Multifractal detrended fluctuation analysis of nonstationary time series

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

Characterization of Sleep Stages by Correlations of Heartbeat Increments

We study correlation properties of the magnitude and the sign of the increments in the time intervals between successive heartbeats during light sleep, deep sleep, and REM sleep using the detrended fluctuation analysis method. We find short-range anticorrelations in the sign time series, which are strong during deep sleep, weaker during light sleep and even weaker during REM sleep. In contrast, we find long-range positive correlations in the magnitude time series, which are strong during REM sleep and weaker during light sleep. We observe uncorrelated behavior for the magnitude during deep sleep. Since the magnitude series relates to the nonlinear properties of the original time series, while the signs series relates to the linear properties, our findings suggest that the nonlinear properties of the heartbeat dynamics are more pronounced during REM sleep. Thus, the sign and the magnitude series provide information which is useful in distinguishing between the sleep stages.Comment: 7 pages, 4 figures, revte

Effect of Trends on Detrended Fluctuation Analysis

Detrended fluctuation analysis (DFA) is a scaling analysis method used to estimate long-range power-law correlation exponents in noisy signals. Many noisy signals in real systems display trends, so that the scaling results obtained from the DFA method become difficult to analyze. We systematically study the effects of three types of trends -- linear, periodic, and power-law trends, and offer examples where these trends are likely to occur in real data. We compare the difference between the scaling results for artificially generated correlated noise and correlated noise with a trend, and study how trends lead to the appearance of crossovers in the scaling behavior. We find that crossovers result from the competition between the scaling of the noise and the ``apparent'' scaling of the trend. We study how the characteristics of these crossovers depend on (i) the slope of the linear trend; (ii) the amplitude and period of the periodic trend; (iii) the amplitude and power of the power-law trend and (iv) the length as well as the correlation properties of the noise. Surprisingly, we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws -- i.e. long-range power-law dependence of the position of the crossover on the parameters of the trends. We show that the DFA result of noise with a trend can be exactly determined by the superposition of the separate results of the DFA on the noise and on the trend, assuming that the noise and the trend are not correlated. If this superposition rule is not followed, this is an indication that the noise and the superimposed trend are not independent, so that removing the trend could lead to changes in the correlation properties of the noise.Comment: 20 pages, 16 figure

Effect of nonstationarities on detrended fluctuation analysis

Detrended fluctuation analysis (DFA) is a scaling analysis method used to quantify long-range power-law correlations in signals. Many physical and biological signals are ``noisy'', heterogeneous and exhibit different types of nonstationarities, which can affect the correlation properties of these signals. We systematically study the effects of three types of nonstationarities often encountered in real data. Specifically, we consider nonstationary sequences formed in three ways: (i) stitching together segments of data obtained from discontinuous experimental recordings, or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts -- a ``cutting'' procedure commonly used in preparing data prior to signal analysis; (ii) adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitude, and (iii) generating a signal comprised of segments with different properties -- e.g. different standard deviations or different correlation exponents. We compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities.Comment: 17 pages, 10 figures, corrected some typos, added one referenc

Investigation of differences in follicular penetration of particle-and nonparticle-containing emulsions by laser scanning microscopy

Hair follicles represent a long-term storage of topically applied drugs and cosmetics in the skin. Analyzing the penetration of particle-and nonparticle-containing formulations by laser scanning microscopy, it was found, surprisingly, that particles at a size similar to the thickness of the keratin cells of the hair penetrate more efficiently into the hair follicles. These results were obtained from in vitro and in vivo investigations. It seems that the moving hairs in the follicles act as a geared pump because of the zigzag structure of the surface of the hairs. This pumping effect probably pushes particles with the corresponding size deep into the hair follicles