Enhanced scaling crossover detection in long-range correlated time series

We have established the Markov model for long range correlated time series (LRCS), by analyzing their evolutionary characteristics, then defined a physical effective correlation length (ECL) of the LRCS, which reflects the predictability of the LRCS, and find that the ECL has a better power law relation with the long range correlated exponent (LRCE) of the LRCS. We apply the power law relation between ECL and LRCE to the daily maximum temperature series (DMTS) at 740 stations in China for the period 1960–2005, calculate the ECL of the DMTS, and the results show the remarkable regional distributive features that the ECL is about 10–14 days in west, northwest and northern China and about 5–10 days in east, southeast and southern China. Namely, the predictability of the DMTS is higher in central-west China than in east and southeast China.

The observable outputs of a great variety of complex dynamical systems form long-range correlated time series with scale invariance behavior. Important properties of such time series are related to the statistical behavior of the first-passage time (FPT), i.e., the time required for an output variable that defines the time series to return to a certain value. Experimental findings in complex systems have attributed the properties of the FPT probability distribution and the FPT mean value to the specifics of the particular system. However, in a previous work we showed (Carretero-Campos, Phys Rev E 85:011139, 2012) that correlations are a unifying factor behind the variety of findings for FPT, and that diverse systems characterized by the same degree of correlations in the output time series exhibit similar FPT properties. Here, we extend our analysis and study the FPT properties of long-range correlated time series with crossovers in the scaling, similar to those observed in many experimental systems. To do so, first we introduce an algorithm able to generate artificial time series of this kind, and study numerically the statistical properties of FPT for these time series. Then, we compare our results to those found in the output time series of real systems and we demonstrate that, independently of the specifics of the system, correlations are the unifying factor underlying key FPT properties of systems with output time series exhibiting crossovers in the scaling.

The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long-range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long-range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long-range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.

We analyze the stochastic function C(n)(i) identical with y(i)-y(n)(i), where y(i) is a long-range correlated time series of length N(max) and y(n)(i) identical with (1/n) Sigma(n-1)(k=0)y(i-k) is the moving average with window n. We argue that C(n)(i) generates a stationary sequence of self-affine clusters C with length l, lifetime tau, and area s. The length and the area are related to the lifetime by the relationships l approximately tau(psi(l)) and s approximately tau(psi(s)), where psi(l)=1 and psi(s)=1+H. We also find that l, tau, and s are power law distributed with exponents depending on H: P(l) approximately l(-alpha), P(tau) approximately tau(-beta), and P(s) approximately s(-gamma), with alpha=beta=2-H and gamma=2/(1+H). These predictions are tested by extensive simulations on series generated by the midpoint displacement algorithm of assigned Hurst exponent H (ranging from 0.05 to 0.95) of length up to N(max)=2(21) and n up to 2(13).

Enhanced Scaling Crossover Detection in Long-Range Correlated Time Series

By establishing the Markov model for a long-range correlated time series (LRCS) and analysing its evolutionary characteristics, this paper defines a physical effective correlation length (ECL) τ, which reflects the predictability of the LRCS. It also finds that the ECL has a better power law relation with the long-range correlated exponent γ of the LRCS: τ = K exp(−γ/0.3) + Y, (0 < γ < 1) — the predictability of the LRCS decays exponentially with the increase of γ. It is then applied to a daily maximum temperature series (DMTS) recorded at 740 stations in China between the years 1960–2005 and calculates the ECL of the DMTS. The results show the remarkable regional distributive feature that the ECL is about 10–14 days in west, northwest and northern China, and about 5–10 days in east, southeast and southern China. Namely, the predictability of the DMTS is higher in central-west China than in east and southeast China. In addition, the ECL is reduced by 1–8 days in most areas of China after subtracting the seasonal oscillation signal of the DMTS from its original DMTS; however, it is only slightly altered when the decadal linear trend is removed from the original DMTS. Therefore, it is shown that seasonal oscillation is a significant component of daily maximum temperature evolution and may provide a basis for predicting daily maximum temperatures. Seasonal oscillation is also significant for guiding general weather predictions, as well as seasonal weather predictions.

Various fields within biological and psychological inquiry recognize the significance of exploring long-range temporal correlations to study phenomena. However, these fields face challenges during this transition, primarily stemming from the impracticality of acquiring the considerably longer time series demanded by canonical methods. The Bayesian Hurst-Kolmogorov (HK) method estimates the Hurst exponents of time series-quantifying the strength of long-range temporal correlations or "fractality"-more accurately than the canonical detrended fluctuation analysis (DFA), especially when the time series is short. Therefore, the systematic application of the HK method has been encouraged to assess the strength of long-range temporal correlations in empirical time series in behavioral sciences. However, the Bayesian foundation of the HK method fuels reservations about its performance when artifacts corrupt time series. Here, we compare the HK method's and DFA's performance in estimating the Hurst exponents of synthetic long-range correlated time series in the presence of additive white Gaussian noise, fractional Gaussian noise, short-range correlations, and various periodic and non-periodic trends. These artifacts can affect the accuracy and variability of the Hurst exponent and, therefore, the interpretation and generalizability of behavioral research findings. We show that the HK method outperforms DFA in most contexts-while both processes break down for anti-persistent time series, the HK method continues to provide reasonably accurate H values for persistent time series as short as N=64 samples. Not only can the HK method detect long-range temporal correlations accurately, show minimal dispersion around the central tendency, and not be affected by the time series length, but it is also more immune to artifacts than DFA. This information becomes particularly valuable in favor of choosing the HK method over DFA, especially when acquiring a longer time series proves challenging due to methodological constraints, such as in studies involving psychological phenomena that rely on self-reports. Moreover, it holds significance when the researcher foreknows that the empirical time series may be susceptible to contamination from these processes.

We systematically study the scaling properties of the magnitude and sign of the fluctuations in correlated time series, which is a simple and useful approach to distinguish between systems with different dynamical properties but the same linear correlations. First, we decompose artificial long-range power-law linearly correlated time series into magnitude and sign series derived from the consecutive increments in the original series, and we study their correlation properties. We find analytical expressions for the correlation exponent of the sign series as a function of the exponent of the original series. Such expressions are necessary for modeling surrogate time series with desired scaling properties. Next, we study linear and nonlinear correlation properties of series composed as products of independent magnitude and sign series. These surrogate series can be considered as a zero-order approximation to the analysis of the coupling of magnitude and sign in real data, a problem still open in many fields. We find analytical results for the scaling behavior of the composed series as a function of the correlation exponents of the magnitude and sign series used in the composition, and we determine the ranges of magnitude and sign correlation exponents leading to either single scaling or to crossover behaviors. Finally, we obtain how the linear and nonlinear properties of the composed series depend on the correlation exponents of their magnitude and sign series. Based on this information we propose a method to generate surrogate series with controlled correlation exponent and multifractal spectrum.

Previous studies indicated that nonlinear properties of Gaussian distributed time series with long-range correlations, u(i), can be detected and quantified by studying the correlations in the magnitude series |u(i)|, the "volatility." However, the origin for this empirical observation still remains unclear and the exact relation between the correlations in u(i) and the correlations in |u(i)| is still unknown. Here we develop analytical relations between the scaling exponent of linear series u(i) and its magnitude series |u(i)|. Moreover, we find that nonlinear time series exhibit stronger (or the same) correlations in the magnitude time series compared with linear time series with the same two-point correlations. Based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series; the nonlinear multifractal time series is generated by multiplying a long-range correlated time series (that represents the magnitude series) with uncorrelated time series [that represents the sign series sgn (u(i))]. We apply our techniques on daily deep ocean temperature records from the equatorial Pacific, the region of the El-Ninõ phenomenon, and find: (i) long-range correlations from several days to several years with 1/f power spectrum, (ii) significant nonlinear behavior as expressed by long-range correlations of the volatility series, and (iii) broad multifractal spectrum.

Could network analysis of horizontal visibility graphs be faithfully used to infer long-term memory properties in real-world time series?

Early warning indicators of the collapse of the Atlantic Meridional Overturning Circulation (MOC) have up to now mostly been based on temporal correlations in single time series. Here, we propose new indicators based on spatial correlations in the time series of the Atlantic temperature field. To demonstrate the performance of these indicators, we use a meridional‒depth model of the MOC for which the critical conditions for collapse can be explicitly computed. An interaction network approach is used to monitor changes in spatial correlations in the model temperature time series as the critical transition is approached. The new early warning indicators are based on changes in topological properties of the network, in particular changes in the distribution functions of the degree and the clustering coefficient.

Effect of noise on fractal structure

Recently there has been much attention devoted to exploring the complicated possibly chaotic dynamics in pseudoperiodic time series. Two methods [Zhang, Phys. Rev. E 73, 016216 (2006); Zhang and Small, Phys. Rev. Lett. 96, 238701 (2006)] have been forwarded to reveal the chaotic temporal and spatial correlations, respectively, among the cycles in the time series. Both these methods treat the cycle as the basic unit and design specific statistics that indicate the presence of chaotic dynamics. In this paper, we verify the validity of these statistics to capture the chaotic correlation among cycles by using the surrogate data method. In particular, the statistics computed for the original time series are compared with those from its surrogates. The surrogate data we generate is pseudoperiodic type (PPS), which preserves the inherent periodic components while destroying the subtle nonlinear (chaotic) structure. Since the inherent chaotic correlations among cycles, either spatial or temporal (which are suitably characterized by the proposed statistics), are eliminated through the surrogate generation process, we expect the statistics from the surrogate to take significantly different values than those from the original time series. Hence the ability of the statistics to capture the chaotic correlation in the time series can be validated. Application of this procedure to both chaotic time series and real world data clearly demonstrates the effectiveness of the statistics. We have found clear evidence of chaotic correlations among cycles in human electrocardiogram and vowel time series. Furthermore, we show that this framework is more sensitive to examine the subtle changes in the dynamics of the time series due to the match between PPS surrogate and the statistics adopted. It offers a more reliable tool to reveal the possible correlations among cycles intrinsic to the chaotic nature of the pseudoperiodic time series.

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