Non linear time series analysis 1
A methodological note on non-linear time series analysis: Is Collins and De Luca (1993)'s open- and closed-loop model a statistical artifact? Delignières, D., Deschamps, T., Legros, A. & Caillou, N. EA 2991 “Sport, Performance, Health”, University Montpellier I, France (Journal of Motor Behavior, 2003, vol.35, n°1, 86-96) Abstract: In this paper we re-examine, theoretically and empirically, the method proposed by Collins and De Luca (1993) for the analysis of center-of-pressure trajectories. Our main argument is that this approach is not adapted to the analysis of bounded time series, and leads to statistical artifacts (non-linearity of stabilogram-diffusion plots, underestimation of the diffusion process for long-term intervals). The open- and closed-loop model developed by the authors appears as a direct consequence of these statistical problems. Applying more classical methods, such as the Rescaled Range Analysis or the Detrented Fluctuation Analysis, we show that center-of-pressure trajectories can be modeled as a continuous, persistent fractional Brownian motion. More specifically, these trajectories behave like 1/f noise, an ubiquitous features in adaptive biological systems. Key words: time series analysis, stochastic processes, center-of-pressure profile, correlated noise, self-organized criticality. In human movement studies, a number of variables measured in steady state conditions were classically conceived as randomly varying around a stable mean value (e.g., heartbeats, step duration,...). In other words, measurement fluctuations were viewed as unmeaning white noise, and generally eliminated by averaging (Slifkin & Newell, 1998). In these traditional approaches, the dynamics of the measured variables was clearly ignored, i.e. the temporal ordering of the series, the magnitude and direction of displacements between successive points, etc... During the last decade, a number of attempts were made to analyze more specifically the dynamics of such time series (e.g. Collins & De Luca, 1993; Hausdorff, Peng, Ladin, Wei & Goldberger, 1995; Peng, Havlin, Stanley & Goldberger, 1995). These authors used a set of time series analysis methods, designed to reveal the hidden fractal properties of such stochastic processes. For example, Hausdorff et al. (1995) showed that the apparently noisy fluctuations in stride duration during walking display complex fractal properties: when healthy subjects walked at their preferred pace, stride variability was not simply attributable to uncorrelated random fluctuations, but exhibited long-range power-law correlations and self-similarity, indicative of a fractal process. The aim of the present paper is to discuss some methodological and statistical problems, concerning the method proposed by Collins and De Luca (1993), for the analysis of center-of-pressure trajectories during quiet stance. Their method, as well as other mathematical techniques widely used in this domain, will be fully exposed in the following paragraphs and in the Appendix.
Corresponding author : Pr Didier Delignières, EA 2991, Faculty of Sport and Physical Education Sciences, University Montpellier I, 700, avenue du Pic saint Loup, 34090 Montpellier, France. Tel: +33 4 67 41 57 54, Fax: +33 4 67 41 57 50, Email:
[email protected]
The point departure of these approaches was the seminal work of Einstein (1905), who studied Brownian motion (an integration of white noise) and showed that the variance of the displacement was in this case proportional of the expended time: Var(∆x)= 2D∆t,
(1)
where the parameter D is the diffusion coefficient. D is an average measure of the stochastic activity of the white noise process which caused the displacement, and is directly related to its frequency and/or amplitude. The variance of displacement can be expressed as: Var(∆x) = E((∆x – E(∆x))²) = E(∆x²) - E(∆x)² = E(∆x²)
(2)
as, in the case of Browian motion, E(∆x) = 0. E(∆x²) can be estimated by the empirical mean, computed over a given time interval, 1. The Einstein relation takes then the following form: = 2D∆t,
(3)
signifying that, on the average, the displacement is proportional to the square root of the time. The basic relation given by Eq. 1 was generalized by Mandelbrot et van Ness (1968), to account for a family of stochastic processes they termed fractional Brownian motion. They proposed the following scaling law: Var(∆x) ∝ ∆t2H
(4)
where the scaling exponent H can be any real number in the range 0 < H 1 s), with Hl exponents suggesting anti-persistence (0 < Hl < 0.5). These results were revealed by an inflexion of the double logarithmic plot, which was modeled with two distinct straight lines intersecting approximately at ∆t = 1 s. Similar results were obtained by Treffner and Kelso (1995, 1999), in an experiment where participants attempted to actively balance an inverted pendulum (an aluminum rod). Collins and De Luca (1993) proposed also an estimation of the diffusion coefficient D (see Eq. 3), based on an assessment on the slope of the natural plot of as a function of ∆t. This analysis led also to the identification of two distinct regions, with larger D values for short-term intervals, and smaller D values for long-term intervals. Such an estimation appears nevertheless awkward, as the original relation of Einstein holds only for Brownian motion. These results were interpreted through the hypothesis of a two-component, open- and closed-loop control mechanism. In this view, short-term open loop control refers to exploratory processes (considering persistence as information gathering), and long-term closed loop control to performatory processes (considering antipersistence as adjusting on the basis of obtained information). Subsequent experiments showed systematic evolutions of the parameters of the model (Hs and Hl exponents for short- and long-term processes, coordinates of the point of inflexion), under the 3
Note that the equation Var(x) = holds only in the special case of Brownian motion, where E(∆x) = 0. A better estimation should be Var(x) = - (See Eq. 2, line 2). This correction could be important, in the case of fractional Browian motion (and especially for persistent processes) for the highest ∆t, because the number of estimations included in the averaging processes is necessarily lower. 4 We respect, in this paper, the notation proposed by Collins and De Luca (1993). Hs designates the scaling exponent obtained with their method for the short-term region, and Hl the corresponding exponent for the long-term region. H will represent the theoretical scaling exponent of the Mandelbrot and van Ness (1968)’s relation.
Non linear time series analysis 3 manipulation of factors such as vision, leaning, or haptic touch (Riley, Wong, Mitra & Turvey, 1997; Riley, Mitra, Stoffregen, & Turvey, 1997). Nevertheless, the mapping of biological time series to formal stochastic processes raises a number of methodological problem. Of crucial importance is the fact that a formal fractional Brownian motion is typically unbounded: the fluctuations grow with the time interval length in a power-law way, and the expected displacement increases indefinitely with time. In contrast, biological times series are generally bounded within physiological limits. This is the case, for example, for the aforementioned heart rate and gait time series, as well as for the trajectory of the center-of-pressure, which is obviously bounded within the area of support defined by the subject’s feet. As a consequence, the variance of such biological time series cannot exceed a ceiling value, and, at least beyond a critical time interval (necessary to reach this ceiling value), is independent to time. In other words, the generalized relation of Mandelbrot and van Ness (1968) given by Eq. 4 cannot directly hold for such biological series. An elegant solution for this problem is to study the fractal properties of the integrated time series, rather than those of the original signals (Feder, 1988; Hurst, 1965; Peng et al., 1995). If the original signal is constrained within physiological boundaries, the integrated series is not bounded and exhibits fractal properties that can be quantified on the basis of Eq. 4. Then, mapping the original bounded time series to an integrated signal appeared essential in biological time series analysis. Such a procedure constitutes the first step of the aforementioned Rescaled Range Analysis and Detrended Fluctuation Analysis (with slight methodological differences, see Appendix for details). This explains why a HR/S exponent (or a α scaling exponent) of 0.5 characterizes a white noise process, and not a Brownian motion as in the relation of Mandelbrot and van Ness (1968). If the original signal is a purely random, uncorrelated process, its integration leads to a random walk (Brownian motion), revealed by these typical exponents. This problem was not considered by Collins and De Luca (1993), which applied their method directly on the raw trajectory of the center-of-pressure. One can hypothesize, with this respect, that the flattening of the slope of the double logarithmic plot beyond a critical point, described by the authors, is simply the consequence of the bounded character of this original series. And the length of the short-term region represents the average time the variable needs, from a given point of the allowed space, to reach one of the physiological boundaries. In other words, the two-processes model proposed by the authors to account for their center-ofpressure trajectories could be simply a statistical artifact. Another problem with Collins and De Luca (1993)’s method is related to the oscillatory character of many biological time series, and especially in the case of displacement data. The displacement of the center of
pressure, or the displacement of the bottom of the rod in the experiment of Treffner and Kelso (1995), results of the behavior of a complex system, composed of many interacting oscillatory components. Fourier analysis generally reveals a rich spectrum of periodicities, and in the case of center-of-pressure displacement, Powell and Dzendolet (1984) showed that the most power was concentrated in the region of about 0.3 –0.4 Hz, revealing the presence of low-frequency oscillatory components. Working on the basis of the squared differences between points separated by given time lags, the method of Collins and De Luca (1993) should be sensitive to these periodicities. With an original signal with a main underlying component of frequency f, the stabilogramdiffusion plot should exhibit a wave behavior, with minima corresponding to ∆t = k(1/f), (k ∈{1, 2, 3, …}), and maxima corresponding to ∆t = (k/2)(1/f), (k ∈{1, 2, 3, …}). Such behavior was not evident in the stabilogram-diffusion plots presented by Collins and De Luca (1993), but one could think that the trial averaging realized by the authors has hidden these periodicities. In contrast, the single-trial stabilogram-diffusion plot reported in Treffner and Kelso (1995, see p. 85) clearly exhibited a wave behavior, related, according to the authors, to the eigenfrequency of the rod used in the trial. The R/S analysis and the DFA avoid this problem, as they work on the basis of the global behavior of the variable of interest within each interval (the maximal range for the first method, the detrended standard deviation for the second), and not only on the relative distance between the points corresponding to the temporal boundaries of each interval. To sum up, we think that the application of the method proposed by Collins and De Luca (1993) to a bounded series leads automatically to a non-linear loglog plot. For the smallest time lags considered, the plot should produce a straight line, revealing short-term dependence in position and direction. Beyond a given time lag, a flattening of the slope of the log-log plot should be observed, as , bounded within physiological limits, cannot increase indefinitely. Finally, when the original time series presents an oscillatory behavior, the plot should exhibit a typical waveform, whose periodicities should be related to the eigenfrequencies of the system under study. These expected alterations of the log-log plot are problematic, as the obtaining of a straight line is generally considered as an important test of the fractality of the data (Hurst, 1965; Peng et al., 1995). Before to compare the results obtained by these diverse methods on similar series, some additional points must be discussed, concerning the meaning of their respective exponents. The original relation of Einstein (1905) signifies that, on the average, the displacement in Brownian motion is proportional to the square root of the time (see Eq. 3). As said previously, Brownian motion is considered as a special case of fractional Brownian motion, with H = 0.5 (see Eq. 4). According to
Non linear time series analysis 4 Mandelbrot and van Ness (1968), the scaling exponent H can be any real number in the range 0 < H 0.5 or α>0.5), in which large values are more likely followed by large values, and vice versa, and anti-persistent series (HR/S 1.0). When α = 1.5, the original series is a Brownian motion. Finally, α < 0.5 denotes long-term anti-correlations. Spectral analysis A third classical method for assessing stochastic processes works on the basis of the periodogram obtained by Fourier analysis. The relation of Mandelbrot et van Ness (1968) can be expressed as follows: S(f)∝1/f ß Where f is the frequency and S(f) the correspondent squared amplitude. ß is estimated by calculating the negative slope of the line relating log(S(f) to log f, as: log10S(f) ∝ log10f -ß For a white noise process, ß = 0 (signifying that all squared amplitudes are equivalent, whatever the frequency). ß = 2 for Brownian motion. In the special case of pink noise, ß = 1.0: in other words S(f) is proportional to 1/f.
