Numerical Approximation of Fractal Dimension of Gaussian Stochastic Processes

Freddy H. Marin Sanchez, William Eduardo Alfonso

Basic Science Department, EAFIT University, Carrera 49 # 7 Sur-50, Medellín, Colombia.

DOI: 10.4236/am.2014.512169                

Abstract

In this paper we propose a numerical method to estimate the fractal dimension of stationary Gaussian stochastic processes using the random Euler numerical scheme and based on an analytical formulation of the fractal dimension for filtered stochastic signals. The discretization of continuous time processes through this random scheme allows us to find, numerically, the expected value, variance and correlation functions at any point of time. This alternative method for estimating the fractal dimension is easy to implement and requires no sophisticated routines. We use simulated data sets for stationary processes of the type Random Ornstein Uhlenbeck to graphically illustrate the results and compare them with those obtained whit the box counting theorem.

Keywords

Stationary Gaussian Stochastic Processes, Fractal Dimension, Random Euler Numerical Scheme

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Richardson, L. (1961) The Problem of Contiguity: An Appendix of Statistic of Deadly Quarrels. General Systems Year Book, 61, 139-187.
[2]	Mandelbrot, B. (1967) How Long Is the Coast of Britain Statistical Self-Similarity and Fractional Dimension. Science, 156, 636-638. http://dx.doi.org/10.1126/science.156.3775.636
[3]	Duque, J.C., Betancourt, A. and Marin, F. (2014) An Algorithmic Approach for Simulating Realistic Irregular Lattices. In: Thill, J.C. and Dragicevic, S., Eds., GeoComputational Analysis and Modeling of Regional Systems, Springer Heidelberg, in press.
[4]	Mandelbrot, B. (1975) Stochastic Models for the Earth’s Relief, the Shape and the Fractal Dimension of the Coastlines, and the Number-Area Rule for Islands. Proceedings of the National Academy of Sciences, 72, 3825-3828. http://dx.doi.org/10.1073/pnas.72.10.3825
[5]	Adler, R.J. (1978) Some Erratic Patterns Generated by the Planar Wiener Process. Advances in Applied Probability, 10, 22-27. http://dx.doi.org/10.2307/1427003
[6]	Cullin, W.E. and Datko, M. (1987) The Fractal Geometry of the Soil-Covered Landscape. Earth Surface Processes and Landforms, 12, 369-385. http://dx.doi.org/10.1002/esp.3290120404
[7]	Cullin, W.E. (1988) Dimension and Entropy in the Soil-Covered Landscape. Earth Surface Processes and Landforms, 13, 619-648. http://dx.doi.org/10.1002/esp.3290130706
[8]	Beauvais, A. Montgomery, D. (1996) Influence of Valley Type on the Scaling Properties of River Planforms. Water Resources Research, 32, 1441-1448. http://dx.doi.org/10.1029/96WR00279
[9]	Montgomery, K. (1996) Sinuosity and Fractal Dimension of Meandering Rivers. Area, 28, 491-500.
[10]	Theiler, J. (1990) Estimating Fractal Dimension. JOSA A, 7, 1055-1073. http://dx.doi.org/10.1364/JOSAA.7.001055
[11]	Taylor C. and Taylor, S.J. (1991) Estimating the Dimension of a Fractal. Journal of the Royal Statistical Society. Series B (Methodological), 53, 353-364.
[12]	Allen, M., Brown, G.J. and Miles, N.J. (1995) Measurement of Boundary Fractal Dimensions: Review of Current Techniques. Powder Technology, 84, 1-14. http://dx.doi.org/10.1016/0032-5910(94)02967-S
[13]	Blachowski, A. and Ruebenbauer, K. (2009) Roughness Method to Estimate Fractal Dimension. Acta Physica Polonica A, 115, 636-640.
[14]	Girault, J., Kouame, D. and Ouahabi, A. (2010) Analytical Formulation of the Fractal Dimension of Filtered Stochastic Signals. Signal Processing, 90, 2690-2697. http://dx.doi.org/10.1016/j.sigpro.2010.03.019
[15]	Loeve, M. (1963) Probability Theory. D. Van N. Company, London.
[16]	Soong, T. (1973) Random Differential Equations in Science and Engineering. Academic Press, New York.
[17]	Jodar, L., Cortez, J. and Villafuerte, L. (2007) Mean Square Numerical Solution of Random Differential Equations: Facts and Possibilities. Computers and Mathematics with Applications, 53, 1098-1106. http://dx.doi.org/10.1016/j.camwa.2006.05.030
[18]	Jodar, L. and Villafuerte, L. (2007) Numerical Solution of Random Differential Equations: A Mean Square Approach. Mathematical and Computer Modelling, 45, 757-765. http://dx.doi.org/10.1016/j.mcm.2006.07.017
[19]	Marin, F. and Laniado, H. (2014) Convergence of Analytical Stochastic Processes in Mean Square. 1-9. http://repository.eafit.edu.co/handle/10784/1523
[20]	Agarwall, R. (1992) Difference Equations and Inequalities. Marcel Dekker, New York.
[21]	Barnsley, M. (1988) Fractals Everywhere. Academic Press, London.
[22]	Le Tavernier, E. (1998) La methode de Higuchi pour la dimension fractal. Signal Processing, 65, 115-128. http://dx.doi.org/10.1016/S0165-1684(97)00211-9
[23]	Uhlenbeck, G. and Ornstein, L. (1930) On the Theory of the Brownian Motion. Physical Review, 36, 823-841. http://dx.doi.org/10.1103/PhysRev.36.823
[24]	Marin, F. and Palacio, J. (2013) Gaussian Estimation of One-Factor Mean Reversion Processes. Journal of Probability and Statistics, 2013, 10 p.