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GLOSSARY OF TERMS USED IN TIME SERIES ANALYSIS
OF CARDIOVASCULAR DATA
DETRENDED FLUCTUATION ANALYSIS

Method for quantifying the correlation property in nonstationary time series based on the computation of a scaling exponent d by means of a modified root mean square analysis of a random walk.

To compute d from a time-series x(i) [i=1,..., N], like the interval tachogram, the time series is first integrated:

where M is the average value of the series x(i), and k ranges between 1 and N.


Next, the integrated series y(k) is divided into boxes of equal length n and the least-square line fitting the data in each box, yn(k), is calculated. The integrated time series is detrended by subtracting the local trend yn(k), and the root-mean square fluctuation of the detrended series, F(n) is computed:

F(n) is computed for all time-scales n. Typically, F(n) increases with n, the "box-size". If log F(n) increases linearly with log n, then the slope of the line relating F(n) and n in a log-log scale gives the scaling exponent d.

d is related to the "1/f" spectral slope:

=2d-1

If d=0.5, the time-series x(i) is uncorrelated (white noise).
If d=1.0, the correlation of the time-series is the same of 1/f noise.
If d=1.5, x(i) behaves like Brown noise (random walk)

This parameter was shown to change with aging (Iyengar 1996), to be the best univariable predictor of mortality in patients following acute myocardial infarction (Makikallio 1999) and to be altered before the spontaneous onset of atrial fibrillation (Vikman, 2001).

References:
Iyengar N et al.  (1996) Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics. Am. J. Physiol.
Peng CK, Havlin S, Stanley HE, Goldberger AL (1995) Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series CHAOS, 5 (1), 82-87
Makikallio TH et al. (1999)  Fractal analysis of heart rate dynamics as a predictor of mortality in patients with depressed left ventricular function after acute myocardial infarction. Am J Cardiol.
Vikman S (2001) Differences in heart rate dynamics before the spontaneous onset of long and short episodes of paroxysmal atrial fibrillation. Ann Noninvasive Electrocardiol.
Pikkujämsä SM et al (1999) Cardiac Interbeat Interval Dynamics From Childhood to Senescence: Comparison of Conventional and New Measures Based on Fractals and Chaos Theory Circulation.

Links:
Fractal Mechanisms in Neural Control by Peng C-K, Hausdorff JM, Goldberger AL
DFA Software Download Page from PhysioToolkit

(PC 15-06-2001)

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