Effect of nonlinear filters on detrended fluctuation analysis

We investigate how various linear and nonlinear transformations affect the scaling properties of a signal, using the detrended fluctuation analysis (DFA). Specifically, we study the effect of three types of transforms: linear, nonlinear polynomial and logarithmic filters. We compare the scaling properties of signals before and after the transform. We find that linear filters do not change the correlation properties, while the effect of nonlinear polynomial and logarithmic filters strongly depends on (a) the strength of correlations in the original signal, (b) the power of the polynomial filter and (c) the offset in the logarithmic filter. We further investigate the correlation properties of three analytic functions: exponential, logarithmic, and power-law. While these three functions have in general different correlation properties, we find that there is a broad range of variable values, common for all three functions, where they exhibit identical scaling behavior. We further note that the scaling behavior of a class of other functions can be reduced to these three typical cases. We systematically test the performance of the DFA method in accurately estimating long-range power-law correlations in the output signals for different parameter values in the three types of filters, and the three analytic functions we consider.Comment: 12 pages, 7 figure

Bayesian Blocks, A New Method to Analyze Structure in Photon Counting Data

I describe a new time-domain algorithm for detecting localized structures (bursts), revealing pulse shapes, and generally characterizing intensity variations. The input is raw counting data, in any of three forms: time-tagged photon events (TTE), binned counts, or time-to-spill (TTS) data. The output is the most likely segmentation of the observation into time intervals during which the photon arrival rate is perceptibly constant -- i.e. has a fixed intensity without statistically significant variations. Since the analysis is based on Bayesian statistics, I call the resulting structures Bayesian Blocks. Unlike most, this method does not stipulate time bins -- instead the data themselves determine a piecewise constant representation. Therefore the analysis procedure itself does not impose a lower limit to the time scale on which variability can be detected. Locations, amplitudes, and rise and decay times of pulses within a time series can be estimated, independent of any pulse-shape model -- but only if they do not overlap too much, as deconvolution is not incorporated. The Bayesian Blocks method is demonstrated by analyzing pulse structure in BATSE $\gamma$-ray data. The MatLab scripts and sample data can be found on the WWW at: http://george.arc.nasa.gov/~scargle/papers.htmlComment: 42 pages, 2 figures; revision correcting mathematical errors; clarifications; removed Cyg X-1 sectio

Generalization of entropy based divergence measures for symbolic sequence analysis

Entropy based measures have been frequently used in symbolic sequence analysis. A symmetrized and smoothed form of Kullback-Leibler divergence or relative entropy, the Jensen-Shannon divergence (JSD), is of particular interest because of its sharing properties with families of other divergence measures and its interpretability in different domains including statistical physics, information theory and mathematical statistics. The uniqueness and versatility of this measure arise because of a number of attributes including generalization to any number of probability distributions and association of weights to the distributions. Furthermore, its entropic formulation allows its generalization in different statistical frameworks, such as, non-extensive Tsallis statistics and higher order Markovian statistics. We revisit these generalizations and propose a new generalization of JSD in the integrated Tsallis and Markovian statistical framework. We show that this generalization can be interpreted in terms of mutual information. We also investigate the performance of different JSD generalizations in deconstructing chimeric DNA sequences assembled from bacterial genomes including that of E. coli, S. enterica typhi, Y. pestis and H. influenzae. Our results show that the JSD generalizations bring in more pronounced improvements when the sequences being compared are from phylogenetically proximal organisms, which are often difficult to distinguish because of their compositional similarity. While small but noticeable improvements were observed with the Tsallis statistical JSD generalization, relatively large improvements were observed with the Markovian generalization. In contrast, the proposed Tsallis-Markovian generalization yielded more pronounced improvements relative to the Tsallis and Markovian generalizations, specifically when the sequences being compared arose from phylogenetically proximal organisms.publishedVersionFil: Ré, Miguel Ángel. Universidad Tecnológica Nacional. Facultad Regional Córdoba. Centro de Investigación en Informática para la Ingeniería. Departamento de Ciencias Básicas; Argentina.Fil: Ré, Miguel Ángel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Azad, Rajeev K. University of North Texas. College of Science. Department of Biological Sciences; Estados Unidos de América.Fil: Azad, Rajeev K. University of North Texas. College of Science. Department of Mathematics; Estados Unidos de América.Ciencias de la Información y Bioinformática (desarrollo de hardware va en 2.2 "Ingeniería Eléctrica, Electrónica y de Información" y los aspectos sociales van en 5.8 "Comunicación y Medios"

Multifractal and entropy analysis of resting-state electroencephalography reveals spatial organization in local dynamic functional connectivity

Functional connectivity of the brain fluctuates even in resting-state condition. It has been reported recently that fluctuations of global functional network topology and those of individual connections between brain regions expressed multifractal scaling. To expand on these findings, in this study we investigated if multifractality was indeed an inherent property of dynamic functional connectivity (DFC) on the regional level as well. Furthermore, we explored if local DFC showed region-specific differences in its multifractal and entropy-related features. DFC analyses were performed on 62-channel, resting-state electroencephalography recordings of twelve young, healthy subjects. Surrogate data testing verified the true multifractal nature of regional DFC that could be attributed to the presumed nonlinear nature of the underlying processes. Moreover, we found a characteristic spatial distribution of local connectivity dynamics, in that frontal and occipital regions showed stronger long-range correlation and higher degree of multifractality, whereas the highest values of entropy were found over the central and temporal regions. The revealed topology reflected well the underlying resting-state network organization of the brain. The presented results and the proposed analysis framework could improve our understanding on how resting-state brain activity is spatio-temporally organized and may provide potential biomarkers for future clinical research

Hierarchical structure of cascade of primary and secondary periodicities in Fourier power spectrum of alphoid higher order repeats

<p>Abstract</p> <p>Background</p> <p>Identification of approximate tandem repeats is an important task of broad significance and still remains a challenging problem of computational genomics. Often there is no single best approach to periodicity detection and a combination of different methods may improve the prediction accuracy. Discrete Fourier transform (DFT) has been extensively used to study primary periodicities in DNA sequences. Here we investigate the application of DFT method to identify and study alphoid higher order repeats.</p> <p>Results</p> <p>We used method based on DFT with mapping of symbolic into numerical sequence to identify and study alphoid higher order repeats (HOR). For HORs the power spectrum shows equidistant frequency pattern, with characteristic two-level hierarchical organization as signature of HOR. Our case study was the 16 mer HOR tandem in AC017075.8 from human chromosome 7. Very long array of equidistant peaks at multiple frequencies (more than a thousand higher harmonics) is based on fundamental frequency of 16 mer HOR. Pronounced subset of equidistant peaks is based on multiples of the fundamental HOR frequency (multiplication factor <it>n </it>for <it>n</it>mer) and higher harmonics. In general, <it>n</it>mer HOR-pattern contains equidistant secondary periodicity peaks, having a pronounced subset of equidistant primary periodicity peaks. This hierarchical pattern as signature for HOR detection is robust with respect to monomer insertions and deletions, random sequence insertions etc. For a monomeric alphoid sequence only primary periodicity peaks are present. The 1/<it>f</it>^{<it>β </it>}– noise and periodicity three pattern are missing from power spectra in alphoid regions, in accordance with expectations.</p> <p>Conclusion</p> <p>DFT provides a robust detection method for higher order periodicity. Easily recognizable HOR power spectrum is characterized by hierarchical two-level equidistant pattern: higher harmonics of the fundamental HOR-frequency (secondary periodicity) and a subset of pronounced peaks corresponding to constituent monomers (primary periodicity). The number of lower frequency peaks (secondary periodicity) below the frequency of the first primary periodicity peak reveals the size of <it>n</it>mer HOR, i.e., the number <it>n </it>of monomers contained in consensus HOR.</p

Bayesian Analysis of Isochores.

Statistical identification of isochore structure, the variation in large-scale GC composition (proportion of DNA bases that are G or C as opposed to A or T), of mammalian genomes is a necessary requirement for understanding both the evolution of base composition and the many genomic features such as mutation and recombination rates, which covary with base composition. We have developed a Bayesian method for isochore analysis that we demonstrate to be more accurate than the commonly used binary segmentation approach implemented within the program IsoFinder. The method accounts for both fine-scale and large-scale structure. We adapt direct simulation methods to allow for iid samples from the posterior distribution of our model, and provide an accurate approximation to this that can analyze data from a chromosome in a matter of seconds. We apply our method to human chromosome 1. The resulting estimate of how GC content varies across this region is shown to be a better predictor of local recombination rates than IsoFinder, and we are able to detect regions consistent with the classic definition of isochores that cover 85% of the chromosome. We also show a measure of relative GC content to be particularly predictive of local recombination rates

Towards a deeper understanding of the complex behaviour observed in the distribution of words in written texts

Here we show that the recently reported presence of long-range correlations in the distribution of words along texts is due to the complex distribution of the keywords, while common words are not correlated. Indeed we prove that the degree of long-range correlations of a word at long scales is a good measure of its relevance to the text. Additionally, we develop a model able to reproduce the spatial distribution of a word in a text, based on the long-range correlations observed for the word. The model not only reproduces the complex behaviour characterized by the presence of correlations at long scales and the degree of relevance of the word, but also the probability distribution of the inter-occurrences distances in the whole range of scales

Analysis of symbolic sequences using the Jensen-Shannon divergence

We study statistical properties of the Jensen-Shannon divergence D, which quantifies the difference between probability distributions, and which has been widely applied to analyses of symbolic sequences. We present three interpretations of D in the framework of statistical physics, information theory, and mathematical statistics, and obtain approximations of the mean, the variance, and the probability distribution of D in random, uncorrelated sequences. We present a segmentation method based on D that is able to segment a nonstationary symbolic sequence into stationary subsequences, and apply this method to DNA sequences, which are known to be nonstationary on a wide range of different length scales

Comparison of methods for the assessment of nonlinearity in short-term heart rate variability under different physiopathological states

Despite the widespread diffusion of nonlinear methods for heart rate variability (HRV) analysis, the presence and the extent to which nonlinear dynamics contribute to short-term HRV are still controversial. This work aims at testing the hypothesis that different types of nonlinearity can be observed in HRV depending on the method adopted and on the physiopathological state. Two entropy-based measures of time series complexity (normalized complexity index, NCI) and regularity (information storage, IS), and a measure quantifying deviations from linear correlations in a time series (Gaussian linear contrast, GLC), are applied to short HRV recordings obtained in young (Y) and old (O) healthy subjects and in myocardial infarction (MI) patients monitored in the resting supine position and in the upright position reached through head-up tilt. The method of surrogate data is employed to detect the presence and quantify the contribution of nonlinear dynamics to HRV. We find that the three measures differ both in their variations across groups and conditions and in the percentage and strength of nonlinear HRV dynamics. NCI and IS displayed opposite variations, suggesting more complex dynamics in O and MI compared to Y and less complex dynamics during tilt. The strength of nonlinear dynamics is reduced by tilt using all measures in Y, while only GLC detects a significant strengthening of such dynamics in MI. A large percentage of detected nonlinear dynamics is revealed only by the IS measure in the Y group at rest, with a decrease in O and MI and during T, while NCI and GLC detect lower percentages in all groups and conditions. While these results suggest that distinct dynamic structures may lie beneath short-term HRV in different physiological states and pathological conditions, the strong dependence on the measure adopted and on their implementation suggests that physiological interpretations should be provided with caution

Changes of heart rate variability during exercise

We use recent developments in heart rate dynamic estimation to detrend athlete's heart rate measured during effort tests and study how heart rate variability (HRV) changes during exercise, in a sample of 18 young athletes. RR interval standard deviation decays exponentially with the heart rate, and the decay rate is linked with physical fitness