Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study

Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced

Diagnostic System for Low-Speed Bearings

This thesis resolves the current issue of running diagnostics on low-speed bearings and has resulted from urgent industry requirements.The central concept of the design solution is the introduction of a reference element into the sprocket shaft bearing assembly of a chain conveyor used in the ŠKODA AUTO a.s. paint shop.Using two pairs of roller bearings each, this reference element is connected to both the shaft as well as the frame. With this design, the reference element can freely rotate, thereby making it possible to run diagnostics on the bearings.Firstly, this design makes it possible to rotate the reference element at a high speed and - just like for high-speed bearings - establish the frequency of the vibrations, determining the level of damage in doing so. Secondly, it makes it possible to identify and measure the resistance when rotating the reference element.This design solution has been successfully patented at a European level.In the course of this thesis, various designs have been developed, one of which has been realised as a prototype and integrated into the conveyor system at the ŠKODA AUTO a.s. paint shop in Mladá Boleslav.This thesis resolves the current issue of running diagnostics on low-speed bearings and has resulted from urgent industry requirements.The central concept of the design solution is the introduction of a reference element into the sprocket shaft bearing assembly of a chain conveyor used in the ŠKODA AUTO a.s. paint shop.Using two pairs of roller bearings each, this reference element is connected to both the shaft as well as the frame. With this design, the reference element can freely rotate, thereby making it possible to run diagnostics on the bearings.Firstly, this design makes it possible to rotate the reference element at a high speed and - just like for high-speed bearings - establish the frequency of the vibrations, determining the level of damage in doing so. Secondly, it makes it possible to identify and measure the resistance when rotating the reference element.This design solution has been successfully patented at a European level.In the course of this thesis, various designs have been developed, one of which has been realised as a prototype and integrated into the conveyor system at the ŠKODA AUTO a.s. paint shop in Mladá Boleslav

Deterministic/stochastic wavelet decomposition for recovery of signal from noisy data

In a series of recent articles on nonparametric regression, Donoho and Johnstone developed wavelet-shrinkage methods for recovering unknown piecewise-smooth deterministic signals from noisy data. Wavelet shrinkage based on the Bayesian approach involves specifying a prior distribution on the wavelet coefficients, which is usually assumed to have a distribution with zero mean. There is no a priori reason why all prior means should be 0; indeed, one can imagine certain types of signals in which this is not a good choice of model. In this article, we take an empirical Bayes approach in which we propose an estimator for the prior mean that is `plugged into\u27 the Bayesian shrinkage formulas. Another way we are more general than previous work is that we assume that the underlying signal is composed of a piecewise-smooth deterministic part plus a zero-mean stochastic part; that is, the signal may contain a reasonably large number of nonzero wavelet coefficients. Our goal is to predict this signal from noisy data. We also develop a new estimator for the noise variance based on a geostatistical method that considers the behavior of the variogram near the origin. Simulation studies show that our method (DecompShrink) outperforms the well-known VisuShrink and SureShrink methods for recovering a wide variety of signals. Moreover, it is insensitive to the choice of the lowest-scale cut-off parameter, which is typically not the case for other wavelet-shrinkage methods