This is page 1 Printer Opaque this Empirical Bayesian Spatial Prediction Using Wavelets(6)

ABSTRACT Wavelet shrinkage methods, introduced by Donoho and Johnstone

1. Empirical Bayesian Spatial Predict

ion Using Wavelets

5

Donoho and Johnstone (1994) proposed the hard-thresholding and softthresholding strategies. For a wavelet coe cient wj;k and a threshold, the hard-thresholding value is given by

TH (wj;k )=8<

wj;k; if jwj;k j>; 0; if jwj;k j;

and the soft-thresholding value is given by

wj;k?; if wj;k>; if jwj;k j; TS (wj;k )=: 0; wj;k+; if wj;k<?:For these thresholding rules, the choice of the threshold parameter is important. The VisuShrink method proposed by Donoho and Johnstone p (1994) uses the universal threshold= 2 log n for all levels. Donoho and Johnstone (1995) also proposed the SureShrink method by selecting the threshold parameter in a level-by-level fashion. For a given resolution level j, the threshold is chosen to minimize Stein's unbiased risk estimate (SURE), provided the wavelet representation at that level is not too sparse; the sparsity condition is given by? 1 2X1 (wj;k )2 2j k=0 2j

1+ j j=2: 23 2

=

Otherwise, the threshold j= 2 log(2j ) is chosen. In practice, as suggested by Donoho and Johnstone (1994, 1995), the scaling-function coe cients wJ0 are not shrunk. Usually the noise parameter is unknown, in which case Donoho et al. (1995) proposed a robust estimator, the median of absolute deviations (MAD) of wavelet coe cients at the highest resolution: median j wJ?1;k? median (wJ?1;k )j: (6)~= MAD fwJ?1;k g 0:6745 A Bayesian wavelet shrinkage rule is obtained by specifying a certain prior for both and 2 based on (4). Vidakovic (1998) assumes that f j;k g are independent and identically t-distributed with n degrees of freedom and is independent of f j;k g with an exponential distribution. However, their wavelet shrinkage rule, either based on the posterior mean or via a Bayesian hypotheses testing procedure, requires numerical integration. Chipman et al. (1997) also assume an independent prior for f j;k g. Since a signal is likely to have a sparse wavelet distribution with a heavy tail, they consider a mixture of two zero-mean normal components for f j;k g; one has a very small variance and the other has a large variance. Treating 2 as a hyperparameter, their shrinkage rule based on the posterior mean??

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(6)全文阅读和word下载服务。