Absolute and Relative Timing in Bimanual Coordination : Contribution of Serial Correlation Analysis and Implications for Modeling

The aim of this paper was to evaluate the performances of ARFIMA modelling for detecting long-range dependence and estimating fractal exponents. More specifically, our aim was to test the procedure proposed by Wagenmakers, Farrell and Ratcliff (2005), and to compare the results obtained with the Akaike Information Criterion (AIC) and the Bayes Information Criterion (BIC). The present studies show that ARFIMA modelling is able to adequately detect long-range dependence in simulated fractal series. Conversely, this method tends to produce a non-negligible rate of false detections in pure ARMA series. Generally, ARFIMA modelling presents a bias favouring the detection of long-range dependence. AIC and BIC gave dissimilar results, due to the different weights attributed by the two criteria to accuracy and parsimony. Finally, ARFIMA modelling provides good estimates of fractal exponents, and could adequately complement classical methods, such as spectral analysis, detrended fluctuation analysis or rescaled range analysis. Introduction Some recent experiments, generally underlain by the framework of dynamical systems theory, tried to analyze the dynamical structure of time series of psychological or behavioural variables. A quite intriguing result in these experiments was the recurrent discovery of fractal properties in the studied series. Fractals were evidenced, for example, in selfesteem (Delignières, Fortes, & Ninot, 2004), in mood (Gottschalk, Bauer, & Whybrow, 1995), in serial reaction time (Gilden, 1997; van Orden, Holden, & Turvey, 2003), in finger tapping (Gilden, Thornton, & Mallon, 1995; Delignières, Lemoine, & Torre, 2004), in stride duration during walking (Hausdorff, Peng, Ladin, Wei, & Goldberger, 1995), or in relative phase in a bimanual coordination task (Schmidt, Beek, Treffner, & Turvey, 1991). All these variables were previously conceived as highly stable over time, and fluctuations in successive measurements were considered as randomly distributed, and uncorrelated in time. As a consequence, a sample of repeated measures was assumed to be normally distributed around its mean value, and noise could be discarded by averaging. This methodological standpoint was implicitly adopted in most classical psychological research (for a deeper analysis, see Gilden, 2001; Slifkin & Newell, 1998). In other words, temporal ordering of data points was ignored and the possible correlation structure of fluctuations was clearly neglected. Fractal analysis focuses, in contrast, on the timeevolutionary properties of data series and on their correlation structure. Fractal processes are characterized by a complex pattern of correlations appearing following multiple interpenetrated time scales. In such processes, the value at a particular time is related not just to immediately preceding values, but also to fluctuations in the remote past. Such series are said to present long-term memory, or long-range dependence. This property is typically revealed by a very slow decay over time of the autocorrelation function, which tends to follow a power law. Detecting fractal properties in empirical time series could have important theoretical implications. Fractals are considered as the natural outcome of complex dynamical systems behaving at the frontier of chaos (Bak & Chen, 1991; Marks-Tarlow, 1999). Psychological variables should then be conceived as the macroscopic and dynamical products of complex systems composed of multiple interconnected elements. Moreover, psychological and behavioural Send correspondence to: Kjerstin Torre Faculty of Sport Sciences – University Montpellier 1 EA 2991, “Motor Efficiency and Deficiency” 700 Avenue du Pic Saint Loup 34090 Montpellier – France Email: [email protected]