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In numerous inverse problems, state-of-the-art solving strategies involve training neural networks from ground truth and associated measurement datasets that, however, may be expensive or impossible to collect. Recently, self-supervised learning techniques have emerged, with the major advantage of no longer requiring ground truth data. Most theoretical and experimental results on self-supervised learning focus on linear inverse problems. The present work aims to study self-supervised learning for the non-linear inverse problem of recovering audio signals from clipped measurements. An equivariance-based selfsupervised loss is proposed and studied. Performance is assessed on simulated clipped measurements with controlled and varied levels of clipping, and further reported on standard real music signals. We show that the performance of the proposed equivariance-based self-supervised declipping strategy compares favorably to fully supervised learning while only requiring clipped measurements alone for training.

In this paper, we characterize the convergence of the (rescaled logarithmic) empirical spectral distribution of wavelet random matrices. We assume a moderately high-dimensional framework where the sample size $n$, the dimension $p(n)$ and, for a fixed integer $j$, the scale $a(n)2^j$ go to infinity in such a way that $\lim_{n \rightarrow \infty}p(n)\cdot a(n)/n = \lim_{n \rightarrow \infty} o(\sqrt{a(n)/n})= 0$. We suppose the underlying measurement process is a random scrambling of a sample of size $n$ of a growing number $p(n)$ of fractional processes. Each of the latter processes is a fractional Brownian motion conditionally on a randomly chosen Hurst exponent. We show that the (rescaled logarithmic) empirical spectral distribution of the wavelet random matrices converges weakly, in probability, to the distribution of Hurst exponents.

Self-supervised methods have recently proved to be nearly as effective as supervised methods in various imaging inverse problems, paving the way for learning-based methods in scientific and medical imaging applications where ground truth data is hard or expensive to obtain. This is the case in magnetic resonance imaging and computed tomography. These methods critically rely on invariance to translations and/or rotations of the image distribution to learn from incomplete measurement data alone. However, existing approaches fail to obtain competitive performances in the problems of image super-resolution and deblurring, which play a key role in most imaging systems. In this work, we show that invariance to translations and rotations is insufficient to learn from measurements that only contain low-frequency information. Instead, we propose a new self-supervised approach that leverages the fact that many image distributions are approximately scale-invariant, and that enables recovering high-frequency information lost in the measurement process. We demonstrate throughout a series of experiments on real datasets that the proposed method outperforms other self-supervised approaches, and obtains performances on par with fully supervised learning.

Scale-free dynamics, formalized by selfsimilarity, provides a versatile paradigm massively and ubiquitously used to model temporal dynamics in real-world data. However, its practical use has mostly remained univariate so far. By contrast, modern applications often demand multivariate data analysis. Accordingly, models for multivariate selfsimilarity were recently proposed. Nevertheless, they have remained rarely used in practice because of a lack of available robust estimation procedures for the vector of selfsimilarity parameters. Building upon recent mathematical developments, the present work puts forth an efficient estimation procedure based on the theoretical study of the multiscale eigenstructure of the wavelet spectrum of multivariate selfsimilar processes. The estimation performance is studied theoretically in the asymptotic limits of large scale and sample sizes, and computationally for finite-size samples. As a practical outcome, a fully operational and documented multivariate signal processing estimation toolbox is made freely available and is ready for practical use on real-world data. Its potential benefits are illustrated in epileptic seizure prediction from multi-channel EEG data., Comment: 14 pages, 8 figures

Understanding extreme events and their probability is key for the study of climate change impacts, risk assessment, adaptation, and the protection of living beings. Forecasting the occurrence probability of extreme heatwaves is a primary challenge for risk assessment and attribution, but also for fundamental studies about processes, dataset and model validation, and climate change studies. In this work we develop a methodology to build forecasting models which are based on convolutional neural networks, trained on extremely long climate model outputs. We demonstrate that neural networks have positive predictive skills, with respect to random climatological forecasts, for the occurrence of long-lasting 14-day heatwaves over France, up to 15 days ahead of time for fast dynamical drivers (500 hPa geopotential height fields), and also at much longer lead times for slow physical drivers (soil moisture). This forecast is made seamlessly in time and space, for fast hemispheric and slow local drivers. We find that the neural network selects extreme heatwaves associated with a North-Hemisphere wavenumber-3 pattern. The main scientific message is that most of the time, training neural networks for predicting extreme heatwaves occurs in a regime of lack of data. We suggest that this is likely to be the case for most other applications to large scale atmosphere and climate phenomena. For instance, using one hundred years-long training sets, a regime of drastic lack of data, leads to severely lower predictive skills and general inability to extract useful information available in the 500 hPa geopotential height field at a hemispheric scale in contrast to the dataset of several thousand years long. We discuss perspectives for dealing with the lack of data regime, for instance rare event simulations and how transfer learning may play a role in this latter task., Comment: 34 pages, 12 figures

Monitoring the Covid19 pandemic constitutes a critical societal stake that received considerable research efforts. The intensity of the pandemic on a given territory is efficiently measured by the reproduction number, quantifying the rate of growth of daily new infections. Recently, estimates for the time evolution of the reproduction number were produced using an inverse problem formulation with a nonsmooth functional minimization. While it was designed to be robust to the limited quality of the Covid19 data (outliers, missing counts), the procedure lacks the ability to output credibility interval based estimates. This remains a severe limitation for practical use in actual pandemic monitoring by epidemiologists that the present work aims to overcome by use of Monte Carlo sampling. After interpretation of the nonsmooth functional into a Bayesian framework, several sampling schemes are tailored to adjust the nonsmooth nature of the resulting posterior distribution. The originality of the devised algorithms stems from combining a Langevin Monte Carlo sampling scheme with Proximal operators. Performance of the new algorithms in producing relevant credibility intervals for the reproduction number estimates and denoised counts are compared. Assessment is conducted on real daily new infection counts made available by the Johns Hopkins University. The interest of the devised monitoring tools are illustrated on Covid19 data from several different countries.

Monitoring the evolution of the Covid19 pandemic constitutes a critical step in sanitary policy design. Yet, the assessment of the pandemic intensity within the pandemic period remains a challenging task because of the limited quality of data made available by public health authorities (missing data, outliers and pseudoseasonalities, notably), that calls for cumbersome and ad-hoc preprocessing (denoising) prior to estimation. Recently, the estimation of the reproduction number, a measure of the pandemic intensity, was formulated as an inverse problem, combining data-model fidelity and space-time regularity constraints, solved by nonsmooth convex proximal minimizations. Though promising, that formulation lacks robustness against the limited quality of the Covid19 data and confidence assessment. The present work aims to address both limitations: First, it discusses solutions to produce a robust assessment of the pandemic intensity by accounting for the low quality of the data directly within the inverse problem formulation. Second, exploiting a Bayesian interpretation of the inverse problem formulation, it devises a Monte Carlo sampling strategy, tailored to a nonsmooth log-concave a posteriori distribution, to produce relevant credibility intervalbased estimates for the Covid19 reproduction number. Clinical relevance Applied to daily counts of new infections made publicly available by the Health Authorities for around 200 countries, the proposed procedures permit robust assessments of the time evolution of the Covid19 pandemic intensity, updated automatically and on a daily basis.

Presents a study which aims to investigate the extent to which classroom student adversity, school-provided resources, and school climate directly relate to participants' reports of depressive and anxious symptoms and career optimism, and within this model, the relative influence of each indicator. The authors predict that classroom student adversity, resources, and school climate will each be directly related to symptoms of clinical depression and anxiety and career optimism. They also predict that classroom student adversity will be the strongest relative predictor of each outcome compared to resources and school climate. They investigate the extent to which resources and school climate each interact with classroom student adversity to influence participants' outcomes and predict that higher satisfaction with resources and more positive school climate will each buffer the hypothesized negative effects of classroom student adversity. Undergraduate seniors in a teacher-training program at a public university in the Southwestern United States were recruited to participate in a longitudinal study that followed them from the last year of training through their first year of teaching. In conclusion, all new teachers experience job demands upon entering the profession, and these demands no doubt have steep implications for their career success and longevity. [This paper was published in "Teaching & Teacher Education" v87 p1-13 2020.]

This work focuses on a parameter-free joint piecewise smooth image denoising and contour detection. Formulated as the minimization of a discrete Mumford-Shah functional and estimated via a theoretically grounded alternating minimization scheme, the bottleneck of such a variational approach lies in the need to fine-tune their hyperparameters, while not having access to ground truth data. To that aim, a Stein-like strategy providing optimal hyperparameters is designed, based on the minimization of an unbiased estimate of the quadratic risk. Efficient and automated minimization of the estimate of the risk crucially relies on an unbiased estimate of the gradient of the risk with respect to hyperparameters. Its practical implementation is performed using a forward differentiation of the alternating scheme minimizing the Mumford-Shah functional, requiring exact differentiation of the proximity operators involved. Intensive numerical experiments are performed on synthetic images with different geometry and noise levels, assessing the accuracy and the robustness of the proposed procedure. The resulting parameter-free piecewise-smooth estimation and contour detection procedure, not requiring prior image processing expertise nor annotated data, can then be applied to real-world images.

Daily pandemic surveillance, often achieved through the estimation of the reproduction number, constitutes a critical challenge for national health authorities to design countermeasures. 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 Sant{\'e}-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.

In this paper, we construct the wavelet eigenvalue regression methodology in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. Building upon the asymptotic and large scale properties of wavelet random matrices in high dimensions, the wavelet eigenvalue regression is shown to be consistent and, under additional assumptions, asymptotically Gaussian in the estimation of the fractal structure of the system. We further construct a consistent estimator of the effective dimension $r$ of the system that significantly increases the robustness of the methodology. The estimation performance over finite samples is studied by means of simulations., Comment: 33 pages, 3 figures. Minor revision. Companion to arXiv:2102.05761

Because of the impact of extreme heat waves and heat domes on society and biodiversity, their study is a key challenge. We specifically study long-lasting extreme heat waves, which are among the most important for climate impacts. Physics driven weather forecast systems or climate models can be used to forecast their occurrence or predict their probability. The present work explores the use of deep learning architectures, trained using outputs of a climate model, as an alternative strategy to forecast the occurrence of extreme long-lasting heatwaves. This new approach will be useful for several key scientific goals which include the study of climate model statistics, building a quantitative proxy for resampling rare events in climate models, study the impact of climate change, and should eventually be useful for forecasting. Fulfilling these important goals implies addressing issues such as class-size imbalance that is intrinsically associated with rare event prediction, assessing the potential benefits of transfer learning to address the nested nature of extreme events (naturally included in less extreme ones). We train a Convolutional Neural Network, using 1000 years of climate model outputs, with large-class undersampling and transfer learning. From the observed snapshots of the surface temperature and the 500 hPa geopotential height fields, the trained network achieves significant performance in forecasting the occurrence of long-lasting extreme heatwaves. We are able to predict them at three different levels of intensity, and as early as 15 days ahead of the start of the event (30 days ahead of the end of the event)., Comment: 21 pages, 5 figures

The overarching goal of the present work is to contribute to the understanding of the relations between fetal heart rate (FHR) temporal dynamics and the well-being of the fetus, notably in terms of predicting cardiovascular decompensation (CVD). It makes uses of an established animal model of human labor, where fourteen near-term ovine fetuses subjected to umbilical cord occlusions (UCO) were instrumented to permit regular intermittent measurements of metabolites, pH, and continuous recording of electrocardiogram (ECG) and systemic arterial blood pressure (to identify CVD) during UCO. ECG-derived FHR was digitized at the sampling rate of 1000 Hz and resampled to 4Hz, as used in clinical routine. We focused on four FHR variability features which are tunable to temporal scales of FHR dynamics, robustly computable from FHR sampled at $4$Hz and within short-time sliding windows, hence permitting a time-dependent, or local, analysis of FHR which helps dealing with signal noise. Results show the sensitivity of the proposed features for early detection of CVD, correlation to metabolites and pH, useful for early acidosis detection and the importance of coarse time scales (2.5 to 8 seconds) which are not disturbed by the low FHR sampling rate. Further, we introduce the performance of an individualized self-referencing metric of the distance to healthy state, based on a combination of the four features. We demonstrate that this novel metric, applied to clinically available FHR temporal dynamics alone, accurately predicts the time occurrence of CVD which heralds a clinically significant degradation of the fetal health reserve to tolerate the trial of labor., Comment: 34 pages (preprint version) 10 figures

In this paper, we characterize the asymptotic and large scale behavior of the eigenvalues of wavelet random matrices in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. We show that the $r$ largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge in probability to scale-invariant functions in the high-dimensional limit. By contrast, the remaining $p-r$ eigenvalues remain bounded in probability. Under additional assumptions, we show that the $r$ largest log-eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference.

We present a novel multiview canonical correlation analysis model based on a variational approach. This is the first nonlinear model that takes into account the available graph-based geometric constraints while being scalable for processing large scale datasets with multiple views. It is based on an autoencoder architecture with graph convolutional neural network layers. We experiment with our approach on classification, clustering, and recommendation tasks on real datasets. The algorithm is competitive with state-of-the-art multiview representation learning techniques., Comment: 4 pages, 3 figures, submitted

Numerous fields of nonlinear physics, very different in nature, produce signals and images, that share the common feature of being essentially constituted of piecewise homogeneous phases. Analyzing signals and images from corresponding experiments to construct relevant physical interpretations thus often requires detecting such phases and estimating accurately their characteristics (borders, feature differences, ...). However, situations of physical relevance often comes with low to very low signal to noise ratio precluding the standard use of classical linear filtering for analysis and denoising and thus calling for the design of advanced nonlinear signal/image filtering techniques. Additionally, when dealing with experimental physics signals/images, a second limitation is the large amount of data that need to be analyzed to yield accurate and relevant conclusions requiring the design of fast algorithms. The present work proposes a unified signal/image nonlinear filtering procedure, with fast algorithms and a data-driven automated hyperparameter tuning, based on proximal algorithms and Stein unbiased estimator principles. The interest and potential of these tools are illustrated at work on low-confinement solid friction signals and porous media multiphase flows.

Penalized Least Squares are widely used in signal and image processing. Yet, it suffers from a major limitation since it requires fine-tuning of the regularization parameters. Under assumptions on the noise probability distribution, Stein-based approaches provide unbiased estimator of the quadratic risk. The Generalized Stein Unbiased Risk Estimator is revisited to handle correlated Gaussian noise without requiring to invert the covariance matrix. Then, in order to avoid expansive grid search, it is necessary to design algorithmic scheme minimizing the quadratic risk with respect to regularization parameters. This work extends the Stein's Unbiased GrAdient estimator of the Risk of Deledalle et al. to the case of correlated Gaussian noise, deriving a general automatic tuning of regularization parameters. First, the theoretical asymptotic unbiasedness of the gradient estimator is demonstrated in the case of general correlated Gaussian noise. Then, the proposed parameter selection strategy is particularized to fractal texture segmentation, where problem formulation naturally entails inter-scale and spatially correlated noise. Numerical assessment is provided, as well as discussion of the practical issues., Comment: Submitted to SIAM Imaging Science

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., Comment: To appear in 2021 in Applied and Computational Harmonic Analysis

Origin-Destination matrix estimation is a keystone for traffic representation and analysis. Traditionally estimated thanks to traffic counts, surveys and socio-economic models, recent technological advances permit to rethink the estimation problem. Road user identification technologies, such as connected GPS, Bluetooth or Wifi detectors bring additional information, that is, for a fraction of the users, the origin, the destination and to some extend the itinerary taken. In the present work, this additional information is used for the estimation of a more comprehensive traffic representation tool: the link-origin-destination matrix. Such three-dimensional matrices extend the concept of traditional origin-destination matrices by also giving information on the traffic assignment. Their estimation is solved as an inverse problem whose objective function represents a trade-off between important properties the traffic has to satisfy. This article presents the theory and how to implement such method on real dataset. With the case study of Brisbane City where over 600 hundreds Bluetooth detectors have been installed it also illustrates the opportunities such link-origin-destination matrices create for traffic analysis.

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., Comment: 18 pages, 3 figures

PageRank for Semi-Supervised Learning has shown to leverage data structures and limited tagged examples to yield meaningful classification. Despite successes, classification performance can still be improved, particularly in cases of fuzzy graphs or unbalanced labeled data. To address such limitations, a novel approach based on powers of the Laplacian matrix $L^\gamma$ ($\gamma > 0$), referred to as $L^\gamma$-PageRank, is proposed. Its theoretical study shows that it operates on signed graphs, where nodes belonging to one same class are more likely to share positive edges while nodes from different classes are more likely to be connected with negative edges. It is shown that by selecting an optimal $\gamma$, classification performance can be significantly enhanced. A procedure for the automated estimation of the optimal $\gamma$, from a unique observation of data, is devised and assessed. Experiments on several datasets demonstrate the effectiveness of both $L^\gamma$-PageRank classification and the optimal $\gamma$ estimation., Comment: Submitted to Applied Network Science (special issue on machine learning with graphs)

Information theoretic measures (entropies, entropy rates, mutual information) are nowadays commonly used in statistical signal processing for real-world data analysis. The present work proposes the use of Auto Mutual Information (Mutual Information between subsets of the same signal) and entropy rate as powerful tools to assess refined dependencies of any order in signal temporal dynamics. Notably, it is shown how two-point Auto Mutual Information and entropy rate unveil information conveyed by higher order statistic and thus capture details of temporal dynamics that are overlooked by the (two-point) correlation function. Statistical performance of relevant estimators for Auto Mutual Information and entropy rate are studied numerically, by means of Monte Carlo simulations, as functions of sample size, dependence structures and hyper parameters that enter their definition. Further, it is shown how Auto Mutual Information permits to discriminate between several different non Gaussian processes, having exactly the same marginal distribution and covariance function. Assessing higher order statistics via multipoint Auto Mutual Information is also shown to unveil the global dependence structure fo these processes, indicating that one of the non Gaussian actually has temporal dynamics that ressembles that of a Gaussian process with same covariance while the other does not.

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 work 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., Comment: 10 pages

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 temporally large trip databases has limited the analysis of BSS users behavior in the long term. This article studies the V'elo'v - a BSS located in Lyon, France - subscribers commitment in the long term and the evolution of their usage over time. Using a 5-year dataset covering 121,000 long-term distinct users, we show the heterogeneous individual trajectories masked by the overall system stability. Users follow two main trajectories: about 60% remain in the system for at most one year, showing a low median activity (47 trips); the remaining 40% correspond to more active users (median activity of 96 trips in their first year) that remain continuously active for several years (mean time = 2.9 years). This latter class exhibits a relatively stable activity, decreasing slightly over the years. We show that middle-aged, male and urban users are over represented among the 'stable' users., Comment: 11 pages, 7 figures

Intrapartum fetal heart rate (FHR) monitoring constitutes a reference tool in clinical practice to assess the baby 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 \emph{Approximate Entropy} (ApEn) or \emph{Sample Entropy} (SampEn), have regularly been reported as significant features for intrapartum FHR analysis. We explore how Information Theory, and especially {\em 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.

In this contribution, we extend the methodology proposed in Abry and Didier (2017) to obtain the first joint estimator of the real parts of the Hurst eigenvalues of $n$-variate OFBM. The procedure consists of a wavelet 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 (entrywise) 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.

In the mid-90's, 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 aims first 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 3-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 multifractal-like 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.

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.

Multifractal analysis has become a standard signal processing tool,for which a promising new formulation, the p-leader multifractal formalism, has recently been proposed. It relies on novel multiscale quantities, the p-leaders, defined as local l^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 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., Comment: 10 pages, accepted in IEEE Transactions in Signal Processing

Self-similarity is widely considered the reference framework for modeling the scaling properties of real-world data. However, most theoretical studies and their practical use have remained univariate. Operator Fractional Brownian Motion (OfBm) was recently proposed as a multivariate model for self-similarity. Yet it has remained seldom used in applications because of serious issues that appear in the joint estimation of its numerous parameters. While the univariate fractional Brownian motion requires the estimation of two parameters only, its mere bivariate extension already involves 7 parameters which are very different in nature. The present contribution proposes a method for the full identification of bivariate OfBm (i.e., the joint estimation of all parameters) through an original formulation as a non-linear wavelet regression coupled with a custom-made Branch & Bound numerical scheme. The estimation performance (consistency and asymptotic normality) is mathematically established and numerically assessed by means of Monte Carlo experiments. The impact of the parameters defining OfBm on the estimation performance as well as the associated computational costs are also thoroughly investigated.

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 of maintaining computational effciency. The proposed procedure relies on formally connecting a Bayesian framework to a l2-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 in computational load.

A mixed Gaussian fractional process $\{Y(t)\}_{t \in {\Bbb R}} = \{PX(t)\}_{t \in {\Bbb R}}$ is a multivariate stochastic process obtained by pre-multiplying a vector of independent, Gaussian fractional process entries $X$ by a nonsingular matrix $P$. It is interpreted that $Y$ is observable, while $X$ is a hidden process occurring in an (unknown) system of coordinates $P$. Mixed processes naturally arise as approximations to solutions of physically relevant classes of multivariate fractional SDEs under aggregation. We propose a semiparametric two-step wavelet-based method for estimating both the demixing matrix $P^{-1}$ and the memory parameters of $X$. The asymptotic normality of the estimators is established both in continuous and discrete time. Monte Carlo experiments show that the finite sample estimation performance is comparable to that of parametric methods, while being very computationally efficient. As applications, we model a bivariate time series of annual tree ring width measurements, and establish the asymptotic normality of the eigenstructure of sample wavelet matrices.

Origin-Destination Matrix (ODM) estimation is a classical problem in transport engineering aiming to recover flows from every Origin to every Destination from measured traffic counts and a priori model information. In addition to traffic counts, the present contribution takes advantage of probe trajectories, whose capture is made possible by new measurement technologies. It extends the concept of ODM to that of Link dependent ODM (LODM), keeping the information about the flow distribution on links and containing inherently the ODM assignment. Further, an original formulation of LODM estimation, from traffic counts and probe trajectories is presented as an optimisation problem, where the functional to be minimized consists of five convex functions, each modelling a constraint or property of the transport problem: consistency with traffic counts, consistency with sampled probe trajectories, consistency with traffic conservation (Kirchhoff's law), similarity of flows having close origins and destinations, positivity of traffic flows. A primal-dual algorithm is devised to minimize the designed functional, as the corresponding objective functions are not necessarily differentiable. A case study, on a simulated network and traffic, validates the feasibility of the procedure and details its benefits for the estimation of an LODM matching real-network constraints and observations.

Generalizability theory (GT) offers a useful framework for estimating the reliability of a measure while accounting for multiple sources of error variance. The purpose of this study was to use GT to examine multiple sources of variance in and the reliability of school-level teacher and high school student behaviors as observed using the tool, Assessing School Settings: Interactions of Students and Teachers (ASSIST). The study took place in the context of a high school randomized controlled trial (Maryland Safe and Supportive Schools [MDS3]--58 high school randomized controlled trials funded by the U.S. Department of Education and the William T. Grant Foundation) of Positive Behavioral Interventions and Supports (PBIS) (Sugai & Horner 2006). PBIS is a school-based decision-making framework designed to enhance students' social and academic functioning. Baseline data (collected prior to intervention) from six Maryland high schools provided the analytic sample for the study. Two raters worked in tandem to observe 25 classrooms/occasions in each of the six high schools using the ASSIST, resulting in a partially crossed design in which observations were nested within schools and crossed by rater. School interactions were assessed using the ASSIST, an observational measure of teachers' behavior management strategies and student behavior. Generalizability studies (G-studies) and decision studies (D-studies) were conducted using GENOVA software. Variations in the number of raters used to conduct observations and the number of observations occurring within a school had direct implications for the reliability of ASSIST tally items. Results indicate that raters can be trained to observe selected behaviors with high levels of agreement, and that studies using the ASSIST may be well-positioned to capture school-level differences for the behaviors of teacher proactive behavior expectations, teacher reactive behavior management, student disruption, and student profanity. This study demonstrates GT as a useful tool when evaluating and planning for the use of observational measures in large-scale program evaluations. Tables and figures are appended.

Multifractal analysis studies signals, functions, images or fields via the fluctuations of their local regularity along time or space, which capture crucial features of their temporal/spatial dynamics. It has become a standard signal and image processing tool and is commonly used in numerous applications of different natures. In its common formulation, it relies on the H\"older exponent as a measure of local regularity, which is by nature restricted to positive values and can hence be used for locally bounded functions only. In this contribution, it is proposed to replace the H\"older exponent with a collection of novel exponents for measuring local regularity, the $p$-exponents. One of the major virtues of $p$-exponents is that they can potentially take negative values. The corresponding wavelet-based multiscale quantities, the $p$-leaders, are constructed and shown to permit the definition of a new multifractal formalism, yielding an accurate practical estimation of the multifractal properties of real-world data. Moreover, theoretical and practical connections to and comparisons against another multifractal formalism, referred to as multifractal detrended fluctuation analysis, are achieved. The performance of the proposed $p$-leader multifractal formalism is studied and compared to previous formalisms using synthetic multifractal signals and images, illustrating its theoretical and practical benefits. The present contribution is complemented by a companion article studying in depth the theoretical properties of $p$-exponents and the rich classification of local singularities it permits.

Multifractal analysis aims to characterize signals, functions, images or fields, via the fluctuations of their local regularity along time or space, hence capturing crucial features of their temporal/spatial dynamics. Multifractal analysis is becoming a standard tool in signal and image processing, and is nowadays widely used in numerous applications of different natures. Its common formulation relies on the measure of local regularity via the H\"older exponent, by nature restricted to positive values, and thus to locally bounded functions or signals. It is here proposed to base the quantification of local regularity on $p$-exponents, a novel local regularity measure potentially taking negative values. First, the theoretical properties of $p$-exponents are studied in detail. Second, wavelet-based multiscale quantities, the $p$-leaders, are constructed and shown to permit accurate practical estimation of $p$-exponents. Exploiting the potential dependence with $p$, it is also shown how the collection of $p$-exponents enriches the classification of locally singular behaviors in functions, signals or images. The present contribution is complemented by a companion article developing the $p$-leader based multifractal formalism associated to $p$-exponents.

Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the H\"older exponent, is not feasible. We present a multifractal analysis based on another quantity, the p-exponent, which can take arbitrarily large negative values. We investigate some mathematical properties of this exponent, and show how it allows us to model the idea of "lacunarity" of a singularity at a point. We finally adapt the wavelet based multifractal analysis in this setting, and we give applications to a simple mathematical model of multifractal processes: Lacunary wavelet series.