###### Abstract
Since most real-world physical systems are nonlinear with an ever-present coupled noise, from a graphical analysis with the appropriate dynamic variables, the underlying dynamic system deterministic structure can be deducible, not apparent in a possible erratic evolution in the time domain. A more complex study, although less visual, with many more variables, requires a mathematical formalism not yet fully consolidated [7]. In physiological terms, typically, in a clinical trial, it is customary to acquire a unique biological signal identifying each physiological response, so that not all the dynamic variables involved in the time evolution of the system under study are available. Thus, each biological signal represents a response of a physiological system; each response distinguishes the time evolution of a dynamic variable. Accordingly, from the scalar measurements that define the acquired biological signal, the different state vectors must be generated in order to recreate the different states that characterize the physiological system dynamics in question. The phase space reconstruction allows faithfully reproducing the evolution of the system under study from a simple biological signal, in the absence of the dynamic variables involved. Usually, observations or measures at regular time intervals, i.e., one every \(\Delta t\) seconds, are an example of what in academic jargon is know as a time series. The most common method to define each state vector is to use delayed versions of the observations as state vector components. In the simplest case, in a two-dimensional coordinate system, as illustrated by Figure 1 for the two representative dynamical systems that we use in this paper to explain the methodology employed, the ordinates' axis represents the measurement value \(x_{n}\) at time \(n\Delta t\) and the abscissas' axis the measurement value \(x_{n-1}\) at time \((n-1)\Delta t\). Hence, each point in the plot identifies the \(\mathbf{x}_{n}=(x_{n-1},x_{n})\) state vector. At the most general level, the number \(m\) of components of the state vector and the delay \(\tau\) between components are variable, and this depends on the observable of the physical system being measured, that is \(\mathbf{x}_{n}=(x_{n-(m-1)\tau},\ldots,x_{n-\tau},x_{n})\). From a sequence of scalar measurements, it can reconstruct every state vector, in \(m\)-dimensional phase space, following the lag or delay method [8]. The state space reconstruction quality depends on the proper estimation of the parameters \(\tau\) and \(m\), considering the time delay embedding theorem [9; 10; 11], which provides the ideal conditions under which to reconstitute a chaotic dynamic system from a sequence of observations of the state of the dynamic system. State or phase space reconstruction, in contrast to traditional signal analysis in the time or frequency domain, constitutes the cornerstone of the time series analysis in terms of nonlinear dynamics [12,13]. It is the first step in the analysis process of the real functional nature that reflects the physical system in question. In doing so, it should pay special attention to determine the optimum values of \(\tau\) and \(m\)[14] efficiently, so that state space reconstruction faithfully allows characterizing the dynamics of the original system. We detail the most common techniques, elegant in their simplicity, which enable a first approximation of the state space reconstruction and then apply them to a particular biological signal, the PPG signal, extracting interesting conclusions that lead to promising future studies. In a typical trajectory, over the phase space, dynamic variables' time evolution can tend to infinity or be confined to a bounded region of the phase space. If the dynamic system is dissipative, once the transient response finishes, its dynamics will tend to a subset of the state space called the attractor. This subset is invariant under the dynamical system's evolution.