This is page 1 Printer Opaque this Empirical Bayesian Spatial Prediction Using Wavelets(2)

ABSTRACT Wavelet shrinkage methods, introduced by Donoho and Johnstone

2

Hsin-Cheng Huang, Noel Cressie

following separable form: (x; y)= (x) (y); x; y 2 IR; and there are three wavelet functions given by, (1) (x; y)= (x) (y); (2) (x; y)= (x) (y); (3) (x; y)= (x) (y): For j; k1; k2 2 Z, write Z 2j (2j x? k1; 2j y? k2 ); j;k1;k2 (x; y ) (m) 2j (m) (2j x? k1; 2j y? k2 ); m= 1; 2; 3: j;k1;k2 (x; y ) Then any function g 2 L2 IR2 can be expanded as g(x; y)=X?

k1;k2

ck1;k2 J0;k1;k2 (x; y)+

1 3 X X Xn

j=J0 k1;k2 m=1

o ) ) d(m1;k2 (m1;k2 (x; y): j;k j;k

Because of this direct connection between one-dimensional wavelets and spatial wavelets, we shall present most of the methodological development in IR. However, in a subsequent section, we do give an application of our wavelet methodology to two-dimensional spatial prediction of an image. Wavelets have proved to be a powerful way of analyzing complicated functional behavior because in wavelet space, most of the"energy" tends to be concentrated in only a few of the coe cients fcJ0;k g; fdj;k g. It is interesting to look at the statistical properties of wavelet expansions; that is, if f (

) is a random function in L2 (IR), what is the law of its wavelet coe cients? We shall formulate this question more speci cally in terms of the discrete wavelet transform, which we shall now discuss. Suppose that we observe Y ( ) at a discrete number n= 2J points; that is, we have data Y= (Y1;:::; Yn ), where Yi= Y (ti ) and ti= i=n; i= 1;:::; n. The discrete wavelet transform matrix Wn of Y is an orthogonal matrix such that (1) w?(wJ0 )0; wJ0 0;:::; w0J?1 0= WnY is a vector of scaling function coe cient at scale J0 and wavelet coe cients at scales J0;:::; J? 1 (Mallat, 1989). Thus, if Y is random, so too is w. In all that is to follow, we shall construct probability models directly for w, although it should be noted that if Y ( ) is a stationary process, then wJ0 and fwj: j= J0;:::; J? 1g are also stationary processes, except for some points near the boundary (Cambanis and Houdre, 1995). We assume the following Bayesian model: wj; 2 Gau(; 2 I ); (2)? j; Gau; ( ); (3)

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