On fractal dimension of waveforms (Q813687)

The author recalls the results of his earlier paper [Complex. Int. 5 (1998; Zbl 0971.42500)] where he revised the method proposed by \textit{Katz} [Comput. Biol. Med. 40, 517--526 (1988)] to estimate the fractal dimension of waveforms. The author pointed out a fundamental flaw of Katz' method that means that according to it, all waveforms have fractal dimension 1 (i.e., they are straight lines) if the number \(N\) of sampled points is large. He also presented (loc. cit.) an alternative procedure, without any relation to Katz' method, which did not suffer from the mentioned flaw. The present note reacts to the paper of \textit{J. Gnitecki} and \textit{Z. Moussavi} [Chaos Solitons Fractals 26, No. 4, 1065--1072 (2005; Zbl 1122.76377)] who stated that the author's approximation formula [loc. cit.] was a modified version of the Katz FD (KSFD). The author argues that this statement is improper and can only stem from careless reading of both Katz [loc. cit.] and the author [loc. cit.], because his equation was derived directly from the Hausdorff dimension after normalizing the signal into a unit square. The author points out that the results of Gnitecki and Moussavi themselves confirm the uselessness of Katz' equation and discusses further disputable points. He concludes: ``Most important however, is that the authors approach is wrong; they incur in a methodological error when comparing algorithms to calculate fractal dimension using empirical signals whose real fractal dimension is unknown; their results per se are thus useless to guess which of the procedures produces a correct result.''