Abstract
The signals in numerous fields usually have scaling behaviors (long-range dependence and self-similarity) which is characterized by the Hurst parameter H. Fractal Brownian motion (FBM) plays an important role in modeling signals with self-similarity and long-range dependence. Wavelet analysis is a common method for signal processing, and has been used for estimation of Hurst parameter. This paper conducts a detailed numerical simulation study in the case of FBM on the selection of parameters and the empirical bias in the wavelet-based estimator which have not been studied comprehensively in previous studies, especially for the empirical bias. The results show that the empirical bias is due to the initialization errors caused by discrete sampling, and is not related to simulation methods. When choosing an appropriate orthogonal compact supported wavelet, the empirical bias is almost not related to the inaccurate bias correction caused by correlations of wavelet coefficients. The latter two causes are studied via comparison of estimators and comparison of simulation methods. These results could be a reference for future studies and applications in the scaling behavior of signals. Some preliminary results of this study have provided a reference for my previous studies.
Highlights
The signals in numerous fields usually have scaling behavior which has been recognized as a key property for data characterization and decision making
According to the multiresolution analysis (MRA), the wavelet coefficients can be calculated by fast pyramidal algorithm
This section focuses on the numerical study of commonly used wavelet-based estimator which still lacks of comprehensive and detailed numerical study on estimation of fractal Brownian motion, especially on its empirical bias and the selection of parameters
Summary
The signals in numerous fields usually have scaling behavior (long-range dependence and self-similarity) which has been recognized as a key property for data characterization and decision making (see e.g., [1,2,3,4,5]). Fractal Brownian motion and its increments (fractional Gaussian noise (FGN)) play important roles in modeling signals with self-similarity and long-range dependence Most studies on this issue are based on FBM. Despite extensive studies of standard wavelet-based estimator proposed by Abry et al, there is still a lack of comprehensive and detailed numerical simulation study on fractal Brownian motion, especially for the selection of parameters and the empirical bias. Combining with results of parameters selection, this paper analyzes the above three causes of empirical bias in the case of FBM via comparison of estimators and comparison of simulation methods.
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