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Daily pandemic surveillance, often achieved through the estimation of the reproduction number, constitutes a critical challenge for national health authorities to design counter-measures. In an earlier work, we proposed to formulate the estimation of the reproduction number as an optimization problem, combining data-model fidelity and space-time regularity constraints, solved by nonsmooth convex proximal minimizations. Though promising, that first formulation significantly lacks robustness against the Covid-19 data low quality (irrelevant or missing counts, pseudo-seasonalities,...) stemming from the emergency and crisis context, which significantly impairs accurate pandemic evolution assessments. The present work aims to overcome these limitations by carefully crafting a functional permitting to estimate jointly, in a single step, the reproduction number and outliers defined to model low quality data. This functional also enforces epidemiology-driven regularity properties for the reproduction number estimates, while preserving convexity, thus permitting the design of efficient minimization algorithms, based on proximity operators that are derived analytically. The explicit convergence of the proposed algorithm is proven theoretically. Its relevance is quantified on real Covid-19 data, consisting of daily new infection counts for 200+ countries and for the 96 metropolitan France counties, publicly available at Johns Hopkins University and Santé-Publique-France. The procedure permits automated daily updates of these estimates, reported via animated and interactive maps. Open-source estimation procedures will be made publicly available. [ABSTRACT FROM AUTHOR]

Among the different indicators that quantify the spread of an epidemic such as the on-going COVID-19, stands first the reproduction number which measures how many people can be contaminated by an infected person. In order to permit the monitoring of the evolution of this number, a new estimation procedure is proposed here, assuming a well-accepted model for current incidence data, based on past observations. The novelty of the proposed approach is twofold: 1) the estimation of the reproduction number is achieved by convex optimization within a proximal-based inverse problem formulation, with constraints aimed at promoting piecewise smoothness; 2) the approach is developed in a multivariate setting, allowing for the simultaneous handling of multiple time series attached to different geographical regions, together with a spatial (graph-based) regularization of their evolutions in time. The effectiveness of the approach is first supported by simulations, and two main applications to real COVID-19 data are then discussed. The first one refers to the comparative evolution of the reproduction number for a number of countries, while the second one focuses on French departments and their joint analysis, leading to dynamic maps revealing the temporal co-evolution of their reproduction numbers. [ABSTRACT FROM AUTHOR]

Bike sharing systems (BSS) have been growing fast all over the world, along with the number of articles analyzing such systems. However the lack of databases at the individual level and covering several years has limited the analysis of BSS users' behavior in the long term. This article gives a first detailed description of the temporal evolution of individual customers. Using a 5-year dataset covering 120,827 distinct year-long subscribers, we show the heterogeneous individual trajectories masked by the overall system stability. Users follow two main trajectories: about half remain in the system for at most one year, showing a low median activity (47 trips); the remaining half corresponds to more active users (median activity of 91 trips in their first year) that remain continuously active for several years (mean time = 2.9 years). We show that users from urban cores, middle-aged and male are over represented among these long-term users, which profit most from the BSS. This provides further support for the view that BSS mostly benefit the already privileged. [ABSTRACT FROM AUTHOR]

The second Signal Processing and Monitoring in Labor workshop gathered researchers who utilize promising new research strategies and initiatives to tackle the challenges of intrapartum fetal monitoring. The workshop included a series of lectures and discussions focusing on: new algorithms and techniques for cardiotocogoraphy (CTG) and electrocardiogram acquisition and analyses; the results of a CTG evaluation challenge comparing state-of-the-art computerized methods and visual interpretation for the detection of arterial cord pH <7.05 at birth; the lack of consensus about the role of intrapartum acidemia in the etiology of fetal brain injury; the differences between methods for CTG analysis "mimicking" expert clinicians and those derived from "data-driven" analyses; a critical review of the results from two randomized controlled trials testing the former in clinical practice; and relevant insights from modern physiology-based studies. We concluded that the automated algorithms performed comparably to each other and to clinical assessment of the CTG. However, the sensitivity and specificity urgently need to be improved (both computerized and visual assessment). Data-driven CTG evaluation requires further work with large multicenter datasets based on well-defined labor outcomes. And before first tests in the clinic, there are important lessons to be learnt from clinical trials that tested automated algorithms mimicking expert CTG interpretation. In addition, transabdominal fetal electrocardiogram monitoring provides reliable CTG traces and variability estimates; and fetal electrocardiogram waveform analysis is subject to promising new research. There is a clear need for close collaboration between computing and clinical experts. We believe that progress will be possible with multidisciplinary collaborative research. [ABSTRACT FROM AUTHOR]

The Fourier transform (or spectral analysis) has become a universal tool for data analysis in many different real-world applications, notably for the characterization of temporal/spatial dynamics in data. The wavelet transform (or multiscale analysis) can be regarded as tailoring spectral estimation to classes of signals or functions defined by scale-free dynamics. The present contribution first formally reviews these connections in the context of multivariate stationary processes, and second details the ability of the wavelet transform to extend multivariate scale-free temporal dynamics analysis beyond second-order statistics (Fourier spectrum and autocovariance function) to multivariate self-similarity and multivariate multifractality. Illustrations and qualitative discussions of the relevance of scale-free dynamics for macroscopic brain activity description using MEG data are proposed. [ABSTRACT FROM AUTHOR]

Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called multifractal spectrum. The practical estimation of the multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the multifractal spectrum to be a concave function. This paper addresses this severe practical limitation and proposes a novel formalism for multifractal analysis that enables nonconcave multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave multifractal spectra. Numerical experiments are conducted on several synthetic multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed multifractal formalism over the state of the art. [ABSTRACT FROM AUTHOR]

Objective: Numerous indices were devised for the statistical characterization of temporal dynamics of heart rate variability (HRV) with the aim to discriminate between healthy subjects and nonhealthy patients. Elaborating on the concepts of (multi)fractal and nonlinear analyses, the present contribution defines and studies formally novel non Gaussian multiscale representations. Methods: A methodological framework for non Gaussian multiscale representations constructed on wavelet p-leaders is developed, relying a priori neither on exact scale-free dynamics nor on predefined forms of departure from Gaussianity. Its versatility in quantifying the strength and nature of departure from Gaussian is analyzed theoretically and numerically. The ability of the representations to discriminate between healthy subjects and congestive heart failure (CHF) patients, and between survivors and nonsurvivor CHF patients, is assessed on a large cohort of 198 subjects. Results: The analysis leads to conclude that i) scale-free and multifractal dynamics are observed, both for healthy subjects and CHF patients, for time scales shorter than ${\text{170 s}}$ ; ii) a circadian evolution of multifractal and non Gaussian properties of HRV is evidenced for healthy subjects, but not for CHF patients; iii) non Gaussian multiscale indices possess high discriminative abilities between survivor and nonsurvivor CHF patients, at specific time scales ($\sim \!{\text{20 s}}$ and $\sim\! {\text{85 s}}$). Conclusions: The non Gaussian multiscale representations provide evidence for the existence of short-term cascade-type multifractal mechanisms underlying HRV for both healthy and CHF subjects. A circadian evolution of this mechanism is only evidenced for the healthy group, suggesting an alteration of the sympathetic-parasympathetic balance for CHF patients. Significance: Results obtained for a large cohort of subjects suggest that the novel non Gaussian indices might robustly quantify crucial information for clinical risk stratification in CHF patients. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we extend the methodology proposed in Abry and Didier (2018) to obtain the first joint estimator of the real parts of the Hurst eigenvalues of n -variate operator fractional Brownian motion (OFBM). The procedure consists of a weighted regression on the log-eigenvalues of the sample wavelet spectrum. The estimator is shown to be consistent for any time reversible OFBM and, under stronger assumptions, also asymptotically normal starting from either continuous or discrete time measurements. Simulation studies establish the finite-sample effectiveness of the methodology and illustrate its benefits compared to univariate-like (entry-wise) analysis. As an application, we revisit the well-known self-similar character of Internet traffic by applying the proposed methodology to 4-variate time series of modern, high quality Internet traffic data. The analysis reveals the presence of a rich multivariate self-similarity structure. [ABSTRACT FROM AUTHOR]

Multifractal analysis has become a standard signal processing tool, for which a promising new formulation, the \boldsymbol p-leader multifractal formalism, has recently been proposed. It relies on novel multiscale quantities, the \boldsymbol p-leaders, defined as local $\ell ^p$ norms of sets of wavelet coefficients located at infinitely many fine scales. Computing such infinite sums from actual finite-resolution data requires truncations to the finest available scale, which results in biased \boldsymbol p -leaders and thus in inaccurate estimates of multifractal properties. A systematic study of such finite-resolution effects leads to conjecture an explicit and universal closed-form correction that permits an accurate estimation of scaling exponents. This conjecture is formulated from the theoretical study of a particular class of models for multifractal processes, the wavelet-based cascades. The relevance and generality of the proposed conjecture is assessed by numerical simulations conducted over a large variety of multifractal processes. Finally, the relevance of the proposed corrected estimators is demonstrated on the analysis of heart rate variability data. [ABSTRACT FROM PUBLISHER]

Abstract: Studies employing functional connectivity-type analyses have established that spontaneous fluctuations in functional magnetic resonance imaging (fMRI) signals are organized within large-scale brain networks. Meanwhile, fMRI signals have been shown to exhibit 1/f-type power spectra — a hallmark of scale-free dynamics. We studied the interplay between functional connectivity and scale-free dynamics in fMRI signals, utilizing the fractal connectivity framework — a multivariate extension of the univariate fractional Gaussian noise model, which relies on a wavelet formulation for robust parameter estimation. We applied this framework to fMRI data acquired from healthy young adults at rest and while performing a visual detection task. First, we found that scale-invariance existed beyond univariate dynamics, being present also in bivariate cross-temporal dynamics. Second, we observed that frequencies within the scale-free range do not contribute evenly to inter-regional connectivity, with a systematically stronger contribution of the lowest frequencies, both at rest and during task. Third, in addition to a decrease of the Hurst exponent and inter-regional correlations, task performance modified cross-temporal dynamics, inducing a larger contribution of the highest frequencies within the scale-free range to global correlation. Lastly, we found that across individuals, a weaker task modulation of the frequency contribution to inter-regional connectivity was associated with better task performance manifesting as shorter and less variable reaction times. These findings bring together two related fields that have hitherto been studied separately — resting-state networks and scale-free dynamics, and show that scale-free dynamics of human brain activity manifest in cross-regional interactions as well. [Copyright &y& Elsevier]

In this contribution, the statistical performance of the wavelet-based estimation procedure for the Hurst parameter is studied for non-Gaussian long-range dependent processes obtained from point transformations of Gaussian processes. The statistical properties of the wavelet coefficients and the estimation performance are compared both for processes having the same covariance but different marginal distributions and for processes having the same covariance and same marginal distributions but obtained from different point transformations, analyzed using mother wavelets with different number of vanishing moments. It is shown that the reduction of the dependence range from long to short by increasing the number of vanishing moments, observed for Gaussian processes, and at the origin of the popularity of the wavelet-based estimator, does not hold in general for non-Gaussian processes. Crucially, it is also observed that the Hermite rank of the point transformation impacts significantly the statistical properties of the wavelet coefficients and the estimation performance and also that processes having identical marginal distributions and covariance function can yet yield significantly different estimation performance. These results are interpreted in the light of central and noncentral limit theorems that are fundamental when dealing with long-range dependent processes. Moreover, it will be shown that, on condition that estimation is performed using a range of scales restricted to the coarsest practically available, an approximate, yet analytical and simple to use in practice, formula can be proposed for the evaluation of the variance of the wavelet-based estimator of the Hurst parameter. [ABSTRACT FROM AUTHOR]

Community shared bicycle systems, such as the Vélo'v program launched in Lyon in May 2005, are public transportation programs that can be studied as a complex system composed of interconnected stations that exchange bicycles. They generate digital footprints that reveal the activity in the city over time and space, making possible a quantitative analysis of movements using bicycles in the city. A careful study relying on nonstationary statistical modeling and data mining allows us to first model the time evolution of the dynamics of movements with Vélo'v, that is mostly cyclostationary over the week with nonstationary evolutions over larger time-scales, and second to disentangle the spatial patterns to understand and visualize the flows of Vélo'v bicycles in the city. This study gives insights on the social behaviors of the users of this intermodal transportation system, the objective being to help in designing and planning policy in urban transportation. [ABSTRACT FROM AUTHOR]

Multifractal random Walks are defined as integrals of infinitely divisible stationary multifractal cascades with respect to fractional Brownian motion. Their key properties are studied, such as finiteness of moments and scaling, with respect to the chosen values of the self-similarity and infinite divisibility parameters. The range of these parameters is larger than that considered previously in the literature, and the cases of both exact and nonexact scale invariance are considered. Special attention is paid to various types of definitions of multifractal random walks. The resulting random walks are of interest in modeling multifractal processes whose marginals exhibit stationarity and symmetry. [ABSTRACT FROM AUTHOR]

Multifractal analysis, which mostly consists of measuring scaling exponents, is becoming a standard technique available in most empirical data analysis toolboxes. Making use of the most recent theoretical results, it is based here on the estimation of the cumulants of the log of the wavelet Leaders, an elaboration on the wavelet coefficients. These log-cumulants theoretically enable discrimination between mono- and multifractal processes, as well as between simple log-normal multifractal models and more advanced ones. The goal of the present contribution is to design nonparametric bootstrap hypothesis tests aiming at testing the nature of the multifractal properties of stochastic processes and empirical data. Bootstrap issues together with six declinations of test designs are analyzed. Their statistical performance (significances, powers, and p-values) are assessed and compared by means of Monte Carlo simulations performed on synthetic stochastic processes whose multifractal properties (and log-cumulants) are known theoretically a priori. We demonstrate that the joint use of wavelet Leaders, log-cumulants, and bootstrap procedures enable us to obtain a powerful tool for testing the multifractal properties of data. This tool is practically effective and can be applied to a single observation of data with finite length. [ABSTRACT FROM AUTHOR]

Heart beat rate and blood pressure, together with baroreflex sensitivity, have become important tools in assessing cardiac autonomic system control and in studying sympathovagal balance. These analyses are usually performed thanks to spectral indices computed from standard spectral analysis techniques. However, standard spectral analysis and its corresponding rigid band-pass filter formulation suffer from two major drawbacks. It can be significantly distorted by non-stationarity issues and it proves unable to adjust to natural intra- and inter-individual variability. Empirical mode decomposition (EMD), a tool recently introduced in the literature, provides us with a signal-adaptive decomposition that proves useful for the analysis of non-stationary data and shows a strong capability to precisely adjust to the spectral content of the analyzed data. It is based on the concept that any complicated set of data can be decomposed into a finite number of components, called intrinsic mode functions, associated with different spectral contributions. The aims of this study were twofold. First, we studied the changes in the sympathovagal balance induced by various pharmacological blockades (phentolamine, atropine and atenolol) of the autonomic nervous system in normotensive rats. Secondly, we assessed the use of EMD for the analysis of the cardiac sympathovagal balance after pharmacological injections. For this, we developed a new (EMD-based) low frequency vs. high frequency spectral decomposition of heart beat variability and systolic blood pressure, we define the corresponding EMD spectral indices and study their relevance to detect and analyze changes accurately in the sympathovagal balance without having recourse to any a priori fixed high-pass/low-pass filters. [ABSTRACT FROM AUTHOR]

Abstract: Based on a wavelet-type expansion of the Rosenblatt process, we introduce and examine two different practical ways to simulate the Rosenblatt process. The synthesis procedures proposed here are obtained by either truncating the series of the approximation term or using the approximation coefficients in the wavelet-type expansion of the Rosenblatt process. Both benefit from the low computational cost usually associated with the discrete wavelet transform. We show that the number of zero moments of a related orthogonal multiresolution analysis plays an important role. We study in detail the wavelet-based simulation in terms of uniform convergence. We also discuss at length the importance of the choices of the initial and final resolutions, the specific case of the simulation on the integer grid as well as the usefulness of the wavelet-based simulation. Matlab routines implementing these synthesis procedures as well as their analysis are available upon request. [Copyright &y& Elsevier]

A method is developed for the automatic detection of the onset of scaling for long-range dependent (LRD) time series and other asymptotically scale-invariant processes. Based on wavelet techniques, it provides the lower cutoff scale for the regression that yields the scaling exponent. The method detects the onset of scaling through the dramatic improvement of a goodness-of-fit statistic taken as a function of this lower cutoff scale. It relies on qualitative features of the goodness-of-fit statistic and on features of the wavelet analysis. The method is easy to implement, appropriate for large data sets and highly robust. It is tested against 34 time series models and found to perform very well. Examples involving telecommunications data are presented. [ABSTRACT FROM AUTHOR]

Presents a study which described a statistical test for determining if scaling components vary over time. Wavelet approach to scaling analysis; Wavelet test for the constancy of scaling; Conclusion.

Presents a joint estimator for the two parameters that define the long-range dependence phenomenon. Basis of the estimator; Accuracy of closed-form expressions; Comparison against other estimators; Application to ethernet data.

Analyzes intermittency in turbulence and determines its relation with the existence of coherent structures in the flow. Time and space measurements of velocity and pressure in a swirling flow; Statistics of the interarrivals of the drops; Use of log-infinitely divisible cascade models.

The aggregation procedure is a natural way to analyse signals which exhibit long-range-dependent features and has been used as a basis for estimation of the Hurst parameter, H. In this paper it is shown how aggregation can be naturally rephrased within the wavelet transform framework, being directly related to approximations of the signal in the sense of a Haar multiresolution analysis. A natural wavelet-based generalization to traditional aggregation is then proposed: ‘a-aggregation’. It is shown that a-aggregation cannot lead to good estimators of H, and so a new kind of aggregation, ‘d-aggregation’, is defined, which is related to the details rather than the approximations of a multiresolution analysis. An estimator of H based on d-aggregation has excellent statistical and computational properties, whilst preserving the spirit of aggregation. The estimator is applied to telecommunications network data. [ABSTRACT FROM AUTHOR]

Introduces a wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter. Information about the wavelet-based dependence phenomenon; Stationarity of the wavelet-based tool; Analysis of the wavelet-based Ethernet data.

The present work investigates the segmentation of textures by formulating it as a strongly convex optimization problem, aiming to favor piecewise constancy of fractal features (local variance and local regularity) widely used to model real-world textures in numerous applications very different in nature. Two objective functions combining these two features are compared, referred to as joint and coupled , promoting either independent or co-localized changes in local variance and regularity. To solve the resulting convex nonsmooth optimization problems, because the processing of large size images and databases are targeted, two categories of proximal algorithms (dual forward-backward and primal-dual), are devised and compared. An in-depth study of the objective functions, notably of their strong convexity, memory and computational costs, permits to propose significantly accelerated algorithms. A class of synthetic models of piecewise fractal texture is constructed and studied. They enable, by means of large-scale Monte-Carlo simulations, to quantify the benefits in texture segmentation of combining local regularity and local variance (as opposed to regularity only) while using strong-convexity accelerated primal-dual algorithms. Achieved results also permit to discuss the gains/costs in imposing co-localizations of changes in local regularity and local variance in the problem formulation. Finally, the potential of the proposed approaches is illustrated on real-world textures taken from a publicly available and documented database. [ABSTRACT FROM AUTHOR]

Highlights • We estimated scale-free human brain dynamics using wavelet-leader formalism. • High-to-low self-similarity defined a fronto-occipital gradient. • The gradient was enhanced in visual motion discrimination compared to resting-state. • Scale-free brain dynamics showed multifractal properties. • Self-similarity decreased whereas multifractality increased from rest to visual task. Abstract Background The temporal structure of macroscopic brain activity displays both oscillatory and scale-free dynamics. While the functional relevance of neural oscillations has been largely investigated, both the nature and the role of scale-free dynamics in brain processing have been disputed. New method Here, we offer a novel method to rigorously enrich the characterization of scale-free brain activity using a robust wavelet-based assessment of self-similarity and multifractality. For this, we analyzed human brain activity recorded with magnetoencephalography (MEG) while participants were at rest or performing a visual motion discrimination task. Results First, we report consistent infraslow (from 0.1 to 1.5 Hz) scale-free dynamics (i.e., self-similarity and multifractality) in resting-state and task data. Second, we observed a fronto-occipital gradient of self-similarity reminiscent of the known hierarchy of temporal scales from sensory to higher-order cortices; the anatomical gradient was more pronounced in task than in rest. Third, we observed a significant increase of multifractality during task as compared to rest. Additionally, the decrease in self-similarity and the increase in multifractality from rest to task were negatively correlated in regions involved in the task, suggesting a shift from structured global temporal dynamics in resting-state to locally bursty and non Gaussian scale-free structures during task. Comparison with existing method(s) We showed that the wavelet leader based multifractal approach extends power spectrum estimation methods in the way of characterizing finely scale-free brain dynamics. Conclusions Altogether, our approach provides novel fine-grained characterizations of scale-free dynamics in human brain activity. [ABSTRACT FROM AUTHOR]

Intrapartum fetal heart rate (FHR) monitoring constitutes a reference tool in clinical practice to assess the baby's health status and to detect fetal acidosis. It is usually analyzed by visual inspection grounded on FIGO criteria. Characterization of intrapartum fetal heart rate temporal dynamics remains a challenging task and continuously receives academic research efforts. Complexity measures, often implemented with tools referred to as approximate entropy (ApEn) or sample entropy (SampEn), have regularly been reported as significant features for intrapartum FHR analysis. We explore how information theory, and especially auto-mutual information (AMI), is connected to ApEn and SampEn and can be used to probe FHR dynamics. Applied to a large (1404 subjects) and documented database of FHR data, collected in a French academic hospital, it is shown that (i) auto-mutual information outperforms ApEn and SampEn for acidosis detection in the first stage of labor and continues to yield the best performance in the second stage; (ii) Shannon entropy increases as labor progresses and is always much larger in the second stage; (iii) babies suffering from fetal acidosis additionally show more structured temporal dynamics than healthy ones and that this progressive structuration can be used for early acidosis detection. [ABSTRACT FROM AUTHOR]

Poisson cluster processes are special point processes that find use in modeling Internet traffic, neural spike trains, computer failure times and other real-life phenomena. The focus of this work is on the various moments and cumulants of Poisson cluster processes, and specifically on their behavior at small and large scales. Under suitable assumptions motivated by the multiscale behavior of Internet traffic, it is shown that all these various quantities satisfy scale free (scaling) relations at both small and large scales. Only some of these relations turn out to carry information about salient model parameters of interest, and consequently can be used in the inference of the scaling behavior of Poisson cluster processes. At large scales, the derived results complement those available in the literature on the distributional convergence of normalized Poisson cluster processes, and also bring forward a more practical interpretation of the so-called slow and fast growth regimes. Finally, the results are applied to a real data trace from Internet traffic. [ABSTRACT FROM AUTHOR]

In the context of change-point detection, addressed by Total Variation minimization strategies, an efficient on-the-fly algorithm has been designed leading to exact solutions for univariate data. In this contribution, an extension of such an on-the-fly strategy to multivariate data is investigated. The proposed algorithm relies on the local validation of the Karush-Kuhn-Tucker conditions on the dual problem. Showing that the non-local nature of the multivariate setting precludes to obtain an exact on-the-fly solution, we devise an on-the-fly algorithm delivering an approximate solution, whose quality is controlled by a practitioner-tunable parameter, acting as a trade-off between quality and computational cost. Performance assessment shows that high quality solutions are obtained on-the-fly while benefiting of computational costs several orders of magnitude lower than standard iterative procedures. The proposed algorithm thus provides practitioners with an efficient multivariate change-point detection on-the-fly procedure. [ABSTRACT FROM PUBLISHER]

The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism, non-standard limit distributions are obtained, and related to a form of ergodicity breaking in the underlying non-homogeneous Hidden Markov Chain. The link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions. [ABSTRACT FROM AUTHOR]

Scale invariance and multifractal analysis constitute paradigms nowadays widely used for real-world data characterization. In essence, they amount to assuming power law behaviors of well-chosen multiresolution quantities as functions of the analysis scale. The exponents of these power laws, the scaling exponents, are then measured and involved in classical signal processing tasks. Yet, the practical estimation of such exponents implies the selection of a range of scales where the power law behaviors hold, a difficult task with yet crucial impact on performance. In the present contribution, a nonparametric bootstrap based procedure is devised to achieve scaling range automated selection. It is shown to be effective and relevant in practice. Its performance, benefits and computational costs are assessed by means of Monte Carlo simulations. It is applied to synthetic multifractal processes and shown to yield robust and accurate estimation of multifractal parameters, despite various difficulties such as noise corruption or inter-subject variability. Finally, its potential is illustrated at work for the analysis of adult heart rate variability on a large database. [ABSTRACT FROM AUTHOR]

When generating Gaussian stationary random fields, a standard method based on circulant matrix embedding usually fails because some of the associated eigenvalues are negative. The eigenvalues can be shown to be nonnegative in the limit of increasing sample size. Computationally feasible large sample sizes, however, rarely lead to nonnegative eigenvalues. Another solution is to extend suitably the covariance function of interest so that the eigenvalues of the embedded circulant matrix become nonnegative in theory. Though such extensions have been found for a number of examples of stationary fields, the method depends on nontrivial constructions in specific cases. In this work, the embedded circulant matrix is smoothed at the boundary by using a cutoff window or overlapping windows over a transition region. The windows are not specific to particular examples of stationary fields. The resulting method modifies the standard circulant embedding, and is easy to use. It is shown that this straightforward approach works for many examples of interest, with the overlapping windows performing consistently better. The method even outperforms in the cases where extending the covariance leads to nonnegative eigenvalues in theory, in the sense that the transition region is considerably smaller. The Matlab code implementing the method is included in the online supplementary materials and also publicly available atwww.hermir.org. [ABSTRACT FROM AUTHOR]

Inspired from modern out-of-equilibrium statistical physics models, a matrix product based framework is defined and studied, that permits the formal definition of random vectors and time series whose desired joint distributions are a priori prescribed. Its key feature consists of preserving the writing of the joint distribution as the simple product structure it has under independence, while inputing controlled dependencies amongst components: This is obtained by replacing the product of probability densities by a product of matrices of probability densities. It is first shown that this matrix product model can be remapped onto the framework of Hidden Markov Models. Second, combining this dual perspective enables us both to study the statistical properties of this model in terms of marginal distributions and dependencies (a stationarity condition is notably devised) and to devise an efficient and accurate numerical synthesis procedure. A design procedure is also described that permits the tuning of model parameters to attain targeted statistical properties. Pedagogical well-chosen examples of times series and multivariate vectors aim at illustrating the power and versatility of the proposed approach and at showing how targeted statistical properties can actually be prescribed. [ABSTRACT FROM AUTHOR]

We consider a generalized version of the random energy model in which the energy of each configuration is given by the sum of N independent contributions (local energies) with finite variances but otherwise arbitrary statistics. Using the large deviation formalism, we find that the glass transition generically exists when local energies have a smooth distribution. In contrast, if the distribution of the local energies has a Dirac mass at the minimal energy (e.g., if local energies take discrete values), the glass transition ceases to exist if the number of energy levels grows sufficiently fast with system size. This shows that the statistical independence of energy levels does not imply the existence of a glass transition. [ABSTRACT FROM AUTHOR]

The study of functional brain-heart interplay has provided meaningful insights in cardiology and neuroscience. Regarding biosignal processing, this interplay involves predominantly neural and heartbeat linear dynamics expressed via time and frequency domain-related features. However, the dynamics of central and autonomous nervous systems show nonlinear and multifractal behaviours, and the extent to which this behaviour influences brain-heart interactions is currently unknown. Here, we report a novel signal processing framework aimed at quantifying nonlinear functional brain-heart interplay in the non-Gaussian and multifractal domains that combines electroencephalography (EEG) and heart rate variability series. This framework relies on a maximal information coefficient analysis between nonlinear multiscale features derived from EEG spectra and from an inhomogeneous point-process model for heartbeat dynamics. Experimental results were gathered from 24 healthy volunteers during a resting state and a cold pressor test, revealing that synchronous changes between brain and heartbeat multifractal spectra occur at higher EEG frequency bands and through nonlinear/complex cardiovascular control. We conclude that significant bodily, sympathovagal changes such as those elicited by coldpressure stimuli affect the functional brain-heart interplay beyond second-order statistics, thus extending it to multifractal dynamics. These results provide a platform to define novel nervous-system-targeted biomarkers. [ABSTRACT FROM AUTHOR]

Abstract: This contribution aims at studying the behavior of the classical sample moment estimator, , as a function of the number of available samples n, in the case where the random variables X are positive, have finite moments at all orders and are naturally of the form with the tail of Y behaving like . This class of laws encompasses and generalizes the classical example of the log-normal law. This form is motivated by a number of applications stemming from modern statistical physics or multifractal analysis. Borrowing heuristic and analytical results from the analysis of the Random Energy Model in statistical physics, a critical moment q c (n) is defined as the largest statistical order q up to which the sample mean estimator correctly accounts for the ensemble average , for a given n. A practical estimator for the critical moment q c (n) is then proposed. Its statistical performance are studied analytically and illustrated numerically in the case of i.i.d. samples. A simple modification is proposed to explicitly account for correlation among the observed samples. Estimation performance are then carefully evaluated by means of Monte-Carlo simulations in the practical case of correlated time series. [Copyright &y& Elsevier]

Using a renormalization approach, we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power qn that varies monotonically with the sample size n. Under these conditions, a non-standard class of max-stable limit distributions, which mirror the classical ones, emerges. Furthermore, a transition mechanism between the classical and the non-standard limit distributions is brought to light. If qn grows slower than a characteristic function q* n, the standard limit distributions are recovered, while if qn behaves asymptotically as λq*n, nonstandard limit distributions emerge. [ABSTRACT FROM AUTHOR]

Abstract: The problem of synthesizing multivariate stationary series Y[n]=(Y 1[n],…,Y P [n]) T , , with prescribed non-Gaussian marginal distributions, and a targeted covariance structure, is addressed. The focus is on constructions based on a memoryless transformation Y p [n]=f p (X p [n]) of a multivariate stationary Gaussian series X[n]=(X 1[n],…,X P [n]) T . The mapping between the targeted covariance and that of the Gaussian series is expressed via Hermite expansions. The various choices of the transforms f p for a prescribed marginal distribution are discussed in a comprehensive manner. The interplay between the targeted marginal distributions, the choice of the transforms f p , and on the resulting reachability of the targeted covariance, is discussed theoretically and illustrated on examples. Also, an original practical procedure warranting positive definiteness for the transformed covariance at the price of approximating the targeted covariance is proposed, based on a simple and natural modification of the popular circulant matrix embedding technique. The applications of the proposed methodology are also discussed in the context of network traffic modeling. Matlab codes implementing the proposed synthesis procedure are publicly available at http://www.hermir.org. [Copyright &y& Elsevier]

Abstract: A fast and exact procedure for the numerical synthesis of stationary multivariate Gaussian time series with a priori prescribed and well controlled auto- and cross-covariance functions is proposed. It is based on extending the circulant embedding technique to the multivariate case and can be viewed as a modification and variation around the Chan and Wood algorithm proposed earlier to solve the same problem. The procedure is shown to yield time series possessing exactly the desired covariance structure, when sufficient conditions are satisfied. Such conditions are discussed theoretically and examined on several examples of multivariate time series models. Issues related to prescribing a priori the spectral structure rather than the covariance one are also discussed. Matlab routines implementing this procedure are publicly available at http://www.hermir.org. [Copyright &y& Elsevier]

A research area of increasing interest consists of studying the benefits of using spectral analysis to screen neurogenic erectile dysfunctions. Our hypothesis is that spectral analysis consists of a non-invasive and simple procedure to investigate such patients. Subjects were allocated into two groups: control, no erectile dysfunction (n = 17), and patients with erectile dysfunction (n = 15). RR intervals (RRI), systolic blood pressure, diastolic blood pressure and baroreflex sensitivity recordings were performed continuously in the supine position, followed by the seated position, and finally standing position. In the supine position, the control group had a higher RRI and a lower diastolic blood pressure. For frequency domain analyses of RRI in the supine position, the erectile dysfunction group had a higher normalized low-frequency (LF) index and low-frequency/high-frequency (LF/HF) ratio while showing a lower normalized HF index. In the seated position, the erectile dysfunction group presented a higher LF/HF ratio. Regarding systolic blood pressure, the erectile dysfunction group showed lower normalized LF and higher normalized HF indices only in the supine position. The α index in HF was lower in the erectile dysfunction group in the three positions. Spectral analyses of cardiac sympathovagal drive constitute a fruitful non-invasive approach to evaluate alterations in cardiovascular autonomic modulation in neurogenic erectile dysfunction patients. Copyright © 2010 S. Karger AG, Basel [ABSTRACT FROM AUTHOR]

Multiplicative processes, multifractals, and more recently also infinitely divisible cascades have seen increased popularity in a host of applications requiring versatile multiscale models, ranging from hydrodynamic turbulence to computer network traffic, from image processing to economics. The methodologies prevalent as of today rely to a large extent on iterative schemes used to produce infinite detail and repetitive structure across scales. While appealing, due to their simplicity, these constructions have limited applicability as they lead by default to power law progression of moments through scales, to non- stationary increments and often to inherent log-periodic scaling which favors an exponential set of scales. This paper studies and develops a wide class of infinitely divisible cascades (IDC), thereby establishing the first reported cases of controllable scaling of moments in non-power-law font Embedded in the framework of IDC, these processes exhibit stationary increments and scaling over a continuous range of scales. Criteria for convergence, further statistical properties, as well as MATLAB routines are provided. [ABSTRACT FROM AUTHOR]

We introduce a new approach to the modeling of network traffic, consisting of a semi-experimental methodology combining models with data and a class of point processes (cluster models) to represent the process of packet arrivals in a physically meaningful way. Wavelets are used to examine second-order statistics, and particular attention is paid to the modeling of long-range dependence and to the question of scale invariance at small scales. We analyze in depth the properties of several large traces of packet data and determine unambiguously the influence of network variables such as the arrival patterns, durations, and volumes of transport control protocol (TCP) flows and internal flow structure. We show that session-level modeling is not relevant at the packet level. Our findings naturally suggest the use of cluster models. We define a class where TCP flows are directly modeled, and each model parameter has a direct meaning in network terms, allowing the model to be used to predict traffic properties as networks and traffic evolve. The class has the key advantage of being mathematically tractable, in particular, its spectrum is known and can be readily calculated, its wavelet spectrum deduced, interarrival distributions can be obtained, and it can be simulated in a straightforward way. The model reproduces the main second-order features, and results are compared against a simple black box point process alternative. Discrepancies with the model are discussed and explained, and enhancements are outlined. The elephant and mice view of traffic flows is revisited in the light of our findings. [ABSTRACT FROM AUTHOR]

Many factors are involved in the development of orthostatic intolerance after real or simulated weightlessness. The aim of our study was to compare the effects of 7-day head-down bed rest (HDBR) in eight women and eight men on the spontaneous baroreflex sensitivity (standard spectral method and new time–frequency algorithm) during lower body negative pressure (LBNP) tests.Results obtained before HDBR have shown in women, compared to men, higher heart rate, lower blood pressure, higher parasympathetic modulation at rest and greater decrease in baroreflex sensitivity with greater increase in sympathetic activity during LBNP. After HDBR, we observed in both men and women a dramatic decrease in orthostatic tolerance (7.0 min at R+1 vs. 10.0 min, p<0.05, at BDC-1 in men; 5.4 vs. 9.0 min, p<0.05, in women) together with a decrease in plasma volume (−9.1±0.9% in men, −9.5±1.4% in women) and in spontaneous baroreflex sensitivity without gender effect. After HDBR, at the highest level of LBNP, diastolic blood pressure increased in men (+5.6±1.3 mm Hg) and decreased in women (−1.0±2.7 mm Hg) with a gender difference (p<0.05). This result suggests impaired vasoconstriction in women after HDBR.Neither endocrine response nor alterations to the cardiac baroreflex can explain gender differences in orthostatic tolerance after HDBR as reported by previous studies. Further studies need to be conducted in order to obtain a more precise analysis of gender difference in arteriolar vasoconstriction after HDBR. The time frequency method we developed to study changes in spontaneous baroreflex might be applied to the analysis of LBNP tests. [Copyright &y& Elsevier]

Multifractal analysis, that quantifies the fluctuations of regularities in time series or textures, has become a standard signal/image processing tool. It has been successfully used in a large variety of applicative contexts. Yet, successes are confined to the analysis of one signal or image at a time (univariate analysis). This is because multivariate (or joint) multifractal analysis remains so far rarely used in practice and has barely been studied theoretically. In view of the myriad of modern real-world applications that rely on the joint (multivariate) analysis of collections of signals or images, univariate analysis constitutes a major limitation. The goal of the present work is to theoretically ground multivariate multifractal analysis by studying the properties and limitations of the most natural extension of the univariate formalism to a multivariate formulation. It is notably shown that while performing well for a class of model processes, this natural extension is not valid in general. Based on the theoretical study of the mechanisms leading to failure, we propose alternative formulations and examine their mathematical properties. [ABSTRACT FROM AUTHOR]

Abstract We show how a joint multifractal analysis of a collection of signals unravels correlations between the locations of their pointwise singularities. The multivariate multifractal formalism, reformulated in the general setting supplied by multiresolution quantities, provides a framework which allows to estimate joint multifractal spectra. General results on joint multifractal spectra are derived, and illustrated by the theoretical derivation and practical estimation of the joint multifractal spectra of simple mathematical models, including correlated binomial cascades. [ABSTRACT FROM AUTHOR]

Background: Multifractal analysis of human heartbeat dynamics has been demonstrated to provide promising markers of congestive heart failure (CHF). Yet, it crucially builds on the interpolation of RR interval series which has been generically performed with limited links to CHF pathophysiology. Objective: We devise a novel methodology estimating multifractal autonomic dynamics from heartbeat-derived series defined in the continuous time. We hypothesize that markers estimated from our novel framework are also effective for mortality prediction in severe CHF. Methods: We merge multifractal analysis within a methodological framework based on inhomogeneous point process models of heartbeat dynamics. Specifically, wavelet coefficients and wavelet leaders are computed over measures extracted from instantaneous statistics of probability density functions characterizing and predicting the time until the next heartbeat event occurs. The proposed approach is tested on data from 94 CHF patients aiming at predicting survivor and nonsurvivor individuals as determined after a four years follow up. Results and Discussion: Instantaneous markers of vagal and sympatho-vagal dynamics display power-law scaling for a large range of scales, from $\simeq 0.5$ to $\simeq 100$  s. Using standard support vector machine algorithms, the proposed inhomogeneous point-process representation-based multifractal analysis achieved the best CHF mortality prediction accuracy of 79.11% (sensitivity 90.48%, specificity 67.74%). Conclusion: Our results suggest that heartbeat scaling and multifractal properties in CHF patients are not generated at the sinus-node level, but rather by the intrinsic action of vagal short-term control and of sympatho-vagal fluctuations associated with circadian cardiovascular control especially within the very low frequency band. These markers might provide critical information in devising a clinical tool for individualized prediction of survivor and nonsurvivor CHF patients. [ABSTRACT FROM AUTHOR]

Forming valid predictions about the environment is crucial to survival. However, whether humans are able to form valid predictions about natural stimuli based on their temporal statistical regularities remains unknown. Here, we presented subjects with tone sequences with pitch fluctuations that, over time, capture long-range temporal dependence structures prevalent in natural stimuli. We found that subjects were able to exploit such naturalistic statistical regularities to make valid predictions about upcoming items in a sequence. Magnetoencephalography (MEG) recordings revealed that slow, arrhythmic cortical dynamics tracked the evolving pitch sequence over time such that neural activity at a given moment was influenced by the pitch of up to seven previous tones. Importantly, such history integration contained in neural activity predicted the expected pitch of the upcoming tone, providing a concrete computational mechanism for prediction. These results establish humans' ability to make valid predictions based on temporal regularities inherent in naturalistic stimuli and further reveal the neural mechanisms underlying such predictive computation. [ABSTRACT FROM AUTHOR]

Piecewise constant denoising can be solved either by deterministic optimization approaches, based on the Potts model, or by stochastic Bayesian procedures. The former lead to low computational time but require the selection of a regularization parameter, whose value significantly impacts the achieved solution, and whose automated selection remains an involved and challenging problem. Conversely, fully Bayesian formalisms encapsulate the regularization parameter selection into hierarchical models, at the price of high computational costs. This contribution proposes an operational strategy that combines hierarchical Bayesian and Potts model formulations, with the double aim of automatically tuning the regularization parameter and maintaining computational efficiency. The proposed procedure relies on formally connecting a Bayesian framework to a $\ell _2$ -Potts functional. Behaviors and performance for the proposed piecewise constant denoising and regularization parameter tuning techniques are studied qualitatively and assessed quantitatively, and shown to compare favorably against those of a fully Bayesian hierarchical procedure, both in accuracy and computational load. [ABSTRACT FROM PUBLISHER]

While scale invariance is commonly observed in each component of real world multivariate signals, it is also often the case that the inter-component correlation structure is not fractally connected, i.e., its scaling behavior is not determined by that of the individual components. To model this situation in a versatile manner, we introduce a class of multivariate Gaussian stochastic processes called Hadamard fractional Brownian motion (HfBm). Its theoretical study sheds light on the issues raised by the joint requirement of entry-wise scaling and departures from fractal connectivity. An asymptotically normal wavelet-based estimator for its scaling parameter, called the Hurst matrix, is proposed, as well as asymptotically valid confidence intervals. The latter are accompanied by original finite sample procedures for computing confidence intervals and testing fractal connectivity from one single and finite size observation. Monte Carlo simulation studies are used to assess the estimation performance as a function of the (finite) sample size, and to quantify the impact of omitting wavelet cross-correlation terms. The simulation studies are shown to validate the use of approximate confidence intervals, together with the significance level and power of the fractal connectivity test. The test performance and properties are further studied as functions of the HfBm parameters. [ABSTRACT FROM AUTHOR]

Background: The fetal heart rate (FHR) is commonly monitored during labor to detect early fetal acidosis. FHR variability is traditionally investigated using Fourier transform, often with adult predefined frequency band powers and the corresponding LF/HF ratio. However, fetal conditions differ from adults and modify spectrum repartition along frequencies. Aims: This study questions the arbitrariness definition and relevance of the frequency band splitting procedure, and thus of the calculation of the underlying LF/HF ratio, as efficient tools for characterizing intrapartum FHR variability. Study Design: The last 30 minutes before delivery of the intrapartum FHR were analyzed. Subjects: Case-control study. A total of 45 singletons divided into two groups based on umbilical cord arterial pH: the Index group with pH ≤ 7.05 (n = 15) and Control group with pH > 7.05 (n = 30). Outcome Measures: Frequency band-based LF/HF ratio and Hurst parameter. Results: This study shows that the intrapartum FHR is characterized by fractal temporal dynamics and promotes the Hurst parameter as a potential marker of fetal acidosis. This parameter preserves the intuition of a power frequency balance, while avoiding the frequency band splitting procedure and thus the arbitrary choice of a frequency separating bands. The study also shows that extending the frequency range covered by the adult-based bands to higher and lower frequencies permits the Hurst parameter to achieve better performance for identifying fetal acidosis. Conclusions: The Hurst parameter provides a robust and versatile tool for quantifying FHR variability, yields better acidosis detection performance compared to the LF/HF ratio, and avoids arbitrariness in spectral band splitting and definitions. [ABSTRACT FROM AUTHOR]