Papers by Anestis Antoniadis
Journal of Multivariate Analysis
We consider the prediction problem of a continuous-time stochastic process on an entire time-inte... more 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.
Journal of Multivariate Analysis, 2003
We consider the prediction problem of a continuous-time stochastic process on an entire time-inte... more 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.
Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatial... more 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.
Biometrika, 2001
We propose a wavelet shrinkage methodology for univariate natural exponential families with quadr... more We propose a wavelet shrinkage methodology for univariate natural exponential families with quadratic variance functions, covering the Gaussian, Poisson, gamma, binomial, negative binomial and generalised hyperbolic secant distributions. Simulation studies for Poisson and binomial data are used to illustrate the usefulness of the proposed methodology, and comparisons are made with other methods available in the literature. We also present applications to datasets arising from high-energy astrophysics and from epidemiology.
Journal of The American Statistical Association, 2001
In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametric... more 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-thresholdin g and soft-thresholding estimators of Donoho and Johnstone are speci c 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 ef ciently solve penalized least squares problems, nonlinear regularized Sobolev interpolators (NRS I) are proposed as initial estimators, which are shown to have good sampling properties. The NRS I is further ameliorated by regularized one-step estimators, which are the one-step estimators of the penalized least squares problems using the NRS I as initial estimators. The graduated nonconvexit y algorithm is also introduced to handle penalized least squares problems. The newly introduced approache s are illustrated by a few numerical examples.
Regularization of Wavelets Approximations
In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametric... more In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametricregression functions when sampling points are not uniformly spaced. The approach canapply readily to many other statistical contexts. Various new penalty functions are proposed.The hard-thresholding and soft-thresholding estimators of Donoho and Johnstone (1994) arespecic members of nonlinear regularized wavelet estimators. They correspond to the lowerand upper bound of a
Journal of Neuroscience Methods, 2003
Segmentation (tissue classification) of medical images obtained from a magnetic resonance (MR) sy... more Segmentation (tissue classification) of medical images obtained from a magnetic resonance (MR) system is a primary step in most applications of medical image post-processing. This paper describes nonparametric discriminant analysis methods to segment multispectral MR images of the brain. Starting from routinely available spin-lattice relaxation time, spin-spin relaxation time, and proton density weighted images (T 1 w, T 2 w, P D w), the proposed family of statistical methods is based on: (i) a transform of the images into components that are statistically independent from each other; (ii) a nonparametric estimate of probability density functions of each tissue starting from a training set; (iii) a classic Bayes 0-1 classification rule. Experiments based on a computer built brain phantom (brainweb) and on eight real patient data sets are shown. A comparison with parametric discriminant analysis is also reported. The capability of nonparametric discriminant analysis in improving brain tissue classification of parametric methods is demonstrated. Finally, an assessment of the role of multispectrality in classifying brain tissues is discussed.
Computational Statistics & Data Analysis, 2006
Two dimensional reduction regression methods to predict a scalar response from a discretized samp... more 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.
INDEPENDENT COMPONENT DISCRIMINANT ANALYSIS
... Anestis Antoniadis and Gérard Grégoire LMC-IMAG, Université Joseph Fourier BP 53, 38041 Greno... more ... Anestis Antoniadis and Gérard Grégoire LMC-IMAG, Université Joseph Fourier BP 53, 38041 Grenoble Cedex 09 (France) E-mail (Antoniadis): anestis.antoniadis@ imag.fr E-mail (Grégoire): [email protected] Abstract. ...
This paper describes a wavelet method for the estimation of density and hazard rate functions fro... more 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 speci c 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 possess pointwise and global mean square consistency, obtain the best possible asymptotic MISE convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The rst 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 exibility, provides a new and interesting interpretation. R 0 f(t)g(t) dt. 7
Statistics & Probability Letters, 1997
Common wavelet-based methods for nonparametric regression estimation are difficult to apply when ... more Common wavelet-based methods for nonparametric regression estimation are difficult to apply when the design is random. This paper proposes a modification of the linear wavelet estimator, called the binned wavelet estimator leading to a fast C(n) method with asymptotic properties identical with those of linear wavelet estimators under a fixed equidistant design. (~) 1997 Elsevier Science B.V.
Journal of The American Statistical Association, 1994
The theory of wavelets is a developing branch of mathematics with a wide range of potential appli... more 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. However, 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 analogues 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 practical consequences of this instability are assessed.
Journal of Nonparametric Statistics, 2002
The objective of this paper is to contribute to the methodology available for dealing with the de... more 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.
Biometrika, 1997
In this paper we discuss how to use wavelet decompositions to select a regression model. The meth... more 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 nonparametnc regression.
The objective of this paper is to contribute to the methodology available for dealing with the de... more 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.
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Papers by Anestis Antoniadis