This is page 1 Printer Opaque this Empirical Bayesian Spatial Prediction Using Wavelets(12)

ABSTRACT Wavelet shrinkage methods, introduced by Donoho and Johnstone

1. Empirical Bayesian Spatial Prediction Using Wavelets

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for j= J0;:::; J? 1; k= 0;:::; 2j? 1, where? 1? 2 (14)^j= max 2j wj?^ j 0 wj?^ j?^ 2; 0; and^2 is given by (9). Note that the wavelet coe cients are usually sparse (i.e., most of the coe cients are essentially zero), and have a distribution which is highly non-Gaussian with heavy tails. We are able to consider a Gaussian distribution in (3) because the non-Gaussian components are accounted

for by the mean .

If comes from a temporal process (i.e., d= 1), it is natural to assume a Gaussian autoregressive moving average (ARMA) model, independently for each j; j= J0;:::; J? 1. If is a d-dimensional process, we could specify a Gaussian Markov random eld model, independently for each j; j= J0;:::; J? 1. If one further assumes that the wavelet coe cients are also independent within each scale with var( j )? j I; j= J0;:::; J? 1, then ( ) becomes= 2 2 2 a diagonal matrix and= J0;:::; J?1 0 . Therefore, from (12), the DecompShrink rule based on this simple model can be written as: 2^jk=^jk+ 2^j 2 (wjk?^jk ); (13)^j+^

Though the discrete wavelet transform is an excellent decorrelator for a wide variety of stochastic processes, it does not yield completely uncorrelated wavelet coe cients. A natural way to describe this structure is to use scale-dependent multiscale models for covariances ( ). These models take the dependencies, both within scales and across scales, into account. Moreover, if is convolved with noise, the optimal predictor of can be computed e ciently using the change-of-scale Kalman- lter algorithm (Huang and Cressie, 1997b). A multiscale model consists of a series of processes, j; j= J0;:::; J? 1, with the following Markovian structure: j= Aj j?1+ j; j= J0+ 1;:::; J? 1; where? 2 N 0; J0 I; J0? 2 N 0; j I; j= J0+ 1;:::; J? 1; j all random vectors are indepentent from one another, and AJ0+1;:::; AJ?1 are deterministic matrices describing the causal relations between scales.

4.3 Scale-Dependent Models

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