This is page 1 Printer Opaque this Empirical Bayesian Spatial Prediction Using Wavelets(5)

ABSTRACT Wavelet shrinkage methods, introduced by Donoho and Johnstone

4

Hsin-Cheng Huang, Noel Cressie

discussion of di erent covariance models for ( ), although in all the applications of this chapter we use a simple model that assumes independence and homoskedasticity within a scale but heteroskedasticity (and independence) across scales. Estimation of is also discussed in Section 4. Based on estimates^ (Section 3),^2 and^ (Section 4), we use the empirical Bayes spatial predictor, 0^ S Wn^?^;^ 2;^ to make inference on the unobserved signal S . Section 5 contains a small simulation study showing the value of our empirical Bayesian spatial prediction approach applied to a few test functions and an application to a two-dimensional image. Discussion and conclusions are given in Section 6.

2 Wavelet ShrinkageIn a series of papers, Donoho and Johnstone (1994, 1995, 1998), and Donoho et al. (1995) developed the wavelet shrinkage method for reconstructing signals from noisy data, where the noise is assumed to be Gaussian white noise. The wavelet shrinkage method proceeds as follows. First, the data Y are transformed using a discrete wavelet transform, yielding the empirical wavelet coe cients w. Next, to suppress the noise, the empirical wavelet coe cients are\shrunk" toward zero based on a shrinkage rule. Usually, wavelet shrinkage is carried out by thresholding the wavelet coe cients; that is, the wavelet coe cients that have an absolute value below a prespeci ed threshold are replaced by zero. Finally, the processed empirical wavelet coe cients are transformed back to the original domain using the inverse wavelet transform. In practice, the discrete wavelet transform and its inverse transform can be computed very quickly in only O(n) operations using the pyramid algorithm (Mallat, 1989). With a properly chosen shrinkage method, Donoho and Johnstone (1994, 1995, 1998), and Donoho et al. (1995) show that the resulting estimate of the unknown function is nearly minimax over a large class of function spaces and for a wide range of loss functions. More importantly, it is computationally fast, and it is automatically adaptive to the smoothness of the corresponding true function, without the need to adjust a\bandwidth" as in the kernel smoothing method. Of course, the crucial step of this procedure is the choice of a thresholding (or shrinkage) method. A number of approaches have been proposed including minimax (Donoho and Johnstone, 1994, 1995, 1998), cross-validation (Nason, 1995, 1996), hypotheses testing (Abramovich and Benjamini, 1995, 1996; Ogden and Parzen, 1996a, 1996b), and Bayesian methods (Vidakovic, 1998; Clyde et al., 1996, 1998; Chipman et al., 1997; Crouse et al., 1998; Abramovich et al., 1998; Ruggeri and Vidakovic, 1999).

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新工程科技This is page 1 Printer Opaque this Empirical Bayesian Spatial Prediction Using Wavelets(5)全文阅读和word下载服务。