Function approximation methods based on frames or other overcomplete dictionaries of approximating functions offer advantages over the orthogonal schemes due to the fact that the associated redundancy may lead to better de-noising and reconstruction power. Wavelet packets represent special wavelet frames; they combine overcompleteness with high time-frequency localization power through an optimal frequency-then-time segmentation. Compared to cosine packets, which enable optimal adaptation through time-then-frequency segmentation, wavelet packets show a different time-frequency resolution trade-off that might be useful for analyzing some kinds of non-stationary phenomena. We study the properties of covariance non-stationary stochastic processes whose realizations are observed at very high frequencies; the data are supplied by time series of a stock market return index. For these complex processes the effectiveness of wavelet and cosine packets is explored by implementing entropic optimization, greedy approximation techniques and dimension reduction methods.
Periodic wavelet frames and time–frequency localization
