Comparison of different methods for computing scaling parameter in the presence of trends.

In a large number of analysis of biological signals, such as ElectroEncephaloGrams or Heart Rate Variability, phenomena such as long range correlation or long term memory or self similarity or more generally scaling phenomena have been observed and are often used as a basis for diagnosis and the assertion of major physiological facts. It is therefore an important issue to be able to perform a relevant analysis (detection, estimation) of such phenomena on empirical data. For a reliable use of long-range correlation analysis, it is essential to be able to distinguish between trends (whatever their origin - external, apparatus - or internal but related to other biological mechanism such as breathing...) and long-range fluctuations intrinsic to the data. In this paper we compare estimates of the Hurst parameter based on i) Detrending Fluctuation Analysis, ii) wavelet analysis, iii) discrete variations. We discuss their capabilities for a reliable estimation of the scaling exponent, depending on the signal length and for different trends, using simulated and real data.

[1]  A. Kolmogorov.  The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[2]  Jeffrey M. Hausdorff,et al.  Physionet: Components of a New Research Resource for Complex Physiologic Signals". Circu-lation Vol , 2000 .

[3]  Patrice Abry,et al.  Wavelet-based spectral analysis of 1/f processes , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[4]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

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

[6]  Patrice Abry,et al.  Wavelets for the Analysis, Estimation, and Synthesis of Scaling Data , 2002 .

[7]  Jeffrey M. Hausdorff,et al.  Fractal dynamics in physiology: Alterations with disease and aging , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[8]  J. Coeurjolly,et al.  Estimating the Parameters of a Fractional Brownian Motion by Discrete Variations of its Sample Paths , 2001 .

[9]  B. Mandelbrot,et al.  Fractals: Form, Chance and Dimension , 1978 .

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

[11]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

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

[13]  A. Wood,et al.  Simulation of Stationary Gaussian Processes in [0, 1] d , 1994 .

[14]  Murad S. Taqqu,et al.  Testing for long‐range dependence in the presence of shifting means or a slowly declining trend, using a variance‐type estimator , 1997 .

[15]  Gabriel Lang,et al.  Quadratic variations and estimation of the local Hölder index of a gaussian process , 1997 .