The design and evaluation of the Simple Self-Similar Sequences Generator

This paper describes a new algorithm for the generation of pseudo random numbers with approximate self-similar structure. The Simple Self-Similar Sequences Generator (4SG) elaborates on an intuitive approach to obtain a fast and accurate procedure, capable of reproducing series of points exhibiting the property of persistence and anti-persistence. 4SG has a computational complexity of O(n) and memory requirements of the order of log"2(N), where N is the number of points to be generated. The accuracy of the algorithm is evaluated by means of computer-based simulations, recurring to several Hurst parameter estimators, namely Variance Time (VT) and the Wavelets-based estimator. The Hosking and the Wavelets-based methods for the generation of self-similar series were submitted to the same tests the 4SG was analysed with, providing for a basis for comparison of several performance aspects of the algorithm. Results show that the proposal embodies a good candidate not only for on-demand emulation of arbitrarily long self-similar sequences, but also for fast and efficient online simulations.

[1]  Bruce J. West,et al.  Fractal response of physiological signals to stress conditions, environmental changes, and neurodegenerative diseases , 2007, Complex..

[2]  Anil K. Bera,et al.  Efficient tests for normality, homoscedasticity and serial independence of regression residuals , 1980 .

[3]  Oliver Rose,et al.  Estimation of the Hurst Parameter of Long-Range Dependent Time Series , 1996 .

[4]  Wayne Luk,et al.  Gaussian random number generators , 2007, CSUR.

[5]  Vern Paxson,et al.  Fast, approximate synthesis of fractional Gaussian noise for generating self-similar network traffic , 1997, CCRV.

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

[7]  C. Peng,et al.  Mosaic organization of DNA nucleotides. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  E. H. Lloyd,et al.  The expected value of the adjusted rescaled Hurst range of independent normal summands , 1976 .

[9]  Richard G. Lathrop,et al.  Identifying structural self-similarity in mountainous landscapes , 1992, Landscape Ecology.

[10]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[11]  John N. Tsitsiklis,et al.  Introduction to Probability , 2002 .

[12]  L. Oxley,et al.  Estimators for Long Range Dependence: An Empirical Study , 2009, 0901.0762.

[13]  Pierre L'Ecuyer,et al.  TestU01: A C library for empirical testing of random number generators , 2006, TOMS.

[14]  Patrice Abry,et al.  A Wavelet-Based Joint Estimator of the Parameters of Long-Range Dependence , 1999, IEEE Trans. Inf. Theory.

[15]  Ilkka Norros.  Studies on a model for connectionless traffic, based on fractional Brownian motion , 1992 .

[16]  Mine C aglar.  Simulation of Fractional Brownian Motion with Micropulses , 2022 .

[17]  J. Hosking.  Modeling persistence in hydrological time series using fractional differencing , 1984 .

[18]  Amund Kvalbein,et al.  An empirical comparison of generators for self similar simulated traffic , 2007, Perform. Evaluation.

[19]  O. D. Jones,et al.  Estimating the Hurst index of a self-similar process via the crossing tree , 2004, IEEE Signal Processing Letters.

[20]  Jürgen P. Kropp,et al.  Trend assessment: applications for hydrology and climate research , 2005 .

[21]  S. Rambaldi,et al.  An accurate fractional Brownian motion generator , 1994 .

[22]  Odemir Martinez Bruno,et al.  Fractal dimension applied to plant identification , 2008, Inf. Sci..

[23]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

[24]  V. Pipiras.  Wavelet-based simulation of fractional Brownian motion revisited , 2005 .

[25]  B. C. Brookes,et al.  Information Sciences , 2020, Cognitive Skills You Need for the 21st Century.

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

[27]  Bruce J. West,et al.  Fractal response of physiological signals to stress conditions, environmental changes, and neurodegenerative diseases: Essays and Commentaries , 2007 .

[28]  Walter Willinger,et al.  Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level , 1997, TNET.

[29]  Michalis Faloutsos,et al.  Long-range dependence ten years of Internet traffic modeling , 2004, IEEE Internet Computing.

[30]  Laurent E. Calvet,et al.  A Multifractal Model of Asset Returns , 1997 .

[31]  Krzysztof Pawlikowski,et al.  Fast self-similar teletraffic generation based on FGN and wavelets , 1999, ICON.

[32]  A simple construction of the Fractional Brownian motion , 2002, math/0210272.

[33]  David W. Petr,et al.  Characterization and Modeling of Long-Range Dependent Telecommunication Traffic , 2002 .

[34]  Michel Mandjes,et al.  ON SPECTRAL SIMULATION OF FRACTIONAL BROWNIAN MOTION , 2003, Probability in the Engineering and Informational Sciences.

[35]  Walter Willinger,et al.  Proof of a fundamental result in self-similar traffic modeling , 1997, CCRV.