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Relative roughness: an index for testing the suitability of the monofractal model.

Marmelat V, Torre K, Delignières D - Front Physiol (2012)

Bottom Line: The problem is that fractal methods can be applied to all types of series, and they always give a result, that one can then erroneously interpret in the context of the monofractal framework.An acceptable level of RR, however, is a necessary but not sufficient condition for considering series as long-range correlated.Specific methods should be used in complement for testing for the effective presence of long-range correlations in empirical series.

View Article: PubMed Central - PubMed

Affiliation: Movement to Health, University Montpellier 1 Montpellier, France.

ABSTRACT
Fractal analyses have become very popular and have been applied on a wide variety of empirical time series. The application of these methods supposes that the monofractal framework can offer a suitable model for the analyzed series. However, this model takes into account a quite specific kind of fluctuations, and we consider that fractal analyses have been often applied to series that were completely outside of its relevance. The problem is that fractal methods can be applied to all types of series, and they always give a result, that one can then erroneously interpret in the context of the monofractal framework. We propose in this paper an easily computable index, the relative roughness (RR), defined as the ratio between local and global variances, that allows to test for the applicability of fractal analyses. We show that RR is confined within a limited range (between 1.21 and 0.12, approximately) for long-range correlated series. We propose some examples of empirical series that have been recently analyzed using fractal methods, but, with respect to their RR, should not have been considered in the monofractal model. An acceptable level of RR, however, is a necessary but not sufficient condition for considering series as long-range correlated. Specific methods should be used in complement for testing for the effective presence of long-range correlations in empirical series.

No MeSH data available.


Related in: MedlinePlus

Top row: example series of fractional Brownian motions (fBm) for three typical values of the scaling exponent. The central graph represents an ordinary Brownian motion (H = 0.5). The left graph shows an anti-persistent fBm (H = 0.25) and the right graph a persistent fBm (H = 0.75). The corresponding fractional Gaussian noises series (fGn) are displayed in the bottom row.
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Figure 1: Top row: example series of fractional Brownian motions (fBm) for three typical values of the scaling exponent. The central graph represents an ordinary Brownian motion (H = 0.5). The left graph shows an anti-persistent fBm (H = 0.25) and the right graph a persistent fBm (H = 0.75). The corresponding fractional Gaussian noises series (fGn) are displayed in the bottom row.

Mentions: which signifies that the standard deviation of the process is a power function of the time interval (Δt) over which it was computed. H is the Hurst exponent and can be any real number in the range 0 < H < 1. Anti-persistent series are characterized by H < 0.5, and persistent series by H > 0.5. Brownian motion corresponds to the special case H = 0.5 and constitutes the frontier between anti-persistent and persistent fBm. Eq. 2 expressed the so-called diffusion property of fBm processes. With respect to the standard diffusion of Brownian motion (standard deviation is proportional to the square root of time), anti-persistent fBm are said to be under-diffusive, and persistent fBm over-diffusive. We present in Figure 1 (top row) three example fBm series, for three contrasted H exponents.


Relative roughness: an index for testing the suitability of the monofractal model.

Marmelat V, Torre K, Delignières D - Front Physiol (2012)

Top row: example series of fractional Brownian motions (fBm) for three typical values of the scaling exponent. The central graph represents an ordinary Brownian motion (H = 0.5). The left graph shows an anti-persistent fBm (H = 0.25) and the right graph a persistent fBm (H = 0.75). The corresponding fractional Gaussian noises series (fGn) are displayed in the bottom row.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3376770&req=5

Figure 1: Top row: example series of fractional Brownian motions (fBm) for three typical values of the scaling exponent. The central graph represents an ordinary Brownian motion (H = 0.5). The left graph shows an anti-persistent fBm (H = 0.25) and the right graph a persistent fBm (H = 0.75). The corresponding fractional Gaussian noises series (fGn) are displayed in the bottom row.
Mentions: which signifies that the standard deviation of the process is a power function of the time interval (Δt) over which it was computed. H is the Hurst exponent and can be any real number in the range 0 < H < 1. Anti-persistent series are characterized by H < 0.5, and persistent series by H > 0.5. Brownian motion corresponds to the special case H = 0.5 and constitutes the frontier between anti-persistent and persistent fBm. Eq. 2 expressed the so-called diffusion property of fBm processes. With respect to the standard diffusion of Brownian motion (standard deviation is proportional to the square root of time), anti-persistent fBm are said to be under-diffusive, and persistent fBm over-diffusive. We present in Figure 1 (top row) three example fBm series, for three contrasted H exponents.

Bottom Line: The problem is that fractal methods can be applied to all types of series, and they always give a result, that one can then erroneously interpret in the context of the monofractal framework.An acceptable level of RR, however, is a necessary but not sufficient condition for considering series as long-range correlated.Specific methods should be used in complement for testing for the effective presence of long-range correlations in empirical series.

View Article: PubMed Central - PubMed

Affiliation: Movement to Health, University Montpellier 1 Montpellier, France.

ABSTRACT
Fractal analyses have become very popular and have been applied on a wide variety of empirical time series. The application of these methods supposes that the monofractal framework can offer a suitable model for the analyzed series. However, this model takes into account a quite specific kind of fluctuations, and we consider that fractal analyses have been often applied to series that were completely outside of its relevance. The problem is that fractal methods can be applied to all types of series, and they always give a result, that one can then erroneously interpret in the context of the monofractal framework. We propose in this paper an easily computable index, the relative roughness (RR), defined as the ratio between local and global variances, that allows to test for the applicability of fractal analyses. We show that RR is confined within a limited range (between 1.21 and 0.12, approximately) for long-range correlated series. We propose some examples of empirical series that have been recently analyzed using fractal methods, but, with respect to their RR, should not have been considered in the monofractal model. An acceptable level of RR, however, is a necessary but not sufficient condition for considering series as long-range correlated. Specific methods should be used in complement for testing for the effective presence of long-range correlations in empirical series.

No MeSH data available.


Related in: MedlinePlus