A semi-continuous box counting method for fractal dimension measurement of short single dimension temporal signals - Preliminary study

Box counting method allows to measure the eventual fractal dimension (D) of a single dimension temporal signal. However its accuracy varies as a function of the frequency sampling (Fs) and the duration of the tested signal (Sd). Consequently, as it is impossible to highly increase Fs, this method is not suitable for short physical signals D measurement. Thus, we designed a semi-continuous box counting method (SCBC) allowing a better approach of the small scales of the signal, especially useful in case of short single dimension temporal signal. Let N = number of samples of the tested signal. SCBC provides with the first M points of the graph log - log owing to the dyadic division of boxes at large scales up to a certain box size S-M,. such as S-M = 2(M)/Fs. Then, at smaller scales, for each successive point the box size decreases by 1/Fs, that provides the - log with a large number of points. Thus, when N/S(M+x)/Fs is not a whole number, the analyzed signal is peripherally and symmetrically reduced in abscissa and ordinate, so that a whole number of boxes is obtained. But these truncated samples are then reintroduced for designing following boxes. Using SCBC we measured D of mathematical signals which D is known, and compared these results to those obtained using the classic dyadic box counting method.