A comparative study of validity ranges of some fractal methods of time series analysis

Each fractal method of time series analysis gives origin to a parameter that characterizes them. For example, β is the resulting parameter of the application of spectral analysis, α for detrended fluctuation analysis (DFA), D for Higuchi's method and Ha for semivariogram, to mention only some of them. Diverse linear relations between the mentioned parameters have been reported; some of these relations have been obtained from theoretical approaches by assuming time series properties such as their self-similarity. The obtained relations are linear in all cases; nevertheless, these relations only have a limited validity range. In this work, we determine the relation between each one of the parameters ß, α and D for a suitable value range. This is done by means of numerical experiments, consisting of the generation of self-affine and fractal synthetic series. Each one of these sets has a minimum of 500 series with pre-established values of ß, D and Ha. In addition, these are constructed in a suitable value range for each generation method. Later, we apply fractal methods to each one of the three sets of series and the relations between the parameters are obtained. It is observed that all the relations have a linear part as it is expected, but there exist differences in the validity range depending on the generation method.