Wavelet analysis of matrix–valued time–series

Abstract

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.

m-Band orthogonal vector-valued multiwavelets for vector-valued signals

“…For instance, in the application of feature recognition in EEG (electroenceph alography) data, it is necessary to model and analyze vector-valued signals which are also called multichannel signal with possibly intricate correlations among some of these channels [5]. In voice privacy systems a number of signals may need to be transmitted from one point to another, intermixing of the signals before transmission via matrix-valued filters, combine with perfect reconstruction, adds greatly to the likelihood of secure communications [10]. Fluid flows in oceanography and aerodynamics are usually represented as two-dimensional or three-dimensional vector fields in 2-D or 3-D space [11].…”

Wavelet Analysis of Quaternion-Valued Time-Series

“…Many interesting papers on this topic have been published (see Refs. 7,[9][10][11][12]14,19,[21][22][23], for technical books we refer the reader to Refs. 3 and 4.…”

Matrix-Valued and Quaternion Wavelets

“…[30]) concentrate on satisfying the conditions of Proposition 4. As a result, the ad-hoc form of the scaling filters limits the achievable properties.…”

“…If det Unlike [30], we will not include this technical su cient condition in the design. However, in this form it is simple to calculate and check for a given scaling filter, and we found that all constructions in this paper satisfy it.…”

“…This is the case in [8], [9], where only filters of length 3 are considered, in [12], [30] where ad-hoc constructions centered on the eigenvalues of the scaling filter transfer function are used, and in [13] where a symmetry condition is imposed. More general methods of designing MVWs include multichannel lifting schemes [2], [3], the spectral factorisation of interpolatory vector subdivision schemes [11], and directly solving the system of equations imposed by the design constraints [16], [19].…”