Conclusion and Summary

In this paper we show that the DFA method performs better than the standard R/S analysis to quantify the scaling behavior of noisy signals for a wide range of correlations, and we estimate the range of scales where the performance of the DFA method is optimal. We consider different types of trends superposed on correlated noise, and study how these trends affect the scaling behavior of the noise. We demonstrate that there is a competition between a trend and a noise, and that this competition can lead to crossovers in the scaling. We investigate the features of these crossovers, their dependence on the properties of the noise and the superposed trend. Surprisingly, we find that crossovers which are a result of trends can exhibit power-law dependences on the parameters of the trends. We show that these crossover phenomena can be explained by the superposition of the separate results of the DFA method on the noise and on the trend, assuming that the noise and the trend are not correlated, and that the scaling properties of the noise and the apparent scaling behavior of the trend are known. Our work may provide some help to differentiate between different types of crossovers -- e.g. crossovers which separate scaling regions with different correlation properties may differ from crossovers which are an artifact of trends. The results we present here could be useful for identifying the presence of trends and to accurately interpret correlation properties of noisy data.