# SepFree NMF: A Toolbox for Analyzing the Kinetics of Sequential Spectroscopic Data
## 4 Numerical Examples
### Raman Experiment-Crystallization of Paracetamol in Methanol  
In this example, we show that the determinant of \(\mathcal{S}\), defined in (12), is an effective measure for the physical meaningful number of compounds. Crystallization of paracetamol in methanol (Figure 7) shows a solvated paracetamol in methanol, a metastable intermediate phase, and a crystalline phase. For detailed explanations, including the experimental settings, see Appendix A. Raman shifts characterize different paracetamol phases (solution phase, metastable intermediate phase and form II). After applying the SepFree NMF method, the role that the obtained compounds have in the reaction can be identified by the position of the peaks in matrix \(W\). Firstly, the singular value decomposition of the sequentially measured unprocessed spectral data taken as matrix suggests that the number of compounds can probably be  
Figure 6: Reconstructed component-spectra of the compounds \(W\) on the right and reaction kinetics \(H\) on the for noise level \(\eta=0.4\) and \(30\%\) increased interference of the synthetic Raman dataset. Parameter set \([-0.001,-10.0,1.0,0.0,0.0]\). Minimal memory effect \(\det(\mathcal{S})=0.0005\). Figure 7: Raman spectrum of the crystallization of paracetamol in methanol. The intensities of the Raman shifts are presented as function of delay time \(t\) and Raman shift \(\
u\) (cm\({}^{-1}\)). \(r=3,4,5\). Thus, SepFree NMF is tested then for these \(r\) values. The parameters used for the objective function are \(\alpha=0.1\), \(\beta=10000.0\), \(\gamma=50.0\), \(\delta=1.0\), \(\mu=1.0\). The results of the application of the SepFree NMF are shown in Figure 8 for \(r=3\), in Figure 9 for \(r=4\), and in Figure 10 for \(r=5\). In each figure, the left plot shows the row of the matrix \(H\) as function of the delay time \(t\); the right plot displays the peak intensities of \(W\) as function of the Raman shift (cm\({}^{-1}\)). The sum of the relative concentrations \(\  
This question can be answered with _yes_ by Table 1. In this table, the values of \(\det(\mathcal{S})\) for \(r=3,4,5\) are given. It can be noticed that for \(r=3\) the determinant has the highest value, in accordance with the above considerations on the kinetics. The SepFree NMF yields also a transition matrix \(K(\tau)\) for the process [6]. Examining the transition matrix allows to understand the kinetic model underlying the reaction mechanism [18]. The crystallization process of this dataset is a sequential process. The matrix \(K(\tau)\) resulting from the decomposition  
\[K(\tau)= \begin{pmatrix}0.94&0.16&-0.1\\ 0.&1.,&0.\\ 0.01&-0.&0.99\end{pmatrix}  \tag{16}\]  
shows also a sequential process \(C\to A\to B\) and it shows that \(B\) is an absorbing state of the Markovian process. This means that when the process reaches \(B\), it does not leave that condition anymore, as also shown in Figure 11. The first compound for the analysis of the matrix \(K(\tau)\) is \(C\) since \(C\) is the most abundant at time \(t=0\), see Figure 8.  
In the supplementary materials, we provide a similar analysis for the decomposition of ranks 4 and 5, showing that the kinetic model described is not sequential. Recognizing the chemical moieties from the Raman shifts is advantageous for validating the number of compounds estimated with the minimal memory effect, as shown in the following analysis. To show that the results given by the decomposition of rank 3 are indeed in agreement with the experiment, we plot each peak intensity of \(W\) (Figure 8) with the peaks of form II and metastable amorphous form II. The values of the Raman peaks are reported in the supplementary materials. Spectrum \(C\) (Figure 12) shows only the peaks of the metastable amorphous phase and two clear peaks in the 2800-3000 cm\({}^{-1}\) region, which can be assigned to methanol [25]. Spectrum \(A\) (Figure 13) presents the peaks characteristic of the metastable amorphous phase and solvent, as methanol Raman shifts are still present, in agreement with [25].