Wavelet Estimation Research Articles

Wavelet estimation of norms for a probability density with negatively dependent biased data

This article considers wavelet estimations for density functions in Besov spaces based on negatively dependent biased random samples. More precisely, wavelet estimators are defined and their L p ( 1 ≤ p < ∞ ) convergence rates are provided when the parameter μ is known and unknown, respectively.

Two new Bayesian-wavelet thresholds estimations of elliptical distribution parameters under non-linear exponential balanced loss

The estimation of mean vector parameters is very important in elliptical and spherically models. Among different methods, the Bayesian and shrinkage estimation are interesting. In this paper, the estimation of p-dimensional location parameter for p-variate elliptical and spherical distributions under an asymmetric loss function is investigated. We find generalized Bayes estimator of location parameters for elliptical and spherical distributions. Also we show the minimaxity and admissibility of generalized Bayes estimator in class of We introduce two new shrinkage soft-wavelet threshold estimators based on Huang shrinkage wavelet estimator (empirical) and Stein’s unbiased risk estimator (SURE) for elliptical and spherical distributions under non-linear exponential-balanced loss function. At the end, we present a simulation study to test the validity of the class of proposed estimators and physicochemical properties of the tertiary structure data set that is given to test the efficiency of this estimators in denoising.

Uniform almost sure convergence rate of wavelet estimator for regression model with mixed noise

ABSTRACT This article considers a non parametric estimation problem in a regression model with mixed noise. A wavelet estimator is proposed by using projection of wavelet coefficients estimators. Uniform almost sure convergence rate of this wavelet estimator is derived under some mild conditions. It should be pointed out that the convergence rate of the wavelet estimator coincides with the optimal strong uniform convergence rate of non parametric estimations. Finally, simulation studies illustrate the good performances of the wavelet estimator.

Multi‐scale elastic full‐waveform inversion in a hybrid domain based on deconvolution objective function

AbstractHigh‐precision P‐wave and S‐wave velocity models are very important for seabed exploration. Elastic full‐waveform inversion is an effective way to obtain the velocity models. Full‐waveform inversion is strongly dependent on source wavelet, and inaccurate wavelet estimation will severely affect the inversion results. Furthermore, elastic full‐waveform inversion is a highly non‐linear problem and has larger calculation and memory cost. Based on the deconvolution method, we develop a multi‐scale source‐independent elastic full‐waveform inversion method in a hybrid domain to alleviate the non‐linearity, to improve computation efficiency and decrease memory cost and to reduce the influence of source wavelet. We reconstruct the deconvoluted objective function and further derive the adjoint source and gradient formulas for elastic full‐waveform inversion with the adjoint‐state method. We can obtain good inversion results by using this objective function, adjoint source and gradient even if the source wavelet is unknown. In order to illustrate the influence of different reference traces on the objective function, we provide the theoretical formula of the objective function changing with the variation of reference traces. Experiments show that the deconvolution objective function cannot completely eliminate the influence of inaccurate source wavelet on the inversion, and the inversion effect depends on the selection of reference trace. Different inversion results of the Marmousi2 model indicate that the minimum offset trace is the best reference trace. Comparison of the inverted Marmousi2 model by the proposed method and the conventional elastic full‐waveform inversion method using correct wavelet shows that the two inversion results are very close, which proves the effectiveness of the proposed method and indicates a potential application of the method in seabed exploration.

Wavelet Estimation of a Density From Observations of Almost Periodically Correlated Process Under Positive Quadrant Dependence

In this paper, we construct a new wavelet estimator of density for the component of a finite mixture under positive quadrant dependence. Our sample is extracted from almost periodically correlated processes. To evaluate our estimator we will determine a convergence speed from an upper bound for the mean integrated squared error (MISE). Our result is compared to the independent case which provides an optimal convergence rate.

Block thresholding wavelet estimation of copula density based on NSD assumption

In this article, we consider the wavelet estimation of copula density for negatively superadditive dependent random variables. Using wavelet methods, we propose and develop a new estimation procedure for this problem. In particular, a BlockShrink estimator is constructed and we prove that it enjoys powerful mean integrated squared error properties over Besov balls. The main result is prepared to display the performance of the wavelet-based estimator and a simulation study to compare the behavior of the proposed wavelet estimator with the kernel copula density estimator are also given. Finally, we consider a real life application in hydrology for rainfall intensity data.

Wavelet estimations of the derivatives of variance function in heteroscedastic model

&lt;abstract&gt;&lt;p&gt;This paper studies nonparametric estimations of the derivatives $ r^{(m)}(x) $ of the variance function in a heteroscedastic model. Using a wavelet method, a linear estimator and an adaptive nonlinear estimator are constructed. The convergence rates under $ L^{\tilde{p}} (1\leq \tilde{p} &amp;lt; \infty) $ risk of those two wavelet estimators are considered with some mild assumptions. A simulation study is presented to validate the performances of the wavelet estimators.&lt;/p&gt;&lt;/abstract&gt;

Wavelet estimations of the derivatives of variance function in heteroscedastic model

&lt;abstract&gt;&lt;p&gt;This paper studies nonparametric estimations of the derivatives $ r^{(m)}(x) $ of the variance function in a heteroscedastic model. Using a wavelet method, a linear estimator and an adaptive nonlinear estimator are constructed. The convergence rates under $ L^{\tilde{p}} (1\leq \tilde{p} &amp;lt; \infty) $ risk of those two wavelet estimators are considered with some mild assumptions. A simulation study is presented to validate the performances of the wavelet estimators.&lt;/p&gt;&lt;/abstract&gt;

Wavelet Estimation of Function Derivatives from a Multichannel Deconvolution Model

This paper considers a multichannel deconvolution model with Gaussian white noises. The goal is to estimate the d -th derivatives of an unknown function in the model. For super-smooth case, we construct an adaptive linear wavelet estimator by wavelet projection method. For regular-smooth case, we provide an adaptive nonlinear wavelet estimator by hard-thresholded method. In order to measure the global performances of our estimators, we show upper bounds on convergence rates using the L p -risk ( 1 ≤ p &lt; ∞ ).

Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of random variables with infinite moment and its applications

In this work, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of widely orthant dependent random variables is established under mild conditions including infinite variance. As applications, some new results such as the Cesàro strong law of large numbers, the strong consistency of the least squares estimator in multiple linear regression models, and the strong consistency of the wavelet estimator in nonparametric regression models are presented. Simulation studies are also provided to support the theoretical results.

Bayesian Linear Seismic Inversion Integrating Uncertainty of Noise Level Estimation and Wavelet Extraction

Seismic impedance inversion is an important method to identify the spatial characteristics of underground rock physical properties. Seismic inversion results and uncertainty evaluation are the important scientific basis for risk decision-making in oil and gas development. Under the assumption that the impedance and the error of the observed seismic data meet the Gaussian distribution or log–Gaussian distribution, the Bayesian linear seismic inversion can analytically obtain the posterior probability distribution of impedance. However, errors from observation, calculation, model and other factors can lead to an inaccurate and incomplete uncertainty evaluation. In this paper, the noise variance is used to represent the noise level of seismic data and the uncertainties from seismic wavelet extraction and noise level estimation are considered in inversion. Assuming that the probability distribution of the noise variance meets the inverse gamma distribution and the seismic wavelet meets the Gaussian distribution, we could obtain the conditional distribution for one variable given another analytically using well-log data and seismic data. In order to integrate the uncertainty from noise level estimation and wavelet extraction into the seismic impedance inversion, the Gibbs sampler algorithm was applied to draw a set of realizations of noise variance and wavelet. For each realization, the corresponding posterior probability model of impedance was achieved by Bayesian linear inversion and the final posterior probability of the impedance model was obtained by integrating all the single posterior probabilities for each pair of wavelet and noise variance. Synthetic and real data experiments showed that the uncertainties of seismic wavelet extraction and noise level estimation have an important influence on inversion results and their uncertainties. The proposed method could effectively integrate the uncertainty of wavelet and noise estimation to obtain a more accurate and comprehensive uncertainty evaluation. Under the assumption that the model meets the linear relationship and the parameters meet some specified distribution, the proposed method has high calculation efficiency. However, it also loses some accuracy when the assumptions are not completely consistent with the actual situation.

Asymptotic properties of the wavelet estimator in non parametric regression model with martingale difference errors

This work mainly investigates the asymptotic properties for the wavelet estimator in a non parametric regression model with martingale difference random errors. The moment consistency, strong consistency, complete consistency, the rate of strong consistency and the asymptotic normality for the wavelet estimator are established under some mild conditions, which were not obtained before. Some numerical analysis is carried out to support the theoretical results.

Efficient source-independent full waveform dual-parameter inversion of common-offset GPR data

Full waveform inversion (FWI) of common-offset ground penetrating radar (GPR) data can make up for the defects of the traditional prospecting technology such as destructive, cumbersome, and low efficiency, which also provides a robust and high-precision tool for quantitative assessment of subsurface prospecting. Nonetheless, conventional GPR-FWI is of limited practicality since the deficiency of accurate source wavelet and tremendous computational cost. We develop an efficient source-independent inversion algorithm based on the convolution method to eliminate the influence of source wavelet and enhance the algorithm efficiency to inverse field data. To suppress the noises induced by the convolution and cross-correlation operations, the time window is introduced in the reference trace which is selected automatically by singular value decomposition (SVD). The stochastic strategy reduces the computational cost by source subsampling. The synthetic model of a complex lens body demonstrates that source-independent FWI with time window has more tolerant to the initial model and noise, and the stochastic strategy improves the efficiency with ensured inversion accuracy. In the field GPR data of a sand trough containing two anomalous bodies, our algorithm has good adaptability and can accurately reconstruct the distribution of the permittivity and conductivity of the subsurface media without accurate estimation of the source wavelet. The results above indicate that the proposed algorithm in this study is valid for common-offset GPR data and improves the inversion efficiency by 1.24 to 3.10 times, which shows great potential to promote qualitative assessment in real-time subsurface exploration.

Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes

In this study, we look at the wavelet basis for the nonparametric estimation of density and regression functions for continuous functional stationary processes in Hilbert space. The mean integrated squared error for a small subset is established. We employ a martingale approach to obtain the asymptotic properties of these wavelet estimators. These findings are established under rather broad assumptions. All we assume about the data is that they are ergodic, but beyond that, we make no assumptions. In this paper, the mean integrated squared error findings in the independence or mixing setting were generalized to the ergodic setting. The theoretical results presented in this study are (or will be) valuable resources for various cutting-edge functional data analysis applications. Applications include conditional distribution, conditional quantile, entropy, and curve discrimination.

Source-Independent Waveform Inversion Method for Ground Penetrating Radar Based on Envelope Objective Function

For the full waveform inversion, it is necessary to provide an accurate source wavelet for forwarding modeling in the iteration. The source wavelet estimation method based on deconvolution technology can solve this problem to some extent, but we find that the estimated source wavelet is not accurate and needs to be manually corrected repeatedly in the iteration. This process is highly operator-intensive, and the update process is time-consuming and increases the potential for errors. We propose a source-independent waveform inversion (SIEWI) scheme for cross-hole GPR data, and use the envelope objective function combined with this method to effectively reduce the nonlinearity of inversion. The residual field used by SIEWI to construct the gradient inherits the characteristics of the envelope wavefield. Compared with full waveform inversion (FWI), SIEWI is more robust and less sensitive to frequency components and inaccurate source wavelet. To avoid cycle jumping, the multi-scale strategy effectively utilizes the properties of convolutional wavefields. In one iteration, the wavefield is decomposed into multiple frequency bands through multiple convolutions in the time domain to construct a multi-scale inversion strategy that preferentially inverts low-frequency information.

Spatial transcriptomics dimensionality reduction using wavelet bases

Background: Spatially resolved transcriptomics (ST) measures gene expression along with the spatial coordinates of the measurements. The analysis of ST data involves significant computation complexity. In this work, we propose a gene expression dimensionality reduction algorithm that retains spatial structure. Methods: We combine the wavelet transformation with matrix factorization to select spatially-varying genes. We extract a low-dimensional representation of these genes. We adopt an Empirical Bayes perspective, imposing regularization through the prior distribution of factor genes. Additionally, we visualize the extracted representations, providing an overview of global spatial patterns. We illustrate the performance of our methods through spatial structure recovery and gene expression reconstruction using a simulation and real data analysis. Results: In real data experiments, our method identifies spatial structure of gene factors and outperforms regular decomposition regarding reconstruction error. We find a connection between the fluctuation of gene patterns and wavelet estimates, and this allows us to provide smoother visualizations. We develop the package and share the workflow generating reproducible quantitative results and gene visualization. The package is available at https://github.com/OliverXUZY/waveST. Conclusions: We have proposed a pipeline for dimensionality reduction that respects spatial structure. Both simulations and real data experiments demonstrate that wavelet and shrinkage techniques show positive results in spatially resolved transcriptomics data. We highlight the idea of combining image processing techniques and statistical methods for application in a spatial genomics context

Pointwise Wavelet Estimations for a Regression Model in Local Hölder Space

This paper considers an unknown functional estimation problem in a regression model with multiplicative and additive noise. A linear wavelet estimator is first constructed by a wavelet projection operator. The convergence rate under the pointwise error of linear wavelet estimators is studied in local Hölder space. A nonlinear wavelet estimator is provided by the hard thresholding method in order to obtain an adaptive estimator. The convergence rate of the nonlinear estimator is the same as the linear estimator up to a logarithmic term. Finally, it should be pointed out that the convergence rates of two wavelet estimators are consistent with the optimal convergence rate on pointwise nonparametric estimation.

Continuous wavelet estimation for multivariate fractional Brownian motion

In this paper, we propose a method using continuous wavelets to study the multivariate fractional Brownian motion through the deviations of the transformed random process to find an efficient estimate of Hurst exponent using eigenvalue regression of the covariance matrix. The results of simulations experiments shown that the performance of the proposed estimator was efficient in bias but the variance get increase as signal change from short to long memory the MASE increase relatively. The estimation process was made by calculating the eigenvalues for the variance-covariance matrix of Meyer’s continuous wavelet details coefficients.

Using copula information in wavelet estimation of bivariate density function based on censorship observations

This article discusses the nonparametric estimation of a bivariate density function using copula information under right censoring. We propose an adaptive estimator based on wavelet methods and the formulae for the asymptotic mean integrated squared error(MISE) is used to get the near optimal rate on a large functional class of regular densities. In particular, the asymptotic formulae for MISE in the context of kernel density estimators is derived in the case of censoring. Finally, the consistency of the proposed estimators is established and its effectiveness is validated through a numerical simulations.

Stability and Sample-Based Approximations of Composite Stochastic Optimization Problems

Optimization under uncertainty and risk is ubiquitous in business, engineering, and finance. Typically, we use observed or simulated data in our decision models, which aim to control risk, and result in composite risk functionals. The paper addresses the stability of the decision problems when the composite risk functionals are subjected to measure perturbations at multiple levels of potentially different nature. We analyze data-driven formulations with empirical or smoothing estimators such as kernels or wavelets applied to some or to all functions of the compositions and establish laws of large numbers and consistency of the optimal values and solutions. This is the first study to propose and analyze smoothing in data-driven composite optimization problems. It is shown that kernel-based and wavelet estimation provide less biased estimation of the risk compared with the empirical plug-in estimators under some assumptions.