# Multifractal Characterization and Modeling of Blood Pressure Signals
## 3 Results and Discussion  
### Multifractal Characterization of the BPd Signal  
In describing the multifractal characteristics of the BPd signal at hand we evaluate the DFA3 for timescales higher than \(s>150\) until \(s=\lfloor N/5\rfloor\), where the scaling function \(F_{q}(s)\) is close to be a straight line for all exponents \(q\) and timescales \(s\)--see Figure 5a. We report the following main multifractal indices:  
* Multifractal strength of the singularity spectrum \(f(\widetilde{\alpha})\): \[\Delta_{\widetilde{\alpha}}=\widetilde{\alpha}_{+}-\widetilde{\alpha}_{-},\] (17) where \(\widetilde{\alpha}_{+},\widetilde{\alpha}_{-}\) are the two extreme values at the two ends of the multifractal spectrum support, respectively;
* The asymmetry of the singularity spectrum that is evaluated through a suitable asymmetry index \(A_{\widetilde{\alpha}}\), that reads as [63]: \[A_{\widetilde{\alpha}}=\frac{\Delta\widetilde{\alpha}_{L}-\Delta\widetilde{ \alpha}_{R}}{\Delta\widetilde{\alpha}_{L}+\Delta\widetilde{\alpha}_{R}},\] (18) with \(\Delta\widetilde{\alpha}_{L}=\widetilde{\alpha}^{*}-\widetilde{\alpha}_{-}\) and \(\Delta\widetilde{\alpha}_{R}=\widetilde{\alpha}_{+}-\widetilde{\alpha}^{*}\), and where \(\widetilde{\alpha}^{*}\) is the \(\widetilde{\alpha}\) value at maximum of the singularity spectrum \(f(\widetilde{\alpha})\), i.e., the box counting dimension. The main results of our investigation are reported in Table 1. In particular, the shape parameters of the multifractal spectrum are provided, such as \(\widetilde{\alpha}_{-}\), \(\widetilde{\alpha}_{+}\), \(\widetilde{\alpha}^{*}\), \(\Delta_{\widetilde{\alpha}}\) and \(A_{\widetilde{\alpha}}\), together with the Hurst exponent \(\mathrm{H}=h(2)\) and the correlation dimension \(\tau(2)\). As concerns the multifractal strength, we find a higher value \(\Delta_{\widetilde{\alpha}}=0.752\) and a symmetric spectrum, with a small \(A_{\widetilde{\alpha}}=0.011\), that together with the correlation dimension \(\tau(2)\) indicate a strong multifractality of the BPd signal. Finally, the Hurst exponent \(\mathrm{H}=0.880\) indicates the presence of long-range correlations. As concerns the multifractal nature, it is possible to distinguish from multifractality due to the wideness of the PDF [64] and multifractality due to the two different correlations in small- and large-scale fluctuations. These two kinds of multifractality can be assessed by evaluating two time series derived from the original one: the shuffled time series and the surrogate one. The former is computed by simply shuffling at random the time indices, while the latter is obtained by changing the phases, computed through the discrete Fourier transform (DFT) of the original signal, drawing from a uniform distribution in \((-\pi,\pi)\)[65]. In the last case, one can demonstrate that the PDF tends to be normally distributed but correlations do not change, destroying intrinsic non-linearity. The shuffling procedure will destroy all long-range correlations and the corresponding shuffled time series will exhibit monofractal scaling. Conversely, the multifractality due to the wideness of the PDF is not affected by the shuffling procedure. Since the shuffling of time series destroys the long-range correlations, if the multifractality is due only to long-range correlations, then one should expect a constant value \(h_{\text{shuffle}}=0.5\). If both types of multifractality are present, the shuffled and the surrogate series will show weaker multifractality with respect to the original series. Figure 5: Results of the MFDFA3 on the BPd signal. Panel (**a**) reports the \(F_{q}(s)\) vs. \(s\) diagram varying the timescale \(s\) in a wide range of values. Results of the MFDFA3 on the BPd signal for the original (normal), shuffled and surrogate time series. Panel (**b**) shows the \(h(q)\) vs. \(q\) diagram. Circles denotes the modified multiplicative model fitted to the data.