In fractal geometry, the Higuchi dimension (or Higuchi fractal dimension (HFD)) is an approximate value for the box-counting dimension of the graph of a real-valued function or time series. This value is obtained via an algorithmic approximation so one also talks about the Higuchi method. It has many applications in science and engineering and has been applied to subjects like characterizing primary waves in seismograms,[1] clinical neurophysiology[2] and analyzing changes in the electroencephalogram in Alzheimer's disease.[3]

Formulation of the method

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The original formulation of the method is due to T. Higuchi.[4] Given a time series   consisting of   data points and a parameter   the Higuchi Fractal dimension (HFD) of   is calculated in the following way: For each   and   define the length   by

 

The length   is defined by the average value of the   lengths  ,

 

The slope of the best-fitting linear function through the data points   is defined to be the Higuchi fractal dimension of the time-series  .

Application to functions

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For a real-valued function   one can partition the unit interval   into   equidistantly intervals   and apply the Higuchi algorithm to the times series  . This results into the Higuchi fractal dimension of the function  . It was shown that in this case the Higuchi method yields an approximation for the box-counting dimension of the graph of   as it follows a geometrical approach (see Liehr & Massopust 2020[5]).

Robustness and stability

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Applications to fractional Brownian functions and the Weierstrass function reveal that the Higuchi fractal dimension can be close to the box-dimension.[4][5] On the other hand, the method can be unstable in the case where the data   are periodic or if subsets of it lie on a horizontal line (see Liehr & Massopust 2020[5]).

References

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  1. ^ Gálvez-Coyt, Gonzalo; Muñoz-Diosdado, Alejandro; Peralta, José A.; Balderas-López, José A.; Angulo-Brown, Fernando (June 2012). "Parameters of Higuchi's method to characterize primary waves in some seismograms from the Mexican subduction zone". Acta Geophysica. 60 (3): 910–927. Bibcode:2012AcGeo..60..910G. doi:10.2478/s11600-012-0033-9. ISSN 1895-6572. S2CID 129794825.
  2. ^ Kesić, Srdjan; Spasić, Sladjana Z. (2016-09-01). "Application of Higuchi's fractal dimension from basic to clinical neurophysiology: A review". Computer Methods and Programs in Biomedicine. 133: 55–70. doi:10.1016/j.cmpb.2016.05.014. ISSN 0169-2607. PMID 27393800.
  3. ^ Nobukawa, Sou; Yamanishi, Teruya; Nishimura, Haruhiko; Wada, Yuji; Kikuchi, Mitsuru; Takahashi, Tetsuya (February 2019). "Atypical temporal-scale-specific fractal changes in Alzheimer's disease EEG and their relevance to cognitive decline". Cognitive Neurodynamics. 13 (1): 1–11. doi:10.1007/s11571-018-9509-x. ISSN 1871-4080. PMC 6339858. PMID 30728867.
  4. ^ a b Higuchi, T. (1988-06-01). "Approach to an irregular time series on the basis of the fractal theory". Physica D: Nonlinear Phenomena. 31 (2): 277–283. Bibcode:1988PhyD...31..277H. doi:10.1016/0167-2789(88)90081-4. ISSN 0167-2789.
  5. ^ a b c Liehr, Lukas; Massopust, Peter (2020-01-15). "On the mathematical validity of the Higuchi method". Physica D: Nonlinear Phenomena. 402: 132265. arXiv:1906.10558. doi:10.1016/j.physd.2019.132265. ISSN 0167-2789. S2CID 195584346.