2.1 Cartesian map of a DOE

Definition 2.1.1   Let $\Omega\doteq\left[0 .. a-1\right]\times\left[0 .. b-1\right]\times\left[0 .. c-1\right]$ be the full factorial space (of a design with three factors). The following map is called the associated Euclidean encoding :

$$% latex2html id marker 8381\begin{array}{ccc} \Omega & \ho... ...t(u, v, w\right) & \mapsto & s=u+a v+a b w\end{array}$$

Proposition 2.1.2   Euclidean encoding is a one to one application, whose reciprocal is given by :

$$u=\mathrm{rem}\left(s, a\right) ;... ...a}} ; v=\mathrm{rem}\left(q, b\right) ; w=\displaystyle {\frac{q-v}{b}}$$

The generalization to more complicated spaces, i.e. with more factors, is straightforward.

Definition 2.1.3   A Cartesian map with two coordinates $x, y$ is obtained by grouping the factors in two sets. The abscissa is the Euclidean coding of the first group, and the ordinate the Euclidean coding on the second group.

Example 2.1.4   Let us consider $\Omega=[1,2]\times\left[1,2\right]\times\left[1,2,3\right]$. Grouping the factors into $X=\left[1,2\right]$ and $Y=\left[3\right]$, the Cartesian map of DOE
$\omega=\left[[1, 2 1], [1, 2 2], [2 2, 1], [2, 1 1], [3 2, 1]\right]$ is the $4\times3$ matrix :

$$\begin{array}{ccccrcc} & & & x & & & x\\ & & & 0 & & & 3\... ... & & & 1 & . & 1 & .\\ y\!\! & 2 & & . & 1 & . & .\end{array}$$

For example, $\left[a, b, c\right]=\left[1, 2, 2\right]$ is mapped onto $\left(2,1\right)$ since $x=\left(a-1\right)+2\times\left(b-1\right)=2$, $y=\left(c-1\right)=1$.

Remark 2.1.5   Among other utilities, this Cartesian map shows why the "one factor at a time method" is optimally wrong. Indeed, its representation is:

$$\begin{array}{ccccrcc} & & & x & & & x\\ & & & 0 & & & 3\... ... & & & 1 & . & . & .\\ y\!\! & 2 & & 1 & . & . & .\end{array}$$

Example 2.1.6   Applying the map $\left[a,b,c,d,e\right]\mapsto\left(b+3 a+6 d, c+3 e\right)$ to the DOE described FIG. 1.1 leads to :

$$\begin{array}{ccccccccccccccccccccccccccc} 0&0&0&0&0&0& &0&0&... ...0&0&0&1&0& &0&0&0&0&0&0& &0&0&1&0&0&0& &0&0&0&0&0&0 \end{array}$$

where the blocking into $3\times6$ submatrices is used to emphasize the repartition of the last two factors. This could also be achieved by map $\left[a,b,c,d,e\right]\mapsto\left(d,e\right)$. Both maps emphasize the fact that the design, when restricted to the last two factors, is a full factorial one.

Scilab 2.1.7   The command recmap([xlist],[ylist],data) LISTING  draws the full or restricted Cartesian map of a design. Examples are given TAB. 2.1.

TAB. 2.1: Various maps of a given DOE