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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: relative phase series collected during a bimanual coordination task. The task was performed following an in-phase mode. The left graph represents a series of discrete relative phases (DRP, 1044 data points, computed by the point estimate method at the time of maximal pronation of the right hand). The right graph is a series of continuous relative phase (CRP, 32,000 data points, sampled at 100 Hz, representing approximately 96 consecutive cycles). Second row: DRP (left) and CRP (right) series. These graphs focus on 200 points for DRP and 2000 points for CRP. Third row: average DFA diffusion plots obtained for DRP (left), and CRP (right) series. Bottom row: average log–log power spectra obtained for DRP (left), and CRP (right) series.
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Figure 6: Top Row: relative phase series collected during a bimanual coordination task. The task was performed following an in-phase mode. The left graph represents a series of discrete relative phases (DRP, 1044 data points, computed by the point estimate method at the time of maximal pronation of the right hand). The right graph is a series of continuous relative phase (CRP, 32,000 data points, sampled at 100 Hz, representing approximately 96 consecutive cycles). Second row: DRP (left) and CRP (right) series. These graphs focus on 200 points for DRP and 2000 points for CRP. Third row: average DFA diffusion plots obtained for DRP (left), and CRP (right) series. Bottom row: average log–log power spectra obtained for DRP (left), and CRP (right) series.

Mentions: Torre et al. (2007a) analyzed the fractal properties of both DRP and CRP series. Two example series, collected in trials performed in in-phase mode, are presented in Figure 6 (top row). At first glance, the two series look similar, presenting a weak stationarity around a mean value of 0°. However, the DRP series (left graph) contains only 1044 data points, while the CRP series is composed of 32,000 data points (sampled at 100 Hz, representing approximately 96 consecutive cycles). The graphs in the second row highlight the differences between the two series, by focusing on 200 points for DRP and 2000 points for CRP. The DRP series is composed of discrete points, and differences between adjacent values provide the series with a marked level of roughness. In contrast the CRP series appears as a very smooth motion, with slow oscillations around the mean value.


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

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

Top Row: relative phase series collected during a bimanual coordination task. The task was performed following an in-phase mode. The left graph represents a series of discrete relative phases (DRP, 1044 data points, computed by the point estimate method at the time of maximal pronation of the right hand). The right graph is a series of continuous relative phase (CRP, 32,000 data points, sampled at 100 Hz, representing approximately 96 consecutive cycles). Second row: DRP (left) and CRP (right) series. These graphs focus on 200 points for DRP and 2000 points for CRP. Third row: average DFA diffusion plots obtained for DRP (left), and CRP (right) series. Bottom row: average log–log power spectra obtained for DRP (left), and CRP (right) series.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 6: Top Row: relative phase series collected during a bimanual coordination task. The task was performed following an in-phase mode. The left graph represents a series of discrete relative phases (DRP, 1044 data points, computed by the point estimate method at the time of maximal pronation of the right hand). The right graph is a series of continuous relative phase (CRP, 32,000 data points, sampled at 100 Hz, representing approximately 96 consecutive cycles). Second row: DRP (left) and CRP (right) series. These graphs focus on 200 points for DRP and 2000 points for CRP. Third row: average DFA diffusion plots obtained for DRP (left), and CRP (right) series. Bottom row: average log–log power spectra obtained for DRP (left), and CRP (right) series.
Mentions: Torre et al. (2007a) analyzed the fractal properties of both DRP and CRP series. Two example series, collected in trials performed in in-phase mode, are presented in Figure 6 (top row). At first glance, the two series look similar, presenting a weak stationarity around a mean value of 0°. However, the DRP series (left graph) contains only 1044 data points, while the CRP series is composed of 32,000 data points (sampled at 100 Hz, representing approximately 96 consecutive cycles). The graphs in the second row highlight the differences between the two series, by focusing on 200 points for DRP and 2000 points for CRP. The DRP series is composed of discrete points, and differences between adjacent values provide the series with a marked level of roughness. In contrast the CRP series appears as a very smooth motion, with slow oscillations around the mean value.

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