Mean-centring

The prefixes c and p mean centred and primitive , respectively, where centred refers to when an adsorbate is added in the centre of the primitive unit cell. 

We subtract the mean spectrum from each measured spectrum yp and as a result, the origin of the system of axes is moved into the mean. In the above example, it is into the plane of all spectral vectors. This is called meancentring. Mean-centring is numerically superior to subtraction of one particular spectrum, e.g. the first one. The Matlab program, Main MeanCenter, m, performs mean-centring on the titration data and displays the resulting curve in such a way that we see the zero us,3-component, i.e. the fact that the origin (+) lies in the (us ,i,us >2)-plane. 

Figure 5-20. Mean-centring moved the origin of the system of axes into the centre of the action. This reduces the dimension of the subspace by one.

The argument can be turned around. If mean-centring reduces the rank of the matrix by one, the data set is closed. 

There are numerous publications proposing a glut of data treatment methods prior to PCR/PLS. Well established, tested and essentially universally applied are mean-centring and normalisation of the data. We have seen in Mean Centring, Closure (p.239) that mean centring reduces the dimensionality by one, which of course cannot harm. In PCR/PLS it is also common to normalise to the standard deviation of the signals. Both are implemented in Main PCR. m. 

Mean-centring and normalisation are optional. The PCR (and PLS) algorithm are essentially independent of the nature of pre-treatment of the data, only the centring has to be reversed in the prediction step. In the programs we... 

Now we use the information gathered so far for the prediction of the 10 test samples Ys removed from the complete data set at the veiy beginning. The function PCR PLS pred. m does the work according to equation (5.70). Importantly the mean-centring and normalisation have to be performed in exactly the same way as in the calibration. 

Figure 4.3 Step 1 of the NIPALS algorithm mean centring, starting point and first calculations.

Figure 4.11 Two typical pretreatments to the original spectral data shown in Figure 4.9 (a) mean centring and (b) autoscaling.

Table 4.1 Amount of information explained by PLS models with a different number of factors. Original data correspond to Figure 4.9, mean centred.

Figure 4.16 Typical example of a PRESS plot to select the model dimensionality (data shown in Figure 4.9, mean centred) (a) overall PRESS and (b) behaviour of each sample for each number of factors considered in the model. Four factors seem to be the optimal choice here as the minimum is clearly defined. 

Figure 4.18 PoLiSh strategy to set the number of latent variables to be included in the PLS model. Application of the Durbin-Watson criterion to any series suggests that after latent variable 5, random noise is introduced into the model. This should be the maximum dimensionality to consider in the model (original data in Figure 4.9, mean centred).

Figure 4.22 Example of a control chart to test for outlying samples on the calibration set. Four factors were used to develop the model (original spectra from Figure 4.9, mean centred).

Figure 4.24 Studentised residuals leverage plot to detect outlier samples on the calibration set (a) general rules displayed on a hypothetical case and (b) diagnostics for the data from the worked example (Figure 4.9, mean centred, four factors in the PLS model).

Figure 4.28 Pseudo-univariate representation of the PLS model implemented for the case study (four latent variables and mean-centred data). Reproduced from Ref. [12], with permission from the Royal Society of Chemistry.

Figure 4.29 Plot of the norm of the NAS (mean-centred data) versus the norm of the measured spectra. The slope is a measure of the selectivity of the PLS model, whose value (here 0.83) indicates that approximately 83% of the measured spectra in the case study is used for prediction and that 17% of the measured signal is lost due to presence of the interferences. 

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