Roberto Leonarduzzi

Wavelet leader based multifractal analysis of heart rate variability during myocardial ischaemia

Heart rate variability is a non invasive and indirect measure of the autonomic control of the heart. Therefore, alterations to this control system caused by myocardial ischaemia are reflected in changes in the complex and irregular fluctuations of this signal. Multifractal analysis is a well suited tool for the analysis of this kind of fluctuations, since it gives a description of the singular behavior of a signal. Recently, a new approach for multifractal analysis was proposed, the wavelet leader based multifractal formalism, which shows remarkable improvements over previous methods. In order to characterize and detect ischaemic episodes, in this work we propose to perform a short-time windowed wavelet leader based multifractal analysis. Our results suggest that this new method provides appropriate indexes that could be used as a tool for the detection of myocardial ischaemia.

p-exponent and p-leaders, Part II: Multifractal Analysis. Relations to Detrended Fluctuation Analysis

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 Holder 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 Holder 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.

Extending multifractal analysis to negative regularity: P-exponents and P-leaders

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 Hölder 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 Hölder 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 Hölder and p-exponents for a subclass of processes. Even when Hölder and p-exponents coincide, the p-leader formalism is shown to achieve better estimation performance.

Multifractal Analysis Based on p-Exponents and Lacunarity Exponents

Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the Holder 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.

p-exponent and p-leaders, Part I: Negative pointwise regularity

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 Holder 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.

p-leader based classification of first stage intrapartum fetal HRV

Interpretation and analysis of intrapartum fetal heart rate, enabling early detection of fetal acidosis, remains a challenging signal processing task. Recently, a variant of the wavelet-based multifractal analysis, based on p-exponents and p-leaders, which provides a rich framework for data regularity analysis, has been proposed. The present contribution aims at studying the benefits of using the p-leader multifractal formalism for discrimination of intrapartum fetal heart rate. First, a dependence on p of the multifractal properties of data is evidenced and interpreted. Second, classification between healthy subjects and fetuses suffering from acidosis is shown to have satisfactory performance that increases when p is decreased.

Intrapartum fetal heart rate classification from trajectory in Sparse SVM feature space.

Intrapartum fetal heart rate (FHR) constitutes a prominent source of information for the assessment of fetal reactions to stress events during delivery. Yet, early detection of fetal acidosis remains a challenging signal processing task. The originality of the present contribution are three-fold: multiscale representations and wavelet leader based multifractal analysis are used to quantify FHR variability ; Supervised classification is achieved by means of Sparse-SVM that aim jointly to achieve optimal detection performance and to select relevant features in a multivariate setting ; Trajectories in the feature space accounting for the evolution along time of features while labor progresses are involved in the construction of indices quantifying fetal health. The classification performance permitted by this combination of tools are quantified on a intrapartum FHR large database (≃ 1250 subjects) collected at a French academic public hospital.

Scattering Transform of Heart Rate Variability for the Prediction of Ischemic Stroke in Patients with Atrial Fibrillation

BACKGROUND Atrial fibrillation (AF) is an identified risk factor for ischemic strokes (IS). AF causes a loss in atrial contractile function that favors the formation of thrombi, and thus increases the risk of stroke. Also, AF produces highly irregular and complex temporal dynamics in ventricular response RR intervals. Thus, it is hypothesized that the analysis of RR dynamics could provide predictors for IS. However, these complex and nonlinear dynamics call for the use of advanced multiscale nonlinear signal processing tools. OBJECTIVES The global aim is to investigate the performance of a recently-proposed multiscale and nonlinear signal processing tool, the scattering transform, in predicting IS for patients suffering from AF. METHODS The heart rate of a cohort of 173 patients from Fujita Health University Hospital in Japan was analyzed with the scattering transform. First, p-values of Wilcoxon rank sum tests were used to identify scattering coefficients achieving significant (univariate) discrimination between patients with and without IS. Second, a multivariate procedure for feature selection and classification, the Sparse Support Vector Machine (S-SVM), was applied to predict IS. RESULTS Groups of scattering coefficients, located at several time-scales, were identified as significantly higher (p-value < 0.05) in patients who developed IS than in those who did not. Though the overall predictive power of these indices remained moderate (around 60 %), it was found to be much higher when analysis was restricted to patients not taking antithrombotic treatment (around 80 %). Further, S-SVM showed that multivariate classification improves IS prediction, and also indicated that coefficients involved in classification differ for patients with and without antithrombotic treatment. CONCLUSIONS Scattering coefficients were found to play a significant role in predicting IS, notably for patients not receiving antithrombotic treatment. S-SVM improves IS detection performance and also provides insight on which features are important. Notably, it shows that AF patients not taking antithrombotic treatment are characterized by a slow modulation of RR dynamics in the ULF range and a faster modulation in the HF range. These modulations are significantly decreased in patients with IS, and hence have a good discriminant ability.

Intrapartum Fetal Heart Rate Classification: Cross-Database Evaluation

Fetal Heart Rate (FHR) provides obstetricians with essential information about fetal reactions to stress events during delivery. Early detection of fetal acidosis, enabling timely interventions and prevention of adverse consequences of acidosis for fetuses, remains a challenging task. In particular, the use of different, proprietary and small databases in various published works hinders meaningful comparisons of achieved results. This work relies on the the use of two independent databases in order to asses relevantly acidosis detection performance and to address important issues of knowledge transfer (features, classification model) from one database to the other. Using a large set of features, supervised classification is performed with state-of-the-art sparse support vector machines. It shows that selected features and classification performance are consistent for both databases. Further it quantifies the level of generalization of the achieved results, by making use of one database for learning and the other one for testing.

p-leader Multifractal Analysis and Sparse SVM for Intrapartum Fetal Acidosis Detection

Interpretation and analysis of intrapartum fetal heart rate, enabling early detection of fetal acidosis, remains a challenging signal processing task. Among the many strategies that were used to tackle this problem, scale-invariance and multifractal analysis stand out. Recently, a new and promising variant of multifractal analysis, based on p-leaders, has been proposed. In this contribution, we use sparse support vector machines applied to p-leader multifractal features with a double aim: Assessment of the features actually contributing to classification; Assessment of the contribution of non linear features (as opposed to linear ones) to classification performance. We observe and interpret that the classification rate improves when small values of the tunable parameter p are used.