Asymptotic Properties of the Detrended Fluctuation Analysis of Long-Range-Dependent Processes

In the past few years, a certain number of authors have proposed analysis methods of the time series built from a long-range dependence noise. One of these methods is the detrended fluctuation analysis (DFA), frequently used in the case of physiological data processing. The aim of this method is to highlight the long-range dependence of a time series with trend. In this paper, asymptotic properties of the DFA of the fractional Gaussian noise (FGN) are provided. Those results are also extended to a general class of stationary long-range-dependent processes. As a consequence, the convergence of the semiparametric estimator of the Hurst parameter is established. However, several simple examples also show that this method is not at all robust in the case of trends.

[1]  François Roueff,et al.  On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter , 2005, math/0512635.

[2]  Liudas Giraitis,et al.  CLT and other limit theorems for functionals of Gaussian processes , 1985 .

[3]  Liudas Giraitis,et al.  Rate Optimal Semiparametric Estimation of the Memory Parameter of the Gaussian Time Serieswith Long-Range Dependence - (Now published in 'Journal of Time Series Analysis', 18 (1997), pp.49-60.) , 1997 .

[4]  Tailen Hsing,et al.  Limit theorems for functionals of moving averages , 1997 .

[5]  L. Oxley,et al.  Estimators for Long Range Dependence: An Empirical Study , 2009, 0901.0762.

[6]  C. Peng,et al.  Mosaic organization of DNA nucleotides. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  H. Stanley,et al.  Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. , 1995, Chaos.

[8]  M. A. Arcones,et al.  Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors , 1994 .

[9]  R. Sepulchrez,et al.  Nonlinear Analysis of Cardiac Rhythm Uctuations Using Dfa Method , 1999 .

[10]  A. Philippe,et al.  Generators of long-range dependent processes: A survey , 2003 .

[11]  M. Taqqu.  Weak convergence to fractional brownian motion and to the rosenblatt process , 1975, Advances in Applied Probability.

[12]  Liudas Giraitis,et al.  RATE OPTIMAL SEMIPARAMETRIC ESTIMATION OF THE MEMORY PARAMETER OF THE GAUSSIAN TIME SERIES WITH LONG‐RANGE DEPENDENCE , 1997 .

[13]  R. Dobrushin,et al.  Non-central limit theorems for non-linear functional of Gaussian fields , 1979 .

[14]  W. Willinger,et al.  ESTIMATORS FOR LONG-RANGE DEPENDENCE: AN EMPIRICAL STUDY , 1995 .

[15]  Jean-Marc Bardet,et al.  Statistical study of the wavelet analysis of fractional Brownian motion , 2002, IEEE Trans. Inf. Theory.

[16]  P. Major,et al.  Central limit theorems for non-linear functionals of Gaussian fields , 1983 .

[17]  Yu. A. Davydov,et al.  The Invariance Principle for Stationary Processes , 1970 .

[18]  Murad S. Taqqu,et al.  Weak convergence to fractional brownian motion and to the rosenblatt process , 1975, Advances in Applied Probability.

[19]  Liudas Giraitis,et al.  Adaptive Semiparametric Estimation of the Memory Parameter , 2000 .

[20]  Jeffrey M. Hausdorff,et al.  Self-Organized Biological Dynamics and Nonlinear Control: Fractal mechanisms in neuronal control: human heartbeat and gait dynamics in health and disease , 2000 .

[21]  H. Stanley,et al.  Effect of trends on detrended fluctuation analysis. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Jianbo Gao,et al.  Principal component analysis of 1/fα noise , 2003 .

[23]  P. Lavie,et al.  Correlation differences in heartbeat fluctuations during rest and exercise. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Patrice Abry,et al.  A Wavelet-Based Joint Estimator of the Parameters of Long-Range Dependence , 1999, IEEE Trans. Inf. Theory.

[25]  Masao Masugi.  Detrended fluctuation analysis of IP-network traffic using a two-dimensional topology map , 2004 .

[26]  Pierre-Antoine Absil,et al.  Nonlinear analysis of cardiac rhythm fluctuations using DFA method , 1999 .

[27]  S. Havlin,et al.  Detecting long-range correlations with detrended fluctuation analysis , 2001, cond-mat/0102214.

[28]  Philippe Soulier.  Moment bounds and central limit theorem for functions of Gaussian vectors , 2001 .

[29]  Harvard Medical School,et al.  Effect of nonstationarities on detrended fluctuation analysis. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  G. Krstačić,et al.  Changes in the Hurst exponent of heartbeat intervals during physical activity. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Radhakrishnan Nagarajan,et al.  Minimizing the Effect of sinusoidal Trends in Detrended Fluctuation Analysis , 2005, Int. J. Bifurc. Chaos.

[32]  Jean-Marc Bardet,et al.  Testing for the Presence of Self‐Similarity of Gaussian Time Series Having Stationary Increments , 2000 .

[33]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[34]  Patrice Abry,et al.  Long‐range Dependence: Revisiting Aggregation with Wavelets , 1998 .

[35]  P. Robinson.  Gaussian Semiparametric Estimation of Long Range Dependence , 1995 .

[36]  H. Stanley,et al.  Characterization of sleep stages by correlations in the magnitude and sign of heartbeat increments. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  T. Ferrée,et al.  Fluctuation Analysis of Human Electroencephalogram , 2001, physics/0105029.

[38]  Jean-Marc Bardet,et al.  Adaptive wavelet-based estimator of the memory parameter for stationary Gaussian processes , 2007 .