The construction of matrix-valued filters for multi-resolution analysis of matrixvalued time-series is studied. Several different designs are explicitly derived corresponding to perfect reconstruction orthonormal filter banks. The resulting matrixvalued scaling functions and wavelet functions are calculated. Two of the derived filter classes have variable parameters providing rich classes of designs. A discrete matrix-valued wavelet transform and inverse transform is fully described and applied to a 2 × 2 time-series of daily bond yields. It is demonstrated that this series can be reconstructed to a very high approximation accuracy using just a small percentage of retained coefficient matrices, both when the retained coefficient matrices are chosen by size of their norm, or by transform level. Possible uses of such matrix-valued multi-resolution analysis include privacy systems and scalable and progressive coding schemes.
Many scientific studies require a thorough understanding of the scaling characteristics of observed Drocesses. We derive and iustifv a decomwjition of the usual aoss-covariancc in terms br scale-by-scale wavelet~cro&covarian&, and describe estimators of both quantities along with asymptotic equivalenas. The scale-by-scale decomposition of autocovariance and crosscovariance sequences a n illustrated via a simulation study. A detailed scale analysis of the surface albedo and temperature of pack ice in the Beaufort Sea shows that the differing resolutions of the two sensors can be seen in the wavelet cross-correlations and autocorrelations at small scales, while at larger scales correlation patterns emerged suggesting fractal-like behaviour over a range of scales.
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