Intermittency-Driven Complexity in Signal Processing

Abstract

In this chapter, we first discuss the main motivations that are causing an increasing interest of many research fields and the interdisciplinary effort of many research groups towards the new paradigm of complexity. Then, without claiming to include all possible complex systems, which is much beyond the scope of this review, we introduce a possible definition of complexity. Along this line, we also introduce our particular approach to the analysis and modeling of complex systems. This is based on the ubiquitous observation of metastability of self-organization, which triggers the emergence of intermittent events with fractal statistics. This condition, named fractal intermittency, is the signature of a particular class of complexity here referred to as Intermittency-Driven Complexity (IDC). Limiting to the IDC framework, we give a survey of some recently developed statistical tools for the analysis of complex behavior in multi-component systems and we review recent applications to real data, especially in the field of human physiology. Finally, we give a brief discussion about the role of complexity paradigm in human health and wellness.

Appendix: Diffusion Entropy and Detrended Fluctuation Analysis

Given a diffusive variable X(t); t = 1, 2, … [e.g., the event-driven random walks of the EDDiS algorithm, Eq. (6.6)], we are interested in evaluating the self-similarity index δ of the PDF, defined in Eq. (6.7), and the second moment scaling H, defined in Eq. (6.8). δ and H are evaluated by means of the Diffusion Entropy (DE) analysis [13, 20, 100] and of the Detrended Fluctuation Analysis (DFA) [101], respectively.

1.1 Diffusion Entropy

Given the PDF P(x, t), the DE is defined as the Shannon entropy of the diffusion process:

$$\displaystyle{ S(\Delta t) \equiv -\int _{-\infty }^{+\infty }p(\Delta x,\Delta t)\ln p(\Delta x,\Delta t)d\Delta x\, }$$

(6.11)

where $\Delta t$ here denotes a time lag and not the absolute laboratory time. Using the self-similarity condition (6.7), it is easy to prove that

$$\displaystyle{ S(t) =\delta \ln \Delta t + S_{{\ast}}\, }$$

(6.12)

where S _∗ = −∫ _−∞ ^+∞ F(x)lnF(x)dx. Notice that the scaling is in fact asymptotic, namely it is only exact for t → ∞, and an effective time T _∗ can be introduced as an additional fitting parameter:

$$\displaystyle{ S(\Delta t) =\delta \ln (\Delta t + T_{{\ast}}) + S_{{\ast}}. }$$

(6.13)

It is possible to estimate δ by considering the graph $(\Delta t,S(\Delta t))$ in a log-lin plot and then fitting Eq. (6.13) to the data.

The computation of $S(\Delta t)$ requires the evaluation of the PDF $P(x,\Delta t)$. This is done by considering a moving window of length $\Delta t$, so that the set of pseudo-trajectories X _r (k) = X(r + k) − X(r), with $0 \leq k \leq \Delta t$, r = 1, 2, …, is considered. The pseudo-trajectories all start from X _r (0) = 0, and, for each $\Delta t$, it is possible to evaluate the histogram $P(x,\Delta t)$ of the sequence $X_{1}(\Delta t),X_{2}(\Delta t),\ldots$ and, then, the DE $S(\Delta t)$.

1.2 Detrended Fluctuation Analysis

Given the diffusive variable X(t); t = 1, 2, …, the DFA essentially estimates the second moment of a proper detrended time series $X(t) -\overline{X}(t)$. The detrending can be done with a n-order polynomial function and the most simple algorithm uses a linear detrending by a least-squares straight line fit [101]. The DFA algorithm works as follows: (i) for each discrete time L, the time series X(t) is split into not-overlapping time windows of length L: [kL + 1, kL + L], k = 0, 1, …; (iii) for each time window [kL + 1, kL + L] the local trend is evaluated with a least-squares straight line fit: $\overline{X}_{k,L}(t) = a_{_{k,L}}t + b_{_{k,L}};\ kL <t \leq (k + 1)L$; (iii) the fluctuation is derived in the usual way: $\widetilde{X}_{k,L}(t) = X(t) -\overline{X}_{k,L}(t) = X(t) - a_{_{k,L}}t - b_{_{k,L}};\ kL <t \leq (k + 1)L$; (iv) the mean-square deviation of the fluctuation is calculated over every window:

$$\displaystyle{ F^{2}(k,L) = \frac{1} {L}\sum _{t=kL+1}^{(k+1)L}\widetilde{X}_{ k,L}^{2}(t) = \frac{1} {L}\sum _{t=kL+1}^{(k+1)L}\left (X(t) -\overline{X}_{ k,L}(t)\right )^{2}\; }$$

(6.14)

and, finally, averaged over all the time windows, thus getting F ²(L).

In the case of a self-similar process, it results: $F(L) \sim L^{^{H} }$. The parameter H can be derived by a linear fitting applied to the function z = Hy + C, with z = log(F(L)) and y = log(L). The DFA output is H = 0. 5 for the case of uncorrelated (white) noise (e.g., Brownian motion), where the integrated process X(t) display the typical Gaussian PDF G(x, t) with so-called normal scaling of the variance: 〈X ²〉(t) ∼ t. H ≠ 0. 5 is denoted as anomalous scaling, is a signature of long-range (power-law) correlations and, thus, cooperation and complexity. In particular, H < 0. 5, also denoted as subdiffusion, corresponds to a anti-correlated (anti-persistent) signal, while H > 0. 5, also denoted as superdiffusion, corresponds to a positively correlated (persistent) signal.