Solution 5.5 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant

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  • Published: Jan 1, 2000
  • Channels: Chemometrics & Informatics

1. The scores and loadings for the centred data are given below.

The total sum of squares for the centred data is 11.329, the first eigenvalue is of size 11.130 and the second of size 0.182, totalling 11.312 altogether, or 99.81% of the variance. Notice that there are actually two true components in the mixtures, so the comparatively small size of PC2 does not mean it should be ignored. It would not be completely obvious how many components are in the data from first inspection.

2. The coefficients that relate the scores of the centred data matrix to the centred concentrations are

cA » 2.49 t1 - 8.51 t2

cB » 1.89 t1 + 15.08 t2

If you obtained different coefficients, probably you have not followed the guidelines about centring.

The estimated concentrations are as follows.

These are quite close to the true concentrations.

3. The matrix is as follows.

2.49

1.90

-8.51

15.09

4. The inverse of the rotation matrix is

0.281

-0.035

0.159

0.046

The estimated spectra are given by R-1.P and are as follows.

0.039

0.053

0.078

0.097

0.115

0.130

0.128

0.117

0.060

0.068

0.063

0.057

0.052

0.047

0.054

0.063

5. The two concentration vectors are correlated. Hence when PCA is performed on the centred data matrix, there is only one PC, and it is not possible to predict the concentrations, independently of two different compounds. When performing calibration experiments it is important to ensure that the two concentrations are reasonably uncorrelated.

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