A Comparison of Analytical Methods for the Study of Fractional Brownian Motion

Fractional Brownian motion (FBM) provides a useful model for many physical phenomena demonstrating long-term dependencies and l/f-type spectral behavior. In this model, only one parameter is necessary to describe the complexity of the data, H, the Hurst exponent. FBM is a nonstationary random function not well suited to traditional power spectral analysis however. In this paper we discuss alternative methods for the analysis of FBM, in the context of real-time biomedical signal processing. Regression-based methods utilizing the power spectral density (PSD), the discrete wavelet transform (DWT), and dispersive analysis (DA) are compared for estimation accuracy and precision on synthesized FBM datasets. The performance of a maximum likelihood estimator for H, theoretically the best possible estimator, are presented for reference. Of the regression-based methods, it is found that the estimates provided by the DWT method have better accuracy and precision for H > 0.5, but become biased for low values of H. The DA method is most accurate for H < 0.5 for a 256-point data window size. The PSD method was biased for both H < 0.5 and H > 0.5.