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

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

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

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

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

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 \textrm MADdtrd, baseline level $\beta _0$ , 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. [ABSTRACT FROM AUTHOR]

Intrapartum fetal heart rate monitoring, aiming at early acidosis detection, constitutes an important public health stake. Scattering transform is proposed here as a new tool to analyze intrapartum fetal heart rate (FHR) variability. It consists of a nonlinear extension of the underlying wavelet transform, that thus preserves its multiscale nature. Applied to an FHR signal database constructed in a French academic hospital, the scattering transform is shown to permit to efficiently measure scaling exponents characterizing the fractal properties of intrapartum FHR temporal dynamics, that relate not only to the sole covariance (correlation scaling exponent), but also to the full dependence structure of data (intermittency scaling exponent). Such exponents are found to satisfactorily discriminate temporal dynamics of healthy subjects (from that of nonhealthy ones) and to emphasize the role of the highest frequencies (around and above 1 Hz) in intrapartum FHR variability. This permits us to achieve satisfactory classification performance that improves on those obtained from the analysis of International Federation of Gynecology and Obstetrics (FIGO) criteria, notably by classifying as healthy a number of subjects that were incorrectly classified as nonhealthy by classical clinically used FIGO criteria. Combined to obstetrician annotations, these scaling exponents enable us to sketch a typology of these FIGO-false positive subjects. Also, they permit us to monitor the evolution along time of the intrapartum health status of the fetuses and to estimate an optimal detection time-frame. [ABSTRACT FROM PUBLISHER]

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