# SepFree NMF: A Toolbox for Analyzing the Kinetics of Sequential Spectroscopic Data
## 3 Minimal Memory Effect in Sequentially Measured Spectroscopic Data
### Relation between EDMD and SepFree NMF  
For a better understanding of the meaning that the minimal memory effect has for the decomposition and the studied system, a few considerations on the involved mathematical objects should be made.  
An observable is \"something that can be measured\", so it is a function on the state space. Spectra are optical signals, and so they are observables themselves; sequentially measured spectra are observables depending on the wavenumber and delay time. Considering only the relative-concentration matrix \(H\), each relative concentration \(H^{i}\), \(H^{i}\in\mathbb{R}^{m}\), is a row-vector which contains \(m\) delay-time evaluations of the \(i\)th observable, so of one of the \(r\) compounds. We can think about the row-vector \(H^{i}\) as a specific function for the \(i\)th observable that depends on time. The whole matrix \(H\), \(H\in\mathbb{R}^{r\times m}\), contains a collection of time developments of all the considered \(r\)-observables [21].  
By analyzing the data via the NMF without separability assumption, the transition matrix is given by  
\[(H_{t}^{T})^{-1}H_{t+\tau}^{T}=K(\tau). \tag{8}\]  
Equation (8) means that the transition probability is given by the multiplication of the following:  
1. \(H(t)\), a set of time-dependent functions of the compounds at \(t\);
2. \(H(t+\tau)\), a set of time-dependent functions of the compounds at \(t+\tau\), so evolved in time.  
For the decomposition of sequential spectroscopic data, the collection of the concentrations of the \(r\) compounds contains the set of the time-dependent functions. The formulation in Equation (8) is the extended dynamic mode decomposition (EDMD) formulation for the projection of the Koopman operator [21; 22].  
In the SepFree NMF, the matrix \(H\) is not obtained as a function of the observable; rather, it results from the optimization of \(\chi\), the membership-functions matrix of PCCA+, for the components in the system. The discretized Koopman operator is obtained in Equation (8) in a EDMD manner. The PCCA+ is a specific kind of Galerkin projection (see Section 2.2) [5; 19]. The corresponding transition matrix (the matrix representation of a discretized Koopman operator) is given by  
\[K=\mathcal{S}^{-1}T, \tag{9}\]  
with \(\mathcal{S},T\in\mathbb{R}^{r\times r}\) with \(r\) being the number of identified components [19]. Details about the derivation, the discretization and the computation of the objects in Equation (9) can be found in [19].  
The expression in Equation (9) is the starting point for the derivation of the minimal memory effect [5].  
EDMD computes finite-dimensional approximations of the Koopman operator; for a certain kind of decomposition of the state space, it results in [PERSON]'s method, and thus a Markov state model (MSM) [23]. The MSM is underlying SepFree, thus the relation to EDMD and SepFree NMF gives the foundation to introduce how to compute the minimal memory effect by SepFree NMF. The relation between the two representations of the transition matrix with EDMD and SepFree NMF are analyzed in [18]. The concept of the _minimal memory effect_ is based on the theory of the rebinding effect of [5].