# Multifractal Characterization and Modeling of Blood Pressure Signals
## 2 Data and Methodology
### Multifractal Detrended Fluctuation Analysis  
The multifractal version of DFA is derived from the original DFA version proposed by [PERSON] et al. in [18] for non-stationary time series. The first three steps of the procedure [30] are equivalent, and they are presented in the following. Let \(X=\{x_{k}\}_{k=1}^{N}\) be a time series consisting of \(N\) time samples characterized by a compact support, i.e., \(x_{k}=0\) for a negligible fraction of indices \(k\). The main steps read as follows:  
**Step 1:** Compute the _profile_:  
\[Y(i)=\sum_{k=1}^{i}(x_{k}-\overline{x}),\quad i=1,\ldots,N, \tag{1}\]  
where \(\overline{x}\) is the mean of \(X\), i.e.,  
\[\overline{x}=\frac{1}{N}\sum_{i=1}^{N}x_{i} \tag{2}\]  
**Step 2:**: Divide the profile into \(N_{s}\) non-overlapping segments of equal length \(s\). Recall that \(N\) is the length of the time series; then, the number of segments \(N_{s}\) reads as  
\[N_{s}=\left\lfloor\frac{N}{s}+\frac{1}{2}\right\rfloor \tag{3}\]  
where \(\left\lfloor x+\frac{1}{2}\right\rfloor\) rounds \(x\) to the nearest integer. Since a short part at the end of the profile may be discarded, with \(N\) often not being a multiple of the considered timescale \(s\), the same procedure is repeated starting from the opposite end, in order to not discard any part of the series, yielding \(2N_{s}\) segments. **Step 3:**: Calculate the _local trend_ for each of the \(2N_{s}\) segments by a least-square fit of the series and then determine the variance as follows:  
\[F^{2}(s,\
u)\equiv\frac{1}{s}\sum_{i=1}^{s}\{Y[(\
u-1)s+i]-y_{\
u}(i)\}^{2}, \tag{4}\]  
for each segment \(\
u=1,\ldots,N_{s}\) and:  
\[F^{2}(s,\
u)\equiv\frac{1}{s}\sum_{i=1}^{s}\{Y[N(\
u-N_{s})s+i]-y_{\
u}(i)\}^ {2}, \tag{5}\]  
for each segment \(\
u=N_{s}+1,\ldots,2N_{s}\). The quantity \(y_{\
u}(i)\) is the fitting polynomial of the segment \(\
u\). Depending on the order \(m\) of the polynomial, DFA\(m\) yields different polynomial detrending from the computed profile (i.e., of order \(m\), or \(m-1\) if the original series \(X\) is considered). For example, using linear, quadratic, and cubic polynomials yield different DFA1, DFA2, DFA3 and so on for higher-order analogues [18]. The DFA\(m\) performance in removing polynomial trends can be appreciated in [40]. **Step 4:**: Average over all segments to obtain the \(q\)-th order fluctuation function:  
\[F_{q}(s)\equiv\left\{\frac{1}{2N_{s}}\sum_{\
u=1}^{2N_{s}}\left[F^{2}(s,\
u) \right]^{q/2}\right\}^{1/q}. \tag{6}\]  
The case for \(q=0\) will be discussed below, while for \(q=2\), the standard DFA is obtained. Here, the interest is in how the generalized \(q\)-dependent fluctuation \(F_{q}(s)\) depends on the timescale \(s\) for different values of \(q\). Hence, it is requested to repeat steps 1-4 for several suitable timescales \(s\). The fluctuation function \(F_{q}(s)\) depends on the DFA order \(m\) and will increase by increasing \(s\). By construction, \(F_{q}(s)\) is only defined for \(s\geq m+2\). **Step 5:**: Determine the scaling behavior of the fluctuation function by analyzing the log-log plots \(F_{q}(s)\) vs. \(s\) for each value of \(q\). If the series \(X\) is long-range power-law correlated, \(F_{q}(s)\) will increase, for large values of the timescales \(s\), as power-law, that is:  
\[F_{q}(s)\sim s^{h(q)} \tag{7}\]  
For large values of \(s\), the procedure of determining the scaling behavior becomes statistically unreliable because of the number of segments \(N_{s}\) for the averaging procedure in step 4. Hence, it is advised not to overcome the limit \(s=N/4\) in the fitting procedure to determine \(h(q)\). For timescale \(s\approx 10\), a systematic error can occur even if it can be suitably corrected. The exponent \(h(q)\) may depend on \(q\). The quantity \(h(q)\) is known as the _generalized Hurst exponent_ and for stationary time series \(h(2)\) is identical to the Hurst exponent H [18].