# SepFree NMF: A Toolbox for Analyzing the Kinetics of Sequential Spectroscopic Data
###### Abstract  
This work addresses the problem of determining the number of components from sequential spectroscopic data analyzed by non-negative matrix factorization without separability assumption (SepFree NMF). These data are stored in a matrix \(M\) of dimension \"measured times\" versus \"measured wavenumbers\" and can be decomposed to obtain the spectral fingerprints of the states and their evolution over time. SepFree NMF assumes a memoryless (Markovian) process to underline the dynamics and decomposes \(M\) so that \(M=WH\), with \(W\) representing the components' fingerprints and \(H\) their kinetics. However, the rank of this decomposition (i.e., the number of physical states in the process) has to be guessed from pre-existing knowledge on the observed process. We propose a measure for determining the number of components with the computation of the minimal memory effect resulting from the decomposition; by quantifying how much the obtained factorization is deviating from the Markovian property, we are able to score factorizations of a different number of components. In this way, we estimate the number of different entities which contribute to the observed system, and we can extract kinetic information without knowing the characteristic spectra of the single components. This manuscript provides the mathematical background as well as an analysis of computer generated and experimental sequentially measured Raman spectra. kinetics from experiments; separability assumption; sequential spectroscopic data 1  
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by the colors red, orange and yellow in the overall system at each point in time of the measurement. This situation is displayed in Figure 1.  
In other words, the solution has a memory because its color cannot \"switch\" from red to yellow within the interval \(\tau\); if the solution is red, and it was dark red before, in the next time step it will be more likely orange-red, not yellow. Meanwhile, the particles in the solution can jump from red to yellow or to any color, regardless of the past particle color (Markov). If the Markov property were also met on the level of the projected macrosystem (the solution can be red-orange-yellow), one could find, e.g., an ordinary differential equation (a Markov process) matrix for the evolution of the concentration vectors. A solution of this system for fixed time-step \(\tau\) could be represented by a matrix equation:  
\[h_{t+\tau}^{T}=h_{t}^{T}K(\tau),\]  
where the matrix \(K(\tau)\) usually also depends on further quantities (e.g., temperature at which the reaction takes place), and \(h^{T}(t)\) is the proportion of a state in the reaction. In the analysis of sequentially measured spectroscopic data, \(h\) is associated to a spectral fingerprint, as explained later in this work. The transition from a micro to a macrosystem can have a systematic source of error, because a projected Markov system loses its Markov property. Mathematically, there are two possibilities to model this \"non-Markovianity\":  
1. The process is assumed to have a memory. Not only the current (macro)state of the system, but also past steps of the process development have to be taken into account in order to derive a \"forecast\" for the future. 2. The clustering of microstates into coarser macrostates is assumed to be \"fuzzy\". Then Markovianity can be preserved, but we lose control about what we will denote as a macrostate. Chemically, this article develops option \(B\) and presents a tool to compute the minimal past-time dependency of compounds in a reaction spectrum. A compound is a system's state with an associated spectral fingerprint. The tool considers not only the compounds' dependency on the past, which is called _memory_, but also the dependency in the kinetic development between the compounds themselves. This memory effect has been investigated in the mathematical context (e.g., [1; 2]) and it has been observed in chemistry, for instance in the context of protein folding [3; 4].