Fractals time series scaling

Altemeier et al. [30] measured the fractal characteristics of ventilation and determined that not only are local ventilation and perfusion highly correlated, but they scale as well. Finally, Peng et al. [31] analyzed the BRV time series for 40 healthy adults and found that under supine, resting, and spontaneous breathing conditions, the time series scale. This result implies that human BRV time series have long-range (fractal) correlations across multiple time scales. ... 

In this section the notion of an allometric relation is generalized to include measures of time series. In this view, y is interpreted to be the variance and x the average value of the quantity being measured. The fact that these two central measures of a time series satisfy an allometric relation implies that the underlying time series is a fractal random process and therefore scales. It was first determined empirically that certain statistical data satisfy a power-law relation of the form given by Taylor [17] in Eq. (1), and this is where we begin our discussion of the allometric aggregation method of data analysis. 

It is well established that the exponent in such scaling equations is related to the fractal dimension [21] D of the underlying time series by D = 2 — //, so that... 

A second method for determining the singularity spectrum, the one we use here, is to numerically determine both the mass exponent and its derivative. In this way we calculate the multifractal spectrum directly from the data using Eq. (86). It is clear from Fig. 9b that we obtain the canonical form of the spectrum that is, f(h) is a convex function of the scaling parameter h. The peak of the spectrum is determined to be the fractal dimension, as it should. Here again we have an indication that the interstride interval time series describes a multifractal process. We stress that we are only using the qualitative properties of the spectrum for q < 0, due to the sensitivity of the numerical method to weak singularities. 

The nonlinear form of the mass exponent in Fig. 9a, the convex form of the singularity spectmm/(/i) in Fig. 9b, and the fit to f(q) in Fig. 13, are all evidence that the interstride interval time series are multifractal. This analysis is further supported by the fact that the maxima of the singularity spectra coincide with the fractal dimensions determined using the scaling properties of the time series using the allometric aggregation technique. 

But power-law distributions may lack any characteristic scale. This property prevented the use of power-law distributions in the natural sciences until mathematical introduction of Levy s new probabilistic concepts and the physical introduction of new scaling concepts for thermodynamic functions and correlation functions (see ref. [40]). In financial markets, invariance of time scales means that even a stock expert cannot distinguish in a time series analysis if the charts are, for example, daily, weekly, or monthly. These charts are statistically self-similar or fractal. 

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