Characterizing Detrended Fluctuation Analysis of multifractional Brownian motion

V.A. Setty, A.S. Sharma
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="" style="color: black;">Physica A: Statistical Mechanics and its Applications</a> </i> &nbsp;
The Hurst exponent (H) is widely used to quantify long range dependence in time series data and is estimated using several well known techniques. Recognizing its ability to remove trends the Detrended Fluctuation Analysis (DFA) is used extensively to estimate a Hurst exponent in non-stationary data. Mutifractional Brownian motion (mBm) broadly encompasses a set of models of non-stationary data exhibiting time varying Hurst exponents, H(t) as against a constant H. Recently, there has been a
more &raquo; ... ng interest in time dependence of H(t) and sliding window techniques have been used to estimate a local time average of the exponent. This brought to fore the ability of DFA to estimate scaling exponents in systems with time varying H(t), such as mBm. This paper characterizes the performance of DFA on mBm data with linearly varying H(t) and further test the robustness of estimated time average with respect to data and technique related parameters. Our results serve as a bench-mark for using DFA as a sliding window estimator to obtain H(t) from time series data. I. INTRODUCTION Statistical properties such as trends and correlations of complex phenomena are important in the study of nonequilibrium phenomena such as extreme events.. Due to the non-equilibrium nature of complex driven systems, general statistical analysis tools cannot be readily applied to them. Long range dependence (LRD) in data is a key feature [1] and is studied in data from diverse physical systems such as temperature records, river flows, heart beat variability, space weather etc., [2]- [11] . Rescaled range analysis (R/S ) [12] and fluctuation analysis (FA) [13] are statistical tools developed to estimate the variability of time series through estimation of Hurst exponent, H [14], a statistic which is directly related to the scaling in autocorrelation functions, and, also to the fractal dimension of the time series data. While the scaling exponent, H, is equal to 0.5 for uncorrelated white noise, many natural systems demonstrate values close to 0.7 [15] . These techniques, however, fail to estimate H in non-stationary data. More recently, Detrended Fluctuation Analysis (DFA) [16] , which is widely considered a better technique than either R/S or FA due to its capability to detrend a time series data while estimating H, making it viable for non-stationary systems. With increased use of DFA technique, its limitations in detrending capabilities are evident [17] and there is need for better alternative detrending schemes for data with atypical trends e.g., trends that are not addressable by polynomial detrending [18] . Inspite of its purported shortcomings, DFA is recognized as an efficient Hurst exponent estimation technique because it utilizes detrending to estimate over lesser number of averages than FA. Fractional Brownian motion (fBm), a generalization of Brownian motion, is a quintessential theoretical model for the Hurst effect [19] . Since its discovery, there has been an interest in modeling physical systems as fBm. However, it was quickly realized that imposing a uniform H over the span of the data is in fact a restricting condition as uniform level of LRD in real life data is uncommon. Multifractional Brownian motion (mBm) is a generalization of fBm relaxing this condition [20], allowing for variable degrees of self-similarity with non-stationary increments i.e., H varies as H(t) over the time span of the data. It should be realized that mBm is also multifractal in nature due to multiple fractal dimensions in the system within the time span of the data. Tunability of its local regularity is a valuable property of mBm, realizing which there has been increased interest in modelling various geophysical systems as mBm [21] [22] [23] . Although there is increasing use of DFA as a technique to study LRD in time series data, it is widely recognized that it yields a single Hurst exponent, and thus can not distinguish between multi-fractal and mono-fractal systems, e. g., between mBm and fBm. In fact most systems exhibit time varying H exponent, but the estimates yield a constant value. Further, previous studies show the effect of data size used on the Hurst exponent [24, 25] , thus requiring caution in the interpretation of the estimated values. This is in direct agreement with our study of effect of data size on the Hurst exponent estimated by DFA in mBm data as seen in section IV B. Other schemes such as Multi Fractal Detrended Fluctuation Analysis (MF-DFA) were proposed [26] , though such techniques address the multifractal nature of time series with respect to one fractal dimension at a time and do not provide a solution with respect to estimating the time varying fractal structure of mBm. It is apparent that DFA and other similar techniques were assumed to locally estimate a time averaged Hurst exponent [27] [28] . This assumption underlies
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="">doi:10.1016/j.physa.2014.10.016</a> <a target="_blank" rel="external noopener" href="">fatcat:5izezovagjbvnjyoaeojlrvfoq</a> </span>
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