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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

Relative roughness as a function of the scaling exponent α in simulated fGn and fBm series. Results are given for series lengths of 512, 1024, and 2048 data points.
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Figure 5: Relative roughness as a function of the scaling exponent α in simulated fGn and fBm series. Results are given for series lengths of 512, 1024, and 2048 data points.

Mentions: In order to analyze the evolution of RR according to the strength of serial correlations in the series, we generated exact fractal series with α exponents ranging from 0.1 (highly anti-persistent fGn) to 1.9 (highly persistent fBm), by steps of 0.1, using the algorithm proposed by Davies and Harte (1987). In order to check the effect of series length on RR, we worked on series of 512, 1024, and 2048 data points, which correspond to the series lengths mostly used in the literature. One-hundred series was generated for each α level and each series length. The results are illustrated in Figure 5. As expected RR decreased as correlations increased in the series. RR was about 2.0 for white noise, and anti-persistent fGn series were characterized by values greater than 2.0, up to 2.9 for the most negatively correlated series (α = 0.1). For fBm series RR presented an asymptotical trend toward zero as α increased. As expected, series length affects RR, but this effect is located in a narrow range of anti-persistent fBm (between α = 1.0 and α = 1.4).


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

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

Relative roughness as a function of the scaling exponent α in simulated fGn and fBm series. Results are given for series lengths of 512, 1024, and 2048 data points.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 5: Relative roughness as a function of the scaling exponent α in simulated fGn and fBm series. Results are given for series lengths of 512, 1024, and 2048 data points.
Mentions: In order to analyze the evolution of RR according to the strength of serial correlations in the series, we generated exact fractal series with α exponents ranging from 0.1 (highly anti-persistent fGn) to 1.9 (highly persistent fBm), by steps of 0.1, using the algorithm proposed by Davies and Harte (1987). In order to check the effect of series length on RR, we worked on series of 512, 1024, and 2048 data points, which correspond to the series lengths mostly used in the literature. One-hundred series was generated for each α level and each series length. The results are illustrated in Figure 5. As expected RR decreased as correlations increased in the series. RR was about 2.0 for white noise, and anti-persistent fGn series were characterized by values greater than 2.0, up to 2.9 for the most negatively correlated series (α = 0.1). For fBm series RR presented an asymptotical trend toward zero as α increased. As expected, series length affects RR, but this effect is located in a narrow range of anti-persistent fBm (between α = 1.0 and α = 1.4).

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