Scaling in Non-stationary time series I

Most data processing techniques, applied to biomedical and sociological time series, are only valid for random fluctuations that are stationary in time. Unfortunately, these data are often non stationary and the use of techniques of analysis resting on the stationary assumption can produce a wrong information on the scaling, and so on the complexity of the process under study. Herein, we test and compare two techniques for removing the non-stationary influences from computer generated time series, consisting of the superposition of a slow signal and a random fluctuation. The former is based on the method of wavelet decomposition, and the latter is a proposal of this paper, denoted by us as step detrending technique. We focus our attention on two cases, when the slow signal is a periodic function mimicking the influence of seasons, and when it is an aperiodic signal mimicking the influence of a population change (increase or decrease). For the purpose of computational simplicity the random fluctuation is taken to be uncorrelated. However, the detrending techniques here illustrated work also in the case when the random component is correlated. This expectation is fully confirmed by the sociological applications made in the companion paper. We also illustrate a new procedure to assess the existence of a genuine scaling, based on the adoption of diffusion entropy, multiscaling analysis and the direct assessment of scaling. Using artificial sequences, we show that the joint use of all these techniques yield the detection of the real scaling, and that this is independent of the technique used to detrend the original signal.Comment: 39 pages, 13 figure

Non-Poisson dichotomous noise: higher-order correlation functions and aging

We study a two-state symmetric noise, with a given waiting time distribution $\psi (\tau)$, and focus our attention on the connection between the four-time and the two-time correlation functions. The transition of $\psi (\tau)$ from the exponential to the non-exponential condition yields the breakdown of the usual factorization condition of high-order correlation functions, as well as the birth of aging effects. We discuss the subtle connections between these two properties, and establish the condition that the Liouville-like approach has to satisfy in order to produce a correct description of the resulting diffusion process

Non-Poisson dichotomous noise: higher-order correlation functions and aging

We study a two-state symmetric noise, with a given waiting time distribution $\psi (\tau)$, and focus our attention on the connection between the four-time and the two-time correlation functions. The transition of $\psi (\tau)$ from the exponential to the non-exponential condition yields the breakdown of the usual factorization condition of high-order correlation functions, as well as the birth of aging effects. We discuss the subtle connections between these two properties, and establish the condition that the Liouville-like approach has to satisfy in order to produce a correct description of the resulting diffusion process

Facing Non-Stationary Conditions with a New Indicator of Entropy Increase: The Cassandra Algorithm

We address the problem of detecting non-stationary effects in time series (in particular fractal time series) by means of the Diffusion Entropy Method (DEM). This means that the experimental sequence under study, of size $N$, is explored with a window of size $L << N$. The DEM makes a wise use of the statistical information available and, consequently, in spite of the modest size of the window used, does succeed in revealing local statistical properties, and it shows how they change upon moving the windows along the experimental sequence. The method is expected to work also to predict catastrophic events before their occurrence.Comment: FRACTAL 2002 (Spain

Scaling law of diffusivity generated by a noisy telegraph signal with fractal intermittency

In many complex systems the non-linear cooperative dynamics determine the emergence of self-organized, metastable, structures that are associated with a birth-death process of cooperation. This is found to be described by a renewal point process, i.e., a sequence of crucial birth-death events corresponding to transitions among states that are faster than the typical long-life time of the metastable states. Metastable states are highly correlated, but the occurrence of crucial events is typically associated with a fast memory drop, which is the reason for the renewal condition. Consequently, these complex systems display a power-law decay and, thus, a long-range or scale-free behavior, in both time correlations and distribution of inter-event times, i.e., fractal intermittency. The emergence of fractal intermittency is then a signature of complexity. However, the scaling features of complex systems are, in general, affected by the presence of added white or short-term noise. This has been found also for fractal intermittency. In this work, after a brief review on metastability and noise in complex systems, we discuss the emerging paradigm of Temporal Complexity. Then, we propose a model of noisy fractal intermittency, where noise is interpreted as a renewal Poisson process with event rate rp. We show that the presence of Poisson noise causes the emergence of a normal diffusion scaling in the long-time range of diffusion generated by a telegraph signal driven by noisy fractal intermittency. We analytically derive the scaling law of the long-time normal diffusivity coefficient. We find the surprising result that this long-time normal diffusivity depends not only on the Poisson event rate, but also on the parameters of the complex component of the signal: the power exponent μ of the inter-event time distribution, denoted as complexity index, and the time scale T needed to reach the asymptotic power-law behavior marking the emergence of complexity. In particular, in the range μ &lt; 3, we find the counter-intuitive result that normal diffusivity increases as the Poisson rate decreases. Starting from the diffusivity scaling law here derived, we propose a novel scaling analysis of complex signals being able to estimate both the complexity index μ and the Poisson noise rate rp

Memory beyond memory in heart beating: an efficient way to detect pathological conditions

We study the long-range correlations of heartbeat fluctuations with the method of diffusion entropy. We show that this method of analysis yields a scaling parameter $\delta$ that apparently conflicts with the direct evaluation of the distribution of times of sojourn in states with a given heartbeat frequency. The strength of the memory responsible for this discrepancy is given by a parameter $\epsilon^{2}$, which is derived from real data. The distribution of patients in the ($\delta$, $\epsilon^{2}$)-plane yields a neat separation of the healthy from the congestive heart failure subjects.Comment: submitted to Physical Review Letters, 5 figure

A renewal model for the emergence of anomalous solute crowding in liposomes

A fundamental evolutionary step in the onset of living cells is thought to be the spontaneous formation of lipid vesicles (liposomes) in the pre-biotic mixture. Even though it is well known that hydrophobic forces drive spontaneous liposome formation in aqueous solutions, how the components of the earliest biochemical pathways were trapped and concentrated in the forming vesicles is an issue that still needs to be clarified. In recent years, some authors carried out a set of experiments where a unexpectedly high amount of solutes were found in a small number of liposomes, spontaneously formed in aqueous solution. A great number of empty liposomes were found in the same experiments and the global observed behavior was that of a distribution of solute particles into liposomes in agreement with a inverse power-law function rather than with the expected Poisson distribution. The chemical and physical mechanisms leading to the observed "anomalous solute crowding" are still unclear, but the non-Poisson power-law behavior is associated with some cooperative behavior with strong non-linear interactions in the biochemical processes occurring in the solution. For tackling this issue we propose a model grounding on the Cox's theory of renewal point processes, which many authors consider to play a central role in the description of complex cooperative systems. Starting from two very basic hypotheses and the renewal assumption, we derive a model reproducing the behavior outlined above. In particular, we show that the assumption of a "cooperative" interaction between the solute molecules and the forming liposomes is sufficient for the emergence of the observed power-law behavior. Even though our approach does not provide experimental evidences of the chemical and physical bases of the solute crowding, it suggests promising directions for experimental research and it also provide a first theoretical prediction that could possibly be tested in future experimental investigations

Improving Visual Field Examination of the Macula Using Structural Information

Purpose: To investigate a novel approach for structure-function modeling in glaucoma to improve visual field testing in the macula. Methods: We acquired data from the macular region in 20 healthy eyes and 31 with central glaucomatous damage. Optical coherence tomography (OCT) scans were used to estimate the local macular ganglion cell density. Perimetry was performed with a fundus-tracking device using a 10-2 grid. OCT scans were matched to the retinal image from the fundus perimeter to accurately map the tested locations onto the structural damage. Binary responses from the subjects to all presented stimuli were used to calculate the structure-function model used to generate prior distributions for a ZEST (Zippy Estimation by Sequential Testing) Bayesian strategy. We used simulations based on structural and functional data acquired from an independent dataset of 20 glaucoma patients to compare the performance of this new strategy, structural macular ZEST (MacS-ZEST), with a standard ZEST. Results: Compared to the standard ZEST, MacS-ZEST reduced the number of presentations by 13% in reliable simulated subjects and 14% with higher rates (≥20%) of false positive or false negative errors. Reduction in mean absolute error was not present for reliable subjects but was gradually more important with unreliable responses (≥10% at 30% error rate). Conclusions: Binary responses can be modeled to incorporate detailed structural information from macular OCT into visual field testing, improving overall speed and accuracy in poor responders. Translational Relevance: Structural information can improve speed and reliability for macular testing in glaucoma practice

Power-Law Time Distribution of Large Earthquakes

We study the statistical properties of time distribution of seimicity in California by means of a new method of analysis, the Diffusion Entropy. We find that the distribution of time intervals between a large earthquake (the main shock of a given seismic sequence) and the next one does not obey Poisson statistics, as assumed by the current models. We prove that this distribution is an inverse power law with an exponent $\mu=2.06 \pm 0.01$. We propose the Long-Range model, reproducing the main properties of the diffusion entropy and describing the seismic triggering mechanisms induced by large earthquakes.Comment: 4 pages, 3 figures. Revised version accepted for publication. Typos corrected, more detailed discussion on the method used, refs added. Phys. Rev. Lett. (2003) in pres