Introduction

In recent years, there has been growing evidence indicating that many physical and biological systems have no characteristic length scale and exhibit long-range power-law correlations. Traditional approaches such as the power-spectrum and correlation analysis are suited to quantify correlations in stationary signals [1,2]. However, many signals which are outputs of complex physical and biological systems are nonstationary -- the mean, standard deviation and higher moments, or the correlation functions are not invariant under time translation [1,2]. Nonstationarity, an important aspect of complex variability, can often be associated with different trends in the signal or heterogeneous segments (patches) with different local statistical properties. To address this problem, detrended fluctuation analysis (DFA) was developed to accurately quantify long-range power-law correlations embedded in a nonstationary time series [3,4]. This method provides a single quantitative parameter -- the scaling exponent $\alpha$ -- to quantify the correlation properties of a signal. One advantage of the DFA method is that it allows the detection of long-range power-law correlations in noisy signals with embedded polynomial trends that can mask the true correlations in the fluctuations of a signal. The DFA method has been successfully applied to research fields such as DNA[3,5,6,7,8,9,10,11,12,13,14,15,16], cardiac dynamics [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37], human gait [38], meteorology [39], climate temperature fluctuations [40,41,42], river flow and discharge [43,44], neural receptors in biological systems [45], and economics [46,47,48,49,50,51,52,53,54,55,56,57,58]. The DFA method may also help identify different states of the same system with different scaling behavior -- e.g., the scaling exponent $\alpha$ for heart-beat intervals is different for healthy and sick individuals [17,28] as well as for waking and sleeping states [23,33].

To understand the intrinsic dynamics of a given system, it is important to analyze and correctly interpret its output signals. One of the common challenges is that the scaling exponent is not always constant (independent of scale) and crossovers often exist -- i.e., the value of the scaling exponent $\alpha$ differs for different ranges of scales [17,18,23,59,60]. A crossover is usually due to a change in the correlation properties of the signal at different time or space scales, though it can also be a result of nonstationarities in the signal. A recent work considered different types of nonstationarities associated with different trends (e.g., polynomial, sinusoidal and power-law trends) and systematically studied their effect on the scaling behavior of long-range correlated signals [61]. Here we consider the effects of three other types of nonstationarities which are often encountered in real data or result from ``standard'' data pre-processing approaches.

(i) Signals with segments removed
First we consider a type of nonstationarity caused by discontinuities in signals. Discontinuities may arise from the nature of experimental recordings - e.g., stock exchange data are not recorded during the nights, weekends and holidays [46,47,48,49,50,51,52,53]. Alternatively, discontinuities may be caused by the fact that some noisy and unreliable portions of continuous recordings must be discarded, as often occurs when analyzing physiological signals [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]. In this case, a common pre-processing procedure is to cut out the noisy, unreliable parts of the recording and stitch together the remaining informative segments before any statistical analysis is performed. One immediate problem is how such cutting procedure will affect the scaling properties of long-range correlated signals. A careful consideration should be given when interpreting results obtained from scaling analysis, so that an accurate estimate of the true correlation properties of the original signal may be obtained.

(ii) Signals with random spikes
A second type of nonstationarity is due to the existence of spikes in data, which is very common in real life signals [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. Spikes may arise from external conditions which have little to do with the intrinsic dynamics of the system. In this case, we must distinguish the spikes from normal intrinsic fluctuations in the system's output and filter them out when we attempt to quantify correlations. Alternatively, spikes may arise from the intrinsic dynamics of the system, rather than being an epiphenomenon of external conditions. In this second case, careful considerations should be given as to whether the spikes should be filtered out when estimating correlations in the signal, since such ``intrinsic'' spikes may be related to the properties of the noisy fluctuations. Here, we consider only the simpler case - namely, when the spikes are independent of the fluctuations in the signal. The problem is how spikes affect the scaling behavior of correlated signals, e.g., what kind of crossovers they may possibly cause. We also demonstrate to what extent features of the crossovers depend on the statistical properties of the spikes. Furthermore, we show how to recognize if a crossover indeed indicates a transition from one type of underlying correlations to a different type, or if the crossover is due to spikes without any transition in the dynamical properties of the fluctuations.

(iii) Signals with different local behavior
A third type of nonstationarity is associated with the presence of segments in a signal which exhibit different local statistical properties, e.g., different local standard deviations or different local correlations. Some examples include: (a) 24 hour records of heart rate fluctuations are characterized by segments with larger standard deviation during stress and physical activity and segments with smaller standard deviation during rest [19]; (b) studies of DNA show that coding and non-coding regions are characterized by different types of correlations [5,8]; (c) brain wave analysis of different sleep stages (rapid eye movement [REM] sleep, light sleep and deep sleep) indicates that the signal during each stage may have different correlation properties [62]; (d) heartbeat signals during different sleep stages exhibit different scaling properties[33]. For such complex signals, results from scaling analysis often reveal a very complicated structure. It is a challenge to quantify the correlation properties of these signals. Here, we take a first step toward understanding the scaling behavior of such signals.

We study these three types of nonstationarities embedded in correlated signals. We apply the DFA method to stationary correlated signals and identical signals with artificially imposed nonstationarities, and compare the difference in the scaling results. (i) We find that cutting segments from a signal and stitching together the remaining parts does not affect the scaling for positively correlated signals. However, this cutting procedure strongly affects anti-correlated signals, leading to a crossover from an anti-correlated regime (at small scales) to an uncorrelated regime (at large scales). (ii) For the correlated signals with superposed random spikes, we find that the scaling behavior is a superposition of the scaling of the signal and the apparent scaling of the spikes. We analytically prove this superposition relation by introducing a superposition rule. (iii) For the case of complex signals comprised of segments with different local properties, we find that their scaling behavior is a superposition of the scaling of the different components -- each component containing only the segments exhibiting identical statistical properties. Thus, to obtain the scaling properties of the signal, we need only to examine the properties of each component -- a much simpler task than analyzing the original signal.

The layout of the paper is as follows: In Sec. 2, we describe how we generate signals with desired long-range correlation properties and introduce the DFA method to quantify these correlations. In Sec. 3, we compare the scaling properties of correlated signals before and after removing some segments from the signals. In Sec. 4, we consider the effect of random spikes on correlated signals. We show that the superposition of spikes and signals can be explained by a superposition rule derived in Appendix 7.1. In Sec. 5, we study signals comprised of segments with different local behavior. We systematically examine all resulting crossovers, their conditions of existence, and their typical characteristics associated with the different types of nonstationarity. We summarize our findings in Sec. 6.