Wavelet methods in statistics: Some recent developments and their applications

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

Wavelet regression estimation in nonparametric mixed effect models

AbstractWe show that a nonparametric estimator of a regression function, obtained as solution of a specific regularization problem is the best linear unbiased predictor in some nonparametric mixed effect model. Since this estimator is intractable from a numerical point of view, we propose a tight approximation of it easy and fast to implement. This second estimator achieves the usual optimal rate of convergence of the mean integrated squared error over a Sobolev class both for equispaced and nonequispaced design. Numerical experiments are presented both on simulated and ERP real data

Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging

In the present paper we consider the problem of Laplace deconvolution with noisy discrete non-equally spaced observations on a finite time interval. We propose a new method for Laplace deconvolution which is based on expansions of the convolution kernel, the unknown function and the observed signal over Laguerre functions basis (which acts as a surrogate eigenfunction basis of the Laplace convolution operator) using regression setting. The expansion results in a small system of linear equations with the matrix of the system being triangular and Toeplitz. Due to this triangular structure, there is a common number $m$ of terms in the function expansions to control, which is realized via complexity penalty. The advantage of this methodology is that it leads to very fast computations, produces no boundary effects due to extension at zero and cut-off at $T$ and provides an estimator with the risk within a logarithmic factor of the oracle risk. We emphasize that, in the present paper, we consider the true observational model with possibly nonequispaced observations which are available on a finite interval of length $T$ which appears in many different contexts, and account for the bias associated with this model (which is not present when $T\rightarrow\infty$). The study is motivated by perfusion imaging using a short injection of contrast agent, a procedure which is applied for medical assessment of micro-circulation within tissues such as cancerous tumors. Presence of a tuning parameter $a$ allows to choose the most advantageous time units, so that both the kernel and the unknown right hand side of the equation are well represented for the deconvolution. The methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast, efficient, accurate, usable from a practical point of view and very competitive.Comment: 36 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1207.223

Wavelet Reconstruction of Nonuniformly Sampled Signals

For the reconstruction of a nonuniformly sampled signal based on its noisy observations, we propose a level dependent l1 penalized wavelet reconstruction method. The LARS/Lasso algorithm is applied to solve the Lasso problem. The data adaptive choice of the regularization parameters is based on the AIC and the degrees of freedom is estimated by the number of nonzero elements in the Lasso solution. Simulation results conducted on some commonly used 1_D test signals illustrate that the proposed method possesses good empirical properties

Elastic-Net Regularization in Learning Theory

Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie for the selection of groups of correlated variables. To investigate on the statistical properties of this scheme and in particular on its consistency properties, we set up a suitable mathematical framework. Our setting is random-design regression where we allow the response variable to be vector-valued and we consider prediction functions which are linear combination of elements ({\em features}) in an infinite-dimensional dictionary. Under the assumption that the regression function admits a sparse representation on the dictionary, we prove that there exists a particular ``{\em elastic-net representation}'' of the regression function such that, if the number of data increases, the elastic-net estimator is consistent not only for prediction but also for variable/feature selection. Our results include finite-sample bounds and an adaptive scheme to select the regularization parameter. Moreover, using convex analysis tools, we derive an iterative thresholding algorithm for computing the elastic-net solution which is different from the optimization procedure originally proposed by Zou and HastieComment: 32 pages, 3 figure

Regression in random design and Bayesian warped wavelets estimators

In this paper we deal with the regression problem in a random design setting. We investigate asymptotic optimality under minimax point of view of various Bayesian rules based on warped wavelets and show that they nearly attain optimal minimax rates of convergence over the Besov smoothness class considered. Warped wavelets have been introduced recently, they offer very good computable and easy-to-implement properties while being well adapted to the statistical problem at hand. We particularly put emphasis on Bayesian rules leaning on small and large variance Gaussian priors and discuss their simulation performances comparing them with a hard thresholding procedure

Nonlinear Structural Functional Models

A common objective in functional data analyses is the registration of data curves and estimation of the locations of their salient structures, such as spikes or local extrema. Existing methods separate curve modeling and structure estimation into disjoint steps, optimize different criteria for estimation, or recast the problem into the testing framework. Moreover, curve registration is often implemented in a pre-processing step. The aim of this dissertation is to ameliorate the shortcomings of existing methods through the development of unified nonlinear modeling procedures for the analysis of structural functional data. A general model-based framework is proposed to unify registration and estimation of curves and their structures. In particular, this work focuses on three specific research problems. First, a Sparse Semiparametric Nonlinear Model (SSNM) is proposed to jointly register curves, perform model selection, and estimate the features of sparsely-structured functional data. The SSNM is fitted to chromatographic data from a study of the composition of Chinese rhubarb. Next, the SSNM is extended to the nonlinear mixed effects setting to enable the comparison of sparse structures across group-averaged curves. The model is utilized to compare compositions of medicinal herbs collected from two groups of production sites. Finally, a Piecewise Monotonic B-spline Model (PMBM) is proposed to estimate the locations of local extrema in a curve. The PMBM is applied to MRI data from a study of gray matter growth in the brain