Introduction

Many physical and biological systems exhibit complex behavior characterized by long-range power-law correlations. Traditional approaches such as the power-spectrum and correlation analysis are not suited to accurately quantify long-range correlations in non-stationary signals -- e.g. signals exhibiting fluctuations along polynomial trends. Detrended fluctuation analysis (DFA)[1,2,3,4] is a scaling analysis method providing a simple quantitative parameter -- the scaling exponent $\alpha$ -- to represent the correlation properties of a signal. The advantages of DFA over many methods are that it permits the detection of long-range correlations embedded in seemingly non-stationary time series, and also avoids the spurious detection of apparent long-range correlations that are artifact of non-stationarity. In the past few years, more than 100 publications have utilized the DFA as method of correlation analysis, and have uncovered long-range power-law correlations in many research fields such as cardiac dynamics[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23], bioinformatics[2,1,24,25,26,27,28,29,30,31,32,33,34], economics[35,36,37,38,39,40,41,42,43,44,45,46,47], meteorology[48,49,50], material science[51], ethology[52] etc. Furthermore, the DFA method may help identify different states of the same system according to its different scaling behaviors -- e.g. the scaling exponent $\alpha$ for heart inter-beat intervals is different for healthy and sick individuals[14,53,17,16].

The correct interpretation of the scaling results obtained by the DFA method is crucial for understanding the intrinsic dynamics of the systems under study. In fact, for all systems where the DFA method was applied, there are many issues that remain unexplained. One of the common challenges is that the correlation exponent is not always a constant (independent of scale) and crossovers often exist -- i.e. change of the scaling exponent $\alpha$ for different range of scales[16,35,5]. A crossover usually can arise from a change in the correlation properties of the signal at different time or space scales, or can often arise from trends in the data. In this paper, we systematically study how different types of trends affect the apparent scaling behavior of long-range correlated signals. The existence of trends in times series generated by physical or biological systems is so common that it is almost unavoidable. For example, the number of particles emitted by a radiation source in an unit time has a trend of decreasing because the source becomes weaker[54,55]; the density of air due to gravity has a trend at different altitude [56]; the air temperature in different geographic locations and the water flow of rivers have a periodic trend due to seasonal changes[57,58,59,49,50]; the occurrence rate of earthquakes in certain area has trend in different time period[60]. An immediate problem facing researchers applying scaling analysis to time series is whether trends in data arise from external conditions, having little to do with the intrinsic dynamics of the system generating noisy fluctuating data. In this case, a possible approach is to first recognize and filter out the trends before we attempt to quantify correlations in the noise. Alternatively, trends may arise from the intrinsic dynamics of the system, rather than being an epiphenomenon of external conditions, and thus may be correlated with the noisy fluctuations generated by the system. In this case, careful considerations should be given if trends should be filtered out when estimating correlations in the noise, since such "intrinsic" trends may be related to the local properties of the noisy fluctuations.

Here we study the origin and the properties of crossovers in the scaling behavior of noisy signals, by applying the DFA method first on correlated noise and then on noise with trends, and comparing the difference in the scaling results. To this end, we generate artificial time series -- anticorrelated, white and correlated noise with standard deviation equal to one -- using the modified Fourier filtering method introduced by Makse et al.[63]. We consider the case when the trend is independent of the local properties of the noise (external trend). We find that the scaling behavior of noise with a trend is a superposition of the scaling of the noise and the apparent scaling of the trend, and we derive analytical relations based on the DFA, which we call ``superposition rule''. We show how this ``superposition rule'' can be used to determine if the trends are independent of the noisy fluctuation in real data, and if filtering these trends out will no affect the scaling properties of the data.

The outline of this paper is as follows. In Sec.2, we review the algorithm of the DFA method, and in Appendix 7.1 we compare the performance of the DFA with the classical scaling analysis --Hurst's analysis (R/S analysis)-- and show that the DFA is a superior method to quantify the scaling behavior of noisy signals. In Sec. 3, we consider the effect of a linear trend and we present an analytic derivation of the apparent scaling behavior of a linear trend in Appendix 7.3. In Sec. 4, we study a periodic trend, and in Sec. 5 the effect of power-law trend. We systematically study all resulting crossovers, their conditions of existence and their typical characteristics associated with the different types of trends. In addition, we also show how to use DFA appropriately to minimize or even eliminate the effects of those trends in cases that trends are not choices of the study, that is, trends do not reflect the dynamics of the system but are caused by some ``irrelevant'' background. Finally, Sec. 6 contains a summary.