# Multifractal Characterization and Modeling of Blood Pressure Signals
## 1 Introduction
Notably, \(1/f\) noise is observed in blood pressure measurements [13] together with changing regime after baroreceptor denervation [14]. Within the frequency range where the \(1/f\) noise is observed, no specific frequency is preferred by the overall system, meaning that there are no resonance frequencies or characteristic time constants. As a result, the fine-grained regulation involves many complex factors. In this work, a multifractal characterization of a diastolic blood pressure (BPd) time series for a traumatic brain injury (TBI) patient is conducted. Time series data are collected from the Brain IT dataset [15] in the time span of _circa_ one day and in a suitable frequency range. In fact, monitoring the blood pressure is an important part of the entire assessment and diagnosis of acute brain injuries, as this protects the brain from a secondary brain injury [16]. \"Multifractal\" means that the complete fractal dynamic of a given signal, if any, is not characterized by a single type of singularity (such as in monofractal signals), but it is described by a suitably wide spectrum, characterized via several indices [9]. In other words, within the local approximation through suitable series, such as the Taylor series, for several points in time, we find a family of non-integer, hence fractional, exponents; this is a hallmark of complexity. Moreover, the monofractal characterization through the Hurst exponent H allows us to define which type of persistence, or memory, characterizes the time series. The concept of persistence is related to the auto-correlation of the signal whether its decay as a function of the time lag follows a power-law. **Definition 2** ((Anti-)Persistent signals). : _Uncorrelated signals, such as white noise, are featured by a Hurst exponent \(H=0.5\) with a fast auto-correlation decay. For persistent (positively correlated) signals, we find \(H>0.5\), whereas for anti-persistent signals, we find \(H<0.5\). Signals with \(H>0.5\) are defined as (positively) long-range correlated, meaning that the value of the signal in a given point in time is correlated with past values giving rise to a memory in the system._  
If, for a monofractal signal, the moments of the underlying probability distribution are constants for a wide range of timescales, the same does not hold for multifractal signals, suggesting a complex hierarchical organization of singularities. In this work, to give a complete multifractal characterization of the BPd signal, we use the detrended fluctuation analysis (DFA) [17] and its counterpart, the multifractal detrended fluctuation analysis (MFDFA) [18]. DFA and MFDFA are used in many fields to investigate the complexity of signals from biological and physiological data to finance, engineering and physics [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. As an example, the DFA is found to be very useful in characterizing intracranial pressure (ICP) signals and to predict TBI [34]. The appealing of this methodology is in its widespread applicability on real-world signals. In fact, unlike traditional methods--for details see [35]--useful only for stationary signals, MFDFA gives the possibility to locally detrend datasets containing polynomial trends of a given order and, hence, allowing one to also analyze non-stationary signals. However, as we will see, other types of trends, such as periodic or quasi-periodic trends have to be faced carefully in order to reach stationarity. After the breakthrough of [PERSON] with a series of dimensional and scaling arguments for fluids in the limit of very large Reynolds numbers (fully developed turbulence), binomial multiplicative models--known as _strongly multifractals_ given their wide hierarchy of singularities--gained popularity in explaining natural phenomena [36; 37], not only in fluid dynamics.