Wavelet analysis of matrix-valued time-series

The construction of matrix-valued filters for multi-resolution analysis of matrix-valued time-series is studied. Several different designs are explicitly derived corresponding to perfect reconstruction orthonormal filter banks. The resulting matrix-valued 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 x 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.