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Mathematical Problems in Engineering
Volume 2013, Article ID 102834, 14 pages
Research Article

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

1Department of Electrical Engineering, Center of Research and Advanced Studies (CINVESTAV), 45019 Guadalajara, JAL, Mexico
2Department of Sciences and Engineering, University of Quintana Roo (UQROO), 77019 Chetumal, QROO, Mexico

Received 13 January 2013; Revised 27 May 2013; Accepted 27 May 2013

Academic Editor: Marcelo Moreira Cavalcanti

Copyright © 2013 Leopoldo Estrada Vargas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 the Variance Plot is described, as well as its close association with the popular Logscale Diagram.