 1&-1&-1&-1&0&1&-1... ...55\cr 63\cr 52\cr 42\cr 64\cr 58\cr 52\cr 59 \end{array}\right]\end{displaymath}$