Nonhomogeneous Wavelet Systems in High Dimensions

It is of interest to study a wavelet system with a minimum number of generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in [11] that for any $d\times d$ real-valued expansive matrix M, a homogeneous orthonormal M-wavelet basis can be generated by a single wavelet function. On the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet systems, though much less studied in the literature, play a fundamental role in wavelet analysis and naturally link many aspects of wavelet analysis together. In this paper, we are interested in nonhomogeneous wavelet systems in high dimensions with a minimum number of generators. As we shall see in this paper, a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system with almost all properties preserved. We also show that a nonredundant nonhomogeneous wavelet system is naturally connected to refinable structures and has a fixed number of wavelet generators. Consequently, it is often impossible for a nonhomogeneous orthonormal wavelet basis to have a single wavelet generator. However, for redundant nonhomogeneous wavelet systems, we show that for any $d\times d$ real-valued expansive matrix M, we can always construct a nonhomogeneous smooth tight M-wavelet frame in $L_2(R^d)$ with a single wavelet generator whose Fourier transform is a compactly supported $C^\infty$ function. Moreover, such nonhomogeneous tight wavelet frames are associated with filter banks and can be modified to achieve directionality in high dimensions. Our analysis of nonhomogeneous wavelet systems employs a notion of frequency-based nonhomogeneous wavelet systems in the distribution space. Such a notion allows us to separate the perfect reconstruction property of a wavelet system from its stability in function spaces

Wavelet design by means of multi-objective GAs for motor imagery EEG analysis

Wavelet-based analysis has been broadly used in the study of brain-computer interfaces (BCI), but in most cases these wavelet functions have not been designed taking into account the requirements of this field. In this study we propose a method to automatically generate wavelet-like functions by means of genetic algorithms. Results strongly indicate that it is possible to generate (evolve) wavelet functions that improve the classification accuracy compared to other well-known wavelets (e.g. Daubechies and Coiflets)

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

A New Approach to Detect Spurious Regressions using Wavelets

In this paper, we propose the use of wavelet covariance and correlation to detect spurious regression. Based on Monte Carlo simulation results and experiments with real exchange rate data, it is shown that the wavelet approach is able to detect spurious relationship in a bivariate time series more directly. Using the wavelet approach, it is sufficient to detect a spurious regression between bivariate time series if the wavelet covariance and correlation for the two series are significantly equal to zero. The wavelet approach does not rely on restrictive assumptions which are critical to the Durbin Watson test. Another distinct advantage of the graphical wavelet analysis of wavelet covariance and correlation to detect spurious regression is the simplicity and efficiency of the decision rule compared to the complicated Durbin-Watson decision rules.Wavelet analysis, spurious regression