## 4 Discussion  
The well-known phase space reconstruction served as a starting point for the analysis or modeling of the dynamics of physiological systems reconstructed from a single biological signal. The dynamics that described a PPG signal was deterministic. The embedding dimension \(m\), for healthy young subjects, was five, regardless of the subject and the time interval estimation. The optimal lag or delay time \(\tau_{\text{opt}}\) depended on the subject and calculated time interval. A further study, with individuals of different ages and with different proven pathologies, will be carried out to confirm if the embedding dimension remains the same. In addition, it will be thoroughly examined if the variation of \(\tau\), even of \(m\), in each individual is an indicator of the mood and physical status in which he or she is, either as a result of a sporadic situation, such as an episode of stress, a more severe disease, or the age related evidence for biological and physiological decline. We described, in some detail, the most usual and clear methodology to calculate the phase space reconstruction because we found that in its application to biological signals, it has not been well understood at the physiological level, and its discriminant \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Evaluated Signal**} & \multicolumn{2}{c}{\(\tau\)} & \multicolumn{2}{c}{\(m\)} \\ \cline{2-6}  & **AF** & **MI** & **PCA** & **D\({}_{\mathbf{2}}\)** & **FNN** \\ \hline Subject Number 1 (PPG1) & 52 & 81 & 5 & 9 & 5 \\ Subject Number 2 (PPG2) & 37 & 35 & 6 & 5 & 5 \\ Subject Number 3 (PPG3) & 29 & 30 & 5 & 5 & 5 \\ Subject Number 4 (PPG4) & 44 & 54 & 6 & 5 & 5 \\ Subject Number 5 (PPG5) & 44 & 33 & 6 & 5 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters \(\tau\) and \(m\) of the PPG signals, acquired from five individuals chosen at random, for the different methods already explained, namely autocorrelation function (AF), mutual information (MI), principal component analysis (PCA), correlation dimension (\(\mathbf{D_{2}}\)), and false nearest neighbors (FNN). The parameters are calculated for all the subjects of the experimental research (40 PPG signals), proving that with the FNN method, all of them showed \(m=5\). potential in the clinical setting could not be sufficiently exploited. In this sense, its effectiveness could be corroborated with the most modern state space reconstruction techniques that are less heuristic and with a more consolidated mathematical formalism [40]. The following are available online at [[http://www.mdpi.com/2076-3417/10/4/1430/s1](http://www.mdpi.com/2076-3417/10/4/1430/s1)]([http://www.mdpi.com/2076-3417/10/4/1430/s1](http://www.mdpi.com/2076-3417/10/4/1430/s1)), Video S1: 2D-3D phase diagrams PPG signals. Conceptualization, J.d.P.-C. and A.P.G.-M.; methodology, J.d.P.-C.; software, J.d.P.-C.; validation, J.d.P.-C., D.F.-J., and A.M.U. ; formal analysis, J.d.P.-C., D.F.-J., and A.M.U. ; data curation, A.M.U. ; writing, original draft preparation, J.d.P.-C.; writing, review and editing. J.d.P.-C. and A.P.G.-M.; visualization, J.d.P.-C.; supervision, A.P.G.-M.; project administration, A.M.U. ; funding acquisition, A.P.G.-M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors would like to thank Life Supporting Technologies Group (LST-UPM) for taking part in Project FIS-PI12/00514, from MINECO. Furthermore, they want to thank especially the support given by [PERSON], from the University of Alcala (UAH) and the Spanish National Research Council (CSIC), providing a cluster of servers with which to execute some of the computational algorithms of this work. The cluster consists of ten computing nodes, with a total of 64 cores, an Intel Xeon architecture, and 166 GB of memory. The authors declare no conflict of interest. Figure A2: From a sample PPG signal and for \(\tau=1\): (a) 2D phase diagram; (b) 3D phase diagram; (c) successive-maxima plot; (d) difference plot.