Hybrid local polynomial wavelet shrinkage: wavelet regression with automatic boundary adjustment
Introduction
Consider the following regression model:where xi=(i−1)/n and the εi's are independent and identical N(0,σ2) random errors. The unknown function f, mostly smooth, is suspected to have a few discontinuities or abrupt changes. Under this situation one popular method to estimate f is wavelet shrinkage; see, for examples, the seminal papers Donoho and Johnstone 1994, Donoho and Johnstone 1995.
However, wavelet shrinkage suffers from boundary problems that are caused by the application of the wavelet transformation to a finite signal. Many approaches have been proposed to overcome these problems. Perhaps the most popular approach is to impose some additional constraints, such as periodicity or symmetry, on f. This approach can be easily implemented, but such additional constraints may not always be realistic, especially so for two-dimensional data such as images. More recently, Oh et al. (2001) (see also Lee and Oh, 2003; Naveau and Oh, 2003) propose a simple method called polynomial wavelet regression (PWR) for handling these boundary problems. The idea of PWR is to estimate f with the sum of a set of wavelet basis functions, , and a low-order (global) polynomial, . That is,where is the PWR estimate for f. The hope is that, once is removed from the data yi, the remaining signal hidden in can be well estimated using wavelet regression with say periodic boundary assumption. In practice, one needs to determine the order of the polynomial for . Simulation results from Lee and Oh (2003) suggest the use of BIC (Schwarz, 1978).
The use of PWR for resolving boundary problems works very well if is able to remove the “non-periodicity” in yi. However, due to the global nature of , for those cases when f has complex boundary conditions or has some abrupt changing objects present near the boundaries, PWR does not work well. The goal of this article is to proposal a new method which will also work well under these situations. Fig. 1 provides some illustrative examples. The left column displays various curve estimates for a regression function that can be well estimated using PWR. It can be seen that both the PWR and the proposed method (to be described in the next section) produce good estimates, while a classical wavelet regression estimate with periodic assumption suffers from edge effects. On the other hand, the right column presents a situation that both the PWR and classical wavelet regression fail at the boundaries, but the proposed method is still able to procedure good boundary estimates.
The basic idea behind the proposed method is to introduce a local polynomial regression component to the wavelet shrinkage. Since local polynomial regression produces excellent boundary handling (Fan, 1992; Hastie and Loader, 1993), it is expected that the addition of this component to wavelet shrinkage will result in equally well boundary properties. Results from numerical experiments strongly support this claim. Besides producing promising empirical results, other desirable properties of the proposed method include: it is easy to implement, computationally fast, and can be straightforwardly extended to higher dimensional settings.
The rest of this article is organized as follows. Section 2 presents the proposed method. Results from a simulation study are reported in Section 3. In Section 4 the two-dimensional setting is considered. Conclusion is offered in Section 5.
Section snippets
Hybrid local polynomial wavelet shrinkage
This section presents the proposed method for improving boundary adjustment in wavelet regression. Driven by the fact that local polynomial regression is extremely effective in adapting to boundary conditions (e.g., see Fan, 1992; Hastie and Loader, 1993), we propose replacing the global polynomial fit in PWR with a local polynomial fit . We call the new resulting curve estimate a hybrid estimate:Now the expectation is that, the use of local polynomial
Simulation study
This section reports results from a simulation study that was designed to assess the practical performance of the proposed method. The R-codes used to carry out this simulation are available from the website http://www.stat.ualberta.ca/~heeseok/hybrid.html
Two-dimensional fitting
In higher dimensional problems, such as image denoising, the proportion of boundary observations is much higher than univariate curve estimation problems. Thus the need for boundary adjustment is even stronger. The above proposed method can be extended straightforwardly to such higher dimensional settings. This section present some simulation results obtained from two-dimensional surface fittings.
Displayed in Fig. 5 are the five two-dimensional test functions used in this study. These five
Conclusions
In this paper a hybrid wavelet shrinkage method is proposed for reducing the boundary bias that is commonly found in wavelet shrinkage. The proposed method is based on a coupling of classical wavelet shrinkage and local polynomial regression. The empirical performance of the method was tested on different numerical experiments, including both the univariate and bivariate settings. Results from these experiments illustrate the improvement of the hybrid wavelet shrinkage over the classical
Acknowledgements
The authors are grateful to the reviewer for his/her comments which lead to a more succinct version of the paper. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation under Grant No. 0203901.
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