# Multifractal Characterization and Modeling of Blood Pressure Signals
## 2 Data and Methodology
### Data Pre-Processing
#### 2.2.3 Addressing Non-Stationarities
\] with \(k=4500\) samples;
3. the so-called Fourier-detrended MFDFA [56], which consists of removing the first \(c=60\) sinusoids from the Fourier spectrum, corresponding to slow varying cycles that are sources of residual non-stationarity. The last detrending method, used in several works in the literature dealing with real-word signals [57; 58; 29; 54], has the drawback of generating an apparent crossover located around the timescales corresponding to the cut-off frequency related to \(c\) in the fast Fourier transform (FFT) spectrum [59]. However, this phenomenon is not a real problem here since, as we will see, we are interested in the asymptotic behavior of the scaling function \(F_{q}(s)\), which is the scaling behavior for higher timescales. One can experimentally verify the non-stationarity property by computing the variogram \(G_{k}\) that measures the variance of differences \(k\) time units apart relative to the variance of the differences \(1\) time unit apart [60]. Specifically, for a given time series \(z_{t}\), the variogram is defined as:  
\[G_{k}=\frac{\text{Var}(z_{t+k}-z_{t})}{\text{Var}(z_{t+1}-z_{1})},\quad\text{ for lag $k=1,2,3,\ldots$} \tag{16}\]  
It can be shown that for a stationary process, as \(k\) increases, the graph of \(G_{k}\) vs. \(k\) approximates to an asymptotic stable behavior, while for a non-stationary process, \(G_{k}\) tends to increase monotonically. In Figure 3a, the variogram for the original signal is reported, while in Figure 3b, the one obtained for the processed signal is shown. The general appearance in Figure 3a does seem to indicate that it does not converge to a stable level, confirming the non-stationary nature of the time series under analysis. Instead, in the processed case, we find a clear plateau indicating the removal of an important amount of  
Figure 2: Scaling function \(F_{2}(s)\) computed through the DFA3 for the BPd signal filtered with the Hampel and the low-pass FIR filters, but not pre-treated for addressing non-stationarity. non-stationarity. Furthermore, the residual non-stationarity is addressed directly by the MFDFA detrending procedure given a suitable order polynomial. As noted above, pre-processing of a noisy non-stationary signal can impair the MFDFA analysis due to the appearance of crossovers. As an example, it is well known that the Fourier detrending technique, used to eliminate the low-frequency periodicities, generates a crossover and allows us to measure the multifractal characteristics only after a predefined timescale \(s_{\times}\), where the scaling function can be approximated with a linear regression [59]. As expected, the normal low-pass filtering procedure based on a linear-phase FIR filter (whose delay group is suitably compensated) also distorts the scaling function for lower scales. With the aim of understanding the dynamic of such distortion, we generated two synthetic signals of \(2^{15}\) samples. One was an uncorrelated signal with a Hurst exponent \(\mathrm{H}=0.5\), while the other was a long-range (positively) correlated signal with \(\mathrm{H}=0.8\). The correlated fractional Gaussian noise (fGn) was obtained by the FFT algorithm together with an embedding of the covariance matrix in a circulant matrix--see [61; 62]. It is worth noting that, for both signals, the scaling function \(F_{q}(s)\) is a straight line with an angular coefficient equal to \(\mathrm{H}\). In Figure 4, the main findings for the synthetic uncorrelated and correlated signals after the FIR filtering are reported. In Figure 4a,b, the power spectrum estimate after the low-pass filtering is reported. Figure 4c,d gives an idea of the profiles of both signals, while Figure 4e,f report the scaling behavior computed by DFA1. In both cases the wide crossovers around \(s_{\times}=30\) with a bad Hurst exponent are clearly visible, while the scaling behavior remains the original one for \(s_{\times}\gg 30\), that is, for asymptotic timescales. This fact has to be taken into account in analyzing pre-processed real-world signals, such as the one at hand.