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Wavelet shrinkage estimators are widely applied to analyse datasets in wavelet domain from several fields of science. They typically act by reducing magnitudes of empirical coefficients in a discrete wavelet transformation to estimate wavelet coefficients. In nonparametric regression problems, most of the shrinkage rules are derived from models composed of an unknown function with additive Gaussian noise. Although Gaussian noise assumption is reasonable in several real data analysis, mainly for large sample sizes, it is not general. Contaminated data with positive noise can occur in practice and nonparametric regression models with positive noise bring challenges in wavelet shrinkage point of view. In this sense, this work proposes bayesian shrinkage rules to estimate wavelet coefficients from a nonparametric regression framework with additive and strictly positive noise under exponential and lognormal distributions. Computational aspects are discussed and simulation studies to analyse the performances of the proposed shrinkage rules and compare them with standard techniques are done. An application for winning times Boston Marathon dataset is also provided. [ABSTRACT FROM AUTHOR]

The parameter that is of most interest in applications of stable distributions is the stability index α, which determines how "heavy" the tail is. For applications when the main interest is on the tail, we develop an estimator of α for univariate symmetric stable distributions, based on the characteristic function and on the principle of wavelet shrinkage. We analyze the proposed estimator by the Monte Carlo simulations and prove that it is a consistent estimator. The presence of human-made explosions in a seismic catalog leads to errors in the statistical analysis of seismicity. Here, we also proposed the CWS coefficient, based on the proposed estimator, like an additional classification tool to discriminate between natural tectonic earthquakes and human-made explosions. [ABSTRACT FROM AUTHOR]

Consider the nonparametric curve estimation problem. Wavelet shrinkage methods are applied to the data in the wavelet domain for noise reduction. After denoising, the function can be estimated by wavelet basis expansion. The present paper proposes a Bayesian approach for wavelet shrinkage with the use of a mixture of a point mass function at zero and the logistic distribution symmetric around zero as the prior distribution to the wavelet coefficients of an unknown function in models with additive Gaussian errors. Statistical properties such as bias, classical and Bayesian risks of the rules are analyzed and performances of the proposed rules are obtained in simulations studies involving the Donoho-Johnstone test functions. Application in a mass spectrometry real data set is done. [ABSTRACT FROM AUTHOR]

We can use wavelet shrinkage to estimate a possibly multivariate regression function g under the general regression setup, y = g + ε. We propose an enhanced wavelet-based denoising methodology based on Bayesian adaptive multiresolution shrinkage, an effective Bayesian shrinkage rule in addition to the semi-supervised learning mechanism. The Bayesian shrinkage rule is advanced by utilizing the semi-supervised learning method in which the neighboring structure of a wavelet coefficient is adopted and an appropriate decision function is derived. According to decision function, wavelet coefficients follow one of two prespecified Bayesian rules obtained using varying related parameters. The decision of a wavelet coefficient depends not only on its magnitude, but also on the neighboring structure on which the coefficient is located. We discuss the theoretical properties of the suggested method and provide recommended parameter settings. We show that the proposed method is often superior to several existing wavelet denoising methods through extensive experimentation. [ABSTRACT FROM AUTHOR]

The method of detrended fluctuation analysis (DFA) is useful in revealing the extent of long-range dependence, it has successfully been applied to different fields of interest. In this paper we proposed a smoothed detrended fluctuation analysis method based on the principle of wavelet shrinkage. The procedure is illustrated and compared with the DFA method by Monte Carlo simulations on fractional Gaussian noise models. [ABSTRACT FROM AUTHOR]

In this article, we propose a denoising methodology in the wavelet domain based on a Bayesian hierarchical model using Double Weibull prior. We propose two estimators, one based on posterior mean (Double Weibull Wavelet Shrinker, DWWS) and the other based on larger posterior mode (DWWS-LPM), and show how to calculate them efficiently. Traditionally, mixture priors have been used for modeling sparse wavelet coefficients. The interesting feature of this article is the use of non-mixture prior. We show that the methodology provides good denoising performance, comparable even to state-of-the-art methods that use mixture priors and empirical Bayes setting of hyperparameters, which is demonstrated by extensive simulations on standardly used test functions. An application to real-word dataset is also considered. [ABSTRACT FROM AUTHOR]

During ambulatory monitoring, it is sometimes required to record an electroencephalogram (EEG) and an electrocardiogram (ECG) simultaneously. It would be ideal if both EEG and ECG could be obtained with one measurement. Here, we introduce an algorithm that combines the wavelet shrinkage and variable cosine window operation to separate the EEG and ECG components from an EEG signal recorded with a noncephalic reference (NCR). Evaluation using simulated data and actual measured data showed that accurate frequency analysis of EEG and an R-R detection-based heart rate analysis were feasible with our proposed algorithm, which improved the signal-averaging based algorithm so that ECG components containing ectopic beats can be applied. [ABSTRACT FROM AUTHOR]

In this paper, we have proposed a multilevel soft thresholding technique for noise removal in Daubechies complex wavelet transform domain. Two useful properties of Daubechies complex wavelet transform, approximate shift invariance and strong edge representation, have been explored. Most of the uncorrelated noise gets removed by shrinking complex wavelet coefficients at the lowest level, while correlated noise gets removed by only a fraction at lower levels, so we used multilevel thresholding and shrinkage on complex wavelet coefficients. The proposed method firstly detects strong edges using imaginary components of complex coefficients and then applies multilevel thresholding and shrinkage on complex wavelet coefficients in the wavelet domain at non-edge points. The proposed threshold depends on the variance of wavelet coefficients, the mean and the median of absolute wavelet coefficients at a particular level. Dependence of these parameters makes this method adaptive in nature. Results obtained for one-dimensional signals and two-dimensional images demonstrate an improved denoising performance over other related methods available in the literature. [ABSTRACT FROM AUTHOR]

A wavelet method is proposed for recovering damaged images. The proposed method combines wavelet shrinkage with preprocessing based on a binning process and an imputation procedure that is designed to extend the scope of wavelet shrinkage to data with missing values and perturbed locations. The proposed algorithm, termed as the BTW algorithm is simple to implement and efficient for recovering an image. Furthermore, this algorithm can be easily applied to wavelet regression for one-dimensional (1-D) signal estimation with irregularly spaced data. Simulation studies and real examples show that the proposed method can produce substantially effective results. [ABSTRACT FROM AUTHOR]

Single-index models have found applications in econometrics and biometrics, where multidimensional regression models are often encountered. This article proposes a non-parametric estimation approach that combines wavelet methods for nonequispaced designs with Bayesian models. We consider a wavelet series expansion of the unknown regression function and set prior distributions for the wavelet coefficients and the other model parameters. To ensure model identifiability, the direction parameter is represented via its polar coordinates. We employ ad hoc hierarchical mixture priors that perform shrinkage on wave let coefficients and use Markov chain Monte Carlo methods for a posteriori inference. We investigate an independence-type Metropolis-Hastings algorithm to produce samples for the direction parameter. Our method leads to simultaneous estimates of the link function and of the index parameters. We present results on both simulated and real data, where we look at comparisons with other methods. [ABSTRACT FROM AUTHOR]