1 / 20

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

russell-ryan
Télécharger la présentation

An improved Hurst parameter estimator based on fractional Fourier transform

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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?

More Related