Warped Infinitely Divisible Cascades: Beyond Power Laws. Cascades Infiniment Divisibles Voilées: Au-Delà des Lois de Puissance

Warped Infinitely Divisible Cascades: Beyond Power Laws

Cascades Infiniment Divisibles Voilées: Au-Delà des Lois de Puissance

Pierre Chainais Rudolf Riedi  Patrice Abry 

ISIMA-LIMOS UMR 6158 – Université Clermont II,Aubière France

Depts. of Statistics and of ECE, Rice University, Houston Texas, USA

CNRS UMR 5672, Laboratoire de Physique. ENS Lyon, France

Page: 
27-40
|
Received: 
9 February 2004
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Accepted: 
N/A
| | Citation

OPEN ACCESS

Abstract: 

We address the definitions and synthesis of stochastic processes which possess warped scaling laws that depart from power law behaviors in a controlled manner. We define warped infinitely divisible cascading (IDC) noise,motion and random walk. We provide a theoretical derivation of the scaling behavior of the moments of their increments. We provide numerical simulations of a warped log-Normal cascade to illustrate these results. Algorithms for synthesis and Matlab functions are available from our web pages. 

Résumé

Nous présentons les définitions et synthèses de processus stochastiques respectant des lois d'échelles voilées, qui s'écartent de façon contrôlée d'un comportement en loi de puissance. Nous définissons des bruit, mouvement et marche aléatoire issus de cascades infiniment divisibles (IDC) voilées. Nous étudions analytiquement le comportement des moments des accroissements de ces processus à travers les échelles. Ces résultats théoriques sont illustrés sur l'exemple d'une cascade log-Normale voilée. Les algorithmes de synthèse et les fonctions Matlab utilisés sont disponibles sur nos pages web.

Keywords: 

Fractional Brownian motion,infinitely divisible cascades,multifractal processes,multiplicative cascades, multiscaling,random walk,turbulence

Mots clés 

Mouvement Brownien fractionnaire,cascades infiniment divisibles,processus multifractals,cascades multiplicatives, multiscaling,marche aléatoire,turbulence.

1. Introduction
2. IDC Noise
3. IDC Motion & Random Walk
4. Scaling Behavior of IDC
5. Evolution of the Increments Distributions
6. Numerical Validations
7. Conclusion
  References

[1] A. ARNEODO, F. ARGOUL, E. BACRY, and J.-F ELEZGARAY, J. MUZY. Ondelettes, multifractales et turbulences. Diderot, Editeur des Sciences et des Arts, Paris, 1995. 

[2] A. ARNEODO, E. BACRY, and J.F. MUZY. Random cascade on wavelet dyadic trees. J. Math. Phys., 39(8):4142-4164, 1998. 

[3] E. BACRY, J. DELOUR, and J.F. MUZY. Multifractal random walk. Phys. Rev. E, 64:026103, 2001. 

[4] E. BACRY and J.F. MUZY. Log-infinitely divisible multifractal processes. Comm. in Math. Phys., 236(3):449-475, 2003. 

[5] J. BARRAL and B. MANDELBROT. Multiplicative products of cylindrical pulses. Probab. Theory Relat. Fields, 124:409-430, 2002. 

[6] B. CASTAING,Y. GAGNE, and E. HOPFINGER. Velocity probability density functions of high Reynolds number turbulence. Physica D, 46:177-200, 1990. 

[7] P. CHAINAIS. Cascades log-infiniment divisibles et analyse multirésolution. Application à l'étude des intermittences en turbulence. PhD thesis, E.N.S. Lyon, 2001. 

[8] P. CHAINAIS, R. RIEDI, and P. ABRY. Non scale invariant infinitely divisible cascades. In Proceedings of the 19th Colloquium GRETSI, Paris, 2003. 

[9] P. CHAINAIS, R. RIEDI, and P. ABRY. On non scale invariant infinitely divisible cascades. IEEE Transactions on Information Theory, Vol. 51, no 3, March 2005. 

[10] P. CHAINAIS, R. RIEDI, and P. ABRY. Scale invariant infinitely divisible cascades. In Int. Symp. on Physics in Signal and Image Processing, Grenoble, France, 2003. 

[11] J. DELOUR. Processus aléatoires auto-similaires : applications en turbulence et en finance. PhD thesis, Université de Bordeaux I, 2001. 

[12] U. FRISCH. Turbulence. The legacy of A. Kolmogorov. Cambridge University Press, Cambridge, UK, 1995.

[13] A. N. KOLMOGOROV. A refinement hypothesis concerning the local structure of turbulence in a viscous incompressible fluid at high reynolds number. J. of Fluid Mech., 13:82-85, 1962. 

[14] B. B. MANDELBROT. Intermittent turbulence in self similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid. Mech., 62:331, 1974. 

[15] B. B. MANDELBROT and J. W. VAN NESS. Fractional Brownian motion, fractional noises and applications. SIAM Reviews, 10:422437, 1968. 

[16] B.B. MANDELBROT. A multifractal walk down wall street. Scientific American, 280(2):70-73, Feb. 1999. 

[17] P. MANNERSALO, R. RIEDI, and I. NORROS. Multifractal products of stochastic processes: construction and some basic properties. Applied Probability, 34(4):1-16, 2002. 

[18] J.F. MUZY and E. BACRY. Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws. Phys. Rev. E, 66, 2002. 

[19] J.F. MUZY, E. BACRY, and A. ARNEODO. The multifractal formalism revisited with wavelets. Int. J. of Bifurc. and Chaos, 4(2):245301, 1994. 

[20] R.H. RIEDI. Multifractal processes. In: "Theory and applications of long range dependence", eds. Doukhan, Oppenheim and Taqqu (Birkhauser), pages 625-716, 2002. 

[21] D. SCHERTZER and S. LOVEJOY. Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes. J. Geophys. Res., 92:9693, 1987. 

[22] F. SCHMITT and D. MARSAN. Stochastic equations generating continuous multiplicative cascades. Eur. Phys. J. B, 20:3-6, 2001. 

[23] D. VEITCH, P. ABRY, P. FLANDRIN, and P. CHAINAIS. Infinitely divisible cascade analysis of network traffic data. Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing - ICASSP, 2000.