On the automatic selection of the onset of scaling

A method is developed for the automatic detection of the onset of scaling for long-range dependent (LRD) time series and other asymptotically scale-invariant processes. Based on wavelet techniques, it provides the lower cutoff scale for the regression that yields the scaling exponent. The method detects the onset of scaling through the dramatic improvement of a goodness-of-fit statistic taken as a function of this lower cutoff scale. It relies on qualitative features of the goodness-of-fit statistic and on features of the wavelet analysis. The method is easy to implement, appropriate for large data sets and highly robust. It is tested against 34 time series models and found to perform very well. Examples involving telecommunications data are presented.