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Rescaled Range Analysis and Detrended Fluctuation Analysis: Finite Sample Properties and Confidence Intervals
Abstract and Figures
We focus on finite sample properties of two mostly used methods of Hurst exponent H estimation—rescaled range analysis (R/S) and detrended fluctuation analysis (DFA). Even though both methods have been widely applied on different types of financial assets, only several papers have dealt with the finite sample properties which are crucial as the properties differ significantly from the asymptotic ones. Recently, R/S analysis has been shown to overestimate H when compared to DFA. However, we show that even though the estimates of R/S are truly significantly higher than an asymptotic limit of 0.5, for random time series with lengths from 2^9 to 2^17, they remain very close to the estimates proposed by Anis & Lloyd and the estimated standard deviations are lower than the ones of DFA. On the other hand, DFA estimates are very close to 0.5. The results propose that R/S still remains useful and robust method even when compared to newer method of DFA which is usually preferred in recent literature.
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... Following Urquhart (2016) and Bariviera (2017), the Hurst exponent is calculated using the rescaled range analysis (R/S). According to Kristoufek (2010), this method can be represented as an analysis of the rescaled range of a time series for different scales of a given length. In effect, there is a dependence on a distraction (range -R) from different lengths of scale (i). ...
... As pointed out by Kristoufek (2010), the standard deviations for the rescaled range analysis are smaller compared to the detrended fluctuation analysis (DFA) which is a very popular alternative in this case. However, he states that in general, the results of both methods are quite similar. ...
... However, he states that in general, the results of both methods are quite similar. Furthermore, Kristoufek (2010) recommends applying a minimum scale of 16 observations and a minimum length of time series equal to 512 data points in the case of R/S. He argues that too-small scales can lead to an incorrect value of the standard deviation (bias), which is used to rescale the ranges during the estimation of the Hurst exponent. ...
Despite recent studies focused on comparing the dynamics of market efficiency between Bitcoin and other traditional assets, there is a lack of knowledge about whether Bitcoin and emerging markets efficiency behave similarly. This paper aims to compare the market efficiency dynamics between Bitcoin and the emerging stock markets. In particular, this study indicates whether the dynamics of Bitcoin market efficiency mimic those of emerging stock markets. Thus, the paper's contribution emerges from the combination of Bitcoin and emerging markets in the field of dynamics of market efficiency. The dynamics of market efficiency are measured using the Hurst exponent in the rolling window. The study uses daily data for the MSCI Emerging Markets Index and the Bitcoin market over the period 2011–2022. Our results show that there is at most a moderate correlation between the dynamics of Bitcoin and emerging stock markets’ efficiency over the entire study period. The strongest correlations occur mainly in periods of high economic policy uncertainty in the largest Bitcoin mining countries. Therefore, the association between Bitcoin market efficiency and emerging stock markets’ efficiency may strengthen with an increase in economic policy uncertainty. These findings may be useful for investors and portfolio managers in constructing better investment strategies.
... Peng et al. (1994) [22] developed the detrended fluctuation analysis (DFA) to analyze the existence of serial dependence (the statistical self-affinity of a signal), with the advantage of being also possible to be used in non-stationary data. Its main advantage is to avoid spurious detection of long-range dependence due to non-stationary data [25][26][27][28][29]. For a given "Y" time series, the algorithm is described as follows: ...
Climate change is one of the most relevant challenges that the world has to deal with. Studies that aim to understand the behavior of environmental and atmospheric variables and the way they relate to each other can provide helpful insights into how the climate is changing. However, such studies are complex and rarely found in the literature, especially in dealing with data from the Brazilian territory. In this paper, we analyze four environmental and atmospheric variables, namely, wind speed, radiation, temperature, and humidity, measured in 27 Weather Stations (the capital of each of the 26 Brazilian states plus the federal district). We use the detrended fluctuation analysis to evaluate the statistical self-affinity of the time series, as well as the cross-correlation coefficient ρ DCCA to quantify the long-range cross-correlation between stations, and a network analysis that considers the top 10% ρ DCCA values to represent the cross-correlations between stations better. The methodology used in this paper represents a step forward in the field of hybrid methodologies, combining time series and network analysis that can be applied to other regions, other environmental variables, and also to other fields of research. The application results are of great importance to better understand the behavior of environmental and atmospheric variables in the Brazilian territory and to provide helpful insights about climate change and renewable energy production.
... If it scales with a power law (i.e., exhibits a nonlinear relationship), the time series exhibits long-term persistence or anti-persistence. This method can be used to calculate the rescaled range's dependence on smaller observation periods [14], [15]. When partitioning a time series of length N into several shorter series with lengths = /2, /4, etc., the average rescaled range for each value is determined. ...
The global climate has been changing rapidly in recent decades, with significant consequences for the environment and human societies. Understanding the long-term behavior and properties of climate data is crucial for predicting future changes and developing effective mitigation strategies. This study investigates the fractal and stationary properties of global temperature anomaly time series data from 1880 to 2022 using statistical techniques such as the Hurst exponent, rescaled range analysis, detrended fluctuation analysis, augmented Dicky Fuller test, and Kwiatkowski-Phillips-Schmidt-Shin test. The results of the analysis reveal that the global temperature anomaly time series exhibits fractal behavior with a Hurst exponent value of 0.6 during the last 42 years, indicating persistent long-term memory. Additionally, the data show nonstationarity with a significant increasing trend over the entire period of analysis. The authors found evidence of changes in the fractal properties of the data since 1980, possibly due to human-induced climate change. This study provides vital insights into the complexity of global temperature anomaly time series data and highlights the need for continuous tracking and evaluation of climate data to better understand and manage the issues of climate change. The findings have important implications for climate modeling and policy development, highlighting the need for continued efforts to mitigate climate change and its impacts.
... al., 2012). Moreover, it can calculate Hurst Exponent (Krištoufek, 2010;KKW, 2014;Kannan, 2012). . The Hurst Exponent is the measure of the smoothness of fractal time series based on the asymptotic behaviour of the rescaled range of the process. ...
Present study is aimed at investigating the solar faculae area from 1990 to 2007 which partially covered the 22nd and 23rd solar cycle. Rescaled Range Analysis (RRA) and Detrended Fluctuation Analysis (DFA) have been adopted to evaluate the behaviour of nonlinear dynamics of solar faculae area. Results show that the value of Hurst exponent for solar faculae area from RRA and DFA is negatively correlated. It means it is non-persistent and long-range correlated. Obtained result is inaccurate so the only solution is to transform the data into stationary data by taking differencing. RRA is applied on residuals and RRA to evaluate the fractal property of the time series. Solar faculae area investigated in this study is fractal in nature and predictable ass well. Moreover, the time series of solar faculae area is non-linear as established by the Brock – Dechert – Scheinkman (BDS) test results.
... A dimensionless ratio, R/S, in a rescaled range analysis (R/S analysis), was utilised to calculate the rescale range of each subsequence [56,57]. The Hurst index (H) of the R/S analysis is an indicator used to measure the correlation and trend intensity of a time series. ...
... Similarly, LRCs are apparent when 1 < HFD < 2, where HFD → 1 is indicative of weaker LRCs and HFD → 2 is indicative of stronger LRCs [28]. Further, R/S produces the Hurst (H) exponent, which is equivalent to α for measuring LRCs, where 0.5 < H ≤ 1 is indicative of a persistent time series, hence LRCs [29,30]. Measuring LRCs in human performance variables is important because LRCs have been proposed as a sign of healthy physiological systems [31][32][33] as proposed in the Optimal Movement Variability Hypothesis (OMVH) [3,34]. ...
Background:
Walking and running are common forms of locomotion, both of which exhibit variability over many gait cycles. Many studies have investigated the patterns generated from that ebb and flow, and a large proportion suggests human gait exhibits Long Range Correlations (LRCs). LRCs refer to the observation that healthy gait characteristic, like stride times, are positively correlated to themselves over time. Literature on LRCs in walking gait is well known but less attention has been given to LRCs in running gait.
Research question:
What is the state of the art concerning LRCs in running gait?
Methods:
We conducted a systematic review to identify the typical LRC patterns present in human running gait, in addition to disease, injury, and running surface effects on LRCs. Inclusion criteria were human subjects, running related experiments, computed LRCs, and experimental design. Exclusion criteria were studies on animals, non-humans, walking only, non-running, non-LRC analysis, and non-experiments.
Results:
The initial search returned 536 articles. After review and deliberation, our review included 26 articles. Almost every article produced strong evidence for LRCs apparent in running gait and in all running surfaces. Additionally, LRCs tended to decrease due to fatigue, past injury, increased load carriage and seem to be lowest at preferred running speed on a treadmill. No studies investigated disease effects on LRCs in running gait.
Significance:
LRCs seem to increase with deviations away from preferred running speed. Previously injured runners produced decreased LRCs compared to non-injured runners. LRCs also tended to decrease due to an increase in fatigue rate, which has been associated with increased injury rate. Lastly, there is a need for research on the typical LRCs in an overground environment, for which the typical LRCs found in a treadmill environment may or may not transfer.
... The Hurst exponent, H, is given by the slope of this line (Li and Chen, 2001). This method can be used to calculate the R/S's dependence on smaller observation periods (Li and Chen, 2001;Krištoufek, 2010). Then the average R/S is calculated for each value n, where n is the length of a few smaller time series divided from a time series of length N, each with a length of n = N, N / 2, N / 4... ...
... al., 2012). Moreover, it can calculate Hurst Exponent (Krištoufek, 2010;KKW, 2014;Kannan, 2012). . The Hurst Exponent is the measure of the smoothness of fractal time series based on the asymptotic behaviour of the rescaled range of the process. ...
Present study is aimed at investigating the solar faculae area from 1990 to 2007 which partially covered the22nd and 23rd solar cycle. Rescaled Range Analysis (RRA) and Detrended Fluctuation Analysis (DFA) have beenadopted to evaluate the behaviour of nonlinear dynamics of solar faculae area. Results show that the value of Hurstexponent for solar faculae area from RRA and DFA is negatively correlated. It means it is non-persistent and longrange correlated. Obtained result is inaccurate so the only solution is to transform the data into stationary data by takingdifferencing. RRA is applied on residuals and RRA to evaluate the fractal property of the time series. Solar faculae areainvestigated in this study is fractal in nature and predictable ass well. Moreover, the time series of solar faculae area isnon-linear as established by the Brock – Dechert – Scheinkman (BDS) test results
This article aims to analyze the daily fluctuations of the time series of wind speed in some municipalities in the State of Bahia, Brazil from January 2009 to December 2018 with the approach of sliding windows. The analysis will be performed, mainly, with the method known in the literature as Detrended Fluctuation Analysis (DFA) able to identify and measure autocorrelation in non-stationary time series on different time scales (Peng et al., 1994). To meet the proposed objective, we chose five cities of Bahia, by methodological option: Barreiras, Feira de Santana, Guanambi, Salvador and Vitória da Conquista. The results indicated a persistent and statistically significant behavior at the level of 5% for all the studied cities and period. The description with sliding windows (w=365) found a predominance of relative variation above 15%, kurtosis and positive asymmetry. Our findings can be used as one more proposal to model wind speed data and contribute to research related to climatological data.
The most suitable paradigms and tools for investigating the scaling structure of financial time series are reviewed and discussed in the light of some recent empirical results. Different types of scaling are distinguished and several definitions of scaling exponents, scaling and multi-scaling processes are given. Methods to estimate such exponents from empirical financial data are reviewed. A detailed description of the Generalized Hurst exponent approach is presented and substantiated with an empirical analysis across different markets and assets.
Recently introduced Detrended Moving Average (DMA) method is examined and compared with Detrended Fluctuation Analysis (DFA) technique for artificial stochastic Brownian time series of various length L ∼ 10 3 ÷ 10 5 . Our analysis reveals some statistical properties of the Hurst exponent values measured with the use of DFA and DMA methods. Good agreement between DFA and DMA techniques is found for long time series L ∼ 10 5 , however for shorter series two methods are clearly distinguishable. No clear systematic relation previously postulated in literature between DFA and DMA results is found. However, it is shown that on the average, DMA method gives overestimation of the Hurst exponent compared with DFA technique.
We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
For the London Stock Exchange we demonstrate that the signs of orders obey a long-memory process. The autocorrelation function decays roughly as a power law with an exponent of 0.6, corresponding to a Hurst exponent H = 0.7. This implies that the signs of future orders are quite predictable from the signs of past orders; all else being equal, this would suggest a very strong market inefficiency. We demonstrate, however, that fluctuations in order signs are compensated for by anti-correlated fluctuations in transaction size and liquidity, which are also long-memory processes that act to make the returns whiter. We show that some institutions display long-range memory and others don’t.
We confront global and local methods to analyze the financial crash-like events on the Polish financial market from the critical phenomena point of view. These methods are based on the analysis of log-periodicity and the local fractal properties of financial time series in the vicinity of phase transitions (crashes). The whole history (1991–2008) of Warsaw Stock Exchange Index (WIG) describing the largest developing financial market in Europe, is analyzed in a daily time horizon. We find that crash-like events on the Polish financial market are described better by the log-divergent price model decorated with log-periodic behavior than the corresponding power-law-divergent price model. Predictions coming from log-periodicity scenario are verified for all main crashes that took place in WIG history. It is argued that crash predictions within log-periodicity model strongly depend on the amount of data taken to make a fit and therefore are likely to contain huge inaccuracies. Turning to local fractal description, we calculate the so-called local (time dependent) Hurst exponent Hloc for the WIG time series and we find the dependence between the behavior of the local fractal properties of the WIG time series and the crashes appearance on the financial market. The latter method seems to work better than the global approach — both for developing as for developed markets. The current situation on the market, particularly related to the Fed intervention in September’07 and the situation on the market immediately after this intervention is also analyzed from the fractional Brownian motion point of view.
Hurst's empirical law concerning geophysical time series such as annual river flows was framed in terms of an adjusted rescaled range, namely, the range of cumulative sums of deviations of summands from a linearly time-shifted origin, expressed in units of the estimated standard deviation of the full sample. Nonsimulatory theoretical results of the Hurst phenomenon have hitherto been confined to the adjusted range, without rescaling. The present paper derives the expectation of the adjusted rescaled range for independent normal summands.
A fundamental hypothesis of quantitative finance is that stock price variations are independent and can be modeled using Brownian motion. In recent years, it was proposed to use rescaled range analysis and its characteristic value, the Hurst exponent, to test for independence in financial time series. Theoretically, independent time series should be characterized by a Hurst exponent of 1/2. However, finite Brownian motion data sets will always give a value of the Hurst exponent larger than 1/2 and without an appropriate statistical test such a value can mistakenly be interpreted as evidence of long term memory. We obtain a more precise statistical significance test for the Hurst exponent and apply it to real financial data sets. Our empirical analysis shows no long-term memory in some financial returns, suggesting that Brownian motion cannot be rejected as a model for price dynamics.