Wavelet Estimation Research Articles

Wavelet thresholding estimation of density derivatives from a negatively associated size-biased sample

This paper considers wavelet estimation for density derivatives based on negatively associated and size-biased data. We provide upper bounds of nonlinear wavelet estimator on [Formula: see text] risk. It turns out that the convergence rate of the nonlinear estimator is better than that of the linear one.

Remaining useful life prediction for multivariable stochastic degradation systems with non‐Markovian diffusion processes

AbstractMultivariable stochastic degradation system (MSDS) is quite common in industries such as blast furnace ironmaking, vehicle transportation, and aerospace manufacturing. Large‐scale complex equipments may be affected by multiple factors, resulting in not just a single deteriorating performance characteristic. It is difficult to handle unknown failure structures of practical systems by using traditional univariate degradation modeling methods. A novel health index (HI) is constructed to quantitatively analyze the health state for the overall system. Considering the interaction between internal reactions and external environments, the fractional Brownian motion (FBM), a typical non‐Markovian diffusion process, is added for the purpose of reflecting stochastic uncertainties and memory effects. Based on the wavelet estimators and the maximum likelihood estimation (MLE) algorithm, multi‐sensor observations of degradation variables are analyzed simultaneously to identify model parameters. A closed‐form distribution of system‐level remaining useful life (RUL) is obtained with a mild two‐layer approximation. Relevant case studies are then handled that adequately demonstrate the effectiveness and the practical utility of the proposed method.

The adaptivity of thresholding wavelet estimators in heteroscedastic nonparametric model with negatively super-additive dependent errors

In this paper, we consider two estimators, a hard thresholding wavelet estimator and a block thresholding wavelet estimator, for the regression function in heteroscedastic nonparametric model with negatively super-additive dependent (NSD) errors. The random design distribution is known or unknown, and the corresponding adaptive properties of these estimators are investigated over Besov spaces, for the $${L^2}$$ risk. The results indicate that the block thresholding estimator is theoretically and computationally superior to the hard thresholding estimator with the former attains the optimal convergence rates, while the later achieves the nearly optimal convergence rates. Thus the block thresholding estimator provides extensive adaptivity to many irregular function classes even though with the presence of heteroscedastic NSD errors.

Pointwise Optimality of Wavelet Density Estimation for Negatively Associated Biased Sample

This paper focuses on the density estimation problem that occurs when the sample is negatively associated and biased. We constructed a block thresholding wavelet estimator to recover the density function from the negatively associated biased sample. The pointwise optimality of this wavelet density estimation is shown as L p ( 1 ≤ p < ∞ ) risks over Besov space. To validate the effectiveness of the block thresholding wavelet method, we provide some examples and implement the numerical simulations. The results indicate that our block thresholding wavelet density estimator is superior in terms of the mean squared error (MSE) when comparing with the nonlinear wavelet density estimator.

Density log correction for borehole effects and its impact on well-to-seismic tie: Application on a North Sea data set

Reservoir characterization requires accurate elastic logs. It is necessary to guarantee that the logging tool is stable during the drilling process to avoid compromising the measurements of the physical properties in the formation in the vicinity of the well. Irregularities along the borehole may happen, especially if the drilling device is passing through unconsolidated formations. This affects the signals recorded by the logging tool, and the measurements may be more impacted by the drilling mud than by the formation. The caliper log indicates the change in the diameter of the borehole with depth and can be used as an indicator of the quality of other logs whose data have been degraded by the enlargement or shrinkage of the borehole wall. Damaged well-log data, particularly density and velocity profiles, affect the quality and accuracy of the well-to-seismic tie. To investigate the effects of borehole enlargement on the well-to-seismic tie, an analysis of density log correction was performed. This approach uses Doll’s geometric factor to correct the density log for wellbore enlargement using the caliper readings. Because the wavelet is an important factor on the well tie, we tested our methodology with statistical and deterministic wavelet estimations. For both cases, the results using the real data set from the Viking Graben field — North Sea indicated up to a 7% improvement on the correlation between the real and synthetic seismic traces for well-to-seismic tie when the density correction was made.

Intrinsic Wavelet Regression for Curves of Hermitian Positive Definite Matrices

Intrinsic wavelet transforms and wavelet estimation methods are introduced for curves in the non-Euclidean space of Hermitian positive definite matrices, with in mind the application to Fourier spectral estimation of multivariate stationary time series. The main focus is on intrinsic average-interpolation wavelet transforms in the space of positive definite matrices equipped with an affine-invariant Riemannian metric, and convergence rates of linear wavelet thresholding are derived for intrinsically smooth curves of Hermitian positive definite matrices. In the context of multivariate Fourier spectral estimation, intrinsic wavelet thresholding is equivariant under a change of basis of the time series, and nonlinear wavelet thresholding is able to capture localized features in the spectral density matrix across frequency, always guaranteeing positive definite estimates. The finite-sample performance of intrinsic wavelet thresholding is assessed by means of simulated data and compared to several benchmark estimators in the Riemannian manifold. Further illustrations are provided by examining the multivariate spectra of trial-replicated brain signal time series recorded during a learning experiment. Supplementary materials for this article are available online.

Simultaneous Imaging of Primaries and Multiples Based on Wavelet Estimation and Stereographic Imaging Condition

Multiple reflections contain abundant structural information about the subsurface because of their smaller reflection angles and wider coverage. Multiple imaging has increasingly attracted attention. However, the imaging of multiples first needs to separate the multiples, which is a time-consuming and high-cost job. Therefore, the simultaneous imaging of primaries and multiples, which can make good use of multiple reflections with no need for multiple separation, has increasingly attracted attention. However, there are still some challenges to the conventional method, including wavelet selection and image artifacts suppression. In this paper, an improved method of the simultaneous imaging of primaries and multiples is proposed by introducing two creative strategies. First, considering that the wavelet, which is usually estimated by signal analysis under certain assumptions, is crucial to the simultaneous imaging, we propose to estimate the wavelet by iterative SRME (surface-related multiple elimination) using the original data containing multiples based on wave equation theory instead of signal analysis. Second, the stereographic imaging condition is introduced to suppress the crosstalk artifacts in the image. According to the numerical examples and field data test, the feasibility and effectiveness of our approach is verified.

Adaptive log-density estimation

This study examines an adaptive log-density estimation method with an $$\ell _1$$ -type penalty. The proposed estimator is guaranteed to be a valid density in the sense that it is positive and integrates to one. The smoothness of the estimator is controlled in a data-adaptive way via $$\ell _1$$ penalization. The advantages of the penalized log-density estimator are discussed with an emphasis on wavelet estimators. Theoretical properties of the estimator are studied when the quality of fit is measured by the Kullback–Leibler divergence (relative entropy). A nonasymptotic oracle inequality is obtained assuming a near orthogonality condition on the given dictionary. Based on the oracle inequality, selection consistency and minimax adaptivity are proved under some regularity conditions. The proposed method is implemented with a coordinate descent algorithm. Numerical illustrations based on the periodized Meyer wavelets are performed to demonstrate the finite sample performance of the proposed estimator.

Complexity analysis and dynamic characteristics of EEG using MODWT based entropies for identification of seizure onset.

In this paper, complexity analysis and dynamic characteristics of electroencephalogram (EEG) signal based on maximal overlap discrete wavelet transform (MODWT) has been exploited for the identification of seizure onset. Since wavelet-based studies were well suited for classification of normal and epileptic seizure EEG, we have applied MODWT which is an improved version of discrete wavelet transform (DWT). The selection of optimal wavelet sub-band and features plays a crucial role to understand the brain dynamics in epileptic patients. Therefore, we have investigated MODWT using four different wavelets, namely Haar, Coif4, Dmey, and Sym4 sub-bands until seven levels. Further, we have explored the potentials of six entropies, namely sigmoid, Shannon, wavelet, Renyi, Tsallis, and Steins unbiased risk estimator (SURE) entropies in each sub-band. The sigmoid entropy extracted from Haar wavelet in sub-band D4 showed the highest accuracy of 98.44% using support vector machine classifier for the EEG collected from Ramaiah Medical College and Hospitals (RMCH). Further, the highest accuracy of 100% and 94.51% was achieved for the University of Bonn (UBonn) and CHB-MIT databases respectively. The findings of the study showed that Haar and Dmey wavelets were found to be computationally economical and expensive respectively. Besides, in terms of dynamic characteristics, MODWT results revealed that the highest energy present in sub-bands D2, D3, and D4 and entropies in those respective sub-bands outperformed other entropies in terms of classification results for RMCH database. Similarly, using all the entropies, sub-bands D5 and D6 outperformed other sub-bands for UBonn and CHB-MIT databases respectively. In conclusion, the comparison results of MODWT outperformed DWT.

The Berry-Esseen bound of a wavelet estimator in non-randomly designed nonparametric regression model based on ANA errors

Consider the nonparametric regression model Yni = g(tni) + εi, i = 1, 2, …, n, n ≥ 1, where εi, 1 ≤ i ≤ n, are asymptotically negatively associated (ANA, for short) random variables. Under some appropriate conditions, the Berry-Esseen bound of the wavelet estimator of g(⋅) is established. In addition, some numerical simulations are provided in this paper. The results obtained in this paper generalize some corresponding ones in the literature.

A Berry-Ess$\acute{e}$n bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence

By using some inequalities for linearly negative quadrant dependent random variables, Berry-Ess$\acute{e}$en bound of wavelet estimation for a nonparametric regression model is investigated under linear process errors based on linearly negative quadrant dependent sequence. The rate of uniform asymptotic normality is presented and the rate of convergence is near $O(n^{-\frac{1}{6}})$ under mild conditions, which generalizes or extends the corresponding results of Li et al.(2008) under associated random samples in some sense.

Estimation of the seismic wavelet through homomorphic deconvolution and well log data: application on well‐to‐seismic tie procedure

ABSTRACTWavelet estimation and well‐tie procedures are important tasks in seismic processing and interpretation. Deconvolutional statistical methods to estimate the proper wavelet, in general, are based on the assumptions of the classical convolutional model, which implies a random process reflectivity and a minimum‐phase wavelet. The homomorphic deconvolution, however, does not take these premises into account. In this work, we propose an approach to estimate the seismic wavelet using the advantages of the homomorphic deconvolution and the deterministic estimation of the wavelet, which uses both seismic and well log data. The feasibility of this approach is verified on well‐to‐seismic tie from a real data set from Viking Graben Field, North Sea, Norway. The results show that the wavelet estimated through this methodology produced a higher quality well tie when compared to methods of estimation of the wavelet that consider the classical assumptions of the convolutional model.

A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors

Abstract In this paper, we consider the wavelet estimators of a nonparametric regression model based on widely orthant dependent random errors. The moment consistency and the completely consistency for wavelet estimators under some more mild moment conditions are investigated. The results obtained in the paper improve and extend the corresponding ones for dependent random variables. Finally, we provide a numerical simulation to verify the validity of our results.

MISE of wavelet estimators for regression derivatives with biased strong mixing data

Using a wavelet basis, this article considers the mean integrated squared error (MISE) of linear and non linear wavelet estimators for regression derivatives based on biased strong mixing data. It turns out that the convergence rates coincide with those of Chesneau and Shirazi’s (Communication in Statistics-Theory and Methods, 2014), when d = 0 and the random sample is independent.

Generalized Bayesian shrinkage and wavelet estimation of location parameter for spherical distribution under balance-type loss: Minimaxity and admissibility

One of the most important subject in multivariate analysis is parameters estimation. Among different methods, the shrinkage estimation is of interest. In this paper we consider the generalized Bayes shrinkage estimator of location parameter for spherical distribution under balance-type loss. We assume that the random vector having a spherical symmetric distribution with the known scalar variational component. Also, we find minimax and admissible estimator of location parameter based on generalized Bayes estimator. We investigate wavelet generalized Bayes estimator of location under balance-LINEX loss function. At the end, the performance evaluation of the proposed class of estimators is checked through a simulation study.

Full waveform inversion based on a local traveltime correction and zero-mean cross-correlation-based misfit function

Full waveform inversion (FWI) suffers from the cycle skipping problem, because the observed data usually lack low-frequency components or due to errors in the wavelet estimation. In addition, the strong low-frequency non-zero-mean noise can have a large impact on FWI results. Thus, we propose a local waveform traveltime correction scheme to solve the situations when the observed data lack low-frequency components or when the estimation for the wavelet is incorrect. We use a sliding time window, which is used to decrease the traveltime differences between the calculated and observed data to increase the cross-correlation between them. Besides, we propose a zero-mean normalized cross-correlation misfit function to reduce the interference of the low-frequency non-zero-mean noise. Therefore, we propose new approaches to improve FWI results whether the observed data lack low-frequency components or the observed data are contaminated by the non-zero-mean low-frequency noise. Numerical examples on Marmousi model show the feasibility of a FWI based on the zero-mean normalized cross-correlation misfit function and a FWI based on the local traveltime correction method.

The wavelet scaling approach to forecasting: Verification on a large set of Noisy data

AbstractIn the paper, we undertake a detailed empirical verification of wavelet scaling as a forecasting method through its application to a large set of noisy data. The method consists of two steps. In the first, the data are smoothed with the help of wavelet estimators of stochastic signals based on the idea of scaling, and, in the second, an AR(I)MA model is built on the estimated signal. This procedure is compared with some alternative approaches encompassing exponential smoothing, moving average, AR(I)MA and regularized AR models. Special attention is given to the ways of treating boundary regions in the wavelet signal estimation and to the use of biased, weakly biased and unbiased estimators of the wavelet variance. According to a collection of popular forecast accuracy measures, when applied to noisy time series with a high level of noise, wavelet scaling is able to outperform the other forecasting procedures, although this conclusion applies mainly to longer time series and not uniformly across all the examined accuracy measures.

Asymptotic properties of wavelet estimators in heteroscedastic semiparametric model based on negatively associated innovations

Consider the heteroscedastic semiparametric regression model y_{i}=x_{i}beta+g(t_{i})+varepsilon_{i}, i=1, 2, ldots, n, where β is an unknown slope parameter, varepsilon_{i}=sigma_{i}e_{i}, sigma^{2}_{i}=f(u_{i}), (x_{i},t_{i},u_{i}) are nonrandom design points, y_{i} are the response variables, f and g are unknown functions defined on the closed interval [0,1], random errors {e_{i} } are negatively associated (NA) random variables with zero means. Whereas kernel estimators of β, g, and f have attracted a lot of attention in the literature, in this paper, we investigate their wavelet estimators and derive the strong consistency of these estimators under NA error assumption. At the same time, we also obtain the Berry–Esséen type bounds of the wavelet estimators of β and g.

Tempered fractional Brownian motion: Wavelet estimation, modeling and testing

The Davenport spectrum is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for non-scaling low frequency behavior. Like the classical fractional Brownian motion vis-à-vis the Kolmogorov spectrum, tempered fractional Brownian motion (tfBm) is a new model that displays the Davenport spectrum. The autocorrelation of the increments of tfBm displays semi-long range dependence (hyperbolic and quasi-exponential decays over moderate and large scales, respectively), a phenomenon that has been observed in a wide range of applications from wind speeds to geophysics to finance. In this paper, we use wavelets to construct the first estimation method for tfBm and a simple and computationally efficient test for fBm vs tfBm alternatives. The properties of the wavelet estimator and test are mathematically and computationally established. An application of the methodology shows that tfBm is a better model than fBm for a geophysical flow data set.

Review of crosshole ground-penetrating radar full-waveform inversion of experimental data: Recent developments, challenges, and pitfalls

Heterogeneous small-scale high-contrast layers and spatial variabilities of soil properties can have a large impact on flow and transport processes in the critical zone. Because their characterization is difficult and critical, high-resolution methods are required. Standard ray-based approaches for imaging the subsurface consider only a small amount of the measured data and suffer from limited resolution. In contrast, full-waveform inversion (FWI) considers the full information content of the measured data and could yield higher resolution images in the subwavelength scale. In the past few decades, ground-penetrating radar (GPR) FWI and its application to experimental data have matured, which makes GPR FWI an established approach to significantly improve resolution. Several theoretical developments were achieved to improve the application to experimental data from crosshole GPR FWI. We have determined the necessary steps to perform FWI for experimental data to obtain reliable and reproducible high-resolution images. We concentrate on experimental crosshole GPR data from a test site in Switzerland to illustrate the challenges of applying FWI to experimental data and discuss the obtained results for different development steps including possible pitfalls. Thereby, we acknowledge out the importance of a correct time-zero correction of the data, the estimation of the effective source wavelet, and the effect of the choice of starting models. The reliability of the FWI results is investigated by analyzing the fit of the measured and modeled traces, the remaining gradients of the final models, and validating with independently measured logging data. Thereby, we found that special care needs to be taken to define the optimal inversion parameters to avoid overshooting of the inversion or truncation errors.