Alexander S. Balankin
Grupo “Mecánica Fractal”, Instituto Politécnico Nacional, México D.F., México 07738

We propose a new approach for properly analyzing stochastic time series by mapping the dynamics of time series fluctuations onto a suitable nonequilibrium surface-growth problem. In this framework, the fluctuation sampling time interval plays the role of time variable, whereas the physical time is treated as the analog of spatial variable. In this way we found that the fluctuations of many real-world time series satisfy the analog of the Family-Viscek dynamic scaling ansatz. This finding permits to use the powerful tools of kinetic roughening theory to classify, model, and forecast the fluctuations of real-world time series.

PACS numbers: 89.75.Da, 05.45.Tp, 05.40.-a, 83.60.Uv

2 The internal dynamics of many complex real-world systems is often studied through analysis of the fluctuations of the output time series [1,2]. Statistically, the fluctuations of time series ) (t z

are commonly characterized by their log-returns [ ]) ( /) ( ln ) , ( τ τ +

t z t z t v for a fixed sampling time interval τ [3]. A large number of studies have been carried out to analyze the time series of dynamical variables from physical, biological and financial systems (e.g., [1-4,5]). Since the resulting observable variable associated with fluctuations at each moment is the product of log-return magnitude and sign, recent investigations have focused on the study of correlations in the absolute value and sign time series [6]. It was found that the absolute values of log- returns, v t V

) , ( τ , of many real-world time series exhibit long-range power-law correlations [3-6], indicating that the system does not immediately respond to an amount of information flowing in it, but reacts to it gradually over a period of time [7]. Accordingly, the analysis of the scaling properties of the fluctuations has been shown to give important information regarding the underlying processes responsible for the observed macroscopic behavior of real-world system [1-7].

The long-term memory in the time series fluctuations is commonly analyzed through a study of their structure function, defined as [ ] 2 / 1 2 ) , ( ) , ( ) , ( τ τ δ δ τ σ t V t t V t − +

, where the overbar denotes average over all t in time series of length τ − T (T is the length of original time series ) (t z ) and the brackets denote average over different realizations of the time window of size tδ [3]. Alternatively, one can use power-spectrum of log-return time series, defined as ) , ( ) , ( ) , ( τ τ τ q V q V q S −

) ) , where

3 ) exp( ] ) , ( [ ) ( ) , ( 2 / 1 iqt V t V T q V t − −

∑ − τ τ τ ) is the Fourier transform of
) (t V [3-5]. It was found that the structure function and the power spectrum of the absolute log-returns commonly exhibit the power law behavior

ζ δ σ ) ( t ∝ and θ − ∝q S , (1)

where 1 2 + = ζ θ , even when the time series ) (t z looks erratic and its log-returns ) , ( τ t v

are apparently random (see [3-5]). The scaling exponent ζ , also called the Hurst exponent, characterizes the strength of long-range correlations in the fluctuation behavior [8]. It has been shown that the knowledge of ζ is very helpful also for practical purposes [3-7]. However, the scaling behavior (1) characterizes the correlations in a log-return time series treated as an analog of rough interface in (1+1) dimensions (see Refs. [8,9,10]). It’s worth nothing that this model gives very limited information about the studied system that does not permit to determine the universality class of the system, even within the framework of an equilibrium-type theory [11]. Additional information can be obtained from the studies of the probability density function of log-return record [12] and/or the diffusion entropy analysis [13].

A better physical understanding of fluctuation dynamics requires a proper description for correlation properties of local variables on different time sampling intervals [5]. In this work, we propose a novel approach to properly understand the dynamics of time series fluctuations by studying the behavior of absolute log-returns for different sampling intervals. Specifically, we suggest to treat the absolute log-return time series as a moving

4 interface ) , ( τ t V , where the sampling time interval, τ , plays the role of time variable, whereas the physical time, t , plays the role of spatial variable.

In this framework, we performed the dynamic analysis of fluctuations in the time series associated with systems of different nature. Specifically, we studied time series associated with the physical, economic, and informatic systems (see Table 1) early analyzed within other frameworks. Namely, the stress-strain behavior ) (

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