## 2 Materials and Methods
### Phase Space Reconstruction Method
#### 2.3.1 Lag or Delay Time Selection  
In theory, with an unlimited number of measurements without noise, any delay is equally valid. In practice, with real experimental data, the geometrical and dynamic properties that characterize the attractors differ from each other, depending on the delay value chosen [26]. If the sampling time \(\Delta t\) is tiny, the values of consecutive samples are very similar, \(s_{n}\approx s_{n+\tau}\), and with \(\tau=1\), the represented points contain much redundant information. Furthermore, with a small delay, the noise level can blur or hide any local geometric structure. Regardless of the form that the attractor can take, the graphic arrangement of perfectly related pairs of points describes a straight line, as an identity function (a diagonal line of \(45^{\circ}\)) without any meaning, as illustrated in Figure 5a. In order for the distribution of points to provide some geometric significance, the values of the components that make up the coordinate system must be sufficiently independent. In the real state space, in which the coordinates correspond to different variables, this condition is satisfied. In the same vein, a delay too large can be counterproductive. With a significant delay, the dynamic relationship between the values of the variable disappears, so that, as the delay value increases, the geometric structure of the points becomes more complex and diffuse until finally the points are dispersed randomly on the state space (see Figure 5b). In Figure 6, the Lorenz flow, with the correct delay time \(\tau=16\), is represented. The next points describe how to calculate it. The autocorrelation coefficients \(R_{\tau}\) measure the correlation degree of a variable with itself at different instances. Its constituents are autocovariance and variance. \[R_{\tau} = \frac{\text{autocovariance}}{\text{variance}} \tag{5}\] \[= \frac{\frac{1}{N}\sum_{i=1}^{N-\tau}\left(s_{i}-\left\langle s \right\rangle\right)\left(s_{i+\tau}-\left\langle s\right\rangle\right)}{\frac{ 1}{N}\sum_{i=1}^{N}\left(s_{i}-\left\langle s\right\rangle\right)^{2}},\]  
where \(N\) represents the time series length and \(\left\langle s\right\rangle\) the arithmetic mean of all the time series observations. According to Equation (5), with a minimum delay \(\tau=0\), the coordinates of each point are identical (\(s_{n}=s_{n+\tau}\)), and the autocorrelation \(R_{\tau=0}=1\) is maximal. As delay increases, autocorrelation decreases until it eventually is reduced to zero, or as happens in real data, there is a fluctuation around zero, within a narrow margin, due to the noise in the data. In short, the autocorrelation coefficients range from \(+1\) y \(-1\). Maximum values, \(R_{\tau}=\pm 1\), are perfectly correlated data; minimum values, \(R_{\tau}=0\), correspond to uncorrelated data. Figure 5: State space reconstruction from the dynamic variable \(x\) of the Lorenz attractor, with a sampling time \(\Delta t\) of 2 ms: (**a**) with \(\tau=1\); and (**b**) with \(\tau=150\). Figure 6: State space reconstruction from the dynamic variable \(x\) of the Lorenz attractor, with a sampling time \(\Delta t\) of 2 ms and \(\tau=16\). Autocorrelation coefficients, for successive delays, conform to the autocorrelation function. It is also known as the spectral autocorrelation coefficient [33] (it is recommended to use \(\lfloor N/4\rfloor\) delays with a time series of \(N>50\) observations). The graphical representation of \(R_{\tau}\) is known as the correlogram. To a certain extent, a correlogram shows the type of regularity in the data. In the case of trendless and uncorrelated data, 95% of the autocorrelation coefficients are contained, in theory, in a band of \(\pm 2/\sqrt{N}\), around zero. About 5% of them can exceed the indicated limit without losing its condition of uncorrelated data [34]. From the autocorrelation function, it is possible to adopt some criteria for selecting the optimal delay.