Mathematical Incorrectness of So-Called Higuchi‘s Fractal Dimension

MENDEL — Soft Computing Journal, Volume 23, No.gk, .2+2K#2` 202k, Brno, Czech RepublicX ISSN: 1803-3814 (Printed), 2571-3701 (Online) https://doi.org/10.13164/mendel.202k.k.093 Mathematical Incorrectness of So-Called Higuchi‘s Fractal Dimension Dalibor Martiˇsek  Institute of Mathematics, Faculty of Mechanical Engineering,Brno University of Technology, Czech Republic ma[email protected]  Abstract The so-called Higuchi’s method of fractal dimension estimation is widely used and the term Higuchi’s fractal dimension even occurs in many publications. This paper deals with this method from a mathematical point of view. Terms distance and dimension and its basic properties are explained and Higuchi’s dimension according the original source is defined. The definition of Higuchi’s dimension was compared with the mathematical definition of distance and dimension. It is shown, that the definition of Higuchi’s dimension does not satisfy axioms of distance and dimension. The so-called Higuchi’s method and Higuchi’s dimension are mathematically incorrect. Therefore, all results achieved by this method are scientifically unreliable. Keywords: Higuchi’s method, Higuchi’s fractal dimension, Distance, Metric space. Received: 10 May 2022 Accepted: 10 December 2022 Online: 20 December 2022 Published: 20 December 2022 1 Introduction Fractal dimension is any dimension that allows noninteger values. The oldest and the most general among them is the Hausdorff dimension (H-dimension). Nowadays, many other definitions are used as well. Measurement of the fractal dimension plays an important role in many applications in engineering and science. The fractal dimension is measured either from time series or from digital images. The fractal dimension of various structures has been studied recently. [4] deals with the relation between the fractal dimension of surfaces of metal samples and their wear resistance coefficient as well as with the relation between the volume of pores in ceramics and the Hausdorff dimension of the pore boundary. [14] and [13] perform Hausdorff dimension analysis on fracture surfaces of porous materials such as hydrated cement pastes. The Hausdorff dimension of a fracture surface gives us information on the material and its properties. [5] studies the properties of different fracture surfaces (of metals, ceramics, rocks etc.) and their fractal properties. [7] studied the fracture surfaces of aluminium alloys subjected to four different heat treatments to find that their fractal dimensions were almost identical. [26] showed that the fractal dimension can be a measure of toughness in metals. Many researchers have suggested using the fractal dimension to quantify rock joint roughness - see [6,27,31,28,12] for example. There exist also many papers which deal with fractal dimension for other various purposes: natural phenomena [25], medicine [15], clinical neurophysiology especially [24], seismology [16], computer science [19] or communications technology [29]. Unfortunately, some published papers contain many inaccuracies and even mathematical nonsenses caused by insufficient mathematical foundations from which their authors deduce their conclusions – see [13,29,21, 30,8] for example. A special group of incorrect papers are texts which deal with so called Higuchi’s method or Higuchi’s fractal dimension – see [29,20,17,23,32, 2,22]. Research based on this incorrrect method was published even in physical [3], fractal [18] and chaos [1] journals. 2 Materials and Methods Each definition of a fractal dimension is based on the concept of distance or length of abscissa. In mathematics, they may be abstract terms and they can be defined in many ways. However, each distance or length must satisfy a few simple properties. Formally, metric space is defined in mathematics: A metric space is an ordered pair (M, L) where Mis a set and L:M × M → Ris a mapping such that for any x;y;z∈ M, the following holds: a) L(x;y)≥0 b) L(x;y) = 0 if and only if x=y c) L(x;y) = L(y;x) d) L(x;z)≤L(x;y) + L(y;z) Mapping Lis called distance function or metrics on set M. If we speak of any dimension, we obviously presume its integer value – objects are one-, twoor three dimensional. It is so called topological dimension. Fractal dimension is an arbitrary dimension which allows noninteger values. For its computing, we must measure a series of its approximations. For example, perimeter 93 MENDEL — Soft Computing Journal, Volume 23, No.gk, .2+2K#2` 202k, Brno, Czech RepublicX Figure 1: The principle of the so-called Higuchi’s method. Figure 2: The Higuchi’s method applied to a whole straight line gives incorrect result. of a circle can be measured approximately with the series of hypothetical folding rules – polygons with shortenig segments. Each segment shortening in half means approximately twice as many segments. If we denote length of segments as Lkand corresponding number as Nkthen Nk·L1 k≈constant. The perimeter of a circle is one-dimensional. For measuring of its interior, square grid can be used by analogy. Each shortening of square side Lkto one half means approximately quadruplicate number of squares Nk, i.e. Nk·L2 k≈constant. The interior of a circle is two-dimensional. For the interior of a ball is Nk·L3 k≈constant by analogy. It is threedimensional. Generally, this dimension is defined as the number D, for which is Nk·LD k≈constant However, this dimension is not topological because number Dcan be non-integer for many geometric shapes. D= 1.2618... for the so-called Koch curve, D= 1.5849... for the so-called Sierpinski triangle, etc. It is possible to proof that each fractal dimension of each set is greater or equal to its topological dimension. Topological dimension of Koch curve and also Sierpinski triangle is equal to one for example. See [9,10,11] for more information about these problems. Higuchi’s method [20] is a common method of the estimation of the fractal dimension of a curve The Higuchi’s method is defined only for functions X(t), whose values are known only in points t= 1,2, . . . , M. An approximation Xm kof the graph of function X(t) are polygons going through points Xm k:X(m); X(m+k); X(m+ 2k); . . . Xm+jN−m k·kk;m= 1; 2, . . . ;k(1) and the “length” of the polygon is defined as Lm(k)=PM−m n ·k i=1 |X(m+ik)−X(m+(i−1)k)|·M−1 M−m n·k2 (2) where ⌊·⌋ stands for the nearest lower integer [29]. The principle of this method is illustrated in Fig. 1 where ∆yi=|X(m+ik)−X(m+ (i−1)k)|(3) for simplicity. The values of Lm(k) are averaged over mto obtain function ⟨L(k)⟩. Then Higuchi claims that if ⟨L(k)⟩is proportional to k−D, then the curve is a fractal with dimension D. However, each fractal dimension must work with concept of length or distance according to points a) – d) in previous text and must be equal or greater than topological dimension. We have verified whether the Higuchi’s dimension has these properties or not. 3 Results Higuchi tested his method on artificially generated noise (see Fig. 1) with good results. However, with the fractal dimension close to one (see Fig. 2), we get meaningless results. If the Higuchi’s method is applied to a constant time series X(1) = X(2) = · · · =X(M) we obtain ∆yi= 0 for each iin (3), therefore, sum of ∆yiis equal to zero as well and Lm(k) = 0 for each m, k in (2) and L(AB) = 0 although points A, B are not identical. It is contrary to the condition b) in Sec. 2. Moreover, if ⟨L(k)⟩is proportional to k−D(as Higuchi claims) then must be D= 0 and fractal dimension would be 94 MENDEL — Soft Computing Journal, Volume 23, No.gk, .2+2K#2` 202k, Brno, Czech RepublicX Martišek: Mathematical Incorrectness of So-Called Higuchi‘s Fractal Dimension smaller than topological dimension (it is equal to one). It means that so called Higuchi’s dimension is contrary to known properties of a fractal dimension. 4 Conclusion As was shown in previous section, so called Higuchi’s method for fractal dimension measurement and also the term Higuchi’s dimension is mathematically incorrect. Therefore, all results achieved by this method must be considered scientifically unreliable. Acknowledgement: The author acknowledges support from Private Institute of Applied Mathematics, ˇ Slapanice, Czech Republic. References [1] Ahammer, H., Sabathiel, N., and Reiss, M. A. Is a two-dimensional generalization of the higuchi algorithm really necessary? Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 7 (2015), 073104. [2] Bachmann, M., Lass, J., Suhhova, A., and Hinrikus, H. Spectral asymmetry and higuchi’s fractal dimension measures of depression electroencephalogram. Computational and mathematical methods in medicine 2013 (2013). [3] Bordin, L., Creminelli, P., Mirbabayi, M., and Nore˜ na, J. Tensor squeezed limits and the higuchi bound. Journal of cosmology and astroparticle physics 2016, 09 (2016), 041. [4] Borodich, F. M. Fractals and fractal scaling in fracture mechanics. International Journal of Fracture 95, 1 (1999), 239–259. [5] Bouchaud, E., Lapasset, G., and Planes, J. Fractal dimension of fractured surfaces: a universal value? EPL (Europhysics Letters) 13, 1 (1990), 73. [6] Brown, S. R., and Scholz, C. H. Broad bandwidth study of the topography of natural rock surfaces. Journal of Geophysical Research: Solid Earth 90, B14 (1985), 12575–12582. [7] Cervantes-De la Torre, F., Gonz´ alezTrejo, J. I., Real-Ramirez, C. A., and Hoyos-Reyes, L. F. Fractal dimension algorithms and their application to time series associated with natural phenomena. In Journal of Physics: Conference Series (2013), vol. 475, IOP Publishing, p. 012002. [8] Den Outer, A., Kaashoek, J., and Hack, H. Difficulties with using continuous fractal theory for discontinuity surfaces. In International journal of rock mechanics and mining sciences & geomechanics abstracts (1995), vol. 32, Elsevier, pp. 3–9. [9] Edgar, G. A., and Edgar, G. A. Measure, topology, and fractal geometry, vol. 2. Springer, 2008. [10] Falconer, K. Fractal geometry: mathematical foundations and applications. John Wiley & Sons, 2004. [11] Falconer, K. J. The geometry of fractal sets. No. 85. Cambridge university press, 1986. [12] Ficker, T. Fractal properties of joint roughness coefficients. International Journal of Rock Mechanics and Mining Sciences 94 (2017), 27–31. [13] Ficker, T., Len, A., Chmel´ ık, R., Lovicar, L., Martiˇ sek, D., and Nˇ emec, P. Fracture surfaces of porous materials. EPL (Europhysics Letters) 80, 1 (2007), 16002. [14] Ficker, T., Martiˇ sek, D., and Jennings, H. M. Roughness of fracture surfaces and compressive strength of hydrated cement pastes. Cement and Concrete Research 40, 6 (2010), 947– 955. [15] Fuss, F. K. A robust algorithm for optimisation and customisation of fractal dimensions of time series modified by nonlinearly scaling their time derivatives: Mathematical theory and practical applications. Computational and Mathematical Methods in Medicine 2013 (2013). [16] G´ alvez-Coyt, G., Mu˜ noz-Diosdado, A., Peralta, J. A., Balderas-L´ opez, J. A., and Angulo-Brown, F. Parameters of higuchi’s method to characterize primary waves in some seismograms from the mexican subduction zone. Acta Geophysica 60, 3 (2012), 910–927. [17] Gomolka, R. S., Kampusch, S., Kaniusas, E., Th¨ urk, F., Sz´ eles, J. C., and Klonowski, W. Higuchi fractal dimension of heart rate variability during percutaneous auricular vagus nerve stimulation in healthy and diabetic subjects. Frontiers in physiology 9 (2018), 1162. [18] Grace Elizabeth Rani, T., and Jayalalitha, G. Complex patterns in financial time series through higuchi’s fractal dimension. Fractals 24, 04 (2016), 1650048. [19] G¨ uc¸l¨ u, U., G¨ uc¸l¨ ut¨ urk, Y., and Loo, C. K. Evaluation of fractal dimension estimation methods for feature extraction in motor imagery based brain computer interface. Procedia Computer Science 3 (2011), 589–594. [20] Higuchi, T. Approach to an irregular time series on the basis of the fractal theory. Physica D: Nonlinear Phenomena 31, 2 (1988), 277–283. [21] Huang, S., Oelfke, S., and Speck, R. Applicability of fractal characterization and modelling to rock joint profiles. In International journal of rock mechanics and mining sciences & geomechanics abstracts (1992), vol. 29, Elsevier, pp. 89–98. [22] Kalauzi, A., Boji´ c, T., and Vuckovic, A. Modeling the relationship between higuchi’s fractal dimension and fourier spectra of physiological signals. Medical & biological engineering & computing 50, 7 (2012), 689–699. 95 MENDEL — Soft Computing Journal, Volume 23, No.gk, .2+2K#2` 202k, Brno, Czech RepublicX [23] Kesic, S., Nikolic, L. M., Savi´ c, A. G., Petkovi´ c, B., and Spasi´ c, S. Ouabain modulation of snail br neuron bursting activity after the exposure to 10 mt static magnetic field revealed by higuchi fractal dimension. General physiology and biophysics 33, 3 (2014), 335–344. [24] Kesi´ c, S., and Spasi´ c, S. Z. Application of higuchi’s fractal dimension from basic to clinical neurophysiology: A review. Computer methods and programs in biomedicine 133 (2016), 55–70. [25] Mandelbrot, B. B., and Mandelbrot, B. B. The fractal geometry of nature, vol. 1. WH freeman New York, 1982. [26] Mandelbrot, B. B., Passoja, D., Paullay, A. J., et al. Fractal character of fracture surfaces of metals. Nature 308, 5961 (1984), 721–722. [27] Miller, S., McWilliams, P., and Kerkering, J. Ambiguities in estimating fractal dimensions of rock fracture surfaces. In Rock Mechanics Contributions and Challenges: Proceedings of the 31st US Symposium (2020), CRC Press, pp. 471– 478. [28] Odling, N. Natural fracture profiles, fractal dimension and joint roughness coefficients. Rock mechanics and rock engineering 27, 3 (1994), 135– 153. [29] Phothisonothai, M., Arita, Y., and Watanabe, K. Effects of time windowing for extraction of expression from japanese speech: Higuchi’s fractal dimension. In 2013 13th International Symposium on Communications and Information Technologies (ISCIT) (2013), IEEE, pp. 665–668. [30] Poon, C., Sayles, R., and Jones, T. Surface measurement and fractal characterization of naturally fractured rocks. Journal of Physics D: Applied Physics 25, 8 (1992), 1269. [31] Power, W. L., and Tullis, T. E. Euclidean and fractal models for the description of rock surface roughness. Journal of Geophysical Research: Solid Earth 96, B1 (1991), 415–424. [32] Topc¸u, C¸ ., Bedelo˘ glu, M., Akg¨ ul, A., Sever, R., ¨ Ozkan, ¨ O., ¨ Ozkan, ¨ O., Uysal, H., Polat, ¨ O., and C¸ olak, ¨ O. H. Higuchi fractal dimension analysis of surface emg signals and determination of active electrode positions. In 2014 18th National Biomedical Engineering Meeting (2014), IEEE, pp. 1–4. 96