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Fractal dimensions for rainfall time series

Author

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  • Breslin, M.C.
  • Belward, J.A.

Abstract

Fractals are objects which have a similar appearance when viewed at different scales. Such objects have detail at arbitrarily small scales, making them too complex to be represented by Euclidean space. They are assigned a dimension which is non-integer. Some natural phenomena have been modelled as fractals with success; examples include geologic deposits, topographical surfaces and seismic activity. In particular, time series data has been represented as a curve with dimension between one and two. There are many different ways of defining fractal dimension. Most are equivalent in the continuous domain, but when applied in practice to discrete data sets lead to different results. Three methods for estimating fractal dimension are evaluated for accuracy. Two standard algorithms, Hurst's rescaled range analysis and the box-counting method, are compared with a recently introduced method which has not yet been widely used. It will be seen that this last method offers superior efficiency and accuracy, and it is recommended for fractal dimension calculations for time series data. We have applied these fractal analysis techniques to rainfall time series data from a number of gauge locations in Queensland. The suitability of fractal analysis for rainfall time series data is discussed, as is the question of how the theory might aid our interpretation of rainfall data.

Suggested Citation

  • Breslin, M.C. & Belward, J.A., 1999. "Fractal dimensions for rainfall time series," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 48(4), pages 437-446.
    • Handle: RePEc:eee:matcom:v:48:y:1999:i:4:p:437-446
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    Citations

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    Cited by:

    1. Francisco Gerardo Benavides-Bravo & Dulce Martinez-Peon & Ángela Gabriela Benavides-Ríos & Otoniel Walle-García & Roberto Soto-Villalobos & Mario A. Aguirre-López, 2021. "A Climate-Mathematical Clustering of Rainfall Stations in the Río Bravo-San Juan Basin (Mexico) by Using the Higuchi Fractal Dimension and the Hurst Exponent," Mathematics, MDPI, vol. 9(21), pages 1-11, October.
    2. 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.
    3. Yolanda Caballero & Ramón Giraldo & Jorge Mateu, 2022. "A spatial randomness test based on the box-counting dimension," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(3), pages 499-524, September.
    4. Dong Liu & Mingjie Luo & Qiang Fu & Yongjia Zhang & Khan Imran & Dan Zhao & Tianxiao Li & Faiz Abrar, 2016. "Precipitation Complexity Measurement Using Multifractal Spectra Empirical Mode Decomposition Detrended Fluctuation Analysis," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 30(2), pages 505-522, January.
    5. Balkissoon, Sarah & Fox, Neil & Lupo, Anthony, 2020. "Fractal characteristics of tall tower wind speeds in Missouri," Renewable Energy, Elsevier, vol. 154(C), pages 1346-1356.
    6. Miao Yu & Dong Liu & Jean Dieu Bazimenyera, 2013. "Diagnostic Complexity of Regional Groundwater Resources System Based on time series fractal dimension and Artificial Fish Swarm Algorithm," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 27(7), pages 1897-1911, May.
    7. Morales Martínez, Jorge Luis & Segovia-Domínguez, Ignacio & Rodríguez, Israel Quiros & Horta-Rangel, Francisco Antonio & Sosa-Gómez, Guillermo, 2021. "A modified Multifractal Detrended Fluctuation Analysis (MFDFA) approach for multifractal analysis of precipitation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 565(C).
    8. Pinto, Erveton P. & Pires, Marcelo A. & Matos, Robert S. & Zamora, Robert R.M. & Menezes, Rodrigo P. & Araújo, Raquel S. & de Souza, Tiago M., 2021. "Lacunarity exponent and Moran index: A complementary methodology to analyze AFM images and its application to chitosan films," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 581(C).
    9. Borys, Przemyslaw, 2020. "Long term Hurst memory that does not die at long observation times—Deterministic map to describe ion channel activity," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    10. Dong Liu & Mingjie Luo & Qiang Fu & Yongjia Zhang & Khan M. Imran & Dan Zhao & Tianxiao Li & Faiz M. Abrar, 2016. "Precipitation Complexity Measurement Using Multifractal Spectra Empirical Mode Decomposition Detrended Fluctuation Analysis," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 30(2), pages 505-522, January.

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