An improved surrogate method for detecting the presence of chaos in gait

It has been suggested that the intercycle variability present in the time series of biomechanical gait data is of chaotic nature. However, the proper methodology for the correct determination of whether intercycle fluctuations in the data are deterministic chaos or random noise has not been identified. Our goal was to evaluate the pseudoperiodic surrogation (PPS) [Small et al., 2001. Surrogate test for pseudoperiodic time series data. Physical Review Letters 87(18), 188,101–188,104], and the surrogation algorithms of Theiler et al. [1992. Testing for nonlinearity in time series: the method of surrogate data. Physica D 58(1–4), 77–94] and of Theiler and Rapp [1996. Re-examination of the evidence for low-dimensional, nonlinear structure in the human electroencephalogram. Electroencephalography and Clinical Neurophysiology 98, 213–222], to determine which is the more robust procedure for the verification of the presence of chaos in gait time series. The knee angle kinematic time series from six healthy subjects, generated from a 2-min walk, were processed with both algorithms. The Lyapunov exponent (LyE) and the approximate entropy (ApEn) were calculated from the original data and both surrogates. Paired t-tests that compared the LyE and the ApEn values revealed significant differences between both surrogated time series and the original data, indicating the presence of deterministic chaos in the original data. However, the Theiler algorithm affected the intracycle dynamics of the gait time series by changing their overall shape. This resulted in significantly higher LyE and ApEn values for the Theiler-surrogated data when compared with both the original and the PPS-generated data. Thus, the discovery of significant differences was a false positive because it was not based on differences in the intercycle dynamics but rather on the fact that the time series was of a completely different shape. The PPS algorithm, on the other hand, preserved the intracycle dynamics of the original time series, making it more suitable for the investigation of the intercycle dynamics and the identification of the presence of chaos in the gait time series.