Multiscale Higuchi's fractal dimension method

The aim of this study is to introduce a new method for the evaluation of complexity properties of time series by extending Higuchi's fractal dimension (HFD) over multiple scales. Multiscale Higuchi's fractal dimension (MSHG) is presented and demonstrated on a number of stochastic time series and chaotic time series, starting with the examination of the selection of the effective scaling filter among several widely used filtering methods and then diving into the application of HFD through the scales obtained by coarse-graining procedure. Moreover, on the basis of MSHG, fractal dimension and Hurst exponent relationship are studied by employing MSHG method with computation of Hurst value in multiple scales, simultaneously. Consequently, it is found that the proposed method, MSHG produces remarkable results by exposing unique complexity features of time series in multiple scales. It is also discovered that MSHG with multiscale Hurst exponent calculation leads to revelation of distinguishing patterns between verifying stochastic time series and diverging chaotic time series. In light of these findings, it can be inferred that the proposed methods can be utilized for the characterization and classification of time series in terms of complexity.