## 2 Materials and Methods
### Fractal Dimension (FD)  
Among the various algorithms available to measure the FD, we selected those specially suited to time series analyses that don't need previous modelling of the system. Two of those algorithms are [PERSON] [49] and [PERSON] [50] named after their authors. We used the former as the main reference, and a modification proposed by [PERSON] [51] on the original version developed by [PERSON] [50]. [PERSON] was our first choice because it has been reported to be more accurate [52] but in most of these studies, the algorithm compared to was the original one developed by [PERSON] himself. [PERSON]'s improvement is theoretically sound and has not been tested in many experiments, so we considered interesting to test as an alternative.  
The algorithm proposed by [PERSON] [49] measures the FD of discrete time sequences directly from the time series \(\{x_{1},x_{2},...,x_{n}\}\). The algorithm calculates the length \(L_{m}(k)\) for each value of m and k, where m is initial time \(\{m=1,2,...,k\}\) and k is time interval \(\{k=1,2,...,k_{max}\}\). N is the lenght of the sampled signal.  
\[L_{m}(k)=\frac{\sum_{i=1}^{N-m}|x(m+i_{k})-x(m+(i-1)k)|(n-1)}{\left|\frac{N-m} {k}\right|k} \tag{1}\]  
After that, a sum of all the lengths \(L_{m}(k)\) for each k is determined by:  
\[L(k)=\sum_{m=1}^{k}L_{m}\left(k\right) \tag{2}\]  
And finally, the slope of the curve \(\ln\bigl{(}L(k)\bigr{)}/\ln(1/k)\) is estimated using the best fit by linear least squares. The result is the Higuchi FD.  
On the other hand, [PERSON] [50] proposed a normalized formula of the FD:  
\[FD=\frac{\log(n)}{\log(n)+\log(d/L)} \tag{3}\]  
where the length L and the extension d of the curve are normalized using the average step a = L/n using (4) and (5).  
\[L =\sum_{i=1}^{n}l_{i,i+1} \tag{4}\] \[d =\max\bigl{\{}l_{i,j}\bigr{\}} \tag{5}\]  
Nevertheless, given that the input signal is a mono-dimensional waveform, the length and the extension can be rewritten using [PERSON]'s approach [53]. A simple and efficient way to do this is directing these two magnitudes in their own dimension as it was done by [PERSON] [51]. One by one, the extension on the Y axis is the range of y\({}_{k}\) as seen in:  
\[d=\max\{y_{k}\}-\min\{y_{k}\} \tag{6}\]  
and the length L is the sum of all the increments in modulus as in:  
\[L=\sum_{k=1}^{n}|y_{k+1}-y_{k}| \tag{7}\]This latter calculation is what is called the [PERSON]'s variation of [PERSON]'s algorithm [51].  
Once the FD algorithms were selected, it was extremely important to choose the window size to be used for the calculations. The fractal window goes through the signal, _i.e._, the object to analyze. This sliding window has a fixed size during each analyzing period. In order to calculate the FD's features of the signal there are no restrictions other than the total waveform length to the window size to be used. Each signal was analyzed using three fixed, but configurable, window lengths: 320 points, 640 points and 1280 points. The third window size of 1280 points was tested because previous studies that take into account the window size of similar dimension estimations [52,54] suggest that a bigger window could be useful in some cases. Since the FD is a tool intended to capture the dynamics of the system, with a short window the estimation would be very local and adapting fast to the changes in the waveform. When the window is longer, some details will be lost but the FD anticipates better the characteristics of the signal [8].  
Each of the three experimental cases, C1, C2 and C3, were treated by [PERSON], [PERSON], and the [PERSON]'s variation of [PERSON]'s algorithm using window lengths of 320, 640 and 1280 points.