We develop a method for the multifractal characterization of nonstationary
time series, which is based on a generalization of the detrended fluctuation
analysis (DFA). We relate our multifractal DFA method to the standard partition
function-based multifractal formalism, and prove that both approaches are
equivalent for stationary signals with compact support. By analyzing several
examples we show that the new method can reliably determine the multifractal
scaling behavior of time series. By comparing the multifractal DFA results for
original series to those for shuffled series we can distinguish multifractality
due to long-range correlations from multifractality due to a broad probability
density function. We also compare our results with the wavelet transform
modulus maxima (WTMM) method, and show that the results are equivalent.Comment: 14 pages (RevTex) with 10 figures (eps
We examine the Detrended Fluctuation Analysis (DFA), which is a well-established method for the detection of long-range correlations in time series. We show that deviations from scaling that appear at small time scales become stronger in higher orders of DFA, and suggest a modified DFA method to remove them. The improvement is necessary especially for short records that are affected by non-stationarities. Furthermore, we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of DFA.
The human organism is an integrated network where complex physiological systems, each with its own regulatory mechanisms, continuously interact, and where failure of one system can trigger a breakdown of the entire network. Identifying and quantifying dynamical networks of diverse systems with different types of interactions is a challenge. Here we develop a framework to probe interactions among diverse systems, and we identify a physiological network. We find that each physiological state is characterized by a specific network structure, demonstrating a robust interplay between network topology and function. Across physiological states, the network undergoes topological transitions associated with fast reorganization of physiological interactions on time scales of a few minutes, indicating high network flexibility in response to perturbations. The proposed system-wide integrative approach may facilitate the development of a new field, Network Physiology.
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