Long-range correlations in human standing

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M. Duarte, V.M. Zatsiorsky / Physics Letters A 283 (2001) 124–128 125tionless standing, the authors hypothesized that theprocess should saturate at the region of 10 to 30 s,i.e., no long-range correlations for time intervals above30 s were expected during quiet stance; but the authorsrecognized that longer time series were necessary fora final conclusion [3]. Recently, an analysis of quietstanding by wavelet statistics has shown different scalingregions (multifractal) in short time series [4].One shortcoming in the studies of human standingis that quiet stance tasks with the instruction “to stayas still as possible” are limited to only a few minutesdue to fatigue. On the contrary, natural standing overprolonged periods, longer than few minutes, is a verycommon situation in daily life, such as standing inline, or standing while talking with somebody. Thisstanding is characterized by repeated changes in thebody position, which are self-induced, performed almostunconsciously, and cannot be generally associatedwith any external source. Such natural standing isnot fatiguing and can be reproduced in the laboratoryby a standing task during which a person is allowedto do any movement without any constraint, except tostay within the limited area of the force plate. This unconstrainedstanding in healthy adults was examinedby Duarte and Zatsiorsky [5] who observed very lowfrequencies in the displacement of COP during unconstrainedstanding, a typical signature of a long-rangecorrelation process or long-memory process. Recently,Duarte and Zatsiorsky [6] used the classical rescaledadjusted range analysis or R/S statistic [15] to describethe fractal properties of the COP time seriesduring natural standing. The average fractal or Hurstexponent (H )was0.35 ± 0.06. In this Letter, weused more robust methods and performed a surrogatetest to quantify the long-range correlation phenomenaduring prolonged periods of unconstrained standing.Long-range correlation processes present very longtermfluctuations in addition to very short-term fluctuations,where the dependence of data farther apart ishigher than it is expected for independent data. Thisphenomenon has been observed in very diverse areasof nature. Related to biological systems and specificallyto humans, it has been observed in stride intervalsduring walking [7], human cognition [8], intervalsbetween rhythmic tapping movements [9], heartbeatvariability [10,11] and certain DNA sequences[12,13].2. MethodsIn this study, ten healthy subjects (28 ± 5yr,1.79 ±0.09 m, and 78 ± 14 kg) were asked to stand in an unconstrainedupright bipedal posture on a 40 × 90 cmforce plate for 30 min (the methods were described indetail elsewhere [5]). We analyzed the COP trajectoriesfor the anterior–posterior (a–p) and medial–lateral(m–l) direction. Examples of stabilograms (the mappingof COP a–p versus COP m–l) and a exemplarytime series are shown in Fig. 1.The stochastic model of fractional Gaussian noise(or the integrated function, fractional Brownian motion)is an example of a self-similar process (like afractal) appropriate to model such long-range correlationphenomena. This model has been successfullyused in many of the above examples and it was usedhere. Several heuristic methods have been suggestedto estimate long-range correlations since the classicalR/S statistics proposed by Hurst in hydrology context(for a review of models and methods, see [14]).Two methods were used to estimate long-range correlations:power spectral analysis (PSA) [14], performedin the frequency domain, and detrended fluctuationanalysis (DFA) [10], performed in the time domain.The PSA method involves plotting the power spectraof a sample path on a log–log scale. If the samplepath presents long-range correlations over a rangeof frequencies then significant linear regression in thelog–log plot captures the relation between frequencyand power of the type 1/f −β ,wheref is the frequencyand β is the scaling exponent. The exponentβ is 0 for white noise, −1 for1/f noise, and −2 forBrownian noise.The DFA method is a modification of the root-meansquare analysis of a random walk [15] and it is lesssensitive than the PSA method to possible nonstationaritiesand noise in the data such as the COP displacementduring unconstrained standing. For a randomwalk process, the net displacement of a walker afterk steps is y(k) = ∑ ki=1 u(i),whereu(i) is a samplepath of a random walk process or a Brownian motion.Using the DFA method, the integrated sample path, y,of total length N is divided into windows of length l;each window steps forward by an interval l/s, wheres is the overlapping parameter set to 2 in the presentstudy, and N/(l/s) is the total number of windows.For each window, the local trend is estimated by a lin-
126 M. Duarte, V.M. Zatsiorsky / Physics Letters A 283 (2001) 124–128Fig. 1. Position of the subjects on the force plate and axes convention (a). Two examples of stabilograms during a 30 min PUS: multi-region (b)and single-region standing (c). Exemplary COP time series (d).ear least-squares fit, ŷ, and the curve is subsequentlydetrended. The variance of each window’s detrendedsample path is calculated. The root of the mean variancesof all windows of length l is the detrended fluctuation,F ,ofthewalker:√F(l)=√ l/sNN/(l/s)∑n=11ll∑ [ ] 2.y(k)−ŷ(k)k=1The root-mean square fluctuations increase with l,andif there is a linear relationship in a log–log scale of Fwith l, the data obey a power-law function, F(l)∼ l α ,where α is the scaling exponent quantifying the longrangecorrelations. For white noise, the exponent αis 0.5, for 1/f noise it is 1, and 1.5 for Brownian noise.For data with infinite length, the exponents α and β arerelated by the expression β = 1 − 2α [16].3. Results and discussionLong-range correlations for the COP data wereanalyzed for lags/window lengths greater than 10 sand up to one-third (10 min) of the 30 min data. Thelower limit was selected to ignore the already reportedshort-range correlations for quiet stance in the rangeof 1 s [3]. The upper limit was chosen to obtainthree independent sets of data, in order to increase thestatistical power of the results. The fitted line for eachtrial in the region of 10 s to 10 min representing thescaling exponent (slopes) obtained by the DFA andPSA methods are shown in Fig. 2. The α exponentswere 0.98 ± 0.17 and 1.01 ± 0.26 for the a–p andm–l directions, respectively. The β exponents were−1.00 ± 0.42 and −1.27 ± 0.63 for the a–p andm–l directions, respectively. These results satisfiedthe relation β = 1 − 2α and indicated the presenceof nontrivial long-range correlations. Because thereported values were less than the values for Brownianmotion, the correlations were in fact anti-correlations(negative correlations).The variability of α (expressed by the mean standarddeviations) was about 2.5 times lower than thevariability of β. As expected, this indicated that theDFA method yielded more robust results that were lesssensitive to nonstationarities. The average value of αfor the COP data in the a–p and m–l directions was1.00 ± 0.22 (range: 0.68–1.47), which is equivalent to
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