Synthesis of multifractional Gaussian noises based on variable-order fractional operators

In this paper, a synthesis method, which is based on variable-order fractional operators, for multifractional Gaussian noises (mGn) is proposed by studying the relationship of white Gaussian noise (wGn), mGn, and multifractional Brownian motion (mBm). Furthermore, a synthesis method for multifractional @a@?stable processes, the generalization of mGn, is proposed in order to more accurately characterize the processes with local scaling characteristics and heavy tailed distributions. Synthetic examples of mGn and multifractional @a@?stable noises are provided for illustration.

[1]  Ming Li.  Fractal Time Series—A Tutorial Review , 2010 .

[2]  Azer Bestavros,et al.  Self-similarity in World Wide Web traffic: evidence and possible causes , 1996, SIGMETRICS '96.

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

[4]  K. Falconer.  The Local Structure of Random Processes , 2003 .

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

[6]  S. C. Lim,et al.  Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  M. Ortigueira,et al.  On the relation between the fractional Brownian motion and the fractional derivatives , 2008 .

[8]  Dov Ingman,et al.  Application of Differential Operator with Servo-Order Function in Model of Viscoelastic Deformation Process , 2005 .

[9]  Hu Sheng,et al.  On mean square displacement behaviors of anomalous diffusions with variable and random orders , 2010 .

[10]  Krzysztof Burnecki,et al.  Levy Stable Processes. From Stationary to Self-Similar Dynamics and Back. An Application to Finance , 2004 .

[11]  Bertram Ross,et al.  Fractional integration operator of variable order in the holder spaces Hλ(x) , 1995 .

[12]  Adam Loverro,et al.  Fractional Calculus : History , Definitions and Applications for the Engineer , 2004 .

[13]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

[14]  Yangquan Chen,et al.  FARIMA with stable innovations model of Great Salt Lake elevation time series , 2011, Signal Process..

[15]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .

[16]  Carlos F.M. Coimbra,et al.  The variable viscoelasticity oscillator , 2005 .

[17]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[18]  Anne Lohrli.  Chapman and Hall , 1985 .

[19]  Renaud Marty,et al.  Invariance principle, multifractional gaussian processes and long-range dependence , 2008 .

[20]  Carlos F.M. Coimbra,et al.  On the Selection and Meaning of Variable Order Operators for Dynamic Modeling , 2010 .

[21]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[22]  Murad S. Taqqu,et al.  Robustness of whittle-type estimators for time series with long-range dependence , 1997 .

[23]  Rolando Danganan Navarro,et al.  The Identification of Long Memory Process in the Asean-4 Stock Markets by Fractional and Multifractional Brownian Motion , 2006 .

[24]  Murad S. Taqqu,et al.  Fractional Ornstein-Uhlenbeck Lévy processes and the Telecom process: Upstairs and downstairs , 2005, Signal Process..

[25]  Yangquan Chen,et al.  An improved Hurst parameter estimator based on fractional Fourier transform , 2010, Telecommun. Syst..

[26]  Benjamin M. Tabak,et al.  Time-varying long-range dependence in US interest rates , 2007 .

[27]  Chien-Cheng Tseng,et al.  Design of variable and adaptive fractional order FIR differentiators , 2006, Signal Process..

[28]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[29]  D. W. Allan,et al.  A statistical model of flicker noise , 1966 .

[30]  R. Peltier,et al.  Multifractional Brownian Motion : Definition and Preliminary Results , 1995 .

[31]  Stefan Samko,et al.  Fractional integration and differentiation of variable order , 1995 .

[32]  Martin Schechter.  The invariance principle , 1979 .

[33]  Manuel Duarte Ortigueira,et al.  Introduction to fractional linear systems. Part 1. Continuous-time case , 2000 .

[34]  Dov Ingman,et al.  Constitutive Dynamic-Order Model for Nonlinear Contact Phenomena , 2000 .

[35]  Carlos F.M. Coimbra,et al.  Mechanics with variable‐order differential operators , 2003 .

[36]  I. Podlubny.  Fractional differential equations , 1998 .

[37]  Manuel Duarte Ortigueira.  INTRODUCTION TO FRACTIONAL LINEAR SYSTEMS I: Continuous-Time case 1 , 2000 .

[38]  Jean-Christophe Pesquet,et al.  Synthesis of bidimensional alpha-stable models with long-range dependence , 2002, Signal Process..

[39]  Nikita Zaveri,et al.  Fractional Order Signal Processing of Electrochemical Noises , 2008 .

[40]  Demetris Koutsoyiannis,et al.  Climate change, the Hurst phenomenon, and hydrological statistics , 2003 .

[41]  Tom T. Hartley,et al.  Fractional-order system identification based on continuous order-distributions , 2003, Signal Process..

[42]  A. Kolmogorov.  Wienersche spiralen und einige andere interessante Kurven in Hilbertscen Raum, C. R. (doklady) , 1940 .

[43]  Yangquan Chen,et al.  On distributed order integrator/differentiator , 2011, Signal Process..

[44]  S. C. Lim,et al.  Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type , 2001 .

[45]  Martijn Gough.  Climate change , 2009, Canadian Medical Association Journal.