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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]

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]

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]

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]

In the mid 1990s, it was shown that the statistics of aggregated time series from Internet traffic departed from those of traditional short range-dependent models, and were instead characterized by asymptotic self-similarity. Following this seminal contribution, over the years, many studies have investigated the existence and form of scaling in Internet traffic. This contribution first aims at presenting a methodology, combining multiscale analysis (wavelet and wavelet leaders) and random projections (or sketches), permitting a precise, efficient and robust characterization of scaling, which is capable of seeing through non-stationary anomalies. Second, we apply the methodology to a data set spanning an unusually long period: 14 years, from the MAWI traffic archive, thereby allowing an in-depth longitudinal analysis of the form, nature, and evolutions of scaling in Internet traffic, as well as network mechanisms producing them. We also study a separate three-day long trace to obtain complementary insight into intra-day behavior. We find that a biscaling (two ranges of independent scaling phenomena) regime is systematically observed: long-range dependence over the large scales, and multifractallike scaling over the fine scales. We quantify the actual scaling ranges precisely, verify to high accuracy the expected relationship between the long range dependent parameter and the heavy tail parameter of the flow size distribution, and relate fine scale multifractal scaling to typical IP packet inter-arrival and to round-trip time distributions. [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]

After the seminal work by Taqqu et al. relating self-similarity to heavy-tailed distributions, a number of research articles verified that aggregated Internet traffic time series show self-similarity and that Internet attributes, like Web file sizes and flow lengths, were heavy-tailed. However, the validation of the theoretical prediction relating self-similarity and heavy tails remains unsatisfactorily addressed, being investigated using either numerical or network simulations, or from uncontrolled Web traffic data. Notably, this prediction has never been conclusively verified on real networks using controlled and stationary scenarios, prescribing specific heavy-tailed distributions, and estimating confidence intervals. With this goal in mind, we use the potential and facilities offered by the large-scale, deeply reconfigurable and fully controllable experimental Grid5000 instrument, combined with state-of-the-art estimators, to investigate the prediction's observability on real networks. To this end, we organize a large number of controlled traffic circulation sessions on a nationwide real network involving 200 independent hosts. We use a FPGA-based measurement system to collect the corresponding traffic at packet level. We then estimate both the self-similarity exponent of the aggregated time series and the heavy-tail index of flow-size distributions, independently. Not only do our results complement and validate, with a striking accuracy, some conclusions drawn from a series of pioneering studies, but they also bring in new insights on the controversial role of certain components of real networks. [ABSTRACT FROM AUTHOR]

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]

Assessing world-wide financial integration constitutes a recurrent challenge in macroeconometrics, often addressed by visual inspections searching for data patterns. Econophysics literature enables us to build complementary, data-driven measures of financial integration using graphs. The present contribution investigates the potential and interests of a novel 3-step approach that combines several state-of-the-art procedures to (i) compute graph-based representations of the multivariate dependence structure of asset prices time series representing the financial states of 32 countries world-wide (1955–2015); (ii) compute time series of 5 graph-based indices that characterize the time evolution of the topologies of the graph; (iii) segment these time evolutions in piece-wise constant eras, using an optimization framework constructed on a multivariate multi-norm total variation penalized functional. The method shows first that it is possible to find endogenous stable eras of world-wide financial integration. Then, our results suggest that the most relevant globalization eras would be based on the historical patterns of global capital flows, while the major regulatory events of the 1970s would only appear as a cause of sub-segmentation. [ABSTRACT FROM AUTHOR]