Series of motor outputs generated by cyclic movements are typically complex, suggesting that the correlation function of the time series spans over a large number of consecutive samples. Famous examples include inter-stride intervals, heartbeat variability, spontaneous neural firing patterns or motor synchronization with external pacing. Long-range correlations are potentially important for fundamental research, as the neural and biomechanical mechanisms generating these correlations remain unknown, and for clinical applications, given that the loss of long-range correlation may be a marker of disease. However, no systematic approach or robust analysis methods have yet been used to support the study of correlation functions in physiological series. This study investigates four selected methods (the Hurst exponent, the power spectral density analysis, the rate of moment convergence and the multiscale entropy methods). We present the result of each analysis performed on artificial computer-generated series in which the auto-correlation function is known, and then on time series extracted from gait and upper limb rhythmic movements. Our results suggest that combined analysis using the Hurst exponent and the power spectral density is suitable for rather short series (512 points). The rate of moment convergence directly supports the power spectral density analysis, and the multiscale entropy further confirms the presence of long-range correlation, although this method seems more appropriate for longer series. The proposed methodology increases the level of confidence in the hypothesis that physiological series are long-memory processes, which is of prime importance for future fundamental and clinical research.
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