Solution 5.4 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant

1. For a model of the form , the following coefficients are obtained

b0	4.4	4.4	4.4	4.4	4.4	4.4	4.4	4.4
b1	12.66	9.85	10.67	14.17	10.31	8.66	8.75	11.11

Note that b₀ is simply the average concentration of A.

2. The predicted concentrations at each wavelength together with the root mean square errors are given below. Notice that the r.m.s. should, ideally, be divided by the number of degrees of freedom (8) rather than the number of objects.

Wavelengths 3, 4 and 5 appear best for prediction.

3. The predicted spectrum is as follows.

0.046

0.068

0.088

0.087

0.109

0.132

0.120

0.071

Hence the predicted concentrations are as follows.

2.41
4.83
4.11
6.05
3.83
4.50
4.66
9.65
3.42
4.58

giving a root mean square error of 1.47. Note that it is probably best to divide by 7 rather than 8 in this case. However there are not dramatic differences according to whether 6, 7 or 8 are used for division and the quality of predictions could be assessed graphically. The prediction is slightly worse than at the three best wavelengths using univariate methods.

4. The two spectra are as follows.

0.023	0.036	0.055	0.069	0.094	0.100	0.081	0.040
0.043	0.058	0.060	0.033	0.028	0.057	0.070	0.056

The predicted concentrations together with root mean square errors (dividing by 6 as two components used in the model) are as follows.

A	B
0.39	-0.29
-0.08	0.26
0.62	-0.81
0.28	-0.38
-0.50	0.29
0.89	-1.07
-0.90	0.95
-0.30	0.58
0.13	-0.35
0.00	-0.06
Error
0.66	0.77

The errors are now quite small, for compound A comparable to the best wavelength.

5. The scores are given below.

-0.646	-0.043
-0.013	-0.162
-0.156	0.183
0.352	0.146
-0.284	-0.186
-0.071	0.076
-0.028	0.086
1.275	-0.098
-0.380	-0.080
-0.048	0.078

6. The concentration estimates for compound A are as follows.

1 component	2 components
1.61	1.27
4.35	3.07
3.73	5.17
5.92	7.07
3.17	1.71
4.09	4.70
4.28	4.96
9.89	9.12
2.76	2.13
4.19	4.81

The root mean square residual errors are 1.19 for 1 component and 0.51 for 2 components.

7. In MLR it is only necessary to know the concentrations of one compound for effective calibration, but it is best to use 2 PLS components. MLR tries to fit the entire x matrix but if only one compound is known the information of the other compound is mixed up. In PLS although it is important to know how many components are in a mixture, it is only necessary to know the concentrations of the calibrant.