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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

Detrended Fluctuation Analysis. The exponent α is determined as the slope of the log–log diffusion plot.
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Figure 3: Detrended Fluctuation Analysis. The exponent α is determined as the slope of the log–log diffusion plot.

Mentions: This scaling law is exploited by the Detrended Fluctuation Analysis (DFA) that reveals α as the slope of the log–log diffusion plot (Figure 3). The fGn/fBm continuum is characterized by exponents α ranging from 0 to 2 (see Figure 4). Note that the scaling law expressed in Eq. 4 just derives from the original definition of fBm (Eq. 1). If the series x(i) is a fGn, y(i) is the corresponding fBm and α is the Hurst exponent. If x(i) is a fBm, y(i) belongs to another family of over-diffusive processes, characterized by exponents α ranging from 1 to 3, and in that case α = H + 1.


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

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

Detrended Fluctuation Analysis. The exponent α is determined as the slope of the log–log diffusion plot.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 3: Detrended Fluctuation Analysis. The exponent α is determined as the slope of the log–log diffusion plot.
Mentions: This scaling law is exploited by the Detrended Fluctuation Analysis (DFA) that reveals α as the slope of the log–log diffusion plot (Figure 3). The fGn/fBm continuum is characterized by exponents α ranging from 0 to 2 (see Figure 4). Note that the scaling law expressed in Eq. 4 just derives from the original definition of fBm (Eq. 1). If the series x(i) is a fGn, y(i) is the corresponding fBm and α is the Hurst exponent. If x(i) is a fBm, y(i) belongs to another family of over-diffusive processes, characterized by exponents α ranging from 1 to 3, and in that case α = H + 1.

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