Estimation of Fractal Dimension

References.

Prigarin, S. M., Hahn, K., and Winkler, G. (2007) Evaluation of two numerical methods to measure the Hausdorff dimension of the fractional Brownian motion, Preprint 07-01, IBB, GSF Neuherberg, 28p. ~~***~~

Abstract — The aim of the paper is to study by Monte Carlo simulation statistical properties of two numerical methods (the extended counting method and the variance counting method) developed to estimate the Hausdorff dimension of a time series and applied to the fractional Brownian motion.

Key words: fractal set, Hausdorff dimension, extended counting method, variance counting method, generalized Wiener process, fractional Brownian motion.

Mathematical Subject Classification: 28A80, 62M10, 65C05.

Prigarin, S. M., Hahn, K., and Winkler, G. (2008) Comparative analysis of two numerical methods to measure Hausdorff dimension of the fractional Brownian motion, Numerical Analysis and Applications, V.1, No.2, P.163-178

Abstract — Using Monte Carlo simulation techniques, we look at statistical properties of two numerical methods (the extended counting method and the variance counting method) developed to estimate the Hausdorff dimension of a time series and applied to the fractional Brownian motion.

DOI: 10.1134/S1995423908020079

Key words: fractal set, Hausdorff dimension, extended counting method, variance counting method, generalized Wiener process, fractional Brownian motion.

Prigarin, S. M., Hahn, K., and Winkler, G. (2009), Variational dimension of random sequences and its application, Numerical Analysis and Applications, V.2, No.4, P.352-363

Abstract — A concept of variational dimension is introduced for a random sequence with stationary increments. In Gaussian case the variational dimension in the limit coincides with the Hausdorff dimension of a proper random process. Applications of the concept are illustrated by examples of neurological data and network traffic analysis.

Key words: random sequences with stationary increments, variational dimension, Hausdorff dimension, fractal, self-similarity, data analysis.

Mathematical Subject Classification: 28A80, 62M10, 65C05.

Prigarin, S. M., Hahn, K., and Winkler, G. (2010) On variational dimension of random fields, Preprint 10-01, IBB, GSF Neuherberg, 16p. ~~***~~

Abstract — A concept of variational dimension is introduced for random fields. For a class of Gaussian fields with homogeneous increments the variational dimension converges to the Hausdorff dimension. Several examples are presented to illustrate that the concept of variational dimension can be used to construct effective numerical methods to study fractal properties of random fields.

Key words: random fields, variational dimension, Hausdorff dimension, self-similarity, fractal analysis.

Mathematical Subject Classification: 28A80, 60G60, 65C05.

Prigarin, S. M., Hahn, K., and Winkler, G. (2011) Estimation of fractal dimension of random fields on the basis of variance analysis of increments, Numerical Analysis and Applications, V.4, No.1, P.71-80

Abstract — This paper deals with estimation of fractal dimension of realizations of random fields. Numerical methods are based on analysis of variance of increments. It is proposed to study fractal properties with the use of a specific characteristic of random fields called a “variational dimension.” For a class of Gaussian fields with homogeneous increments the variational dimension converges to the Hausdorff dimension. Several examples are presented to illustrate that the concept of variational dimension can be used to construct effective computational methods.

Prigarin, S. M., Sandau, K., Kazmierczak, M., and Hahn, K. (2013) Estimation of fractal dimension: a survey with numerical experiments and software description, International Journal of Biomathematics and Biostatistics, Vol.2, No.1, January-June 2013, P.167-180 ~~***~~

Abstract — A unifying discussion of basic concepts for the numerical estimation of fractal dimensions and multifractal spectra is presented. For fractal dimensions focus is on box-counting methods, multifractal spectra are introduced by generalized dimensions. Due to limited resolution of empirical data, a discrepancy between analytical and numerical estimation of fractal dimensions can be observed, the corresponding problem of validation is discussed. Numerical estimation is supported by implemented software, which can be downloaded. Numerical experiments are presented and simplify the application of the software, which contains numerous examples.

Key words: fractal dimension, box-counting method, extended counting method, generalized dimensions, multifractal spectrum, numerical methods and software.

Mathematical Subject Classification: 28A80, 65Z05.