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GLOSSARY OF TERMS USED IN TIME SERIES ANALYSIS
OF CARDIOVASCULAR DATA
HURST EXPONENT

Measure of the smoothness of fractal time series based on the asymptotic behaviour of the rescaled range of the process.

The Hurst exponent, H, is defined as:

H:=log(R/S)/log(T)
where T is the duration of the sample of data, and R/S the corresponding value of rescaled range.
In this way Hurst generalized an equation valid for the Brownian motion in order to include a broader class of time series. In fact, Einstein studied the properties of the Brownian motion and found that the distance R covered by a particle undergoing random collisions is directly proportional to the square-root of time T:
R=k*T0.5
where k is a constant which depends on the time-series. The generalization proposed by Hurst was:
R/S=k*TH
where H is the Hurst exponent.
If H=0.5, the behaviour of the time-series is similar to a random walk;
if H<0.5, the time-series covers less "distance" than a random walk (i.e., if the time-series increases, it is more probable that then it will decrease, and vice-versa);
if H>0.5, the time-series covers more "distance" than a random walk (if the time-series increases, it is more probable that it will continue to increase).

Given a time series x(n), n=1,....N, H can be estimated by taking the slope of (R/S) plotted vs. n in a log-log scale.
H is related to the fractal dimension D: H=E+1-D where E is the Euclidean dimension (E=0 for a point, 1 for a line, 2 for a surface). For one-dimensional signals, H=2-D
H is also related to the "1/f" spectral slope:

 =2H-1 

It can be also estimated by means of the Detrended Fluctuation Analysis.

References:
Bassingthwaighte JB, Raymond GM (1994) Evaluating rescaled range analysis for time series Ann Biomed Eng
Fischer R, Akay M. (1996) A comparison of analytical methods for the study of fractional Brownian motion. Ann Biomed Eng.
DePetrillo PB et al  (1999) Determining the Hurst exponent of fractal time series and its application to electrocardiographic analysis. Comput Biol Med.

Links
Estimating the Hurst Exponent Value:Why is the Hurst Exponent Interesting? by I.Kaplan

(PC 30-08-2000)

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