# Multiblock Analysis Applied to TD-NMR of Butters and Related Products
## 3 Results and Discussion
### Multi Block Calibrations
On the other hand, it can be seen in Figure 3d that, in this example, SO-CovSel is much more robust to temperature change than SO-PLS. Figure 4 shows the b-coefficients of the SO-PLS models superimposed on the mean spectra of the three product classes. These coefficients are convoluted with the T\({}_{\rm{IL}}\) and T\({}_{\rm{IS}}\) signals to calculate an estimate of _Fw_. Thus, the analysis of their shape allows a better understanding of which parts of the signal are useful for calibration, and how they are used by the model. Figure 3: Predictions of _Fw_ on the test set by SO-PLS (**a**,**c**) and by SO-CovSel (**b**,**d**) for 6 \({}^{\circ}\)C (**a**,**b**) and for 15 \({}^{\circ}\)C (**c**,**d**). Figure 4a shows the SO-PLS b-coefficients of T\({}_{\rm 1L}\) signal acquired at 6 \({}^{\circ}\)C. These calibration coefficients show two positive peaks, at 8 and 300 ms, and two negative peaks, at 65 and 1100 ms. Except for the peak at 8 ms, the coefficients values are very close to the T\({}_{\rm 1L}\) relaxation time values determined by the Inverse Laplace Transform of the T\({}_{\rm 1L}\) signal (data not shown). The b-coefficients for T\({}_{\rm 1S}\) (Figure 4b), do not show any peak. They have the same shape as the average T\({}_{\rm 1S}\) signal. They contribute to the estimation of \(F_{w}\) by calculating the overall signal strength. Figure 4 also shows the relaxation times selected by SO-CovSel. As a reminder, a multiple linear regression of the signals measured at these times provides an estimate of \(Fw\), according to the performances reported in Table 2. On the T\({}_{\rm 1L}\) signal (Figure 4a), the variables selected, in decreasing order of importance, are: 4000, 269, 49, 5, 1035, 10 and 128 ms. All these variables are located at times such as overall height, peak height and/or depth, or sigmoid curvature, that allow the model to take into account the overall shape of the signal. In the T\({}_{\rm 1S}\) signal (Figure 4b), only one variable is selected, at 4000 ms. It allows the model to take into account the global intensity of the T\({}_{\rm 1S}\) signal. Figure 4c shows the SO-PLS b-coefficients of the T\({}_{\rm 1L}\) signal acquired at 15 \({}^{\circ}\)C. These coefficients show two positive peaks at 30 and 400 ms and three negative peaks at 8, 150 and 2500 ms. For the T\({}_{\rm 1S}\) signal (Figure 4d), the b-coefficients show a negative peak at 350 and a positive peak at 4000 ms. They allow the model to take into account the slope of the right-hand side of the signal. Variables selected by SO-CovSel on the T\({}_{\rm 1L}\) signal at 15 \({}^{\circ}\)C are shown in the same figure. The variables selected, in descending order of importance, are 5000, 474, 76, 5, 1553, 19, 207 and 2949 ms. As before, all these  
Figure 4: SO-PLS b-coefficients and variables selected by SO-CovSel, superimposed to the product mean spectra. Left (**a,c**): For T\({}_{\rm 1L}\) signals. Right (**b,d**): For T\({}_{\rm 1S}\) signals. Top (**a,b**): T = 6 \({}^{\circ}\)C. Bottom (**c,d**): T = 15 \({}^{\circ}\)C. Horizontal scale is logarithmic. Vertical dashed red lines show the relaxation times selected by CovSel. variables are located close to the peaks of the SO-PLS b-coefficients. They allow for the calculation of signal shape parameters, such as the overall height, the height and/or depth of the peaks, or the curvature of the sigmoid. On the T\({}_{\rm 1S}\) signal (Figure 4d), only one variable is selected, at 4146 ms. It allows the model to take into account the overall intensity of the T\({}_{\rm 1S}\) signal. This intensity is related to the nature of the product (see Figure 2) and especially to the quantity of solid matter. All these observations demonstrate that the best models use complex information in the T\({}_{\rm 1L}\) signal, related to its shape, and very simple information in the T\({}_{\rm 1S}\) signal, related to its intensity.