Abstract. Texture analysis consists of a classical and everlasting task in image processing, often involved in a broad range of applications, possibly very different in nature. For large classes of textures, scale-free (or fractal) spatial dynamics as well as anisotropy constitute key properties. However, the local (pixelwise) joint estimation of anisotropy and scale-free attributes constitutes a difficult challenge, as both consist of nonlocal properties. Yet accurately detecting variations of these attributes across the image is often crucial. The overarching goal of the present work is thus to propose an inverse problem formulation for the analysis of piecewise homogeneous textures, grounded jointly on scale-free dynamics and anisotropy, and to study its performance for local assessment of textures. The formulation combines several major contributions. First, piecewise homogeneous Gaussian fields are defined as texture mixture models, with prescribed scale-free and anisotropy properties. Second, multiband complex wavelet coefficients, implemented via (nondecimated) dual-tree fast algorithms, are theoretically shown to be sensitive to both scale-free dynamics and anisotropy. Notably, it is shown that the squared-modulus of the wavelet coefficients behaves locally (or pixelwise) approximately as power-laws, with both scaling exponent and intercept jointly sensitive to anisotropy and scale-free dynamics. Thus, a key originality of the present work is to propose to perform local analysis of intrinsically nonlocal properties without having recourse to local averages, which would significantly impair accurate segmentation or accurate local characterization. Third, these local power-law-like behaviors, combined with regularization terms enforcing piecewise homogeneity, are embedded into a minimization procedure, solved by an optimization algorithm, whose convergence conditions are theoretically well-studied, and practically implemented through an efficient proximal algorithm due to strong convexity of the resulting minimization problem. Segmentation performance achieved by the proposed procedure is quantified and compared with respect to the difficulty of the segmentation task, for synthetic piecewise homogeneous Gaussian fields. The potential of the proposed texture analysis tool is also illustrated at work on real-world textures.

Assessing the intensity of a epidemic, such as the COVID19 pandemic, during the epidemic outbreak, constitutes a significant technical challenge with high societal stakes. Elaborating on classical epidemiological models, this work aims to define a hierarchical Bayesian model that permits the robust estimation of the temporal evolution of the pandemic intensity despite highly corrupted daily new infection counts. It also outputs uncertainty assessment, in the form of credibility intervals robust to the priors choice, accounting for uncertainties on model parameters. The estimation is performed by carefully designed Monte Carlo samplers. The relevance of the proposed estimation procedure is illustrated on real COVID-19 pandemic data for several countries and periods, made available from the Johns Hopkins University repository.

Generating surrogate data using Deep Neural Network (DNN) has become a classic task in image processing, while DNN time series synthesis is less often considered. The present work addresses issues related to the DNN synthesis of time series, with complex, scalefree time nonreversible temporal dynamics, using Wassertein Generative Adversarial Network. Instead of proposing yet another overperforming architecture, it discusses, first, synthesis quality quantitative assessment and, second, architecture designs that both reduce, for the Generator, the number of trainable parameters by a factor of 10000 (compared to state-of-the-art architectures), at no expense in performance cost, and permit to generate time series of size longer than that of the training set, without retraining. This works can thus be considered a contribution towards sustainable Artificial Intelligence.

The present work addresses the segmentation of textures characterized by anisotropy and scale-free statistics, two generic properties of use to model numerous real-world applications. This is achieved by proposing to combine a complex dual-tree multi-scale (wavelet) analysis within an inverse problem formulation aiming to estimate anisotropy and scale-free local parameters and to group them into piecewise homogeneous patches, jointly and in one single step. To minimize the corresponding functional, a primal-dual proximal convergent algorithm is devised and accelerated by taking advantage of the strong convexity of the data-fidelity term. Segmentation performance are assessed as function of the complexity of the task by means of Monte Carlo simulations conducted over synthetic textures, defined from anisotropic scale-free stochastic models.

Multivariate selfsimilarity has become a classical tool to analyze collections of time series recorded jointly on one same system. Often, it amounts to estimating as many scaling exponents as time series. However, this leaves open the important question how many such scaling exponents are actually different. Elaborating on earlier work aiming to test the hypothesis that all exponents are equal, we intend here to count the number of different scaling exponents from a single finite size multivariate time series. To this end, we devise an original clustering procedure that combines a wavelet domain block multivariate bootstrap scheme with a test strategy for a reduced set of multiple hypotheses on the pairwise equality of scaling exponents that are relevant to clustering. Monte Carlo simulations, making use of synthetic multivariate selfsimilar processes, assess the relevance and performance of the proposed procedure under different scenarios and demonstrate that the proposed method yields practically satisfactory cluster number and size estimations.

Multifractal analysis provides a global description for the spatial fluctuations of the strengths of the pointwise regularity of image amplitudes. A global image characterization leads to robust estimation, but is blind to and corrupted by small regions in the image whose multifractality differs from that of the rest of the image. Prior detection of such zones with anomalous multifractality is thus crucial for relevant analysis, and their delineation of central interest in applications, yet has never been achieved so far. The goal of this work is to devise and study such a multifractal anomaly detection scheme. Our approach combines three original key ingredients: i) a recently proposed generic model for the statistics of the multiresolution coefficients used in multifractal estimation (wavelet leaders), ii) an original surrogate data generation procedure for simulating a hypothesized global multifractality and iii) a combination of multiple hypothesis tests to achieve pixel-wise detection. Numerical simulations using synthetic multifractal images show that our procedure is operational and leads to good multifractal anomaly detection results for a range of target sizes and parameter values of practical relevance.

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.

The aim of this work is to investigate the ability of deep learning (DL) architectures to learn temporal dynamics in multivariate time series. The methodology consists in using well known synthetic stochastic processes for which changes in joint temporal dynamics can be controlled. This permits to compare deep learning against classical machine learning techniques relying on documented hand-crafted wavelet-based features. First, we assess the performance of several different DL architectures and show the relevance of convolutional neural networks (CNN). Second, we test the robustness of CNN performance in classifying subtle changes in multivariate temporal dynamics with respect to learning conditions (dataset size, time series sample size, transfer learning).

The self-similarity paradigm enables the analysis of scale-free temporal dynamics and has been widely used in a large set of real-world applications. However, in a multivariate setting, delays amongst components significantly impair the estimation of scale-free parameters. The first framework for the modeling, detection and estimation of delay parameters and for the joint estimation of scale-free parameters is proposed here. It is assumed that a single realization of a multivariate, self-similar time series is available. Use is made of ℂ-valued wavelets and, based on the imaginary part of the wavelet coherence, an original bootstrap-based delay estimation procedure based is constructed. Moreover, a consistent wavelet eigenanalysis-based semiparametric estimation for scale-free parameters that accounts for delay is defined. Monte Carlo experiments conducted over various instances of the model show that the proposed methodology enables the detection of delays with high probability and provides very satisfactory estimates of the delay and scale-free parameters.

Self-similarity has become a well-established modeling framework in several fields of application and its multivariate formulation is of ever-increasing importance in the Big Data era. Multivariate Hurst exponent estimation has thus received a great deal of attention recently, with wavelet eigenvalue-based regression becoming a focal point. The present work tackles the issue of the presence of significant finite-sample bias in wavelet eigenvalue regression stemming from the eigenvalue repulsion effect, whose origin and impact are analyzed and quantified. Furthermore, an original wavelet domain bias reduction technique is developed assuming a single multivariate time series is available. The protocol consists of a bootstrap resampling scheme that preserves the joint covariance structure of multivariate wavelet coefficients. Extensive numerical simulations show that this proposed method is effective in counteracting the bias at the price of a small increase in variance. This leads to wavelet eigenanalysis-based estimation of multivariate Hurst exponents with significantly improved finite-sample performance than earlier state-of-the-art formulations.

Texture segmentation still constitutes an on-going challenge, especially when processing large-size images. Recently, procedures integrating a scale-free (or fractal) wavelet-leader model allowed the problem to be reformulated in a convex optimization framework by including a TV penalization. In this case, the TV penalty plays a prominent role with respect to the data fidelity term, which makes the approach costly in terms of memory and computation cost. The present contribution aims to investigate the potential of recent block-coordinate dual and primal-dual proximal algorithms for overcoming this numerical issue. Our study shows that a key ingredient in the success of the proposed block-coordinate approaches lies in the design of the blocks of variables which are updated at each iteration. Numerical experiments conducted over synthetic textures having piece-wise constant fractal properties confirm our theoretical analysis. The proposed lattice block design strategy is shown to yield significantly lower memory and computational requirements.

Multifractal analysis, notably with its recent wavelet-leader based formulation, has nowadays become a reference tool to characterize scale-free temporal dynamics in time series. It proved successful in numerous applications very diverse in nature. However, such successes remained restricted to univariate analysis while many recent applications call for the joint analysis of several components. Surprisingly, multivariate multifractal analysis remained mostly overlooked. The present contribution aims at defining a wavelet-leader based framework for multivariate multifractal analysis and at studying its properties and estimation performance. To better understand what properties of multivariate data are actually captured in multivariate multifractal analysis, a multivariate multifractal model is used as representative paradigm and permits to show that multivariate multifractal analysis puts in evidence transient and local dependencies that are not well quantified or even evidenced by the classical Pearson correlation coefficient.

As part of the Historic Photographic Paper Classification Challenge, a multitude of approaches to quantifying paper texture similarity have been developed. These approaches have yielded encouraging results when applied to very controlled datasets containing photomicrographs of familiar specimens. In this paper, we report on the k-nearest neighbors classification performance of two multiscale analysis-based texture similarity approaches when applied to a much larger reference collection of silver gelatin photographic papers. The clusters for this data set were derived from a visual sorting experiment conducted by art conservators and paper experts later extended through crowd-sourcing. The results show that these texture similarity approaches, when combined with a simple k-nearest neighbors classification algorithm, yield workable performances with accuracy of up to 69%. We discuss this outcome in the context of available data and the cross-validation procedure used, then provide suggestions for improvement.

Texture segmentation constitutes a task of utmost importance in statistical image processing. Focusing on the broad class of monofractal textures characterized by piecewise constancy of the statistics of their multiscale representations, recently shown to be versatile enough for real-world texture modeling, the present work renews this recurrent topic by proposing an original approach enrolling jointly scale-free and local variance descriptors into a convex, but non smooth, minimization strategy. The performance of the proposed joint approach are compared against disjoint strategies working independently on scale-free features and on local variance on synthetic piecewise monofractal textures. Performance are also compared for multiphase flow image characterization, a topic of crucial importance in geophysics as well as in industrial processes. Applied to large-size images (above two million pixels), the proposed approach is shown to significantly improve state-of-the-art strategies by permitting the detection of the smallest gas bubbles and by offering a better understanding of multiphase flow structures.

Texture characterization of natural images using the mathematical framework of multifractal analysis (MFA) enables the study of the fluctuations in the regularity of image intensity. Although successfully applied in various contexts, the use of MFA has so far been limited to the independent analysis of a single image, while the data available in applications are increasingly multivariate. This paper addresses this limitation and proposes a joint Bayesian model and associated estimation procedure for multifractal parameters of multivariate images. It builds on a recently introduced generic statistical model that enabled the Bayesian estimation of multifractal parameters for a single image and relies on the following original key contributions: First, we develop a novel Fourier domain statistical model for a single image that permits the use of a likelihood that is separable in the multifractal parameters via data augmentation. Second, a joint Bayesian model for multivariate images is formulated in which prior models based on gamma Markov random fields encode the assumption of the smooth evolution of multifractal parameters between the image components. The design of the likelihood and of conjugate prior models is such that exploitation of the conjugacy between the likelihood and prior models enables an efficient estimation procedure that can handle a large number of data components. Numerical simulations conducted using sequences of multifractal images demonstrate that the proposed procedure significantly outperforms previous univariate benchmark formulations at a competitive computational cost.

Scale-free dynamics commonly appear in individual components of multivariate data. Yet, while the behavior of cross-components is crucial in modeling real-world multivariate data, their examination often suggests departures from exact multivariate self-similarity (also termed fractal connectivity). The present paper introduces a multivariate Gaussian stochastic process with Hadamard (i.e., entry-wise) self-similar scale-free dynamics, controlled by a matrix Hurst parameter H, that allows departures from fractal connectivity. The properties of its wavelet coefficients and wavelet spectrum are studied, enabling the estimation of H and of the fractal connectivity parameter. Furthermore, it permits the computation of closed-form confidence intervals for the estimates based on approximate (wavelet) covariances. Finally, these developments enable us to devise a test for fractal connectivity. Monte Carlo simulations are used to assess the accuracy of the proposed approximate confidence intervals and the performance of the fractal connectivity test.

This contribution focuses, within the ℓ1-Potts model, on the automated estimation of the regularization parameter balancing the ℓ1data fidelity term and the TVℓ0penalization. Variational approaches based on total variation gained considerable interest to solve piecewise constant denoising problems thanks to their deterministic setting and low computational cost. However, the quality of the achieved solution strongly depends on the tuning of the regularization parameter. While recent works have tailored various hierarchical Bayesian procedures to additionally estimate the regularization parameter for Gaussian noise, less attention has been granted to Laplacian noise, of interested in numerous applications. This contribution promotes a fast and parameter-free denoising procedure for piecewise constant signals corrupted by Laplacian noise, that includes automated selection of the regularization parameter. It relies on the minimization of a Bayesian-driven criterion whose similarities with the ℓ1-Potts model permit to derive a computationally efficient algorithm.

Among many research efforts devoted to automated art investigations, the problem of quantification of literary style remains current. Meanwhile, linguists and computer scientists have tried to sort out texts according to their types or authors. We use the recently-introduced p-leader multifractal formalism to analyze a corpus of novels written for adults and young adults, with the goal of assessing if a difference in style can be found. Our results agree with the interpretation that novels written for young adults largely follow conventions of the genre, whereas novels written for adults are less homogeneous.

Fetal heart rate (FHR) monitoring is routinely used in clinical practice to help obstetricians assess fetal health status during delivery. However, early detection of fetal acidosis that allows relevant decisions for operative delivery remains a challenging task, receiving considerable attention. This contribution promotes sparse support vector machine classification that permits to select a small number of relevant features and to achieve efficient fetal acidosis detection. A comprehensive set of features is used for FHR description, including enhanced and computerized clinical features, frequency domain, and scaling and multifractal features, all computed on a large (1288 subjects) and well-documented database. The individual performance obtained for each feature independently is discussed first. Then, it is shown that the automatic selection of a sparse subset of features achieves satisfactory classification performance (sensitivity 0.73 and specificity 0.75, outperforming clinical practice). The subset of selected features (average depth of decelerations MADdtrd, baseline level β0, and variability H) receives simple interpretation in clinical practice. Intrapartum fetal acidosis detection is improved in several respects: A comprehensive set of features combining clinical, spectral, and scale-free dynamics is used; an original multivariate classification targeting both sparse feature selection and high performance is devised; state-of-the-art performance is obtained on a much larger database than that generally studied with description of common pitfalls in supervised classification performance assessments.

Bluetooth sensors have recently been developed throughout the world for traffic information gathering. Primarily designed for travel time analysis, this article presents a method for vehicular trajectories retrieval. After a short description of some of the challenges at hand in using Bluetooth data in an urban network, a procedure to extract trip information from such data is proposed. It is further analyzed and illustrated at work on a real dataset collected in Brisbane. Last, this article shows that using spatially constrained shortest path analysis, this trip information, once extracted, can be used for the reconstruction of the trajectories. The performance of the process is assessed using both a simulated dataset and one from the real-world acquired in Brisbane, showing encouraging results, with up to 84% of accurately recovered trajectories.

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.

Texture analysis is an image processing task that can be conducted using the mathematical framework of multifractal analysis to study the regularity fluctuations of image intensity and the practical tools for their assessment, such as (wavelet) leaders. A recently introduced statistical model for leaders enables the Bayesian estimation of multifractal parameters. It significantly improves performance over standard (linear regression based) estimation. However, the computational cost induced by the associated nonstandard posterior distributions limits its application. The present work proposes an alternative Bayesian model for multifractal analysis that leads to more efficient algorithms. It relies on three original contributions: A novel generative model for the Fourier coefficients of log-leaders; an appropriate reparametrization for handling its inherent constraints; a data-augmented Bayesian model yielding standard conditional posterior distributions that can be sampled exactly. Numerical simulations using synthetic multifractal images demonstrate the excellent performance of the proposed algorithm, both in terms of estimation quality and computational cost.

Internet traffic monitoring is a crucial task for network security. Self-similarity, a key property for a relevant description of internet traffic statistics, has already been massively and successfully involved in anomaly detection. Self-similar analysis was however so far applied either to byte or Packet count time series independently, while both signals are jointly collected and technically deeply related. The present contribution elaborates on a recently proposed multivariate self-similar model, Operator fractional Brownian Motion (OfBm), to analyze jointly self-similarity in bytes and packets. A non-linear regression procedure, based on an original Branch & Bound resolution procedure, is devised for the full identification of bivariate OfBm. The estimation performance is assessed by means of Monte Carlo simulations. Further, an Internet traffic anomaly detection procedure is proposed, that makes use of the vector of Hurst exponents underlying the OfBm based Internet data modeling. Applied to a large set of high quality and modern Internet data from the MAWI repository, proof-of-concept results in anomaly detection are detailed and discussed.

Texture analysis can be embedded in the mathematical framework of multifractal (MF) analysis, enabling the study of the fluctuations in regularity of image intensity and providing practical tools for their assessment, wavelet leaders. A statistical model for leaders was proposed permitting Bayesian estimation of MF parameters for images yielding improved estimation quality over linear regression based estimation. This present work proposes an extension of this Bayesian model for patch-wise MF analysis of images. Classical MF analysis assumes space homogeneity of the MF properties whereas here we assume MF properties may change between texture elements and we do not know where the changes are located. This paper proposes a joint Bayesian model for patches formulated using spatially smoothing gamma Markov Random Field priors to counterbalance the increased statistical variability of estimates caused by small patch sizes. Numerical simulations based on synthetic multi-fractal images demonstrate that the proposed algorithm outperforms previous formulations and standard estimators.

Scale invariance has proven a crucial concept in texture modeling and analysis. Isotropic and self-similar fractional Brownian fields (2D-fBf) are often used as the natural reference process to model scale free textures. Its analysis is standardly conducted using the 2D discrete wavelet transform. Generalizations of 2D-fBf were considered independently in two respects: Anisotropy in the texture is allowed while preserving exact self-similarity, analysis then needs to be conducted using the 2D-Hyperbolic wavelet transform; Multifractality enables more versatile scale free models but requires isotropy, analysis is then achieved using wavelet leaders. The present paper proposes a first unifying extension, which is enabled through the following two key contributions: The definition of 2D process that incorporates jointly anisotropy and multi-fractality : The definition of the corresponding analysis tool, the hyperbolic wavelet leaders. Their relevance are studied by numerical simulations using synthetic scale free textures.

Recent advances in wearable and/or biomedical sensing technologies have made it possible to record very long-term, continuous biomedical signals, referred to as biomedical intensive longitudinal data (ILD). To link ILD to clinical applications, such as personalized healthcare and disease prevention, the development of robust and reliable data analysis techniques is considered important. In this review, we introduce multiscale analysis methods for and the applications to two types of intensive longitudinal biomedical signals, heart rate variability (HRV) and spontaneous physical activity (SPA) time series. It has been shown that these ILD have robust characteristics unique to various multiscale complex systems, and some parameters characterizing the multiscale complexity are in fact altered in pathological states, showing potential usability as a new type of ambient diagnostic and/or prognostic tools. For example, parameters characterizing increased intermittency of HRV are found to be potentially useful in detecting abnormality in the state of the autonomic nervous system, in particular the sympathetic hyperactivity, and intermittency parameters of SPA might also be useful in evaluating symptoms of psychiatric patients with depressive as well as manic episodes, all in the daily settings. Therefore, multiscale analysis might be a useful tool to extract information on clinical events occurring at multiple time scales during daily life and the underlying physiological control mechanisms from biomedical ILD.

International audience

Operator self-similarity naturally extends the concepts of univariate self-similarity and scale invariance to multivariate data. Beyond a vector of Hurst parameters, operator self-similarity models also involve a mixing matrix. The present contribution aims at estimating the collection of Hurst parameters in the case where the mixing matrix is not diagonal. To the best of our knowledge, this has never been achieved. In addition, the mixing matrix is also identified. The devised procedure relies on a source separation methodology, since the underlying components of the operator self-similar process are assumed to have a diagonal pre-mixing covariance structure. The principle behind the demixing procedure is illustrated based on synthetic 4-variate operator self-similar processes, with a priori prescribed and controlled Hurst parameters and mixing matrix. Identification and estimation performance for both Hurst parameters and mixing matrices are shown to be very satisfactory, using large size Monte Carlo simulations.

We present a new anomaly detector for data traffic, ‘SMS’, based on combining random projections (sketches) with multiscale analysis, which has low computational complexity. The sketches allow ‘normal’ traffic to be automatically and robustly extracted, and anomalies detected, without the need for training data. The multiscale analysis extracts statistical descriptors, using wavelet leader tools developed recently for multifractal analysis, without any need for timescales to be selected a priori. The proposed detector is illustrated using a large recent dataset of Internet backbone traffic from the MAWI archive, and compared against existing detectors.

In transport networks, Origin-Destination matrices (ODM) are classically estimated from road traffic counts whereas recent technologies grant also access to sample car trajectories. One example is the deployment in cities of Bluetooth scanners that measure the trajectories of Bluetooth equipped cars. Exploiting such sample trajectory information, the classical ODM estimation problem is here extended into a link-dependent ODM (LODM) one. This much larger size estimation problem is formulated here in a variational form as an inverse problem. We develop a convex optimization resolution algorithm that incorporates network constraints. We study the result of the proposed algorithm on simulated network traffic.

This work aims to segment a texture into different regions, each characterized by a priori unknown multifractal properties. The multifractal properties are quantified using the multiscale function C1, jthat quantifies the evolution along analysis scales 2jof the empirical mean of the log of the wavelet leaders. The segmentation procedure is applied to local estimate of C1, j. It involves a multivariate Mumford-Shah relaxation formulated as a convex optimization problem involving a structure tensor penalization and an efficient algorithmic solution based on primal-dual proximal algorithm. The performances are evaluated on synthetic textures.

Texture characterization is a central element in many image processing applications. Multifractal analysis is a useful signal and image processing tool, yet, the accurate estimation of multifractal parameters for image texture remains a challenge. This is due in the main to the fact that current estimation procedures consist of performing linear regressions across frequency scales of the 2D dyadic wavelet transform, for which only a few such scales are computable for images. The strongly non-Gaussian nature of multifractal processes, combined with their complicated dependence structure, makes it difficult to develop suitable models for parameter estimation. Here, we propose a Bayesian procedure that addresses the difficulties in the estimation of the multifractality parameter. The originality of the procedure is threefold. The construction of a generic semiparametric statistical model for the logarithm of wavelet leaders; the formulation of Bayesian estimators that are associated with this model and the set of parameter values admitted by multifractal theory; the exploitation of a suitable Whittle approximation within the Bayesian model which enables the otherwise infeasible evaluation of the posterior distribution associated with the model. Performance is assessed numerically for several 2D multifractal processes, for several image sizes and a large range of process parameters. The procedure yields significant benefits over current benchmark estimators in terms of estimation performance and ability to discriminate between the two most commonly used classes of multifractal process models. The gains in performance are particularly pronounced for small image sizes, notably enabling for the first time the analysis of image patches as small as 64 × 64 pixels.

Scale invariance is a widely used concept to analyze real-world data from many different applications and multifractal analysis has become the standard corresponding signal processing tool. It characterizes data by describing globally and geometrically the fluctuations of local regularity, usually measured by means of the Ho¨lder exponent. A major limitation of the current procedure is that it applies only to locally bounded functions or signals, i.e., to signals with positive regularity. The present contribution proposes to characterize local regularity with a new quantity, the p-exponent, that permits negative regularity in data, a widely observed property in real-world data. Relations to Ho¨lder exponents are detailed and a corresponding p-leader multifractal formalism is devised and shown at work on synthetic multifractal processes, representative of a class of models often used in applications. We formulate a conjecture regarding the equivalence between Ho¨lder and p-exponents for a subclass of processes. Even when Ho¨lder and p-exponents coincide, the p-leader formalism is shown to achieve better estimation performance.

The identification of texture changes is a challenging problem that can be addressed by considering local regularity fluctuations in an image. This work develops a procedure for local regularity estimation that combines a convex optimization strategy with wavelet leaders, specific wavelet coefficients recently introduced in the context of multifractal analysis. The proposed procedure is formulated as an inverse problem that combines the joint estimation of both local regularity exponent and of the optimal weights underlying regularity measurement. Numerical experiments using synthetic texture indicate that the performance of the proposed approach compares favorably against other wavelet based local regularity estimation formulations. The method is also illustrated with an example involving real-world texture.

Intrapartum fetal heart rate monitoring constitutes an important stake aiming at early acidosis detection. Measuring heart rate variability is often considered a powerful tool to assess the intrapartum health status of fetus and has been envisaged using various techniques. In the present contribution, scale invariance parameters, such as the Hurst exponent and the global regularity exponent, are estimated from wavelet coefficients of intrapartum fetal heart rate time series. Their ability to evaluate the health status of fetuses is quantified from a case study database, constituted at a French Academic Hospital in Lyon. Notably, the ability of such parameters to discriminate subjects incorrectly classified according to FIGO rules as abnormal is discussed. Also, the impact of the occurrence of decelerations identified as complicated by obstetricians on the values taken by Hurst parameter is investigated in detail.

Texture segmentation constitutes a classical yet crucial task in image processing. In many applications of very different natures (biomedical, geophysics,...) textures are naturally defined in terms of their local regularity fluctuations, which can be quantified as the variations of local Hölder exponents. Furthermore, such images are often naturally embedded in the class of piece-wise constant local regularity functions. The present contribution aims at proposing and assessing a segmentation procedure for this class of images. Its originality is twofold: First, local regularity is estimated using wavelet leaders, a novel multiresolution quantity recently introduced for multifractal analysis but barely used in local regularity measurement, comparisons against wavelet coefficient based estimation are conducted; Second, the challenging minimal partition problem underlying segmentation is convexified and conducted within a customized proximal framework. The estimation of the number of regions and their target regularity is obtained from a total-variation estimate that enables the actual use of proximal minimization for texture segmentation. Performance is assessed and illustrated on synthetic textures.

Multifractal analysis has matured into a widely used signal and image processing tool. Due to the statistical nature of multifractal processes (strongly non-Gaussian and intricate dependence) the accurate estimation of multifractal parameters is very challenging in situations where the sample size is small (notably including a range of biomedical applications) and currently available estimators need to be improved. To overcome such limitations, the present contribution proposes a Bayesian estimation procedure for the multifractality (or intermittence) parameter. Its originality is threefold: First, the use of wavelet leaders, a recently introduced multiresolution quantity that has been shown to yield significant benefits for multifractal analysis; Second, the construction of a simple yet generic semi-parametric model for the marginals and covariance structure of wavelet leaders for the large class of multiplicative cascade based multifractal processes; Third, the construction of original Bayesian estimators associated with the model and the constraints imposed by multifractal theory. Performance are numerically assessed and illustrated for synthetic multifractal processes for a range of multifractal parameter values. The proposed procedure yields significantly improved estimation performance for small sample sizes.

Textures in images can often be well modeled using self-similar processes while they may simultaneously display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will be first shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2D-discrete wavelet transform with the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axes. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized; this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a nonparametric bootstrap based procedure is described, which provides confidence intervals in addition to the estimates themselves and enables us to construct an isotropy test procedure, which can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis are illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields. As an illustration, we show that a true anisotropy built-in self-similarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed.

End-host profiling by analyzing network traffic comes out as a major stake in traffic engineering. Graphlet constitutes an efficient and common framework for interpreting host behaviors, which essentially consists of a visual representation as a graph. However, graphlet analyses face the issues of choosing between supervised and unsupervised approaches. The former can analyze a priori defined behaviors but is blind to undefined classes, while the latter can discover new behaviors at the cost of difficult a posteriori interpretation. This paper aims at bridging the gap between the two. First, to handle unknown classes, unsupervised clustering is originally revisited by extracting a set of graphlet-inspired attributes for each host. Second, to recover interpretability for each resulting cluster, a synoptic graphlet, defined as a visual graphlet obtained by mapping from a cluster, is newly developed. Comparisons against supervised graphlet-based, port-based, and payload-based classifiers with two datasets demonstrate the effectiveness of the unsupervised clustering of graphlets and the relevance of the a posteriori interpretation through synoptic graphlets. This development is further complemented by studying evolutionary tree of synoptic graphlets, which quantifies the growth of graphlets when increasing the number of inspected packets per host.

Recently, a growing interest in the exploration of the potential of signal or image processing tools for the purposes of art analysis has emerged. The wavelet leader based multifractal analysis consists of a mathematical tool recently introduced in image processing for the characterization of homogeneous textures based on their regularity properties. Here, this novel tool is applied to a set of digitized versions of drawings, made available by the NY Metropolitan Museum of Art, consisting of authentic Bruegel drawings and several imitations. Multifractal attributes are estimated from several patches of each of these drawings, and their ability to discriminate authentic drawings from impostors is investigated by means of subspace projections and quadratic discriminant analysis. Besides showing very satisfactory performance, the achieved discrimination provides interesting insights into the differences between the regularity of the textures of authentic Bruegel drawings versus imitations, potentially relating the fractal properties of the drawings to the artist's drawing style.

Inspired from non-equilibrium statistical physics models, a general framework enabling the definition and synthesis of stationary time series with a priori prescribed and controlled joint distributions is constructed. Its central feature consists of preserving for the joint distribution the simple product structure it has under independence while enabling to input controlled and prescribed dependencies amongst samples. To that end, it is based on products of d-dimensional matrices, whose entries consist of valid distributions. The statistical properties of the thus defined time series are studied in details. Having been able to recast this framework into that of Hidden Markov Models enabled us to obtain an efficient synthesis procedure. Pedagogical well-chosen examples (time series with the same marginal distribution, same covariance function, but different joint distributions) aim at illustrating the power and potential of the approach and at showing how targeted statistical properties can be actually prescribed.

Surrogates are investigated as procedures of synthesis for multi-variate time series with prescribed properties. First it is shown how to prescribe a multivariate covariance function jointly with the (possibly non-Gaussian) marginal distributions. Second, using histogram matching by approximate optimal transport with the Sliced Wasserstein Distance, the surrogate synthesis is extended to prescribe covariance function and joint-distribution of the components. Algorithms are described and justified, and numerical examples are shown. MATLAB codes are publicly available online.

Multifractal analysis describes data as a collection of singularities. However, its classical formulation does not account for their possibly oscillating nature, while, in a number of applications, distinguishing between oscillating and non oscillating singularities may significantly enrich the analysis. This is notably the case in hydrodynamic turbulence, of interest here, where two different important heuristic models contradictorily lead to predict the existence or absence of oscillating singularities. This contribution proposes a wavelet Leader oscillation formalism enabling to evidence the presence of oscillating singularities in real data. It is first validated on synthetic data both with and without oscillating singularities and second applied to high quality ID velocity turbulence data. This constitutes the first quantitative evidence against the presence of oscillating singularities in turbulence data.

Ionosphere consists of a large complex system whose analysis is of major importance, e.g., for climatology or radio-communications. Therefore, studying its variations, usually analyzed in terms of long-term trends versus short-term fluctuations, as well as the mechanisms driving them is of importance. This contribution hence performs a scale-dependent cross-analysis of the F2-region critical frequency data, locally measured at 11 mid-latitude European stations, and 5 global solar and geomagnetic indices. It shows that such Ionospheric variations are correctly described by the superimposition of well-defined long-term cycles with highly correlated fractional Gaussian noise fluctuations. Also, it is shown that mid-latitude European stations display highly correlated variations even for short-term fluctuations and that, while the solar activity mostly drives long-term cycles, short-term fluctuations are essentially controlled by the geomagnetic activity.

Evaluating anomaly detectors is a crucial task in traffic monitoring made particularly difficult due to the lack of ground truth. The goal of the present article is to assist researchers in the evaluation of detectors by providing them with labeled anomaly traffic traces. We aim at automatically finding anomalies in the MAWI archive using a new methodology that combines different and independent detectors. A key challenge is to compare the alarms raised by these detectors, though they operate at different traffic granularities. The main contribution is to propose a reliable graph-based methodology that combines any anomaly detector outputs. We evaluated four unsupervised combination strategies; the best is the one that is based on dimensionality reduction. The synergy between anomaly detectors permits to detect twice as many anomalies as the most accurate detector, and to reject numerous false positive alarms reported by the detectors. Significant anomalous traffic features are extracted from reported alarms, hence the labels assigned to the MAWI archive are concise. The results on the MAWI traffic are publicly available and updated daily. Also, this approach permits to include the results of upcoming anomaly detectors so as to improve over time the quality and variety of labels.

Intrapartum fetal surveillance aims at preventing neonatal morbidity and mortality due to asphyxia. Continuous fetal heart rate monitoring helps in reducing asphyxia yet generates many unnecessary c-sections. In fetal heart rate time series analysis, variability is a key parameter as its decrease is used to identify asphyxia. Multifractal analysis can be envisaged as a new statistical tool enabling to revisit variability analysis. Applied to data collected at an academic hospital, the wavelet Leader based multifractal analysis is shown here to i) permit a rich and relevant characterization of their variability, ii) to achieve a significant discrimination between healthy and non-healthy fetus, iii) to enable this valid discrimination to hold much earlier than during the last 15 min preceding delivery. This hence opens promising tracks to decrease the number of unnecessary c-sections.

After the seminal work by Taqqu 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.

In this work we present a simple toy-model that is able to explain certain empirical observations reported in a set of previous papers by Hohn et al. about the wavelet spectrum of real traffic traces. Therein, the authors found that the wavelet spectrum is substantially invariant to flow scrambling and truncation. Such finding suggested that super-flow structures above the transport layer - i.e., sessions - can be ignored for modeling the packet arrival process. Based on the proposed toy-model, we offer an interpretation framework that goes in the opposite direction, indicating that sessions, not transport-layer flows, should be taken as the main structural entities in simplified on/off models.

Within the framework of long memory multivariate processes, fractal connectivity is a particular model, in which the low frequencies (coarse scales) of the interspectrum of each pair of process components are determined by the autospectra of the components. The underlying intuition is that long memories in each components are likely to arise from a same and single mechanism. The present contribution aims at defining and characterizing a statistical procedure for testing actual fractal connectivity amongst data. The test is based on Fisher's Z transform and Pearson correlation coefficient, and anchored in a wavelet framework. Its performance are analyzed theoretically and validated on synthetic data. Its usefulness is illustrated on the analysis of Internet traffic Packet and Byte count time series.

Image classification often relies on texture characterization. Yet texture characterization has so far rarely been based on a true 2D multifractal analysis. Recently, a 2D wavelet Leader based multifractal formalism has been proposed. It allows to perform an accurate, complete and low computational and memory costs multifractal characterization of textures in images. This contribution describes the first application of such a formalism to a real large size (publicly available) image database, consisting of 25 classes of non traditional textures, with 40 high resolution images in each class. Multifractal attributes are estimated from each image and used as classification features within a standard k nearest neighbor classification procedure. The results reported here show that this Leader based multifractal analysis enables the effective discrimination of different textures, as performances in both classification scores and computational costs compare favorably against those of procedures previously proposed in the literature on the same database.

This contribution aims at performing a longitudinal study of the evolution of the traffic collected every day for seven years on a trans-Pacific backbone link (the MAWI dataset). Long term characteristics are investigated both at TCP/IP layers (packet and flow attributes) and application usages. The analysis of this unique dataset provides new insights into changes in traffic statistics, notably on the persistence of Long Range Dependence, induced by the on-going increase in link bandwidth. Traffic in the MAWI dataset is subject to bandwidth changes, to congestions, and to a variety of anomalies. This allows the comparison of their impacts on the traffic statistics but at the same time significantly impairs long term evolution characterizations. To account for this difficulty, we show and explain how and why random projection (sketch) based analysis procedures provide practitioners with an efficient and robust tool to disentangle actual long term evolutions from time localized events such as anomalies and link congestions. Our central results consist in showing a strong and persistent long range dependence controlling jointly byte and packet counts. An additional study of a 24-hour trace complements the long-term results with the analysis of intraday variabilities.

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.

A new family of distributions, constructed by summing two correlated gamma random variables, is studied. First, a simple closed form expression for their density is derived. Second, the three parameters characterizing such a density are estimated by using the maximum likelihood (ML) principle. Numerical simulations are conducted to compare the performance of the ML estimator against those of the conventional estimator of moments. Finally, a multiresolution multivariate gamma based modeling of Internet traffic illustrates the potential interest of the proposed distributions for the detection of anomalies. Aggregated times series of IP packet counts are split into adjacent non overlapping time blocks. The distribution of the resulting time series are modeled by the proposed multivariate gamma based distributions, over a collection of different aggregation levels. The anomaly detection strategy is based on tracking changes along time of the corresponding multiresolution parameters.

On open and controversial issue in empirical data analysis is to decide whether scaling and multifractal properties observed in empirical data actually exist, or whether they are induced by intricate non stationarities. To contribute to answering this question, we propose a procedure aiming at testing the constancy along time of multifractal attributes estimated over adjacent non overlapping time windows. The procedure is based on non parametric bootstrap resampling and on wavelet Leader estimations for the multifractal parameters.lt is shown, by means of numerical simulations on synthetic multifractal processes, that the proposed procedure is reliable and powerful for discriminating true scaling behavior against non stationarities. We end up with a practical procedure that can be applied to a single finite length observation of data with unknown statistical properties.

We address the problem of fractal dimension estimation of a discrete sample path. After recalling the multiplicity of possible definitions, we focus on the regularity dimension and on the regularization dimension, and report on the common ingredients that underlie these definitions: a scale transform of the signal, and a geometric or statistical measure on the scaled signal. Then, we propose to interchange wavelet transforms, ordinarily used as the scale transform, with empirical mode decomposition (EMD), a recently proposed signal-adaptive transform. The adaptivity of this latter yields estimation performance that overhauls usual wavelet-based techniques. To support our claim, we obtain comprehensive results from a Monte Carlo simulation on fractional Brownian motions.

Multifractal analysis is nowadays commonly used in real-life data analyses and involved in standard signal processing tasks such as detection, identification or classification. In a number of situations, mostly in image processing, the data are available for the analyses only in (possibly severely) quantized versions. The present contribution aims at analyzing the robustness of standard multifractal estimation procedures against quantization. To this end, we analyze the behaviors and statistical performance of these procedures when applied to a large number of realizations of known synthetic multifractal processes subject to various quantization levels. Our study shows that immunity against quantization can be obtained by restricting the range of scales involved in multifractal parameter estimation to the largest ones. Comparing multifractal analyses based on different multiresolution quantities, increments, wavelet coefficients and leaders, we show that wavelets, thanks to their good frequency localization, bring robustness against quantization when increments do not. This study provides the practitioner with a clear guide line to perform multifractal analysis over quantized data.

The goals of the present contribution are twofold. First, we propose the use of a non-Gaussian long-range dependent process to model Internet traffic aggregated time series. We give the definitions and intuition behind the use of this model. We detail numerical procedures that can be used to synthesize artificial traffic exactly following the model prescription. We also propose original and practically effective procedures to estimate the corresponding parameters from empirical data. We show that this empirical model relevantly describes a large variety of Internet traffic, including both regular traffic obtained from public reference repositories and traffic containing legitimate (flash crowd) or illegitimate (DDoS attack) anomalies. We observe that the proposed model accurately fits the data for a wide range of aggregation levels. The model provides us with a meaningful multiresolution (i.e., aggregation level dependent) statistics to characterize the traffic: the evolution of the estimated parameters with respect to the aggregation level. It opens the track to the second goal of the paper: anomaly detection. We propose the use of a quadratic distance computed on these statistics to detect the occurrences of DDoS attack and study the statistical performance of these detection procedures. Traffic with anomalies was produced and collected by us so as to create a controlled and reproducible database, allowing for a relevant assessment of the statistical performance of the proposed (modeling and detection) procedures

Multifractal analysis, which mainly consists in estimating scaling exponents, has become a popular tool for empirical data analysis. Although widely used in different applications, the statistical performance and the reliability of the estimation procedures are still poorly known. Notably, little is known about confidence intervals, though they are of first importance in applications. The present work investigates the potential uses of bootstrap for multifractal estimation. Can bootstrap improve current estimation procedures or be used to obtain reliable confidence intervals? Comparing the statistical performance of different estimators, our major result is to show that bootstrap based procedures provide us both with accurate estimates and reliable confidence intervals. We believe that this brings substantial improvements to practical empirical multifractal analyses

We introduce a new multifractal formalism based on wavelet leaders and study its properties. Comparing it against previously formulated wavelet coefficient based multifractal formalism, we show first that this wavelet leader based formalism allows the multifractal spectrum to be obtained over its entire range, and second that it does not cease to hold when applied to processes embodying unusual chirp-type (or oscillating) singularities (as opposed to the more common cusp-type ones). We illustrate these results and properties on four examples of multifractal deterministic functions or stochastic processes containing a graduation of the major difficulties. We show that this new multifractal formalism benefits from excellent theoretical and practical performance. Matlab routines implementing it are available upon request.

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 nonstationary 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 form. 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.

We study the statistical performance of multiresolution (wavelet based) estimators commonly used for the estimation of the scaling exponents, /spl zeta/(q), of multifractal processes. So far, such studies have been conducted exclusively using the celebrated Mandelbrot's cascades. A new class of processes, compound Poisson cascades, with better statistical properties -stationary increments and continuous scale invariance - has recently been proposed in the literature. Making use of this new type of process, we show that the multiresolution estimators are characterised by a generic and systematic feature: beyond a critical order, q, (which is determined analytically), they fail to estimate the /spl zeta/(q) and present instead a linear behaviour in q. We study in detail this linearisation effect and show that it does not disappear in the limit of infinite observation duration, n, and that the parameters characterising it do not depend on n. We comment on its major practical consequences and on its having been mostly overlooked in applications.

Internet packet data is analysed to determine the relationship between the arrival process of packets, and of TCP flows of packets. Viewed as point processes, second order properties of the two processes are studied using wavelets, and each is found to have long range dependence. A new result is given directly linking flow durations to the onset scale of the long range dependence in the flow process. Using this result, a mechanism is described whereby the flow level structure could, in principle, influence the packet level structure, and it is shown and explained why this is not the case currently. The circumstances under which the flow structure could impact on the packet process, and therefore become important for the modeling of the packet level dynamics, are given.

In addition to the well known long-range dependence in time series of IP bytes and packets, evidence for scaling behaviour has also been found at small scales for these series, separated by a characteristic transition timescale. It is less well known that two scaling regimes are also commonly found in time series describing the arrivals of TCP flows, again with long-range dependence, and with a broadly similar scaling exponent at small scales. The transition timescale is also roughly similar to that found in the IP level case. We investigate the dependencies between the scaling behaviours of the IP and TCP arrival levels at both small and large scales. We also study the origin of scaling at small scales at the IP level. The arrival level process is important to study both for its potential impact on the IP level, and in its own right, for example for web server performance. Our findings are based on gigabytes of high precision packet level data collected at multiple locations. The analysis methodology combines models with real data in a 'semi-experimental' approach which reduces the need for modeling assumptions. Flows and packets are individually manipulated to selectively isolate the components of scaling due to packet dynamics within a TCP flow, the dependencies between flows, their durations and packet counts, and the flow arrival process. The scaling behaviour is analysed using wavelet based methods.

The scaling properties of Internet data are analysed in detail through the unifying viewpoint of infinitely divisible cascades (IDC). From exceptionally precise TCP/IP traffic traces are extracted time series including arrival rate, durations, and interarrival times of TCP connections. We show that IDC offer a pertinent description of these series. Relations between them are investigated, yielding insights on the sources of the scaling and possible modelling approaches.

This work is a contribution to the analysis of the procedure, based on wavelet coefficient partition functions, commonly used to estimate the Legendre multifractal spectrum. The procedure is applied to two examples, a fractional Brownian motion in multifractal time and a self-similar /spl alpha/-stable process, whose sample paths exhibit irregularities that by eye appear very close. We observe that, for the second example, this analysis results in a qualitatively inaccurate estimation of its multifractal spectrum, and a related masking of the /spl alpha/-stable nature of the process. We explain the origin of this error through a detailed analysis of the partition functions of the self-similar /spl alpha/-stable process. Such a study is made possible by the specific properties of the wavelet coefficients of such processes. We indicate how the estimation procedure might be modified to avoid such errors.

Infinitely divisible cascades are a model class previously introduced in the field of turbulence to describe the statistics of velocity fields. In this paper, using a wavelet reformulation of the cascades, we investigate their ability to analyze band model scaling properties of data and compare their fundamental ingredients to those of other scaling model classes such as self-similar and multifractal processes. We also propose an estimation procedure for the propagator or kernel of the cascades. Finally the cascade model is successfully applied to describe Internet TCP network traffic data, bringing new insights into their scaling properties and revealing a pitfall in existing techniques.

An on-line version of the Abry-Veitch (see IEEE GLOBECOM'98, Sydney, Australia,p.3716-21, 1998) wavelet-based estimator of the Hurst parameter is presented. It has very low memory and computational requirements and scales naturally to arbitrarily high data rates, enabling its use in real-time applications such as admission control, and avoiding the need to store huge data sets for off-line analysis. The performance of the estimator as a function of the length of data processed is demonstrated using simulated data. An implementation for 10-Mb/s Ethernet based on standard hardware supporting sampling rates of 1 data point per millisecond is described, and results of its operation presented, as is an implementation for 155-Mb/s asynchronous transfer mode networks. Finally we illustrate the power of on-line measurements by collecting measurements over a period of five months, and using them to look for diurnal trends in scaling properties of the data.

We, study self-similar processes with possibly infinite second order statistics and long-range dependence. To do so, we detail the statistical properties of the wavelet coefficients of /spl alpha/-stable self similar processes, used as a paradigm for those situations. We, then, propose a wavelet based estimator for the self-similarity parameter and analyse its statistical performance both theoretically and numerically. We show that it is unbiased, that its variance decreases as the inverse of the length of the data and that it can be easily implemented.

A joint estimator is presented for the two parameters that define the long-range dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a wavelet-based estimator of the scaling parameter (Abry and Veitch 1998), as well as extending it to include the associated power parameter. An important feature is its conceptual and practical simplicity, consisting essentially in measuring the slope and the intercept of a linear fit after a discrete wavelet transform is performed, a very fast (O(n)) operation. Under well-justified technical idealizations the estimator is shown to be unbiased and of minimum or close to minimum variance for the scale parameter, and asymptotically unbiased and efficient for the second parameter. Through theoretical arguments and numerical simulations it is shown that in practice, even for small data sets, the bias is very small and the variance close to optimal for both parameters. Closed-form expressions are given for the covariance matrix of the estimator as a function of data length, and are shown by simulation to be very accurate even when the technical idealizations are not satisfied. Comparisons are made against two maximum-likelihood estimators. In terms of robustness and computational cost the wavelet estimator is found to be clearly superior and statistically its performance is comparable. We apply the tool to the analysis of Ethernet teletraffic data, completing an earlier study on the scaling parameter alone.

A wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long-range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.

We propose here a multiple-window wavelet transform for the purpose of identifying non-stationary self-similar structures in random processes and estimating the time-varying scaling exponent H(t) that controls the local regularity and correlation of the process. More specifically, our final aim is to be able to track even rapidly varying trajectories (t, H(t)). The solution described here combines analysis obtained from scalograms computed with a set of multi-windows designed so as to satisfy to a decorrelation condition. We derive here the statistics for the estimate of H(t), compare it against numerical simulations and show that we obtain a substantial reduction of variance in estimation, without introducing bias.

The authors show that making use of the discrete wavelet transform to analyse data implies performing a preliminary initialization of the fast pyramidal algorithm. An approximation enabling easy performance of such an initialization is proposed.>