An improved Hurst parameter estimator based on ... - Springer Link

Telecommun Syst (2010) 43: 197–206 DOI 10.1007/s11235-009-9207-4

An improved Hurst parameter estimator based on fractional Fourier transform YangQuan Chen · Rongtao Sun · Anhong Zhou

Published online: 23 October 2009 © Springer Science+Business Media, LLC 2009

Abstract A fractional Fourier transform (FrFT) based estimation method is introduced in this paper to analyze the long range dependence (LRD) in time series. The degree of LRD can be characterized by the Hurst parameter. The FrFTbased estimation of Hurst parameter proposed in this paper can be implemented efficiently allowing very large data set. We used fractional Gaussian noises (FGN) which typically possesses long-range dependence with known Hurst parameters to test the accuracy of the proposed Hurst parameter estimator. For justifying the advantage of the proposed estimator, some other existing Hurst parameter estimation methods, such as wavelet-based method and a global estimator based on dispersional analysis, are compared. The proposed estimator can process the very long experimental time

Submitted June 2008. Revised Oct. 2008. For final submission to “Special Issue on Traffic Modeling, Its Computations and Applications” of Telecommunication Systems. Guest Editors: Professors Ming Li and Pierre Borgnat. Y. Chen () Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical and Computer Engineering, Utah State University, 4160 Old Main Hill, Logan, UT 84322, USA e-mail: [email protected] R. Sun Phase Dynamics, Inc., 1251 Columbia Dr., Richardson, TX 75081, USA e-mail: [email protected] A. Zhou Department of Biological and Irrigational Engineering, Utah State University, 4105 Old Main Hill, Logan, UT 84322-4105, USA e-mail: [email protected]

series locally to achieve a reliable estimation of the Hurst parameter. Keywords Fractional Fourier transform · Fractional Gaussian noise · Hurst parameter · Long-range dependence · Wavelets

1 Introduction The first model for long range dependence was introduced by Mandelbrot and Van Ness (1968) in terms of fractional Brownian motions, and has since been extensively investigated. Long-range dependent (LRD) processes are characterized by their auto-covariance functions. In LRD processes, there is a strong coupling between values at different times. This indicates that the decay of the autocovariance function is hyperbolic and slower than exponential decay, and that the area under the function curve is infinite. Consider a second order stationary time series Y = Y (k) with zero mean as an example. The time series Y is said to be long-range dependent provided its autocovariance rY (k) = Cov(Y (k), Y (0)) = E(Y (k)Y (0)) decays slowly as a power law function of the lag k, so that the series k rY (k) is not summable [1]. According to the research in recent years, financial data [2], communications networks data [3–6], video traffic [7] and biocorrosion noise [8–10] can exhibit long range dependence. The Hurst parameter H [11] characterizes the degree of LRD. The LRD is typically modeled by supposing a power law decay of the spectral density and H is equal to (1 + α)/2 with α being the power of the decay function [3]. A process is said to have long range dependence when 0.5 < H < 1 [12]. Many methods have been proposed for Hurst parameter estimation like wavelet-based [13], local Whittle [14],

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R/S analysis [15, 16], periodogram methods [17], dispersional analysis method [2] and so on. All existing estimation methods have their limitations in terms of estimation accuracy and efficiency. In practice, it is quite important to estimate the Hurst parameter of LRD time series with low bias and high efficiency. In this paper, we propose to use fractional Fourier transform (FrFT) [18, 19] for Hurst parameter estimation. 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. Besides, the FrFT based Hurst estimation method in this paper implements a set of windows over the spectrum to process the LRD time series locally. Local deviation from long range dependence could be visualized by the local analysis. Two popular methods are chosen for comparison with the FrFT based estimator proposed in this paper. The first one is the LASS tool [13] based on wavelet spectrum which has been well developed by Stilian Stoev (2004). Another method utilized the so-called dispersional analysis [22] and globally estimated the Hurst parameter over the whole data set. The dispersional analysis, also known as the Aggregated Variance method [23], averages the differenced time series and calculates the variance of the averaged dataset. The Hurst parameter estimation based on FrFT outperforms these two methods in our comparison experiments. Fractional Gaussian noise (FGN) [24] is used to test the FrFT based estimator with known Hurst parameters. FGN is derived from fractional Brownian motion (FBM) [25] which is a long-range dependent Gaussian process with Hurst parameter H and with stationary increments. The FBM plays a fundamental role in modeling long-range dependence. In practice, it is its increments that are used in modeling. The increments time series GH (k) = BH (k) − BH (k − 1), k ∈ Z of the FBM process BH are called FGN. Figure 1 shows an example of FGN with sample size 1,000,000 and filter length 1,000,000. The rest of the paper is organized as follows. 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. 2, we also discuss the local analysis for estimating the Hurst parameter and the effect of nonstationarity. In Sect. 3, the proposed FrFT-based Hurst parameter estimator is validated and compared using benchmark fractional Gaussian noises. It has been compared with wavelet based method and dispersional analysis method to

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Fig. 1 An example of FGN with pre-determined Hurst parameter of 0.8 and the sample size 1,000,000

illustrate the benefits gained from using the proposed estimation method. Conclusions are drawn in Sect. 4.

2 Hurst parameter estimation based on FrFT 2.1 Definition of long range dependence and Hurst parameter Consider a stationary stochastic process Y = f (u). Y is said to be long-range dependent if it can be modeled by a powerlaw decay of the autocorrelation: rY (τ ) = E[f (u)f (u − τ )] ∼ cY |τ |−γ , τ → ∞,

0 < γ < 1,

(1)

where ‘∼’ means the ratio of the left- and the right-hand sides converges to 1. Imposing the condition (1) on the spectral density sY of Y , as ξ → 0 we get sY (ξ ) ∼ cs |ξ |−α ,

0 < α < 1,

(2)

1  where cs > 0 and sY (ξ ) = (2π)− 2 τ ∈Z eiξ τ rY (τ ). With smoothness assumptions, α and γ have the following relationship

α = 1 − γ.

(3)

Since the Hurst parameter is related with α by H = (1 + α)/2 and α could be estimated by the spectral density sY according to (2), the Hurst parameter H could then be estimated according to sY .

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= F (−ξ ),

(6)

which corresponds to a rotation of the representation axis by 3π/2. If we apply one more F , we should get another rotation over π/2, which brings us back to the original time axis. Therefore, F 4 (f ) = f

Fig. 2 Rotation concept of Fourier transform

2.2 Basics of FrFT Fractional Fourier transform may be considered as a fractional power of the classic Fourier transform [18]. The first idea of fractional power of the Fourier operator appears in 1929 [26]. Like the complex exponentials are the basis functions in Fourier analysis, the chirps (signals sweeping all frequencies in a certain frequency interval) are the basis in fractional Fourier analysis. Since FrFT can provide a richer picture in time-frequency analysis [27] and it is nothing more than a variation of the standard Fourier transform [28], it can improve the performance of some signal processing applications. Like the time and frequency variables in the timefrequency plane, let x be the variable along the x-axis and ξ is the variable along the ξ -axis (Fig. 2). Let xa and ξa represent the rotated variables x and ξ , respectively. We have    cos αF xa = ξa − sin αF

sin αF cos αF

  x , ξ

(4)

where αF = aπ/2 is the rotating angle. It is always assumed that ξa = xa+1 . Therefore, xa and ξa are always orthogonal. If f (x) is a “time” signal of the variable x, it lives on the horizontal axis. Its Fourier transform (FT) F (f (x)) = F (ξ ) is a function of the frequency ξ , and hence it lives on the vertical axis. Fourier transform changes signal f (x) in time domain x to F (ξ ) in frequency domain ξ , which corresponds to a counterclockwise rotation of an angle π/2 in the (x, ξ ) plane. Since applying FT twice to f (x) results in  ∞ 1 2 (F f )(x) = (F (F f ))(x) = √ F (ξ )e−iξ x dξ 2π −∞ = f (−x),

(5)

F 2 is called the parity operator. Thus, the time axis is rotated by an angle π for F 2 . Similarly, it follows that  ∞ 1 3 2 (F f )(ξ ) = (F (F f ))(ξ ) = √ f (−x)e−iξ x dx 2π −∞

or F 4 = I.

(7)

In conclusion, the FT operator corresponds to a rotation of the axis by an angle π/2 in the time-frequency plane. There are six definitions of FrFT [18]. The most intuitive way to define FrFT is by generalizing the rotation concept of classical FT. Like FT corresponds to a rotation in the timefrequency plane over an angle αF = π/2, the FrFT will correspond to a rotation over an arbitrary angle αF = aπ/2 with a ∈ R. For a more formal definition, FrFT can be defined through eigenfunctions [29]. 2.3 FrFT-based Hurst parameter estimation process—main result Now, to derive our main result, let the a-th order of fractional Fourier transform of a signal f (u) be denoted by  ∞ fa (ξ ) = Ka (ξ, u)f (u) du, (8) −∞

where Ka (ξ, u) =√AαF exp[iπ(cot αF ξ 2 − 2 csc αF ξ u + cot αF u2 )], AαF = 1 − i cot αF , αF = aπ/2. Making the change of variable by j = ξ sec αF and denoting the left-hand side of (8) by g(j ) = fa (j/ sec αF ), we can obtain the following result: g(j ) = C(αF )e−iπj sin αF      ∞ j −u 2 × exp iπ f (u) du, (9) 1 −∞ tan 2 αF √ where C(αF ) = 1 − i cot αF exp(iπj 2 ) is a constant that depends on αF only. Note that (9) has certain similar characteristics of a wavelet transform. The continuous wavelet transform [30] is defined as  ∗ WT f (j, k) = f (u)ψj,k (u) du, 2

2

 u−j , ψ ψj,k (u) = 1 k |k| 2 1



(10)

where j, k, ψ(t), ψj,k (t) show scale, scan time, mother wave and wavelet, respectively. Let the mother wave ψ(t) = exp(iπt 2 ), the wavelet transform in (10) can be changed to  1 (11) exp(iπ(u − j )2 /k 2 )f (u) du. WT f (j, k) = 1 2 |k|

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Clearly, (11) has the same form as FrFT in (9) except the 1 scaling factor. Let k = tan 2 αF , (11) becomes 

1

WT f (j, k) =

1

| tan αF | 4

    j −u 2 f (u) du exp iπ 1 tan 2 αF

exp[iπj 2 sin2 αF ]

=

1

C(αF )| tan αF | 4

g(j ),

(12)

which establishes the relationship between FrFT and wavelet transform. For wavelet ψj,k (t), f (u) has the expansion [13]  dj,k (f )ψj,k (u), (13) f (u) = j ∈Z k∈Z

where dj,k (f ) are wavelet coefficients of the function f (u). Let εj denote the mean energy of the wavelet coefficients at the scale j , that is 2 (f )]. εj = E[dj,k

(14)

Given a finite sample Y = f (k), k = 1, 2, . . . , Nj , a triangular array of approximate wavelet coefficients can be obtained by Mallat’s algorithm [14]. Thus, one can estimate log2 (εj ) by using the sample energy of these coefficients: log2

Nj 1  2 2 WT f (j, k) ≈ log2 (εj ) = log2 (E[dj,k (f )]). Nj k=1

√ (ξ ) = 2π eiξ t ψ(t)dt denotes the Fourier transwhere ψ R form of the function ψ . The expression in (18) relates the mean energy εj of the wavelet coefficient dj,k (Y ) to the spectral density of the stationary signal Y (t). For large scales j , the function sY (η/2j ), η ∈ R can be viewed as a zoomed version of the spectral density sY (η) around the zero frequencies. Therefore, the integral in the right-hand side of (18) picks out the spectral behavior of Y . Substituting (2) into (18) yields, as j → ∞  2 (η)|2 |η/2j |−α dη = cf C2j α , Edj,k (Y ) ∼ cf |ψ (19) R

G(j ) = log2 (E[g 2 (j )])   1  C(αF )| tan αF | 4 2 2 = log2 E[d (f )] . j,k exp[iπj 2 sin2 αF ]

(16)

G(j ) ∼ (2H − 1)j + const.

j2  1 [j1 ,j2 ] = 1 H ωj G(j ) + , 2 2

By using the Parseval identity and a change of variables, it can be shown that   2 ψj,k (t) ψj,k (s)rY (t − s)dsdt Edj,k (f ) = R

= =2



= R

j =j1

ωj = 0 and

j = 1, 2, . . . , J − 1. (22)

In (22), G(j + 1) − G(j ) is the local slope of the FrFT spec [j,j +1] should be very close to H at trum. From (20), the H [j1 ,j2 ] in (21) can be expressed large scale j . Therefore, the H as j 2 −1

[j,j +1] , vj H

(23)

j =j1

(2j ξ )|2 sY (ξ ) dξ |ψ

R

(η)|2 sY (η/2j ) dη, |ψ

 j2

Using the above established FrFT based Hurst parameter es [j,j +1] can be given by timation method, H

[j1 ,j2 ] = H

 |ψ j,k (ξ )| sY (ξ ) dξ

j



where the weights ωj ’s are such that j2 j =j1 j ωj = 1.

R 2

R

(21)

j =j1

(17)

R



(20)

By using a linear regression of the log-scale FrFT spectrum G(j ) on the scale j , where 1 < j1 < j2 < J , with J the total number of scales, the Hurst parameter H of Y can be estimated as

[j,j +1] = G(j + 1) − G(j ) + 0.5, H 2

 eiξ τ rY (τ ) dτ.

1 − cot αF ×

2.4 Local analysis

Indeed, 1 sY (ξ ) = √ 2π

√

depends on αF . Therefore, according to H = (1 + α)/2 and the relation2 in (16), ship between log-scale FrFT spectrum and Edj,k the Hurst parameter can be estimated using the fact that

(15) According to (12) and (15), the log-scale FrFT spectrum could be derived as

2 −α R |ψ (η)| |η| dη. αF |1/4 Note that in (16), C(αF )| tan = exp[iπj 2 sin2 αF ] 2 2 1/4 exp(iπj cos αF )| tan αF | , and it only

where C = C(ψ, α) =

(18)

where vj = ωj +1 + · · · + ωj2 . Besides, the stochastic process Y = f (k) (k = 1, . . . , N ) can be divided into some non-overlapping windows for local analysis. Assume that the window size is w (w ≤ N ),

An improved Hurst parameter estimator based on fractional Fourier transform

Fig. 3 Wavelet spectrum of the electrochemical noise on scale j

Yr (r = 1, . . . , N/w) is the time series corresponding to the rth window and the size of the last window is N −w(N/w − 1) ≥ w. Then the FrFT spectrum can be computed within each window to obtain a matrix G, the (j, r)th element of which is defined as

Nj 1  2 gj (Yr ) , (24) Gj (r) = log2 Nj k=1

where gj (Yr ) is the FrFT coefficients of time series Yr . 2.5 Effect of nonstationarity The main reason of utilizing local analysis is that most of the Hurst parameter estimation methods are based on the assumption that the time series is stationary [17]. Nonstationarity will result in poor performance or even fail the estimator [7]. Stove et al. (2004) have shown that nonstationarity effects such as abrupt shifts in the mean would yield a steep wavelet spectrum and overestimate the Hurst parameter [13]. Figures 3 and 4 are wavelet spectrum and FrFT spectrum of the electrochemical noise calculated on scale j , respectively. The testing electrochemical noise is the corrosion potential of the stainless steel electrode with bacteria attached in the artificial saliva called Jenkin’s Solution [16]. The electrode is put in the solution for 24 hours at room temperature. The sampling period is 0.5 second. The Hurst parameter of the biocorrosion data, estimated over the range of scales j , is about 0.89. Figure 3 presents the local wavelet spectrum of the biocorrosion signal in each window as described in (24). Figure 4 presents its local fractional Fourier spectrum in each window. As shown in the plots, both wavelet spectra and FrFT spectra are consistent with long range dependence. However, Fig. 3 shows that at the large scales, there are some obvious variabilities in the

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Fig. 4 FrFT spectrum of the electrochemical noise on scale j

wavelet spectra. When 6 < j < 7, there are also a few variabilities for FrFT spectra in Fig. 4. These variabilities indicate the existence of nonstationarity [13] and may affect the global analysis of long range dependence for Hurst parameter. Nevertheless, compared to wavelet spectrum, FrFT spectrum is more robust to nonstationarity. Therefore, FrFT is more suitable for Hurst parameter estimation of the time series with some parts nonstationary if local analysis is used. In the next section, we will present the detailed validation test for the proposed FrFT based Hurst parameter estimator using the benchmark fractional Gaussian noise.

3 Validation and comparisons using fractional Gaussian noise 3.1 White noise Before implementing the FrFT-based estimator, the right fractional order of FrFT should be considered. It is because the mean value of the local Hurst parameter estimations will be taken as the estimated Hurst parameter, while different fractional orders of FrFT result in different variances of local Hurst parameter estimations. Figure 5 shows the variances of the Hurst parameter estimations with the order of FrFT a varied from 0.1 to 1.0. We use different orders of FrFT to estimate the Hurst parameter of the FGN. Then, the statistic variance of each estimation is calculated correspondingly. In the end, the “order-variance” plot is plotted. The white noise is generated by Matlab using sample size 1,000,000. As shown in Fig. 5, the minimum variance appears at a = 0.6. Therefore, 0.6 will be chosen as the order of FrFT for estimating the Hurst parameter of the white noise. The results are compared with wavelet based local analysis.

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Fig. 6 White noise passing through a first order system

Fig. 5 The variances of Hurst parameter estimations with different orders of FrFT for white noise

The Hurst parameter of a random white noise should be 0.5 [31]. In our experiment, the estimation by the FrFTbased estimator is 0.50977, while the estimation by waveletbased estimator is 0.47122. The results show that the FrFTbased estimator is more accurate.

Fig. 7 Autocorrelation of the output of the first order filter Fig. 8 FGN by fractional integrator

3.2 Fractional Gaussian noise In order to test the accuracy of the Hurst parameter estimation and prevent the bias from the FGN generating algorithm, we use two methods to generate FGN for comparison. 3.2.1 Fractional integrator method Before applying the fractional operator, the simple integer order system shown in Fig. 6 is implemented for an 1 overview. The transfer function of the system is H (s) = s+c −ct with the impulse response h(t) = e u(t) with u(t) the unit step function. Let the input be a continuous-time white noise ω(t) with variance σ 2 . We have the output signal y(t) as follows:  t y(t) = ω(τ )e−c(t−τ ) dτ. (25) −∞

The autocorrelation of the output y(t) is R(τ ) = σ 2 h(τ ) ∗ h(−τ )  ∞ = σ2 ecτ u(τ )e−c(t−τ ) u(t − τ ) dτ −∞

=

σ2 2c

e−c|τ | .

(26)

Since the outputs of a conventional integer order system do not have long-range dependence, their autocorrelation

should decay very fast as an exponential law distribution shown in Fig. 7. In this figure, the solid (blue) line is the autocorrelation of the output of the first order system, while the dashed (red) line is the fitting exponential law function according to (26). The c used for fitting the autocorrelation of the output is almost the same as the c used for generating y(t) and obtaining R(τ ). As explained in [32–34], the FGN can be considered as the output of a fractional integrator. The system can be defined by the transfer function H (s) = s ν = s1β , ν = −β with −ν−1

t impulse response h(t) = (−ν) , ν < 0. β is a real number. The system block diagram is shown in Fig. 8. Let ω(t) be a continuous-time white noise with variance σ 2 , the νth order fractional noise yν (t) can be expressed as  t 1 yν (t) = ω(τ )(t − τ )−ν−1 dτ, (27) (−ν) −∞

where y0 (t) = ω(t). Its autocorrelation function [35], R(τ ) = σ 2

|τ |−2ν−1 . 2(−2ν) cos(νπ)

(28)

An improved Hurst parameter estimator based on fractional Fourier transform

To obtain this function, we compute the following convolution. R(τ ) = σ 2 h(τ ) ∗ h(−τ )  ∞ 1 2 t −ν−1 (t + τ )−ν−1 u(t + τ ) dt =σ 2  (−ν) 0 = σ2

|τ |−2ν−1 B(1 + 2ν, −ν) ,  2 (−ν)

(29)

where B(x, y) is the beta function. As B(x, y) = (x)(y) (x+y) and (z)(1 − z) = sinππz , we obtain (28). Also, we can get its power spectrum S(ξ ) = σ 2 |ξ |2ν ,

(30)

Fig. 9 Autocorrelation of the FGN produced by fractional integrator

Fig. 10 The 1/f noise spectrum

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which has the same form as (2). Therefore we have H = 1+(−2ν) 1+α = 12 − ν. Because 0.5 < H < 1 indicates 2 = 2 long-range dependence, correspondingly −0.5 < ν < 0. The autocorrelation of the output FGN is shown in Fig. 9. The solid (blue) line is its computed autocorrelation and the dashed (red) one is the power law fitting function based on (28). The order ν of both plots are very close, ν = −0.3. As we can see, its autocorrelation decays slowly as a power law function, which proves the existence of long-range dependence. In our LRD experiment, the sample size of the input white noise is 100,000. The spectrum of a sample FGN is illustrated in Fig. 10. With the proper order of FrFT which results in minimum variance of corresponding local estimations, the Hurst parameters of the output fractional Gaussian noises will be estimated according to the FrFT-based Hurst parameter estimator. One hundred FGN with their Hurst parameters in the range of 0.01 to 1.00 with the interval 0.01 will be generated for testing. Besides, local waveletbased method and aggregated variance method will be implemented for comparison. In Fig. 11, the x-axis is the value of Hurst parameters of FGN, the y-axis is the estimated value of the Hurst parameters. There is an ideal estimation plot of y = x for reference. As we can see, the estimation by FrFT-based estimator is the closest to the ideal estimation line except some Hurst parameters very close to 1.0. For 0.5 < H < 1 which indicates long-range dependence, our FrFT-based estimator gives out satisfactory estimations. Besides, as shown in this plot the estimation methods utilizing local analysis including FrFTbased method and wavelet-based method have better performance than that of the aggregated variance method which utilizes global analysis. For 0.01 < H < 0.5, the results of

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Fig. 11 Comparison of three Hurst parameter estimation methods for FGN produced by fractional integrator with 100 Hurst parameters from 0.01 to 1.00

Fig. 12 Comparison of three Hurst parameter estimation methods for FGN produced by symmetric moving average filter with 100 Hurst parameters from 0.01 to 1.00

both three estimation methods are becoming greater than the real value. These results may suggest some bias of the FGN generating method as well as the Hurst parameter estimation methods. Therefore, in the next section, we use symmetric moving average filter to generate the paths of FGN for examination. 3.2.2 Symmetric moving average filter method In this section, the paths of fractional Gaussian noise are generated by using a truncated symmetric moving average filter. The filter coefficients are computed via IFFT of the square root of the FFT of the covariances of the FGN. The moving average is also computed by using the FFT algorithm.

Similar to the experiment in the previous sections, 100 H in the range of 0.01 to 1.00 with the interval 0.01 will be tested for Hurst parameter estimation of FGN generated by moving average filter. The sample size of FGN is 100,000, with generating filter length 200,000. Window size is 5,000 for all the local estimator including FrFT-based and waveletbased. Proper order of FrFT has been tested for the minimum variance of the Hurst parameter estimations. The results of three estimation methods are compared in Fig. 12. As we can see, the FrFT-based estimations are the closest to the ideal line. When the Hurst parameter of FGN approaches to 1.0, the wavelet-based estimations become greater than the actual value and it almost reaches 1.5 at the values of H very close to 1.0. On the other hand, the fluctu-

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ations in the results of aggregated variance method suggest that its estimated values vary more than the other two. Besides, when the real Hurst parameters approach to 1.0, the estimations based on aggregated variance method become much smaller than 1.0. In conclusion, the FrFT-based estimator outperforms the other two in estimating Hurst parameters of FGN produced by symmetric moving average filter. One thing that needs to be mentioned is, for 0.01 < H < 0.50, especially for 0.01 < H < 0.10, all three methods have estimations smaller than the real values, which is opposite to the results in Fig. 11. Therefore, we can tell that the misleading estimations may be caused by the generating methods of FGN. In conclusion, the results presented in this section show that for long-range dependent fractional Gaussian noises, the FrFT-based estimation method performs very well in analyzing their Hurst parameters and gives out better estimations than the other two methods.

4 Conclusions We have presented in detail a new fractional Fourier transform (FrFT) based estimation method to estimate the Hurst parameter for analysis of long range dependence (LRD) in time series. The degree of LRD is characterized by the Hurst parameter. The FrFT-based estimation of Hurst parameter proposed in this paper can be implemented efficiently allowing very large data set. We used fractional Gaussian noises (FGN) which typically possesses long-range dependence with known Hurst parameters to validate the accuracy of the proposed Hurst parameter estimator. For justifying the advantage of the proposed estimator, some other existing Hurst parameter estimation methods, such as waveletbased method and a global estimator based on dispersional analysis, are compared. The proposed estimator can process the very long experimental time series locally to achieve a reliable estimation of the Hurst parameter. Acknowledgement This research was supported in part by USU Skunk Works Research Initiative Grant (“Fractional Order Signal Processing for Bioelectrochemical Sensors”, 2005–2006).

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YangQuan Chen (SM’95–SM’98) received the B.S. degree in industrial automation from the University of Science and Technology of Beijing, Beijing, China, in 1985, the M.S. degree in automatic control from the Beijing Institute of Technology, Beijing, in 1989, and the Ph.D. degree in advanced control and instrumentation from the Nanyang Technological University, Singapore, Singapore, in 1998. He is currently an Associate Professor of electrical and computer engineering at Utah State University, Logan,

and the Director of the Center for Self-Organizing and Intelligent Systems. He is the holder of 13 granted and two pending U.S. patents in various aspects of hard disk drive servomechanics. He has published over 100 journal and book chapter papers, over 200 refereed conference papers, and more than 50 industrial technical reports. He has coauthored two research monographs and five textbooks. His current research interests include robust iterative learning and repetitive control, identification and control of distributed parameter systems with networked movable actuators and sensors, autonomous ground mobile and aerial robots, fractional order dynamic systems control and fractional order signal processing, computational intelligence, and intelligent mechatronic systems. He is a member of AMA, AUVSI, ASME, IEEE, and the American Society for Engineering Education.

Rongtao Sun is an electrical engineer at Phase Dynamics, Inc. He is mainly working on real-time signal processing, analog and digital communication and embedded system design. Before joining Phase Dynamics he worked as a software engineer on video codec at LSI. Sun, Rongtao went to the graduate school of electrical and computer engineering at Utah State University, from which he received a M.S. degree. During the time, he also served as a member of Center for Self-Organizing and Intelligent Systems (CSOIS). His areas of research interests included fractional calculus and fractional order signal processing techniques and applications.

Anhong Zhou is an assistant professor in the Department of Biological and Irrigation Engineering, Utah State University. He received his PhD degree in Bioanalytical Chemistry from Hunan University, China, in 2000. His research interests include the development of biosensors for environmental quality monitoring and biomedical applications, biocorrosion, biological redox, and AFM applications in detecting cellular mechanics. Dr. Zhou is the member of American Chemical Society, Materials Research Society, Electrochemical Society, American Society for Engineering Education, and Institute of Biological Engineering.