Abstract
Classical wavelet thresholding methods suffer from boundary problems caused by the application of the wavelet transformations to a finite signal. As a result, large bias at the edges and artificial wiggles occur when the classical boundary assumptions are not satisfied. Although polynomial wavelet regression and local polynomial wavelet regression effectively reduce the risk of this problem, the estimates from these two methods can be easily affected by the presence of correlated noise and outliers, giving inaccurate estimates. This paper introduces two robust methods in which the effects of boundary problems, outliers, and correlated noise are simultaneously taken into account. The proposed methods combine thresholding estimator with either a local polynomial model or a polynomial model using the generalized least squares method instead of the ordinary one. A primary step that involves removing the outlying observations through a statistical function is considered as well. The practical performance of the proposed methods has been evaluated through simulation experiments and real data examples. The results are strong evidence that the proposed method is extremely effective in terms of correcting the boundary bias and eliminating the effects of outliers and correlated noise.
Highlights
Suppose a noisy data set y1, y2, . . . , yn lives on the fixed design model yi f i n εi; i 1, 2, . . . ; n 2j ; j 1, 2, . . . .The classical model 1.1 assumes that the unknown function, f, is square integrable on the interval 0, 1
Wavelet methods have been intensively used over the last two decades for estimating an unknown function observed in the presence of noise, following the pioneering work presented in the seminal papers of Donoho and Johnstone 1, 2, where the concept of wavelet thresholding was introduced to the statistical literature
When the noise contains a certain amount of structure in the form of a correlation, the variances of the wavelet coefficients will depend on the resolution level of the wavelet decomposition but will be constant at each level
Summary
Suppose a noisy data set y1, y2, . . . , yn lives on the fixed design model yi f i n εi; i 1, 2, . . . ; n 2j ; j 1, 2, . . . .The classical model 1.1 assumes that the unknown function, f, is square integrable on the interval 0, 1. The level-dependent thresholding method proposed by 3 considered the case where the noise is correlated but stationary. Kovac and Silverman 5 have taken these circumstances into consideration and proposed more general thresholding method, which can be used for both stationary and nonstationary noise. Another method was proposed by Wang and Wood 6 , in which the covariance structure of the data has been taken into account. This method is based on estimating the correlation structure of the noise in two different domains, namely, the time domain and the wavelet domain. The estimator of f , fPW , is fPW x fP x fW x
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