On preconditioning the data for the wavelet transform when the sample size is not a power of two

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Abstract
A powerful and efficient method for nonparametric regression involves taking the discrete wavelet transform (DWT) of data, shrinking the resulting wavelet coefficients, and then computing the inverse wavelet transform to get an estimate of the regression function. Currently, most wavelet decomposition software packages require that the original set of data have sample size n equal to a power of two in order to achieve an exact orthogonal wavelet transform. In statistical data analysis, such is rarely the case, so in an effort to broaden the applicability of such methods, various ways of preconditioning data not meeting this restriction are discussed and compared. These results illustrate the important point that wavelet coefficients resulting from preconditioned data should never be thrown blindly into a threshold selection procedure which depends on the coefficients being independent with equal variance. Such procedures can still be used, but great care must be taken to choose an appropriate preconditioning method. Also, the resulting wavelet vector can certainly be variance-corrected (with only rather light computational burden) before a thresholding procedure is applied to it. Some of the correlation can also be removed, though this is certain to be quite computationally expensive. (The coefficients can never, of course, be completely orthogonalized, however, since n < 2J.) It should be pointed out here that the paper by Cohen, Daubechies, and Vial (1993) constructing wavelets on an interval contains the theoretical developments necessary to compute the wavelet transform on data with any sample size n, At this writing, however, the only available, implementation of this scheme (to the author's knowledge) has been only for the case in which n is a power of two.

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