Point Information Gain and Multidimensional Data Analysis

We generalize the Point information gain (PIG) and derived quantities, i.e. Point information entropy (PIE) and Point information entropy density (PIED), for the case of R\'enyi entropy and simulate the behavior of PIG for typical distributions. We also use these methods for the analysis of multidimensional datasets. We demonstrate the main properties of PIE/PIED spectra for the real data on the example of several images, and discuss possible further utilization in other fields of data processing.Comment: 16 pages, 6 figure

Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part one: Sunspot dynamics

In this study, the nonlinear analysis of the sunspot index is embedded in the non-extensive statistical theory of Tsallis. The triplet of Tsallis, as well as the correlation dimension and the Lyapunov exponent spectrum were estimated for the SVD components of the sunspot index timeseries. Also the multifractal scaling exponent spectrum, the generalized Renyi dimension spectrum and the spectrum of the structure function exponents were estimated experimentally and theoretically by using the entropy principle included in Tsallis non extensive statistical theory, following Arimitsu and Arimitsu. Our analysis showed clearly the following: a) a phase transition process in the solar dynamics from high dimensional non Gaussian SOC state to a low dimensional non Gaussian chaotic state, b) strong intermittent solar turbulence and anomalous (multifractal) diffusion solar process, which is strengthened as the solar dynamics makes phase transition to low dimensional chaos in accordance to Ruzmaikin, Zeleny and Milovanov studies c) faithful agreement of Tsallis non equilibrium statistical theory with the experimental estimations of i) non-Gaussian probability distribution function, ii) multifractal scaling exponent spectrum and generalized Renyi dimension spectrum, iii) exponent spectrum of the structure functions estimated for the sunspot index and its underlying non equilibrium solar dynamics.Comment: 40 pages, 11 figure

Quantification of long-range power law correlations among healthy and pathologic subjects using detrended fluctuation analysis and multifractal detrended fluctuation analysis

The healthy heartbeat is traditionally thought to be regulated according to the classical principle of homeostasis whereby physiologic systems operate to reduce variability and achieve an equilibrium-like state. However recent studies reveal that under normal conditions, beat-to-beat fluctuation in heart rate display the kind of long-range correlations typically exhibited by the dynamical system far away from equilibrium. In contrast, heart rate time series from patients with severe congestive heart failure show a breakdown of this long-range correlation behavior. Two different non-linear dynamic methods namely Detrended Fluctuation Analysis (DFA) and Multifractal (MF) DFA are used for the quantification of this correlation property in non-stationary physiological time series and it revealed the presence of long-range power law correlation for the group of healthy subjects while breakdown in the long-range power law correlation for the group of subjects with cardiac heart failure. Application of DFA analysis shows evidence for a crossover phenomenon associated with a change in short(αl) and long(α2) range scaling exponents. For healthy subjects, calculated value of αl and α2 (mean value ± S.D.) are 1.31 ± 0.17 and 1.00 ± 0.07 respectively. For subjects with cardiac heart failure calculated value of ctl and a2 is 0.71 ± 0.20 and 1.24 ± 0.07 respectively i.e. only one scaling exponent is not sufficient to characterize the entire heart-rate time series which resulted into MF-DFA approach. This suggested that there is more than one exponent values needed to characterize the heart rate time series. Multifractal DFA is based on generalization of DFA and a MATLAB code is developed to implement the MF-DFA algorithm and to identify whether the given time series under analysis exhibits multifractality or not by generating more than one exponent values for multifractal signal. The value of a for q\u3eO for healthy is 1.04 ± 0.02 and for CHF is found to be 1.32 ± 0.02 and the value of a for q\u3c0 for healthy subjects is 3.01 ±0.26 and for CHF subjects is found to be 3.53 ± 0.14 (mean value ± S.D.) The student\u27s ttest suggests that p-value is 0.00001 which is less than 0.05 thus the value of a for q \u3c0 and q\u3e0 among healthy subjects and CHF subjects are statistically different. Value of a for q\u3e0 is less than that for q\u3c0. And for q =2 MF-DFA retains monofractal DFA. Thus, MF-DFA is clearly able to discriminate among the healthy and CHF for q\u3c0 as well for q\u3e0. MF-DFA also determines which fluctuations i.e. (small or large) dominate for the given interbeat interval time series because for q\u3c0 the slow fluctuations dominate whereas for q\u3e0 large fluctuations dominate. DFA and MF-DFA were able to discriminate 23 Healthy subjects out of 26 Healthy subjects data sets i.e. true positive specificity is 0.89 and false negative specificity is 0.12 and 9 CHF subjects out of 11 CHF subjects data sets i.e. true positive specificity is 0.82 and false negative specificity is 0.19. These methods may be of use in distinguishing healthy from pathologic data sets based on the difference in the scaling properties

Testing the SOC hypothesis for the magnetosphere

As noted by Chang, the hypothesis of Self-Organised Criticality provides a theoretical framework in which the low dimensionality seen in magnetospheric indices can be combined with the scaling seen in their power spectra and the recently-observed plasma bursty bulk flows. As such, it has considerable appeal, describing the aspects of the magnetospheric fuelling:storage:release cycle which are generic to slowly-driven, interaction-dominated, thresholded systems rather than unique to the magnetosphere. In consequence, several recent numerical "sandpile" algorithms have been used with a view to comparison with magnetospheric observables. However, demonstration of SOC in the magnetosphere will require further work in the definition of a set of observable properties which are the unique "fingerprint" of SOC. This is because, for example, a scale-free power spectrum admits several possible explanations other than SOC. A more subtle problem is important for both simulations and data analysis when dealing with multiscale and hence broadband phenomena such as SOC. This is that finite length systems such as the magnetosphere or magnetotail will by definition give information over a small range of orders of magnitude, and so scaling will tend to be narrowband. Here we develop a simple framework in which previous descriptions of magnetospheric dynamics can be described and contrasted. We then review existing observations which are indicative of SOC, and ask if they are sufficient to demonstrate it unambiguously, and if not, what new observations need to be made?Comment: 29 pages, 0 figures. Based on invited talk at Spring American Geophysical Union Meeting, 1999. Journal of Atmospheric and Solar Terrestrial Physics, in pres

25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.Comment: 139 pages, 28 figures, Review based on ISSI workshops "Self-Organized Criticality and Turbulence" (2012, 2013, Bern, Switzerland