Local Analysis of Long Range Dependence Based on Fractional Fourier Transform

Abstract

The long range dependence (LRD) of stationary process is characterized by the Hurst parameter. In practice, previous methods for estimation of the Hurst parameter might have poor performance when processing the non-stationary time series or trying to distinguish the slight difference between very long stochastic processes. This paper explores the use of fractional Fourier transform (FrFT) for estimating the Hurst parameter. The time series was processed locally to achieve a reliable local estimation of the Hurst parameter. The biocorrosion signal which is very popular in biological engineering was studied as an example to show the long range dependence properties. After comparing with the commonly used wavelet based method and another method based on Matlab's polyfit, the new Hurst parameter estimator proposed in this paper is proved to be more robust for non-stationarity and can show the slight difference clearly between those very long sets of biocorrosion data.Index Terms-biocorrosion signal, fractional Fourier transform, Hurst parameter, long range dependence, parameter estimation

An improved Hurst parameter estimator based on fractional Fourier transform

“…To our best knowledge, the only related reference is [20] where experimental determination of Hurst exponent of the self-affine fractal patterns using optical fractional Fourier transform was attempted. Interestingly, it is shown in this paper that FrFT has a strong relationship with wavelet transform which is believed to be very suitable for analyzing LRD [21]. Since FrFT has been shown to have a computational complexity proportional to the wavelet transform, the performance improvements may come without additional cost.…”

“…In Sect. 2, the FrFT based Hurst parameter estimation method is exploited, and the relationship between the FrFT and wavelet transform has been derived correctly which was presented in [21] with typos. In Sect.…”

An Overview of Fractional Order Signal Processing (FOSP) Techniques

“…In [47] and [48], we proposed to use a fractional Fourier transform (FrFT) based estimator. It uses the spectrum calculated by FrFT for estimation.…”

Modeling and Prediction of Great Salt Lake Elevation Time Series Based on ARFIMA

“…A process is said to have long-range dependence when 0.5 < H < 1 [15]. Many methods have been proposed for Hurst parameter estimation like rescaled range (R/S) analysis [16], aggregated variance method [17], absolute value method [18], variance of residuals method [19], local Whittle method [20], periodogram method [21], wavelet-based method [12] and fractional Fourier transform (FrFT) based method [22]. The following methods will be used for the validation of LRD in GSL elevation time series.…”