Effect of coarse-graining on detrended fluctuation analysis

Several studies have investigated the scaling behavior in naturally occurring biological and physical processes using techniques such as detrended fluctuation analysis (DFA). Data acquisition is an inherent part of these studies and maps the continuous process into digital data. The resulting digital data is discretized in amplitude and time, and shall be referred to as coarse-grained realization in the present study. Since coarse-graining precedes scaling exponent analysis, it is important to understand its effects on scaling exponent estimators such as DFA. In this brief communication, k-means clustering is used to generate coarse-grained realizations of data sets with different correlation properties, namely: anti-correlated noise, long-range correlated noise and uncorrelated noise. It is shown that the coarse-graining can significantly affect the scaling exponent estimates. It is also shown that scaling exponent can be reliably estimated even at low levels of coarse-graining and the number of the clusters required varies across the data sets with different correlation properties.     

Interpretation of heart rate variability via detrended fluctuation analysis and alphabeta filter

Detrended fluctuation analysis (DFA), suitable for the analysis of nonstationary time series, has confirmed the existence of persistent long-range correlations in healthy heart rate variability data. In this paper, we present the incorporation of the alphabeta filter to DFA to determine patterns in the power-law behavior that can be found in these correlations. Well-known simulated scenarios and real data involving normal and pathological circumstances were used to evaluate this process. The results presented here suggest the existence of evolving patterns, not always following a uniform power-law behavior, that cannot be described by scaling exponents estimated using a linear procedure over two predefined ranges. Instead, the power law is observed to have a continuous variation with segment length. We also show that the study of these patterns, avoiding initial assumptions about the nature of the data, may confer advantages to DFA by revealing more clearly abnormal physiological conditions detected in congestive heart failure patients related to the existence of dominant characteristic scales.     

Comparison of detrending methods for fluctuation analysis

We examine several recently suggested methods for the detection of long-range correlations in data series based on similar ideas as the well-established Detrended Fluctuation Analysis (DFA). In particular, we present a detailed comparison between the regular DFA and two recently suggested methods: the Centered Moving Average (CMA) Method and a Modified Detrended Fluctuation Analysis (MDFA). We find that CMA performs the same as DFA in long data with weak trends and is slightly superior to DFA in short data with weak trends. When comparing standard DFA to MDFA we observe that DFA performs slightly better in almost all examples we studied. We also discuss how several types of trends affect different types of DFA. For weak trends in the data, the new methods are comparable with DFA in these respects. However, if the functional form of the trend in data is not a-priori known, DFA remains the method of choice. Only a comparison of DFA results, using different detrending polynomials, yields full recognition of the trends. A comparison with independent methods is recommended for proving long-range correlations.     

Detrending fluctuation analysis based on high-pass filtering

Detrended fluctuation analysis (DFA) is a scaling method commonly used for detecting long-range correlations in non-stationary time series. The DFA method uses a trend based on polynomial fitting to extract and quantify fluctuations at different time scales. Basically, such procedure acts as a (non-dynamical) high-pass filter that removes time series components below a given time scale. As an alternative to the polynomial fitting approach, this paper proposes a DFA method based on well-known high-pass filters (e.g., Butterworth, elliptic, etc.). Numerical results show that the proposed DFA approach yields results similar to traditional DFA method. Maybe, the main advantage of the proposed DFA method is that efficient implementations of high-pass filters are available commercially.     

Non-uniform scaling behavior in ultra-low-frequency (ULF) earthquake-related geomagnetic signals

Investigating highly non-stationary time series, which typically exhibit long-range correlations, is a classic problem in physics. Here, we analyze the scaling properties of the dynamics of ultra-low-frequency (ULF) geomagnetic data (in the frequency range between 1 mHz and 10 Hz) observed at Izu Peninsula in Japan. On the basis of the detrended fluctuation analysis (DFA), deviations from uniform power-law scaling were quantified. Our findings point out to a significant non-uniform scaling behavior in ULF geomagnetic data in relationship with the occurrence of intense seismic clusters.     

Power law and multiscaling properties of the Chinese stock market

We investigate the cumulative probability density function (PDF) and the multiscaling properties of the returns in the Chinese stock market. By using returns data adjusted for thin trading, we find that the distribution has power-law tails at shorter microscopic timescales or lags. However, the distribution follows an exponential law for longer timescales. Furthermore, we investigate the long-range correlation and multifractality of the returns in the Chinese stock market by the DFA and MFDFA methods. We find that all the scaling exponents are between 0.5 and 1 by DFA method, which exhibits the long-range power-law correlations in the Chinese stock market. Moreover, we find, by MFDFA method, that the generalized Hurst exponents h(q) are not constants, which shows the multifractality in the Chinese stock market. We also find that the correlation of Shenzhen stock market is stronger than that of Shanghai stock market.     

Scaling characteristics of ocean wave height time series

Fluctuations in the significant wave height can be quantified by using scaling statistics. In this paper, the scaling properties of the significant wave height were explored by using a large data set of hourly series from 25 monitoring stations located off the west coast of the US. Detrended fluctuation analysis (DFA) was used to investigate the scaling properties of the series. DFA is a robust technique that can be used to detect long-range correlations in nonstationary time series. The significant wave height data was analyzed by using scales from hourly to monthly. It was found that a common scaling behavior can be observed for all stations. A breakpoint in the scaling region around 4-5 days was apparent. Spectral analysis confirms this result. This breakpoint divided the scaling region into two distinct parts. The first part was for finer scales (up to 4 days) which exhibited Brown noise characteristics, while the second one showed 1/f noise behavior at coarser scales (5 days to 1 month). The first order and the second order DFA (DFA1 and DFA2) were used to check the effect of seasonality. It was found that there were no differences between DFA1 and DFA2 results, indicating that there is no effect of trends in the wave height time series. The resulting scaling coefficients range from 0.696 to 0.890 indicating that the wave height exhibits long-term persistence. There were no coherent spatial variations in the scaling coefficients.     

Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes

We investigate how simultaneously recorded long-range power-law correlated multivariate signals cross-correlate. To this end we introduce a two-component ARFIMA stochastic process and a two-component FIARCH process to generate coupled fractal signals with long-range power-law correlations which are at the same time long-range cross-correlated. We study how the degree of cross-correlations between these signals depends on the scaling exponents characterizing the fractal correlations in each signal and on the coupling between the signals. Our findings have relevance when studying parallel outputs of multiple component of physical, physiological and social systems.     

脑电信号的标度分析及其在睡眠状态区分中的应用

脑电信号具有长程幂律相关性及多重分形的标度特性,并随生理病理状态改变.本文首次针对睡眠脑电信号应用单重分形去趋势波动分析(detrended fluctuation analysis,简记为DFA)方法与多重分形奇异谱对睡眠脑电信号的标度特征进行系统的对比研究.发现DFA标度指数α对于不同导联和样本组间的差异较为敏感,随睡眠状态的变化不规律;而多重分形奇异强度区间Δα随睡眠状态的变化更为规律,睡眠Ⅰ期至Ⅳ期不断增大,并且导联间差异和样本组间差异均较小.多重分形Δα参数更适合作为判定睡眠状态的定量参数.     

Levels of complexity in turbulent time series for weakly and high Reynolds number

We use the detrended fluctuation analysis (DFA), the detrended cross correlation analysis (DCCA) and the magnitude and sign decomposition analysis to study the fluctuations in the turbulent time series and to probe long-term nonlinear levels of complexity in weakly and high turbulent flow. The DFA analysis indicate that there is a time scaling region in the fluctuation function, segregating regimes with different scaling exponents. We discuss that this time scaling region is related to inertial range in turbulent flows. The DCCA exponent implies the presence of power-law cross correlations. In addition, we conclude its multifractality for high Reynold’s number in inertial range. Further, we find that turbulent time series exhibit complex features by magnitude and sign scaling exponents.     

Chaotic SVD method for minimizing the effect of exponential trends in detrended fluctuation analysis

The Detrended Fluctuation Analysis (DFA) and its extensions (MF-DFA) have been used extensively to determine possible long-range correlations in self-affine signals. However, recent studies have reported the susceptibility of DFA to trends which give rise to spurious crossovers and prevent reliable estimation of the scaling exponents. In this study, a smoothing algorithm based on the Chaotic Singular-Value Decomposition (CSVD) is proposed to minimize the effect of exponential trends and distortion in the log-log plots obtained by DFA techniques. The effectiveness of the technique is demonstrated on monofractal and multifractal data corrupted with exponential trends.     

Universal scaling behaviors of meteorological variables’ volatility and relations with original records

Volatility series (defined as the magnitude of the increments between successive elements) of five different meteorological variables over China are analyzed by means of detrended fluctuation analysis (DFA for short). Universal scaling behaviors are found in all volatility records, whose scaling exponents take similar distributions with similar mean values and standard deviations. To reconfirm the relation between long-range correlations in volatility and nonlinearity in original series, DFA is also applied to the magnitude records (defined as the absolute values of the original records). The results clearly indicate that the nonlinearity of the original series is more pronounced in the magnitude series.     

Quantifying cross-correlations using local and global detrending approaches

In order to quantify the long-range cross-correlations between two time series qualitatively, we introduce a new cross-correlations test Q_CC(m), where m is the number of degrees of freedom. If there are no cross-correlations between two time series, the cross-correlation test agrees well with the χ²(m) distribution. If the cross-correlations test exceeds the critical value of the χ²(m) distribution, then we say that the cross-correlations are significant. We show that if a Fourier phase-randomization procedure is carried out on a power-law cross-correlated time series, the cross-correlations test is substantially reduced compared to the case before Fourier phase randomization. We also study the effect of periodic trends on systems with power-law cross-correlations. We find that periodic trends can severely affect the quantitative analysis of long-range correlations, leading to crossovers and other spurious deviations from power laws, implying both local and global detrending approaches should be applied to properly uncover long-range power-law auto-correlations and cross-correlations in the random part of the underlying stochastic process.     

Correlation between detrended fluctuation analysis and the Lempel–Ziv complexity in nonlinear time series analysis

We focus on the study of the correlation between the detrended fluctuation analysis (DFA) and the Lempel-Ziv complexity (LZC) in nonlinear time series analysis in this paper. Typical dynamical systems including logistic map and Duffing model are investigated. Moreover, the influences of the Gaussian random noise on both DFA and LZC are analyzed. The results show a high correlation between DFA and LZC, which can quantify the non-stationarity and the nonlinearity of the time series, respectively. With the enhancement of the random component, the exponent α and the normalized complexity index C show increasing trends. In addition, C is found to be more sensitive to the fluctuation in the nonlinear time series than α. Finally, the correlation between DFA and LZC is applied to the feature extraction of vibration signals for a reciprocating compressor gas valve, and an effective fault diagnosis result is obtained.     

Detecting long-range correlations in fire sequences with Detrended fluctuation analysis

The spatial-temporal power-law distributions are found in many natural systems, which have self-similarity and fractal behavior. By analyzing the time series of such systems, we could expect to explore and understand the underlying mechanisms. In this paper, the Detrended fluctuation analysis (DFA) is used to analyze the long-range correlations of forest and urban fires in Japan and China. It is found that the interevent time series of both forest and urban fires have the persistent long-range power-law correlations, and they all have two scaling exponents, α₁ and α₂, which are both bigger than 0.5 and smaller than 1.0, despite the different regions and countries. For forest fires, 0.61<α₁<0.73,0.87<α₂<0.98 and for urban fires, 0.52<α₁<0.61,0.59<α₂<0.88. The result suggests that fires have self-similarity characteristics. The occurrence of forest fires may have connection with the weather fluctuations, which have significant effects on the ignition and have the similar temporal correlations. It is shown that the interval sequences of urban fires closely resemble that of white noise in small timescale, and the correlations are weaker than that of forest fires. Human behavior and human density may affect the long-range correlation in some way. This seems to be helpful to understand the complexity of fire system in temporal aspect.     

Temporal-spatial diversities of long-range correlation for relative humidity over China

Long-range correlations of daily relative humidity anomaly records from 191 weather stations over China during 1951-2000 are analyzed by means of fluctuation analysis (FA) and detrended fluctuation analysis (DFA). The information about trends in the relative humidity records can be obtained by comparing the FA curve with DFA curves. The daily relative humidity fluctuations are found to be power-law correlated and their average scaling exponent is higher than that of the temperature fluctuations, indicating that the relative humidity fluctuations take different statistical behavior from other meteorological quantities and there exists a stronger persistence in the relative humidity fluctuations. Furthermore, it is also found that these power-law scaling properties vary from station to station and show both spatial and temporal diversities, which may be explained by a proposed mechanism.     

Preservation of long range temporal correlations under extreme random dilution

Many natural time series exhibit long range temporal correlations that may be characterized by power-law scaling exponents. However, in many cases, the time series have uneven time intervals due to, for example, missing data points, noisy data, and outliers. Here we study the effect of randomly missing data points on the power-law scaling exponents of time series that are long range temporally correlated. The Fourier transform and detrended fluctuation analysis (DFA) techniques are used for scaling exponent estimation. We find that even under extreme dilution of more than 50%, the value of the scaling exponent remains almost unaffected. Random dilution is also applied on heart interbeat interval time series. It is found that dilution of 70%-80% of the data points leads to a reduction of only 8% in the scaling exponent; it is also found that it is possible to discriminate between healthy and heart failure subjects even under extreme dilution of more than 90%.     

From 1/f noise to multifractal cascades in heartbeat dynamics

We explore the degree to which concepts developed in statistical physics can be usefully applied to physiological signals. We illustrate the problems related to physiologic signal analysis with representative examples of human heartbeat dynamics under healthy and pathologic conditions. We first review recent progress based on two analysis methods, power spectrum and detrended fluctuation analysis, used to quantify long-range power-law correlations in noisy heartbeat fluctuations. The finding of power-law correlations indicates presence of scale-invariant, fractal structures in the human heartbeat. These fractal structures are represented by self-affine cascades of beat-to-beat fluctuations revealed by wavelet decomposition at different time scales. We then describe very recent work that quantifies multifractal features in these cascades, and the discovery that the multifractal structure of healthy dynamics is lost with congestive heart failure. The analytic tools we discuss may be used on a wide range of physiologic signals. (c) 2001 American Institute of Physics.     

北京气候中心气候系统模式对中国四季日平均气温的模拟性能评估

利用去趋势波动分析方法对中国四季日平均气温观测资料进行了研究, 发现四季日平均气温均具有很好的长程相关性特征, 中国西部地区尤其是新疆和西藏的长程相关性较强. 基于观测资料中的这种长程相关性特征, 评估了北京气候中心气候系统模式对中国四季日平均气温的模拟性能, 发现该模式能够较好地反映中国四季日平均气温的长程相关性特征. 总体而言, 模式对春季的日平均气温的长程相关性模拟效果最好, 仅对江南地区的长程相关性的模拟较差; 夏季, 模式模拟误差较大的地区包括中国中东部地区及西藏大部, 其中华北南部、黄淮西部、江南大部、华南等地模拟效果最差; 秋季, 模式对东部沿海及东北大部、华北西南部等地模拟的长程相关性偏强, 而在西北大部模拟的长程相关性明显偏弱; 冬季, 除东部沿海地区模拟的长程相关性偏强外, 全国其余大部分地区接近观测或偏弱, 其中西北、西南、华南北部、江南南部、东北北部偏弱明显, 青藏高原西部偏弱最为显著.