## 2 Materials and Methods
### Principal Component Analysis  
A first approach to the estimation of the parameter \(m\) comes from principal component analysis (PCA), an application of linear algebra. Figure 8a shows the results for the Lorenz attractor. This method can help to extract relevant information from seemingly complex data, a priori reducing the number of dimensions needed to characterize the dynamics present in the data, that is to identify the basis that best represents a noisy dataset, detecting those redundant dimensions that record the same dynamic information [13]. This basis, to a certain extent, filters the noise and improves the recovery of the hidden dynamics in the data. In all, this linear composition aims to find the smallest subspace (hyperplane) that contains roughly the attractor. The PCA method takes up an \(m_{i}\)-dimensional coordinate system (\(m_{i}\) spatial directions) much higher than the \(m\)-dimensional reconstruction space that supposedly characterizes the attractor. The experimenter's intuition plays an important role in deciding on a greater or lesser initial value of \(m_{i}\). This method recognizes the directions of the \(m_{i}\)-dimensional coordinate system that shows more significant variations (variances) in the data, the \(m\)-dimensional reconstruction space, discarding those other directions in which small variations may come from fluctuations of the coupled noise on data. Discarding those less important directions, apart from reducing the effect of noise, particularly white noise, makes it possible for the appearance of a more simplified dynamics from an originally very high-dimensional space. Hence, it enables a dimensional reduction. The main disadvantage of this method is its subjective nature when determining the final value of \(m<m_{i}\), provided that the differences between the principal directions and those directions attributed to noise are not clearly significant. Typically, with real data, or even with an unfortunate initial choice of \(\tau\) and \(m_{i}\), the differences between the variances of the different directions \(m_{i}\) are not always so evident, and it is necessary to define a somewhat arbitrary threshold that classifies the directions as primary (\(m\) principal components) or secondary (coupled noise). Furthermore, there is no guarantee that the reconstruction will always be optimal, since the reconstruction method relies on a non-parametric analysis, and sometimes, it is unable to distinguish between a chaotic signal and the noise itself when they have a similar power spectrum [36]. The correlation dimension involves the most usual measure of dimension, mainly because of its computational efficiency. It is based on spatial correlation [37]. The general procedure includes point counting within a distance \(\varepsilon\), evaluated for each of the points that make up the state space. The normalized total number of points, for a particular distance \(\varepsilon\), is called the correlation sum, the estimator of the correlation integral,  
\[C_{\varepsilon}=\frac{\text{points within a distance }\varepsilon}{N(N-1)}, \tag{8}\]  
where \(N\) is the time series length. For large enough \(N\),  
\[C_{\varepsilon}=\lim_{N\to\infty}\frac{\text{points within a distance }\varepsilon}{N^{2}}, \tag{9}\]  
which applies for an \(N\) value of several hundred data; according to the mathematical formalism,  
\[C_{\varepsilon}=\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N} \Theta\left(\varepsilon-\left\|\mathbf{s}_{i}-\mathbf{s}_{j}\right\|\right), \tag{10}\]  
where \(\Theta\) is the Heaviside step function, so \(\Theta(x)=1\), for \(x\geq 0\), \(\Theta(x)=0\), for \(x<0\), and \(\left\|\cdot\right\|\) symbolizes the maximum norm of one vector. Equation (10) is computed for different values of \(\varepsilon\), with the vectors obtained from the Equation (4) for each particular case of \(m\).