Solution 2.4 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant
Education Article
- Published: Jan 1, 2000
- Channels: Chemometrics & Informatics
1. The equation is given on page 89 of the printed text. The numbers of experiments are 5, 15 and 35 for the three designs.
2. The full {5,3} simplex lattice design is given below.
The first five experiments consist of all the combinations of 3/3 (=1). The next twenty experiments consist of all the combinations of 2/3 and 1/3. The final ten experiments consist of all the combinations of 1/3, 1/3 and 1/3.
3. There are 2^{5}-1 or 31 experiments altogether.
The design is as follows.
Note that there are
- 5 single component blends,
- 10 two component blends,
- 10 three component blends
- 5 four component blends,
- 1 five component blend.
Note also that the one component and three component blends correspond to identical experiments in the {5,3} simplex lattice design.
4. The design is given in Table 2.37 in the printed text and is as follows.
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
2/3 |
1/3 |
0 |
2/3 |
0 |
1/3 |
1/3 |
2/3 |
0 |
0 |
2/3 |
1/3 |
1/3 |
0 |
2/3 |
0 |
1/3 |
2/3 |
1/3 |
1/3 |
1/3 |
The first step is to establish upper bounds (see Table 2.39 in the printed text for an example) so that the new upper and lower bounds become
L |
0.0 |
0.3 |
0.4 | ||||
U |
0.3 |
0.6 |
0.7 |
where the upper bounds of x_{1} equals 1 minus the sum of the lower bounds of x_{2} and x_{3} (=1-0.3-0.4) and so on. Then the new mixture design can be obtained by transforming the original values by the equation x_{new,f} = x_{old,f} (U_{,f} - L_{f}) + L_{f} to give
0.3 |
0.3 |
0.4 |
0.0 |
0.6 |
0.4 |
0.0 |
0.3 |
0.7 |
0.2 |
0.4 |
0.4 |
0.2 |
0.3 |
0.5 |
0.1 |
0.5 |
0.4 |
0.0 |
0.5 |
0.5 |
0.1 |
0.3 |
0.6 |
0.0 |
0.4 |
0.6 |
0.1 |
0.4 |
0.5 |