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An improved Hurst parameter estimator based on fractional Fourier transform

An improved Hurst parameter estimator based on fractional Fourier transform. YangQuan Chen*, Rongtao Sun+, Anhong Zhou#. *Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical & Computer Engineering, Utah State University

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An improved Hurst parameter estimator based on fractional Fourier transform

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  1. An improved Hurst parameter estimator based on fractional Fourier transform YangQuan Chen*, Rongtao Sun+, Anhong Zhou# *Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical & Computer Engineering, Utah State University +Phase Dynamics, Inc. Richardson,TX75081 #Department of Biological and Irrigation Engineering, Utah State University 3RD Int. Symposium on Fractional Derivatives and Their Applications (FDTA07) ASME DETC/CIE 2007, Las Vegas, NV, USA. Sept. 4-7, 2007

  2. Outline • Long-range dependence analysis • Fractional Fourier transform (FrFT) • A fractional Fourier transform based Hurst parameter estimator • An improvement in Hurst parameter estimation

  3. Long-range Dependence (LRD) • The first model was introduced by Mandelbrot and Van Ness (1968) • The auto-covariance of LRD time series decays very slowly, such that it is not summable. • The LRD is typically modeled by supposing a power law decay of the power spectral density (PSD).

  4. Hurst parameter • The Hurst parameter H characterizes the degree of LRD . • Relation: • A stochastic process is said to have long range dependence when 0.5<H<1.

  5. Hurst parameter estimation • Methods: Wavelet-based, local Whittle, R/S analysis, periodogram methods, dispersional analysis • Challenges: Too long to have some parts non-stationary (Stoev, Taqqu, Park and Marron 2004). Slight differences between several long time series.

  6. Fractional Fourier Transform Based Hurst Parameter Estimation • Fractional Fourier transform (FrFT) is a generalization of Fourier transform. • FrFT has a strong relationship with wavelet transform. • In analyzing long-range dependence, it is possible to improve performance by the use of the FrFT. • FrFT has been shown to have a computational complexity proportional to the wavelet transform, the performance improvements may come without additional cost.

  7. Wavelet based Estimator • Definition of wavelet tranform • Relationship between the mean energy and the PSD of Y.

  8. Fractional Fourier Transform • Definition of FrFT:

  9. Relationship Between FrFT & Wavelet • Making the change of variable by and denoting the left hand side of FrFT definition by , we can obtain the following result where • Let the mother wave , the wavelet transform in equation can be change to

  10. Let , this yields • The relationship between the fractional Fourier spectrum and the Hurst parameter:

  11. Estimation of Hurst parameter • By using a linear regression of the FrFT spectrum g(j) on the scales j, the Hurst parameterof Y can be estimated as,

  12. Local analysis • Consider given by , is the local slope of the FrFT spectrum. should be very close to H when j is large. Therefore, the can be expressed as where .

  13. For stochastic process Y, choose the initial window size and divide Y into non-overlapping time series , where is the time series corresponding to the rth window. • Compute the FrFT spectrum of Y within each window and obtain a matrix G where is the FrFT coefficients of the r-th window.

  14. Fractional Gaussian Noise Analysis • Fractional Integrator Method • A Fast Fractional Gaussian Noise Generator • A Fast Fractional Gaussian Noise Generator • A Disaggregation Approach • A Symmetric Moving Average Approach

  15. Fractional Integrator Method • Transfer function: • Impulse response: • Autocorrelation function: • Power spectrum: • Therefore, , for LRD,

  16. Experimental Result • For input white noise of sample size 100,000,

  17. Symmetric moving average filter method • The SMA scheme transforms a white noise sequence Wiinto a process with auto-correlation by taking the weighted average of a number of Wi. • In the SMA process the weights ajare symmetric about a center (a0) that corresponds to the variable Wi, i.e., • The sequence of aj itself,

  18. Experimental Result • The sample size of FGN is 100,000, with generating filter length 200,000.

  19. Conclusions • Fractional Fourier transform is very useful in studying long range dependence. • There are improvements in Hurst parameter estimation • Local estimation provides a richer picture of the data. • Our experiments of FGN validated the FrFT based Hurst parameter estimator and its advantages

  20. Thank you! Questions?

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