Limits...
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 simulated with a one-order auto-regressive model (yi = 0.85yi − 1 + εi, left column), a one-order moving average model (yi = yi − 1 – 0.8εi − 1 + εi, central column), and the Davies–Harte algorithm (fGn with H = 0.9, right column). The corresponding DFA diffusion plots are presented in the bottom row.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3376770&req=5

Figure 8: Top row: example series simulated with a one-order auto-regressive model (yi = 0.85yi − 1 + εi, left column), a one-order moving average model (yi = yi − 1 – 0.8εi − 1 + εi, central column), and the Davies–Harte algorithm (fGn with H = 0.9, right column). The corresponding DFA diffusion plots are presented in the bottom row.

Mentions: We choose these ARMA models and their parameters values because the application of DFA on the series generated by these models yields diffusion plots similar to those obtained with fGn series. We present in Figure 8 one example series of each set, and the corresponding diffusion plots: in all cases a linear slope close to 0.9 is obtained. Obviously the best linear fit is observed for the fGn series, which contains genuine LRC. For the AR series, the diffusion plot presents a slight flattening for long intervals, and conversely the slope tends to increase for long intervals for the MA series. However the diffusion plots obtained the AR and MA series roughly mimic the typical shape expected from long-range correlated series, and could easily lead to erroneous interpretations.


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 simulated with a one-order auto-regressive model (yi = 0.85yi − 1 + εi, left column), a one-order moving average model (yi = yi − 1 – 0.8εi − 1 + εi, central column), and the Davies–Harte algorithm (fGn with H = 0.9, right column). The corresponding DFA diffusion plots are presented in the bottom row.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 8: Top row: example series simulated with a one-order auto-regressive model (yi = 0.85yi − 1 + εi, left column), a one-order moving average model (yi = yi − 1 – 0.8εi − 1 + εi, central column), and the Davies–Harte algorithm (fGn with H = 0.9, right column). The corresponding DFA diffusion plots are presented in the bottom row.
Mentions: We choose these ARMA models and their parameters values because the application of DFA on the series generated by these models yields diffusion plots similar to those obtained with fGn series. We present in Figure 8 one example series of each set, and the corresponding diffusion plots: in all cases a linear slope close to 0.9 is obtained. Obviously the best linear fit is observed for the fGn series, which contains genuine LRC. For the AR series, the diffusion plot presents a slight flattening for long intervals, and conversely the slope tends to increase for long intervals for the MA series. However the diffusion plots obtained the AR and MA series roughly mimic the typical shape expected from long-range correlated series, and could easily lead to erroneous interpretations.

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