Multiscale autoregressive models and wavelets

The multiscale autoregressive (MAR) framework was introduced to support the development of optimal multiscale statistical signal processing. Its power resides in the fast and flexible algorithms to which it leads, While the M AR framework was originally motivated by wavelets, the link between these t wo worlds has been previously established only in the simple case of the Ha ar wavelet. The first contribution of this paper is to provide a unificatio n of the MAR framework and all compactly supported wavelets as well as a ne w view of the multiscale stochastic realization problem. The second contrib ution of this paper is to develop wavelet-based approximate internal MAR mo dels for stochastic processes. This will be done by incorporating a powerfu l synthesis algorithm for the detail coefficients which complements the usu al wavelet reconstruction algorithm for the scaling coefficients. Taking ad vantage of the statistical machinery provided by the MAR framework, we will illustrate the application of our models to sample-path generation and est imation from noisy, irregular, and sparse measurements.