A Study of Wavelet Analysis and Data Extraction from Second-Order Self-Similar Time Series

Statistical analysis and synthesis of self-similar discrete time signals are presented. The analysis equation is formally defined through a special family of basis functions of which the simplest case matches the Haar wavelet. The original discrete time series is synthesized without loss by a linear combination of the basis functions after some scaling, displacement, and phase shift. The decomposition is then used to synthesize a new second-order self-similar signal with a different Hurst index than the original. The components are also used to describe the behavior of the estimated mean and variance of self-similar discrete time series. It is shown that the sample mean, although it is unbiased, provides less information about the process mean as its Hurst index is higher. It is also demonstrated that the classical variance estimator is biased and that the widely accepted aggregated variance-based estimator of the Hurst index results biased not due to its nature (which is being unbiased and has minimal variance) but to flaws in its implementation. Using the proposed decomposition, the correct estimation of theVariance Plotis described, as well as its close association with the popularLogscale Diagram.