Anestis Antoniadis

Wavelets and Statistics

Thresholding of Wavelet Coefficients as Multiple Hypotheses Testing Procedure.- Wavelets, spectrum analysis and 1/f processes.- Variance Function Estimation in Regression by Wavelet Methods.- Locally Self Similar Gaussian Processes.- WaveLab and Reproducible Research.- Extrema Reconstructions and Spline Smoothing: Variations on an Algorithm of Mallat & Zhong..- Identification of Chirps with Continuous Wavelet Transform.- Nonlinear Approximation of Stochastic Processes.- Translation-Invariant De-Noising.- Estimating Wavelet Coefficients.- Nonparametric Supervised Image Segmentation by Energy Minimization using Wavelets.- On the Statistics of Best Bases Criteria.- Discretized Wavelet Density Estimators for Continuous Time Stochastic Processes.- Wavelets and Markov Random Fields in a Bayesian Framework.- Micronde : a Matlab Wavelet Toolbox for Signals and Images.- Wavelet Function Estimation using Cross-Validation.- The Stationary Wavelet Transform and some Statistical Applications.- Wavelet Thresholding: Beyond the Gaussian I.I.D. Situation.- L2(0,1) Weak Convergence of the Empirical Process for Dependent Variables.- Top-Down and Bottom-Up Tree Search Algorithms for Selecting Bases in Wavelet Packet Transforms.- WavBox 4: A Software Toolbox for Wavelet Transforms and Adaptive Wavelet Packet Decompositions.- Using Wavelets for Classifying Human in vivo Magnetic Resonance.- Adaptive Density Estimation.- Wavelets and Regression Analysis.- Readers Guide.

Relation between heart rate variability and training load in middle-distance runners

PURPOSE Monitoring physical performance is of major importance in competitive sports. Indices commonly used, like resting heart rate, VO2max, and hormones, cannot be easily used because of difficulties in routine use, of variations too small to be reliable, or of technical challenges in acquiring the data. METHODS We chose to assess autonomic nervous system activity using heart rate variability in seven middle-distance runners, aged 24.6 +/- 4.8 yr, during their usual training cycle composed of 3 wk of heavy training periods, followed by a relative resting week. The electrocardiogram was recorded overnight twice a week and temporal and frequency indices of heart rate variability, using Fourier and Wavelet transforms, were calculated. Daily training loads and fatigue sensations were estimated with a questionnaire. Similar recordings were performed in a sedentary control group. RESULTS The results demonstrated a significant and progressive decrease in parasympathetic indices of up to -41% (P < 0.05) during the 3 wk of heavy training, followed by a significant increase during the relative resting week of up to +46% (P < 0.05). The indices of sympathetic activity followed the opposite trend, first up to +31% and then -24% (P < 0.05), respectively. The percentage increasing mean nocturnal heart rate variation remained below 12% (P < 0.05). There was no significant variation in the control group. CONCLUSION This study confirmed that heavy training shifted the cardiac autonomic balance toward a predominance of the sympathetic over the parasympathetic drive. When recorded during the night, heart rate variability appeared to be a better tool than resting heart rate to evaluate cumulated physical fatigue, as it magnified the induced changes in autonomic nervous system activity. These results could be of interest for optimizing individual training profiles.

Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on the notion of autoregressive Hilbert processes that represent a generalization of the classical autoregressive processes to random variables with values in a Hilbert space. A careful analysis reveals, in particular, that this approach is related to the theory of function estimation in linear ill-posed inverse problems. In the deterministic literature, such problems are usually solved by suitable regularization techniques. We describe some recent approaches from the deterministic literature that can be adapted to obtain fast and feasible predictions. For large sample sizes, however, these approaches are not computationally efficient.With this in mind, we propose three linear wavelet methods to efficiently address the aforementioned prediction problem. We present regularization techniques for the sample paths of the stochastic process and obtain consistency results of the resulting prediction estimators. We illustrate the performance of the proposed methods in finite sample situations by means of a real-life data example which concerns with the prediction of the entire annual cycle of climatological El Nino-Southern Oscillation time series 1 year ahead. We also compare the resulting predictions with those obtained by other methods available in the literature, in particular with a smoothing spline interpolation method and with a SARIMA model.

Density and hazard rate estimation for right-censored data by using wavelet methods

This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right-censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators have pointwise and global mean-square consistency, obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show that these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its flexibility, provides a new and interesting interpretation.

Estimation and inference in functional mixed-effects models

Functional mixed-effects models are very useful in analyzing functional data. A general functional mixed-effects model that inherits the flexibility of linear mixed-effects models in handling complex designs and correlation structures is considered. A wavelet decomposition approach is used to model both fixed-effects and random-effects in the same functional space, meaning that the population-average curve and the subject-specific curves have the same smoothness property. A linear mixed-effects representation is then obtained that is used for estimation and inference in the general functional mixed-effects model. Adapting recent methodologies in linear mixed-effects and nonparametric regression models, hypothesis testing procedures for both fixed-effects and random-effects are provided. Using classical linear mixed-effects estimation techniques, the linear mixed-effects representation is also used to obtain wavelet-based estimates for both fixed-effects and random-effects in the general functional mixed-effects model. The usefulness of the proposed estimation and hypothesis testing procedures is illustrated by means of a small simulation study and a real-life dataset arising from physiology.

Dimension reduction in functional regression with applications

Two dimensional reduction regression methods to predict a scalar response from a discretized sample path of a continuous time covariate process are presented. The methods take into account the functional nature of the predictor and are both based on appropriate wavelet decompositions. Using such decompositions, prediction methods are devised that are similar to minimum average variance estimation (MAVE) or functional sliced inverse regression (FSIR). Their practical implementation is described, together with their application both to simulated and on real data analyzing three calibration examples of near infrared spectra.

Wavelet regression for random or irregular design

In this paper, wavelet regression estimators are introduced, both in the random and the irregular design cases and without the restriction that the sample size is a power of two. A fast computational algorithm for approximating the empirical counterpart of the scaling and wavelet coefficients, is developed. The convergence rate of the estimator is established. The method is illustrated by some simulations and by a real example.

Cardiac Interbeat Interval Increment for the Identification of Obstructive Sleep Apnea

ROCHE, F., et al.: Cardiac Interbeat Interval Increment for the Identification of Obstructive Sleep Apnea. The prevalence of obstructive sleep apnea syndrome (OSAS) is high in developed countries but its diagnosis is costly. Based on physiological evidence, the frequency component of heart rate variability (HRV) was evaluated as a simple and inexpensive diagnostic tool in OSAS. The predictive accuracy of frequency‐domain HRV variables obtained from 24‐hour ECG Holter monitoring (the power spectral density of the interbeat interval increment of very low frequencies, “VLFIpsd,” and its percentage over the total power spectral density, “%VLFI”), and of established time‐domain HRV variables were analyzed by comparison with respiratory disturbances indexes assessed by complete polysomnography in 124 consecutive patients (98 men aged 53.8 ± 11.2 years) with clinically suspected OSAS. OSAS was present in 54 (43.5%) patients according to standard criteria. Using receiver operating characteristic curve analysis, two of the three most powerful predictors were frequency‐domain variables: %VLFI (W = 0.80, P < 0.0001), and VLFIpsd (W = 0.79, P < 0.0001). Using a multiple logistic regression analysis, %VLFI was the most strongly associated with diseased status (adjusted OR: 8.4; 95% CI: 3.4–19.5). Using an appropriate threshold, %VLFI demonstrated a diagnostic sensitivity of 87%. A 3‐month continuous positive airway pressure treatment significantly improved the same parameter. Frequency‐domain analysis of the interbeat interval increment appears as a powerful tool for OSAS diagnosis and follow‐up. The simplicity of its analysis and of its use makes of it a well‐suited variable for mass screening of OSAS patients.

CLUSTERING FUNCTIONAL DATA USING WAVELETS

We present two methods for detecting patterns and clusters in high dimensional time-dependent functional data. Our methods are based on wavelet-based similarity measures, since wavelets are well suited for identifying highly discriminant local time and scale features. The multiresolution aspect of the wavelet transform provides a time-scale decomposition of the signals allowing to visualize and to cluster the functional data into homogeneous groups. For each input function, through its empirical orthogonal wavelet transform the first method uses the distribution of energy across scales generate a handy number of features that can be sufficient to still make the signals well distinguishable. Our new similarity measure combined with an efficient feature selection technique in the wavelet domain is then used within more or less classical clustering algorithms to effectively differentiate among high dimensional populations. The second method uses dissimilarity measures between the whole time-scale representations and are based on wavelet-coherence tools. The clustering is then performed using a k-centroid algorithm starting from these dissimilarities. Practical performance of these methods that jointly designs both the feature selection in the wavelet domain and the classification distance is demonstrated through simulations as well as daily profiles of the French electricity power demand.

Curvelet-Based Snake for Multiscale Detection and Tracking of Geophysical Fluids

Detection and target tracking have an application to many scientific problems. The approach developed in this paper is motivated by the applications of detection and tracking characteristic deformable structures in geophysical fluids. We develop an integrated detection and tracking method of geophysical fluids based on a discrete curvelet representation of the information characterizing the targets. Curvelets are in some sense geometric wavelets, allowing an optimal sparse representation of two-dimensional piecewise continuous objects with C 2-singularities. The proposed approach first identifies a consistent vortex by a curvelet-based gradient-vector-flow snake and then establishes the motion correspondence of the snaxels between successive time frames by a constructed so-called semi-T or comp-T multiscale motion-estimation method based on the geometric wavelets. Furthermore, a combination of total-variation regularization and cycle-spinning techniques effectively removes false matches and improves significantly the estimation. Numerical experiments at each stage demonstrate the performance of the proposed tracking methodology for temporal oceanographic satellite image sequences corrupted by noise, with weak edges and submitted to large deformations, in comparison to conventional methods