# SepFree NMF: A Toolbox for Analyzing the Kinetics of Sequential Spectroscopic Data
###### Abstract
The interpretation of the compounds' kinetics is more meaningful if the development of each kinetic curve is as independent as possible to the development of the other kinetic curves. A correlated development of kinetics causes an overlap of the kinetic curves. This is because they represent distinct physical system states and they are not chemical components. The tool presented in the following measures the memory of the reaction compounds by scoring the overlap of the kinetic curves, following the theory of [5]. This analysis determines physical states corresponding to the basis vectors of an invariant subspace, as we will explain in the theory section. This way to compute the minimal memory effect is advantageous for spectral-data analysis, since it helps to determine the number of compounds in the reaction. If the number of compounds is too high, two or more compounds' kinetic curves will overlap strongly since they are assigned to the same basis vector, or a linear combination of them. Figure 1: In the micro-system, the solution can assume a gradient of colors between red and yellow (continuum). In the macrosystem, the projection identifies only three states (red, orange, yellow). Knowing the number of reaction compounds is important not only when analyzing the datasets, but also when performing experiments. In fact, the memory measure gives insights into the quality of the sample and the noise levels of the experimental setup. The estimation of the minimal memory effect is possible with the SepFree NMF (separability-assumption free non-negative matrix factorization), a novel method based on the _non-negative matrix factorization without separability assumption_ framework [6]. The non-negative matrix factorization without separability assumption is a matrix-decomposition method already used in the analysis of sequentially measured spectroscopic data. The structure of the datasets is the spectrum, in a certain wavenumber interval, measured for different times after the excitation of the sample. Furthermore, it is applicable also to time-resolved spectra; these investigate the evolution of photo-induced phenomena, as they measure the change of the optical signal (emission, absorption, Raman scattering) as a function of time (e.g., [7; 8; 9]). This work uses the SepFree NMF for the analysis of the sequentially measured spectra. The method examines the sequentially measured spectrum as a matrix \(M\) to decompose into a product of matrix \(H\), describing concentration proportions of the compounds as function of time, and matrix \(W\), which describes the peak intensities of the compounds as function of wavenumber. In particular, the SepFree NMF models the evolution of the concentrations' proportions of the compounds in \(H\) as a memoryless process, called the Markovian process [6]. A transition probability matrix matrix \(K(\tau)\) governs the \(\tau\)-evolution of the columns of \(H\):  
\[H_{t+\tau}^{T}=H_{t}^{T}K(\tau). \tag{1}\]  
The main contribution of this work is to provide a measure for the overlap between the rows of the concentration-proportion matrix \(H\). The SepFree NMF identifies the _physical_ states of the system, not necessarily the chemical ones. In order to understand the difference, think of a reaction in which all the molecules can react into two compounds, A and B, and then go back to the initial state. In a standard multivariate curve resolution analysis [10], the concentration profile of the molecules in A and B evolves as in Figure 2 left. These curves represent the concentrations of A and B in the system as a function of time; they are not between 1 and 0. By SepFree NMF of rank 2, the algorithm returns the amount of system that is either is state I or in state II. Their time development will look as in Figure 2 middle. The first state I is the initial _physical_ state of the system at the beginning of the reaction; all the molecules are in this state at time \(t=0\), so its proportion will be (almost) 1. During the reaction, a second state II arises.