Solution 2.1 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant

1. The design matrix for the raw (uncoded) data is given by

x0	x1	x2	x1x2
1	0.1	2	0.2
1	0.1	4	0.4
1	0.2	2	0.4
1	0.2	4	0.8

2. The coefficients can be calculated by
b = D^-1 .y

given by

b0	35
b1	20
b2	-4
b12	4

so that

3. The design matrix now becomes

x0	x1	x2	x1x2
1	-1	-1	1
1	-1	1	-1
1	1	-1	-1
1	1	1	1

so that

Note the latter equation could have been obtained by a more traditional, non matrix based method of adding and subtracting coefficients, although this is not asked in the problem.

Because the overall uncoded concentration range is 0.1 mM whereas the coded concentration range is 2 units (the difference between –1 and +1), the coefficients for x₁ are much smaller in question 3 compared to question 2. Time covers the same range, so the coefficients are of approximately similar size.