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Dimension of the minimal cover and fractal analysis of time series

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  • Dubovikov, M.M
  • Starchenko, N.V
  • Dubovikov, M.S

Abstract

We develop a new approach to the fractal analysis of time series of various natural, technological and social processes. To compute the fractal dimension, we introduce the sequence of the minimal covers associated with a decreasing scale δ. This results in new fractal characteristics: the dimension of minimal covers Dμ, the variation index μ related to Dμ, and the new multifractal spectrum ζ(q) defined on the basis of μ. Numerical computations performed for the financial series of companies entering Dow Jones Industrial Index show that the minimal scale τμ, which is necessary for determining μ with an acceptable accuracy, is almost two orders smaller than an analogous scale for the Hurst index H. This allows us to consider μ as a local fractal characteristic. The presented fractal analysis of the financial series shows that μ(t) is related to the stability of underlying processes. The results are interpreted in terms of the feedback.

Suggested Citation

  • Dubovikov, M.M & Starchenko, N.V & Dubovikov, M.S, 2004. "Dimension of the minimal cover and fractal analysis of time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 339(3), pages 591-608.
    • Handle: RePEc:eee:phsmap:v:339:y:2004:i:3:p:591-608
    DOI: 10.1016/j.physa.2004.03.025
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    3. 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.
    4. A. Didenko S. & M. Dubovikov M. & B. Poutko A. & А. Диденко С. & М. Дубовиков М. & Б. Путко А., 2015. "Прогнозирование Когерентных Разрывов Волатильности // Forecasting Coherent Volatility Breakouts," Финансы: теория и практика/Finance: Theory and Practice // Finance: Theory and Practice, ФГОБУВО Финансовый университет при Правительстве Российской Федерации // Financial University under The Government of Russian Federation, issue 1, pages 30-36.
    5. Didenko Alexander & Dubovikov Mikhail & Poutko Boris, 2015. "Forecasting coherent volatility breakouts," Вестник Финансового университета, CyberLeninka;Федеральное государственное образовательное бюджетное учреждение высшего профессионального образования «Финансовый университет при Правительстве Российской Федерации» (Финансовый университет), issue 1 (85), pages 30-36.
    6. Batten, Jonathan A. & Ellis, Craig A. & Fethertson, Thomas A., 2008. "Sample period selection and long-term dependence: New evidence from the Dow Jones index," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1126-1140.
    7. Sergey Kamenshchikov & Ilia Drozdov, 2016. "Fractal Optimization of Market Neutral Portfolio," Papers 1612.03698, arXiv.org, revised Dec 2016.
    8. Silva, F.E. & Gonçalves, L.L. & Fereira, D.B.B. & Rebello, J.M.A., 2005. "Characterization of failure mechanism in composite materials through fractal analysis of acoustic emission signals," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 481-494.
    9. Andrey Dmitriev & Vasily Kornilov & Svetlana Maltseva, 2018. "Complexity of a Microblogging Social Network in the Framework of Modern Nonlinear Science," Complexity, Hindawi, vol. 2018, pages 1-11, December.
    10. Jiang, Kai & Liu, Zhifeng & Tian, Yang & Zhang, Tao & Yang, Congbin, 2022. "An estimation method of fractal parameters on rough surfaces based on the exact spectral moment using artificial neural network," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    11. Sergey A. Kamenshchikov, 2014. "Transport catastrophe analysis as an alternative to a fractal description: theory and application to financial crisis time series," Papers 1405.6990, arXiv.org, revised Sep 2014.

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