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Measurements of fractal dimension by box-counting: a critical analysis of data scatter

Author

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  • Buczkowski, Stéphane
  • Hildgen, Patrice
  • Cartilier, Louis

Abstract

The multifractal concept was introduced in the 1980s by Mandelbrot. This theory arose from the analysis of complex and/or discontinuous objects. In this study, we analyzed the data scatter obtained by a modified box-counting method. Considering the curved shape of the data scatter, it is noticeable that there is more than one slope corresponding to different fractal behavior of an object. In this work, to discriminate different fractal dimensions from data scatter obtained by box counting, we suggest a rigorous selection of data points. The results show that large ϵ’s usually characterize the embedding surface of the whole object and that small ϵ’s approximate the dimension of the substructure for discontinuous objects. They also show that a dimension can be associated with a density distribution of singularities.

Suggested Citation

  • Buczkowski, Stéphane & Hildgen, Patrice & Cartilier, Louis, 1998. "Measurements of fractal dimension by box-counting: a critical analysis of data scatter," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 252(1), pages 23-34.
    • Handle: RePEc:eee:phsmap:v:252:y:1998:i:1:p:23-34
    DOI: 10.1016/S0378-4371(97)00581-5
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    Cited by:

    1. Zhenfang He & Yaonan Zhang & Qingchun Guo & Xueru Zhao, 2014. "Comparative Study of Artificial Neural Networks and Wavelet Artificial Neural Networks for Groundwater Depth Data Forecasting with Various Curve Fractal Dimensions," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 28(15), pages 5297-5317, December.

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