Wavelet Estimation Research Articles

Spatially adaptive density estimation by localised Haar projections

A partir d’un echantillon d’une loi de densite $f_{0}:\mathbb{R}\to [0,\infty)$, nous construisons des estimateurs par ondelettes de Haar de $f_{0}$, dont les niveaux de resolution varient et sont construits a partir de tests localises (comme dans l’article Lepski (Ann. Statist. 25 (1997) 927–947)). Nous montrons que ces estimateurs satisfont une inegalite oracle adaptive par rapport a la regularite potentiellement heterogene de $f_{0}$, simultanement pour tout point $x$ dans un intervalle donne, en norme infinie. Les constantes de seuillage utilisees dans les procedures de test peuvent etre choisies en pratique en supposant de maniere idealisee que la vraie densite est localement constante dans un voisinage du point $x$ considere, pratique que nous justifions par un argument de theorie de l’information.

On Simple Wavelet Estimators of Random Signals and Their Small-Sample Properties

In the paper we introduce certain nonparametric estimators of random signals based on the wavelet transform. We consider stochastic signals lying in white noise and extractions with wavelet denoising algorithms utilising the non-decimated discrete wavelet transform and the idea of wavelet scaling. We evaluate properties of these estimators via expansive computer simulations and partially also analytically. Our wavelet estimators of random signals have clear advantages over parametric maximum likelihood methods as far as computational issues are concerned, while at the same time they can compete with these methods in terms of precision of estimation in small samples. An illustrative example concerning smoothing survey data is also provided.

Viterbi sparse spike detection

Accurate interpretation of seismic traveltimes and amplitudes in the exploration and global scales is complicated by the band-limited nature of seismic data. We discovered a stochastic method to reduce a seismic waveform into a most probable constituent spike train. Model waveforms were constructed from a set of candidate spike trains convolved with a source wavelet estimate. For each model waveform, a profile hidden Markov model (HMM) was constructed to represent the waveform as a stochastic generative model with a linear topology corresponding to a sequence of samples. Each match state in the HMM represented a sample in the model waveform, in which the amplitude was represented by a Gaussian distribution. Insert and delete states allowed the underlying source wavelet to dilate or contract, accounting for nonstationarity in the seismic data and errors in the source wavelet estimate. The Gaussian distribution characterizing each sample’s amplitude accounted for random noise. The Viterbi algorithm was employed to simultaneously find the optimal nonlinear alignment between a model waveform and the seismic data and to assign a score to each candidate spike train. The most probable traveltimes and amplitudes were inferred from the alignments of the highest scoring models. The method required no implicit assumptions regarding the distribution of traveltimes and amplitudes; however, in practice, the solution set may be limited to mitigate the nonuniqueness of solutions and to reduce the computational effort. Our analyses found that the method can resolve closely spaced arrivals below traditional resolution limits and that traveltime estimates are robust in the presence of random noise and source wavelet errors. The method was particularly well suited to fine-scale interpretation problems such as thin bed interpretation, tying seismic images to well logs, and the analysis of anomalous waveforms in global seismology.

Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using Lepski et al. (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.

On rate-optimal nonparametric wavelet regression with long memory moving average errors

We consider the wavelet-based estimators of mean regression function with long memory moving average errors and investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, in the presence of long memory non-Gaussian moving average noise, wavelet estimators still achieve nearly optimal convergence rates and provide explicitly the extraordinary local adaptability. The theory is illustrated with some numerical examples.

Asymptotic properties of wavelet-based estimator in nonparametric regression model with weakly dependent processes

In this paper, we consider a nonparametric regression model with replicated observations based on the φ-mixing and the ρ-mixing error’s structures respectively, for exhibiting dependence among the units. The wavelet procedures are developed to estimate the regression function. Under suitable conditions, we obtain expansions for the bias and the variance of wavelet estimator, prove the moment consistency, the strong consistency, the strong convergence rate of it, and establish the asymptotic normality of wavelet estimator.

Online Semiparametric Identification of Lithium-Ion Batteries Using the Wavelet-Based Partially Linear Battery Model

Battery model identification is very important for reliable battery management as well as for battery system design process. The common problem in identifying battery models is how to determine the most appropriate mathematical model structure and parameterized coefficients based on the measured terminal voltage and current. This paper proposes a novel semiparametric approach using the wavelet-based partially linear battery model (PLBM) and a recursive penalized wavelet estimator for online battery model identification. Three main contributions are presented. First, the semiparametric PLBM is proposed to simulate the battery dynamics. Compared with conventional electrical models of a battery, the proposed PLBM is equipped with a semiparametric partially linear structure, which includes a parametric part (involving the linear equivalent circuit parameters) and a nonparametric part [involving the open-circuit voltage (OCV)]. Thus, even with little prior knowledge about the OCV, the PLBM can be identified using a semiparametric identification framework. Second, we model the nonparametric part of the PLBM using the truncated wavelet multiresolution analysis (MRA) expansion, which leads to a parsimonious model structure that is highly desirable for model identification; using this model, the PLBM could be represented in a linear-in-parameter manner. Finally, to exploit the sparsity of the wavelet MRA representation and allow for online implementation, a penalized wavelet estimator that uses a modified online cyclic coordinate descent algorithm is proposed to identify the PLBM in a recursive fashion. The simulation and experimental results demonstrate that the proposed PLBM with the corresponding identification algorithm can accurately simulate the dynamic behavior of a lithium-ion battery in the Federal Urban Driving Schedule tests.

Block‐threshold‐adapted Estimators via a Maxiset Approach

ABSTRACTWe study the maxiset performance of a large collection of block thresholding wavelet estimators, namely the horizontal block thresholding family. We provide sufficient conditions on the choices of rates and threshold values to ensure that the involved adaptive estimators obtain large maxisets. Moreover, we prove that any estimator of such a family reconstructs the Besov balls with a near‐minimax optimal rate that can be faster than the one of any separable thresholding estimator. Then, we identify, in particular cases, the best estimator of such a family, that is, the one associated with the largest maxiset. As a particularity of this paper, we propose a refined approach that models method‐dependent threshold values. By a series of simulation studies, we confirm the good performance of the best estimator by comparing it with the other members of its family.

Wavelet-Based Estimation of Anisotropic Spatiotemporal Long-Range Dependence

In this article, the estimation of spatiotemporal long-range dependence is formulated in the spectral wavelet domain. Sample information is provided by functional spectral data. Their high local singularity at the origin is captured by the wavelet transform. Weak consistency of the spectral wavelet estimators proposed is derived. Two functional estimation algorithms are implemented. A simulation study is developed to illustrate the efficiency of the computational methods derived. An approximation to the empirical convergence rate of the spectral wavelet periodogram is computed in some simulated examples.

On the Performance of Isotropic and Hyperbolic Wavelet Estimators

In this paper, we differentiate between isotropic and hyperbolic wavelet bases in the context of multivariate nonparametric function estimation. The study of the latter leads to new phenomena and non trivial extensions of univariate studies. In this context, we fi rst exhibit the limitations of isotropic wavelet estimators by proving that no isotropic estimator is able to guarantee the reconstruction of a function with anisotropy in an optimal or near optimal way. Second, we show that hyperbolic wavelet estimators are well suited to reconstruct anisotropic functions. In particular, for each considered estimator we focus on the rates at which it can reconstruct functions from anisotropic Besov spaces. We then compute the estimator's maxiset, this is the largest functional spaces over which its risk converges at these rates. Our results furnish novel arguments to understand the primordial role of sparsity and thresholding in multivariate contexts, notably by showing the exposure of linear methods to the curse of dimensionality. Moreover, we propose a block thresholding hyperbolic estimator and show its ability to estimate anisotropic functions at the optimal minimax rate and related, the remaining pertinence of information pooling in high dimensional settings.

Comparison of wavelet estimation methods

Wavelet estimation is a very important task in seismic data processing and analysis such as deterministic deconvolution, seismic-to-well tie, and seismic inversion, among others. We investigated the wavelets estimated from four different methods: (1) the wavelet estimated from the seafloor signal; (2) the wavelet estimated fully from well-log data; (3) the wavelet estimated using seismic and well-log data; and (4) the wavelet estimated from sparse-spike deconvolution. The wavelets estimated from 2-D seismic data using the four methods are quite comparable to one another. The results of the deconvolution and inversion of the 2-D seismic data using the four wavelets show that the wavelet estimated from the seafloor signal can be as effective as those estimated from the more rigorous methods.

A novel seismic wavelet estimation method

Abstract A novel seismic wavelet estimation scheme is proposed based on S transform and Least Squares (LS) method. The scheme can reconstruct the coherent components after estimating major wavelet even when the signal-to-noise ratio (SNR) is low. The main frequency and delay along the event should be estimated. The proposed scheme can estimate seismic source wavelet with minor distortion. It can be realized as follows: firstly uses an S transform to estimate the delay and coarse frequency information, secondly applies a filter designed in S transform domain, thirdly gets the data in time domain by applying an inverse S transform to the filtered data, fourthly uses a windowed FFT unit to get the accuracy frequency estimation, and fifthly reconstructs the seismic source wavelet after estimating the energy based on LS method in time domain. Furthermore, the space–time-variant characteristic of the source wavelet is taken into account. An iteration scheme is designed to estimate all components of wavelet. The simulation results show that the proposed method can estimate the source wavelet effectively when the SNR is higher than 0 dB.

Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises

Motivated by Lounici and Nickl's work (2011), this paper considers the problem of estimation of a densityfbased on an independent and identically distributed sampleY1,…,Ynfromg=f*φ. We show a wavelet optimal estimation for a density (function) over Besov ballBr,qs(L)andLprisk (1≤p<∞) in the presence of severely ill-posed noises. A wavelet linear estimation is firstly presented. Then, we prove a lower bound, which shows our wavelet estimator optimal. In other words, nonlinear wavelet estimations are not needed in that case. It turns out that our results extend some theorems of Pensky and Vidakovic (1999), as well as Fan and Koo (2002).

Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates

In the present paper we consider the problem of estimating a periodic $(r+1)$-dimensional function $f$ based on observations from its noisy convolution. We construct a wavelet estimator of $f$, derive minimax lower bounds for the $L^{2}$-risk when $f$ belongs to a Besov ball of mixed smoothness and demonstrate that the wavelet estimator is adaptive and asymptotically near-optimal within a logarithmic factor, in a wide range of Besov balls. We prove in particular that choosing this type of mixed smoothness leads to rates of convergence which are free of the “curse of dimensionality” and, hence, are higher than usual convergence rates when $r$ is large. The problem studied in the paper is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Indeed, we show that unless the function $f$ is very smooth in the direction of the profiles, very spatially inhomogeneous along the other direction and the number of profiles is very limited, the functional deconvolution solution has a much better precision compared to a combination of $M$ solutions of separate convolution equations. A limited simulation study in the case of $r=1$ confirms theoretical claims of the paper.

Numerical Solutions of Fractional Integrodifferential Equations of Bratu Type by Using CAS Wavelets

A numerical method based on the CAS wavelets is presented for the fractional integrodifferential equations of Bratu type. The CAS wavelets operational matrix of fractional order integration is derived. A truncated CAS wavelets series together with this operational matrix is utilized to reduce the fractional integrodifferential equations to a system of algebraic equations. The solution of this system gives the approximation solution for the truncated limited2k(2M+1). The convergence and error estimation of CAS wavelets are also given. Two examples are included to demonstrate the validity and applicability of the approach.

Wavelet deconvolution in a periodic setting with long-range dependent errors

In this paper, a hard thresholding wavelet estimator is constructed for a deconvolution model in a periodic setting that has long-range dependent noise. The estimation paradigm is based on a maxiset method that attains a near optimal rate of convergence for a variety of Lp loss functions and a wide variety of Besov spaces in the presence of strong dependence. The effect of long-range dependence is detrimental to the rate of convergence. The method is implemented using a modification of the WaveD-package in R and an extensive numerical study is conducted. The numerical study supplements the theoretical results and compares the LRD estimator with a naïve application of the standard WaveD approach.

Combining thresholding rules: a new way to improve the performance of wavelet estimators

In this paper, we address the situation where we cannot differentiate wavelet-based threshold procedures because their sets of well-estimated functions (maxisets) are not nested. As a generic solution, we propose to proceed via a combination of these procedures in order to achieve new procedures which perform better in the sense that the involved maxisets contain the union of the previous ones. Throughout the paper we propose illuminating interpretations of the maxiset results and provide conditions to ensure that this combination generates larger maxisets. As an example, we propose to combine vertical- and horizontal-block thresholding procedures that are already known to perform well. We discuss the limitation of our method, and we check our theoretical results through numerical experiments.

Wavelet penalized likelihood estimation in generalized functional models

The paper deals with generalized functional regression. The aim is to estimate the influence of covariates on observations, drawn from an exponential distribution. The link considered has a semiparametric expression: if we are interested in a functional influence of some covariates, we authorize others to be modeled linearly. We thus consider a generalized partially linear regression model with unknown regression coefficients and an unknown nonparametric function. We present a maximum penalized likelihood procedure to estimate the components of the model introducing penalty based wavelet estimators. Asymptotic rates of the estimates of both the parametric and the nonparametric part of the model are given and quasi-minimax optimality is obtained under usual conditions in literature. We establish in particular that the LASSO penalty leads to an adaptive estimation with respect to the regularity of the estimated function. An algorithm based on backfitting and Fisher-scoring is also proposed for implementation. Simulations are used to illustrate the finite sample behaviour, including a comparison with kernel and splines based methods.

Wavelet Estimation Method for Short-Term Prediction of Deep Pit Settlement

A wavelet estimation method is presented herein to estimate deep pit settlement. In this method, the pit settlement is decomposed into the trend settlement and the stochastic settlement by using Wavelet Analysis based on the characteristic of influencing factor. The model identifier is established by using artificial neural network (ANN), and trained to approximate the trend settlement. Then, the prediction controller developed could be applied for estimating the actual settlement. Finally, the verification examples show that the WIAN is an effective tool for predicting the pit settlement dynamically , high precision could be expected and achieved.

Waveform Inversion of Crosshole Georadar Data: Influence of Source Wavelet Variability and the Suitability of a Single Wavelet Assumption

Waveform-based tomographic imaging of crosshole georadar data is a powerful method to investigate the shallow subsurface because of its ability to provide images of electrical properties in near-surface environments with unprecedented spatial resolution. A critical issue with waveform inversion is the a priori unknown source signal. Indeed, the estimation of the source pulse is notoriously difficult but essential for the effective application of this method. Here, we explore the viability and robustness of a recently proposed deconvolution-based procedure to estimate the source pulse during waveform inversion of crosshole georadar data, where changes in wavelet shape with location as a result of varying near-field conditions and differences in antenna coupling may be significant. Specifically, we examine whether a single, average estimated source current function can adequately represent the pulses radiated at all transmitter locations during a crosshole georadar survey, or whether a separate source wavelet estimation should be performed for each transmitter gather. Tests with synthetic and field data indicate that remarkably good tomographic reconstructions can be obtained using a single estimated source pulse when moderate to strong variability exists in the true source signal with antenna location. Only in the case of very strong variability in the true source pulse are tomographic reconstructions clearly improved by estimating a different source wavelet for each transmitter location.