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Short-Term Load Forecasting (STLF) is a fundamental instrument in the efficient operational management and planning of electric utilities. Emerging smart grid technologies pose new challenges and opportunities. Although load forecasting at the aggregate level has been extensively studied, electrical load forecasting at fine-grained geographical scales of households is more challenging. Among existing approaches, semi-parametric generalized additive models (GAM) have been increasingly popular due to their accuracy, flexibility, and interpretability. Their applicability is justified when forecasting is addressed at higher levels of aggregation, since the aggregated load pattern contains relatively smooth additive components. High resolution data are highly volatile, forecasting the average load using GAM models with smooth components does not provide meaningful information about the future demand. Instead, we need to incorporate irregular and volatile effects to enhance the forecast accuracy. We focus on the analysis of such hybrid additive models applied on smart meters data and show that it leads to improvement of the forecasting performances of classical additive models at low aggregation levels. (C) 2020 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

We congratulate the authors for a very interesting and timely contribution to an important problem: using nonparametric functional data analysis methods for studying spatio-temporal observations with the aim to identify subregions in a spatial domain sharing similar patterns along time. The authors develop an efficient methodology to perform dimensional reduction of spatially dependent functional data based on treelet decompositions and an appropriate bagging Voronoi strategy that allows them to take into account the spatial dependency in the analysed data. Treelets were first introduced in Lee and Nadler (2007) and Lee et al. (2008) as a multi-scale basis that extends wavelets to unordered data. The method is fully adaptive. It returns orthonormal basis functions supported on nested clusters in a hierarchical tree. Unlike other hierarchical methods, the basis and the tree structure are computed simultaneously, and both reflect the internal structure of the data. Combination of treelets with a bagging Voronoi strategy for identifying subregions with similar patterns along time introduces numerical challenges that are efficiently tackled by the authors.

International audience; In this paper we focus on nonparametric estimation of a constrained regression function using penalized wavelet regression techniques. This results into a convex optimization problem under linear constraints. Necessary and sufficient conditions for existence of a unique solution are discussed. The estimator is easily obtained via the dual formulation of the optimization problem. In particular we investigate a penalized wavelet monotone regression estimator. We establish the rate of convergence of this estimator, and illustrate its finite sample performance via a simulation study. We also compare its performance with that of a recently proposed constrained estimator. An illustration to some real data is given.

International audience; 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

International audience; We propose a parametric wavelet thresholding procedure for estimation in the 'function plus independent, identically distributed Gaussian noise' model. To reflect the decreasing sparsity of wavelet coefficients from finer to coarser scales, our thresholds also decrease. They retain the noise-free reconstruction property while being lower than the universal threshold, and are jointly parameterised by a single scalar parameter. We show that our estimator achieves near-optimal risk rates for the usual range of Besov spaces. We propose a crossvalidation technique for choosing the parameter of our procedure. A simulation study demonstrates very good performance of our estimator compared to other state-of-the-art techniques. We discuss an extension to non-Gaussian noise.

We consider the testing problem in a fixed-effects functional analysis of variance model. We test the null hypotheses that the functional main effects and the functional interactions are zeros against the composite nonparametric alternative hypotheses that they are separated away from zero in L2-norm and also possess some smoothness properties. We adapt the optimal (minimax) hypothesis testing procedures for testing a zero signal in a Gaussian "signal plus noise" model to derive optimal (minimax) non-adaptive and adaptive hypothesis testing procedures for the functional main effects and the functional interactions. The corresponding tests are based on the empirical wavelet coefficients of the data. Wavelet decompositions allow one to characterize different types of smoothness conditions assumed on the response function by means of its wavelet coefficients for a wide range of function classes. In order to shed some light on the theoretical results obtained, we carry out a simulation study to examine the finite sample performance of the proposed functional hypothesis testing procedures. As an illustration, we also apply these tests to a real-life data example arising from physiology. Concluding remarks and hints for possible extensions of the proposed methodology are also given.

Wavelet shrinkage and thresholding methods constitute a powerful way to carry out signal denoising, especially when the underlying signal has a sparse wavelet representation. They are computationally fast, and automatically adapt to the smoothness of the signal to be estimated. Nearly minimax properties for simple threshold estimators over a large class of function spaces and for a wide range of loss functions were established in a series of papers by Donoho and Johnstone. The notion behind these wavelet methods is that the unknown function is well approximated by a function with a relatively small proportion of nonzero wavelet coefficients. In this paper, we propose a framework in which this notion of sparseness can be naturally expressed by a Bayesian model for the wavelet coefficients of the underlying signal. Our Bayesian formulation is grounded on the empirical observation that the wavelet coefficients can be summarized adequately by exponential power prior distributions and allows us to establish close connections between wavelet thresholding techniques and Maximum A Posteriori estimation for two classes of noise distributions including heavy-tailed noises. We prove that a great variety of thresholding rules are derived from these MAP criteria. Simulation examples are presented to substantiate the proposed approach.

The objective of this paper is to contribute to the methodology available for dealing with the detection and the estimation of the location of discontinuities in one-dimensional piecewise smooth regression functions observed in white Gaussian noise over an interval. Our approach is nonparametric in nature because the unknown function is not assumed to have any specific form. Our method relies upon a wavelet analysis of the observed signal and belongs to the class of "indirect" methods, where one detects and locates the change points prior to fitting the curve, and then uses ones favorite function estimation technique on each segment to recover the curve. We show that, provided discontinuities can be detected and located with sufficient accuracy, detection followed by wavelet smoothing enjoys optimal rates of convergence.

In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hard-thresholding and soft-thresholding estimators of Donoho and Johnstone are specific members of nonlinear regularized wavelet estimators. They correspond to the lower and upper envelopes of a class of the penalized least squares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax. To efficiently solve penalized least squares problems, nonlinear regularized Sobolev interpolators (NRSI) are proposed as initial estimators, wh...

The authors consider the problem of estimating the density of the radii of spheres in a medium, based on their observed random cross-sections. This problem is known as Wicksell's corpuscle problem. The authors first convert Wicksell's integral equation to a form suitable for the application of thresholding wavelet methods to solve ill-posed integral equations, given noisy data. They then derive the asymptotic properties of their estimators and compare them with other methods available via a Monte Carlo simulation study. They also illustrate their approach with some real data.

It is well-known that multivariate curve estimation suffers from the “curse of dimensionality.” However, reasonable estimators are possible, even in several dimensions, under appropriate restrictions on the complexity of the curve. In the present paper we explore how much appropriate wavelet estimators can exploit a typical restriction on the curve such as additivity. We first propose an adaptive and simultaneous estimation procedure for all additive components in additive regression models and discuss rate of convergence results and data-dependent truncation rules for wavelet series estimators. To speed up computation we then introduce a wavelet version of functional ANOVA algorithm for additive regression models and propose a regularization algorithm which guarantees an adaptive solution to the multivariate estimation problem. Some simulations indicate that wavelets methods complement nicely the existing methodology for nonparametric multivariate curve estimation.

Summary 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.

In this paper we discuss how to use wavelet decompositions to select a regression model. The methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients. Consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector. We then apply the selection rule to specific problems, including testing for pure white noise. The power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression.

The field of nonparametric function estimation has broadened its appeal in recent years with an array of new tools for statistical analysis. In particular, theoretical and applied research on the field of wavelets has had noticeable influence on statistical topics such as nonparametric regression, nonparametric density estimation, nonparametric discrimination and many other related topics. This is a survey article that attempts to synthetize a broad variety of work on wavelets in statistics and includes some recent developments in nonparametric curve estimation that have been omitted from review articles and books on the subject. After a short introduction to wavelet theory, wavelets are treated in the familiar context of estimation of «smooth» functions. Both «linear» and «nonlinear» wavelet estimation methods are discussed and cross-validation methods for choosing the smoothing parameters are addressed. Finally, some areas of related research are mentioned, such as hypothesis testing, model selection, hazard rate estimation for censored data, and nonparametric change-point problems. The closing section formulates some promising research directions relating to wavelets in statistics.

The theory of wavelets is a developing branch of mathematics with a wide range of potential applications. Compactly supported wavelets are particularly interesting because of their natural ability to represent data with intrinsically local properties. They are useful for the detection of edges and singularities in image and sound analysis and for data compression. But most of the wavelet-based procedures currently available do not explicitly account for the presence of noise in the data. A discussion of how this can be done in the setting of some simple nonparametric curve estimation problems is given. Wavelet analogies of some familiar kernel and orthogonal series estimators are introduced, and their finite sample and asymptotic properties are studied. We discover that there is a fundamental instability in the asymptotic variance of wavelet estimators caused by the lack of translation invariance of the wavelet transform. This is related to the properties of certain lacunary sequences. The practi...

Rapport de recherche publié : hal-00942684; 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.

Increasingly, scientific studies yield functional image data, in which the observed data consist of sets of curves recorded on the pixels of the image. Examples include temporal brain response intensities measured by fMRI and NMR frequency spectra measured at each pixel. This article presents a new methodology for improving the characterization of pixels in functional imaging, formulated as a spatial curve clustering problem. Our method operates on curves as a unit. It is nonparametric and involves multiple stages: (i) wavelet thresholding, aggregation, and Neyman truncation to effectively reduce dimensionality; (ii) clustering based on an extended EM algorithm; and (iii) multiscale penalized dyadic partitioning to create a spatial segmentation. We motivate the different stages with theoretical considerations and arguments, and illustrate the overall procedure on simulated and real datasets. Our method appears to offer substantial improvements over monoscale pixel-wise methods. An Appendix which gives some theoretical justifications of the methodology, computer code, documentation and dataset are available in the online supplements.

21 pages; International audience; The objective of this paper is to contribute to the methodology available for extracting and analyzing signal content from protein mass spectrometry data. Data from Matrix-Assisted Laser Desorption Ionization Time-of-Flight (MALDI-TOF) or Surface-Enhanced Laser Desorption and Ionization Time-Of-Flight (SELDI-TOF) spectra require considerable signal pre-processing such as noise removal and baseline level error correction. After removing the noise by an invariant wavelet transform, we develop a background correction method based on penalized spline quantile regression and apply it to MALDI-TOF spectra. The results show that the wavelet transform technique combined with nonparametric quantile regression can handle all kinds of background and low signal-to-background ratio spectra; it requires no prior knowledge about the spectra composition, no selection of suitable background correction points, and no mathematical assumption of the background distribution. We further present a multi-scale based novel spectra alignment methodology useful in a functional analysis of variance method for identifying proteins that are differentially expressed between different type tissues. Our approaches are compared with several existing approaches in the literature and are tested on simulated and real data. The results indicate that the proposed schemes enable accurate diagnosis based on the over-expression of a small number of identified proteins with high sensitivity.

The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in wavelet applications to statistics and to the analysis of experimental data, with many successes in the efficient analysis, processing, and compression of noisy signals and images. This is a selective review article that attempts to synthesize some recent work on ``nonlinear'' wavelet methods in nonparametric curve estimation and their role on a variety of applications. After a short introduction to wavelet theory, we discuss in detail several wavelet shrinkage and wavelet thresholding estimators, scattered in the literature and developed, under more or less standard settings, for density estimation from i.i.d. observations or to denoise data modeled as observations of a signal with additive noise. Most of these methods are fitted into the general concept of regularization with appropriately chosen penalty functions. A narrow range of applications in major areas of statistics is also discussed such as partial linear regression models and functional index models. The usefulness of all these methods are illustrated by means of simulations and practical examples., Comment: Published in at http://dx.doi.org/10.1214/07-SS014 the Statistics Surveys (http://www.i-journals.org/ss/) by the Institute of Mathematical Statistics (http://www.imstat.org)

International audience; 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.

International audience; The paper considers regression problems with univariate design points. The design points are irregular and no assumptions on their distribution are imposed. The regression function is retrieved by a wavelet based reproducing kernel Hilbert space (RKHS) technique with the penalty equal to the sum of blockwise RKHS norms. In order to simplify numerical optimization, the problem is replaced by an equivalent quadratic minimization problem with an additional penalty term. The computational algorithm is described in detail and is implemented with both the sets of simulated and real data. Comparison with existing methods showed that the technique suggested in the paper does not oversmooth the function and is superior in terms of the mean squared error. It is also demonstrated that under additional assumptions on design points the method achieves asymptotic optimality in a wide range of Besov spaces.

International audience; Stable distributions are characterized by four parameters which can be estimated via a number of methods, and although approximate maximum likelihood estimation techniques have been proposed, they are computationally intensive and difficult to implement. This article describes a fast, wavelet-based, regression-type method for estimating the parameters of a stable distribution. Fourier domain representations, combined with a wavelet multiresolution approach, are shown to be effective and highly efficient tools for inference in stable law families. Our procedures are illustrated and compared with other estimation methods using simulated data, and an application to a real data example is explored. One novel aspect of this work is that here wavelets are being used to solve a parametric problem, rather than a nonparametric one, which is the more typical context in wavelet applications.

In this paper we focus on nonparametric estimators in inverse problems for Poisson processes involving the use of wavelet decompositions. Adopting an adaptive wavelet Galerkin discretization, we find that our method combines the well-known theoretical advantages of wavelet--vaguelette decompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution, together with the remarkably simple closed-form expressions of Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann. Statist. 19 (1991) 1347--1369] to the context of log-intensity functions approximated by wavelet series with the use of the Kullback--Leibler distance between two point processes, we also present an asymptotic analysis of convergence rates that justifies our approach. In order to shed some light on the theoretical results obtained and to examine the accuracy of our estimates in finite samples, we illustrate our method by the analysis of some simulated examples., Comment: Published at http://dx.doi.org/10.1214/009053606000000687 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

International audience; The objective of this paper is to contribute to the methodology available for dealing with a very common statistical problem, the estimation of the variance function in heteroscedastic multiple linear regression problems. The variance function is recovered by means of a smoothing nonparametric method, based on wavelet decompositions. The proposed method does not require preliminary or simultaneous estimation of the mean function. The resulting wavelet estimator is shown to be consistent, and is used to improve the estimation of the mean function itself. The method is illustrated with real and simulated data.

DUVERNEY, D., et al.: High Accuracy of Automatic Detection of Atrial Fibrillation Using Wavelet Transform of Heart Rate Intervals. Permanent and paroxysmal AF is a risk factor for the occurrence and the recurrence of stroke, which can occur as its first manifestation. However, its automatic identification is still unsatisfactory. In this study, a new mathematical approach was evaluated to automate AF identification. A derivation set of 30 24-hour Holter recordings, 15 with chronic AF (CAF) and 15 with sinus rhythm (SR), allowed the authors to establish specific RR variability characteristics using wavelet and fractal analysis. Then, a validation set of 50 subjects was studied using these criteria, 19 with CAF, 16 with SR, and 15 with paroxysmal AF (PAF); and each QRS was classified as true or false sinus or AF beat. In the SR group, specificity reached 99.9%; in the CAF group, sensitivity reached 99.2%; in the PAF group, sensitivity reached 96.1%, and specificity 92.6%. However, classification on a patient basis provided a sensitivity of 100%. This new approach showed a high sensitivity and a high specificity for automatic AF detection, and could be used in screening for AF in large populations at risk.

International audience; Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced.

Wavelet-based denoising techniques are well suited to estimate spatially inhomogeneous signals. Waveshrink (Donoho and Johnstone) assumes independent Gaussian errors and equispaced sampling of the signal. Various articles have relaxed some of these assumptions, but a systematic generalization to distributions such as Poisson, binomial, or Bernoulli is missing. We consider a unifying l1-penalized likelihood approach to regularize the maximum likelihood estimation by adding an l1 penalty of the wavelet coefficients. Our approach works for all types of wavelets and for a range of noise distributions. We develop both an algorithm to solve the estimation problem and rules to select the smoothing parameter automatically. In particular, using results from Poisson processes, we give an explicit formula for the universal smoothing parameter to denoise Poisson measurements. Simulations show that the procedure is an improvement over other methods. An astronomy example is given.

International audience; Chirps are signals (or sums of signals) that may be characterized by a local (i.e. time-dependent) amplitude and a local frequency. Time-frequency representations such as wavelet representations are well adapted to the characterization problem of such chirps. Ridges in the modulus of the transform determine regions in the transform domain with a high concentration of energy, and are regarded as natural candidates for the characterization and the reconstruction of the original signal. A couple of algorithmic procedures for the estimation of ridges from the modulus of the (continuous) wavelet transform of one-dimensional signals are described, together with a new reconstruction procedure, using only information of the restriction of the wavelet transform to a sample of points from the ridge. This provides with a very efficient way to code the information contained in the signal.