## 2 Materials and Methods
### Phase Space Reconstruction Method
#### 2.3.1 Lag or Delay Time Selection
Among the various selection criteria, the most commonly used determines as the optimal delay the first zero crossing of the autocorrelation function, as illustrated in Figure 7. As an example of an alternative approach, in climate related attractors [35], other criteria apply the one known as correlation time (when \(R_{\tau}\) decreases to a value of \(1/e\approx 0.37\)). Autocorrelation coefficients consider the degree of the mutual relationship on a linear basis, inappropriate in nonlinear systems [14]. In these situations, it is best to use the mutual information, as it enables determining, on a probabilistic basis, to what extent two values of the same variable, measured at different time instants, relate to each other. For example, if the point coordinates are identical, with \(\tau=0\), these represent the same information, and therefore, one coordinate accurately predicts the other or, otherwise, the amount of information that one coordinate contains about the other, mutual information, is maximal. Whenever there is any relationship between two values of the same variable, acquired at different time instants, one value contains information about the other value. That is, one value helps predict the other value or, in other words, the knowledge of one value reduces the uncertainty of the other value. The uncertainty reduction is called mutual information. If \(\mathbf{X}\) denotes a time series, \(\{s_{n}\}\), ergo, \(s_{1},s_{2},\ldots,s_{N}\), with \(N\) observations, and \(\mathbf{Y}\) the delayed time series, \(\{s_{n+\tau}\}\), ergo, \(s_{1+\tau},s_{2+\tau},\ldots,s_{N}\), with \(N-\tau\) observations, where \(\tau\) indicates the delay; the average mutual information \(I_{\mathbf{Y};\mathbf{X}}\) between both time series can be expressed, in probabilistic terms, as:  
Figure 7: From the dynamic variable \(x\) of the Lorenz attractor: (**a**) autocorrelation function (AF), where the first zero crossing is when \(\tau=409\); and (**b**) mutual information (MI), where the first minimum is with \(\tau=16\). \[I_{\mathbf{Y};\mathbf{X}}=I_{\tau} = \sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{c}}P(x_{i},y_{j})\log\frac{P(x_{i },y_{j})}{P(x_{i})P(y_{j})} \tag{6}\] \[= \sum_{i=1}^{N_{r}}P(s_{i},s_{i+\tau})\log\frac{P(s_{i},s_{i+\tau} )}{P(s_{i})P(s_{i+\tau})},\]  
where \(N_{c}\) is the number of cells containing points, with non-zero probability, and \(N_{r}\) is the number of routes, \(s_{i}s_{i+\tau}\), in the state space. Equation (6), in terms of entropy, is rewritten as:  
\[I_{\mathbf{Y};\mathbf{X}}=H_{\mathbf{Y}}+H_{\mathbf{X}}-H_{\mathbf{X},\mathbf{ Y}}, \tag{7}\]  
where \(H_{\mathbf{X}}\) is the entropy of \(\mathbf{X}\), \(H_{\mathbf{Y}}\) the entropy of \(\mathbf{Y}\), and \(H_{\mathbf{X},\mathbf{Y}}\) the joint entropy of \(\mathbf{X}\) and \(\mathbf{Y}\). Therefore, somehow, the mutual information involves a measure of predictability of the system, that is a measure of the degree of knowledge of \(s_{n+\tau}\) denoted \(s_{n}\). In order to reconstruct an attractor, as faithfully as possible, the minimum mutual information and delay value are required. As with the autocorrelation function, the mutual information decreases as the delay increases, until it eventually becomes zero. The first minimum of the mutual information includes a possible criterion for selecting the optima delay value [8], as shown in Figure 7b. One limitation of this method is that many points are needed if we are to get a consistent result.