554 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 3, MARCH 2002 A Fast Estimation Algorithm on the Hurst Parameter of Discrete-Time Fractional Brownian Motion Yen-Ching Chang and Shyang Chang, Member, IEEE Abstract—The purpose of this paper is to demonstrate that the discrete-time fractional Gaussian noise (DFGN) is a regular process. Based on this property, a fast algorithm with low com- putational cost is proposed to estimate the Hurst parameter , which is an important parameter in fractional Brownian motion (FBM). This algorithm is robust under amplitude shift, invariant to time shift, and unaffected by a scaling factor in power spectral density (PSD). Finally, the computational complexity is also investigated. Index Terms—Discrete-time fractional Brownian motion, dis- crete-time fractional Gaussian noise, fractional Brownian motion, fractional Gaussian noise, Hurst parameter, regular process. I. INTRODUCTION R ECENTLY, interest in describing a vast number of phenomena in nature and biomedical sciences have attracted the attention of signal processing researchers. These objects often exhibit a strong dependence of long-term correla- tions. Based on the ordinary Brownian motion, the fractional Brownian motion (FBM), which is designated as a family of Gaussian random functions, can be invoked for description. Among the properties of FBM, self-similarity [1]–[4] can be frequently observed in natural phenomena and biomedical signals. In order to characterize the property, the fractal dimen- sion or Hurst parameter is usually adopted. These two parameters are connected by the relationship for a line-to-line function [1]. Since the parameter of FBM can describe many compli- cated natural phenomena, it is essential for us to estimate this parameter from realizations of FBM. In recent years, the FBM model and its increment process have been vastly studied in wavelet theory [5]–[9]. The variances of wavelet coefficients for FBM satisfy a power law. The Hurst parameter is then estimated by the power law. However, the estimator will be affected by the chosen wavelet orthonormal basis [8]. In addition, many other estimators have been proposed [10]–[20]. Among these estima- tors, one is the so-called variance method [10]. The Hurst pa- rameter can be estimated from the slope of a line fitted to the data for several lags , where denotes the discrete-time FBM (DFBM). However, the estimator is different for different lags’ range. Some extra effort Manuscript received May 8, 2000; revised November 26, 2001. This work was supported in part by the National Science Council of the R.O.C. under Grant NSC90-2213-E-007-047. The associate editor coordinating the review of this paper and approving it for publication was Dr. Athina Petropulu. The authors are with the Department of Electrical Engineering, Na- tional Tsing Hua University, Hsinchu, Taiwan, R.O.C. (e-mail: shyang@ ee.nthu.edu.tw). Publisher Item Identifier S 1053-587X(02)01339-9. is required for the lower and upper bound of the scale. Another algorithm, which is referred to as the box-counting method [11], [14], [18], [20], can be estimated from the slope of a line fitted to the data for several box sizes [20], where denotes the number of box with size . Similarly, this method will suffer from the problem of selecting the suit- able box sizes. A maximum likelihood estimator (MLE) [12] has also been proposed. This method has no closed form and demands a large amount of time for numerical searching. Moreover, the conver- gence is extremely slow when the data size is large. Hence, the MLE method is not recommended for real time estimation. Due to the nonstationarity of the FBM process, its spectrum, in general, does not exist. However, the (the differential of the FBM) has a spectral density proportional to . This fact suggests that (the FBM) can be regarded as having a “spectral density” proportional to [21]. This further leads to the consideration of fractional Gaussian noise (FGN), which is a stationary process. For real data, one usually sam- ples a continuous-time FBM process. In this paper, the sampled process is referred to as DFBM and its increment discrete-time fractional Gaussian noise (DFGN). Autoregressive moving average (ARMA) models are fre- quently used in the discipline of statistical signal processing [22]. Their parameters can be estimated from non-Bayesian point of view [23]–[25]. If prior information is available, the Bayesian approach can be adopted [26], [27]. However, the order of ARMA models will need special consideration; otherwise, it will provide very poor representations for signals that exhibit long-term dependence between observations. In order to derive reasonable PSD of DFGN and avoid the computational complexity on MLE, an approximate MLE method has been proposed [13]. This approach applies Wold’s decomposition theorem to decompose DFGN into two uncorre- lated processes: One is regular and the other singular [28]–[30]. This algorithm intrinsically is an approximate MLE and faster than MLE since it does not require numerical searching. In this paper, we intend to show that DFGN is regular. According to this property, a relatively fast estimator is pro- posed. It is an approximate MLE but is quicker than that of [13]. This method is extremely less sensitive to and is robust to affine transformation and time shift on the data. Moreover, experimental results show that a smaller variance will result as the data length increases. This algorithm is much more robust and easier to implement than MLE. From the point of view of computational complexity together with robustness, our method is considerably more attractive for the estimation of Hurst parameter . 1053–587X/02$17.00 © 2002 IEEE