Principal Component Analysis of the Fractional Brownian Motion for 0 < H < 0.5

Principal component analysis (PCA) has been proposed for the estimation of the self-similarity parameter H, namely the Hurst parameter of 1/f processes, and an analytical proof is provided only for H/=0.5 in a recent study [I]. In our paper, we extend this study by deriving explicit expressions and presenting an analytical proof for the range of 0 < H < 0.5 (the anti-persistent part of the fractional Brownian motion). We also show via simulations that the accuracy of the estimated H values may decrease considerably as the theoretical H value increases towards the persistent part (0.5<H<1)

[1]  Dimitris G. Manolakis,et al.  Statistical and Adaptive Signal Processing , 2000 .

[2]  G. Wornell.  Wavelet-based representations for the 1/f family of fractal processes , 1993, Proc. IEEE.

[3]  A. Erramilli,et al.  Self-similar traffic generation: the random midpoint displacement algorithm and its properties , 1995, Proceedings IEEE International Conference on Communications ICC '95.

[4]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[5]  Athina P. Petropulu,et al.  Long-range dependence and heavy-tail modeling for teletraffic data , 2002, IEEE Signal Process. Mag..

[6]  Athanasios Papoulis,et al.  Probability, Random Variables and Stochastic Processes , 1965 .

[7]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[8]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[9]  John A. Belward,et al.  Fractal dimensions for rainfall time series , 1999 .

[10]  F. B. Hildebrand.  Advanced Calculus for Applications , 1962 .

[11]  Tayfun Akgül,et al.  IS PCA reliable for the analysis of fractional Brownian motion? , 2005, 2005 13th European Signal Processing Conference.

[12]  W. Willinger,et al.  ESTIMATORS FOR LONG-RANGE DEPENDENCE: AN EMPIRICAL STUDY , 1995 .

[13]  Harry L. Van Trees,et al.  Detection, Estimation, and Modulation Theory, Part I , 1968 .

[14]  Truman Botts.  Advanced Calculus for Applications. Francis B. Hildebrand. Prentice- Hall, Englewood Cliffs, N.J., 1962. ix + 646 pp. Illus. $13 , 1962 .

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  M. Melamed.  Detection , 2021, SETI: Astronomy as a Contact Sport.

[17]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[18]  Jianbo Gao,et al.  Principal component analysis of 1/fα noise , 2003 .

[19]  Jeffrey M. Hausdorff,et al.  Long-range anticorrelations and non-Gaussian behavior of the heartbeat. , 1993, Physical review letters.