This is page 1 Printer Opaque this Empirical Bayesian Spatial Prediction Using Wavelets(3)

ABSTRACT Wavelet shrinkage methods, introduced by Donoho and Johnstone

1. Empirical Bayesian Spatial Prediction Using Wavelets

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where ( ) is an n n covariance matrix with structure (depending on parameters ) to be speci ed. In a like manner to the de nition of w in (1),?? we write= ( J0 )0; 0J0;:::; 0J?1 0 and= ( J0 )0; 0J0;:::; 0J?1 0 . Notice that there are hyperparameters; 2; still to be dealt with in the Bayesian model. A couple of comments are worth making. The rst level of the Bayesian model is the so-called data model that incorporates measurement error; indeed, we can write (2) equivalently as,

w=

+;

(4)

where Gau(0; 2 I ). Hence is the signal, which we do not observe because it is convolved with the noise . Our goal is prediction of, which we assume has a prior distribution given by (3). This prior is di erent from other priors used in the literature, in that we assume it to have nonzero 0 ( J0;:::; 0J?1 ). We regard to be a prior parameter to be speci ed, which represents the large-scale variation in . Thus, we may write=+; (5) where is deterministic and Gau(0; ( )) is the stochastic component representing the small-scale variation. The optimal predictor of is E ( j w), which we would like to transform 0 back to the original data space. The inverse transform of Wn is Wn, since Wn is an orthogonal matrix. Hence (4) becomes0 0 0 Wn w= W n+ W n; 0 and because Wn is white-noise measurement error in the data space, 0 represents the signal that we would like to predict. Because S Wn of linearity, the optimal predictor of S (S1;:::; Sn ) is,

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