Relationship between detrended fluctuation analysis and spectral analysis of heart-rate variability

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Abstract
The recently-introduced technique of detrended fluctuation analysis (DFA) for heart-rate variability appears to yield improved prognostic power in cardiovascular disease through calculation of the fractal scaling exponent α. However, the physiological meaning of α remains unclear. In DFA, the signal is segmented into lengths from 4 to 64 beats. For each segmentation length (n), the individual segments are cumulated, detrended and the sum of the squares (F2) of residuals calculated. α is the slope of log(F) against log(n). We show mathematical equivalence between α calculated by DFA and by a novel alternative method using frequency-weighted power spectra. We show F2 (and thus α) can be obtained from a frequency-weighted power spectrum without DFA. To do this, we cumulate and detrend the Taylor series of individual Fourier components. F2 is found to depend on the relationship between the signal period and segment length. F2 can therefore be expressed in terms of frequency-weighted power spectra. From this, the α coefficient of DFA can then be described in power-spectral terms, which facilitates exploration of its physiological basis. We confirm these findings using samples from 20 healthy volunteers and 40 patients with heart failure.