Short communicationEffect of noise and filtering on largest Lyapunov exponent of time series associated with human walking
Introduction
During recent years, there has been an increasing interest in adopting the nonlinear measure of largest Lyapunov exponent (LyE) to quantify human movement dynamic stability (for a review see Bruijn, Meijer, Beek, and van Dieën, 2013). The LyE measures the exponential rate of divergence of neighboring trajectories of the state space constructed by kinematic data acquired from human motion (Dingwell, 2006, Dingwell and Marin, 2006). The LyE value has been extensively used to quantify human locomotion dynamic stability and thus to estimate fall risk (Bruijn et al., 2013, Kang and Dingwell, 2009, Lockhart and Liu, 2008, McAndrew et al., 2011, Roos and Dingwell, 2011, Su and Dingwell, 2007, Toebes et al., 2012). In biomechanical studies, LyE is commonly calculated using Rosenstein’s algorithm which is suitable for short length time series acquired from experiment (e.g. locomotion time series; Rosenstein et al., 1993). One of the main challenges faced with experimental time series is the measurement noise (Liu et al., 2005, Yang and Wu, 2010). The presence of such noise could have adverse effects on the calculated LyE since it increases the possibility of picking false neighbors in the state space (Yang and Wu, 2010). Nevertheless, studies on the effect of noise on LyE (Rispens et al., 2014, Rosenstein et al., 1993) only considered mathematical dynamical systems (e.g., Lorenz, Rossler), and not signals associated with human movement.
To decrease the effect of noise on LyE results, Rosenstein et al. (1993) suggested filtering the time series before LyE calculation. However, it has been argued that time series filtering may cause possible loss of information at critical points and thus may change the dynamics of the system under study (Kantz and Schreiber, 2004). Since the effect of a prior time series filtering on LyE has not been well investigated, there has been little agreement on what approach should be adopted.
Another area of concern with the application of quantifying LyE in human movement dynamic stability is the minimum time series length required for computations. Several studies have investigated the reliability of LyE for different time series length (Bruijn et al., 2009, Cignetti et al., 2012, Kang and Dingwell, 2006, Riva et al., 2014, Terrier and Reynard, 2014, van Schooten et al., 2013). However, none of these studies documented the effect of noise. As noise affects the determination of neighborhoods (Yang and Wu, 2010), the length of the time series (i.e. size of neighboring trajectories) might also influence the LyE in the presence of noise.
The aims of this study were therefore, to determine (1) the effect of different added noise levels on the LyE value calculated using time series obtained from a passive dynamic walker, (2) the effect of filtering on LyE, and (3) the effect of different time series length on LyE value in the presence of various noise levels.
Section snippets
The passive dynamic walker
In this study, the LyE was calculated for the simplest passive dynamic walker model (Garcia et al., 1998, Kurz and Stergiou, 2007, Roos and Dingwell, 2011). The motion of the model is described by two second-order differential equations:where is the angle of the stance leg with respect to the perpendicular to and is the angle of the swing leg with respect to the stance limb (Fig. 1), t is time and all
Effect of noise on LyE
The results indicates while the curve of noise-free time series has a typical shape of divergence curves and a linear region could be identified, in the curve of noisy data, a linear region could hardly be identified (Fig. 2).
Our results also show that in all number of strides and all noise levels, the value of LyE was lower for noisy than noise-free time series (i.e., negative % errors; Table 1). Moreover, the results demonstrated that, except for the 6-stride case, the value of LyE increases
Discussion
Our results showed while a clear linear region could be identified for the divergence curve of noise-free data, this is not the case for noisy data (Fig. 2). This implies that neighboring trajectories in the state space of the noisy data do not diverge exponentially as time goes on. This might be due to picking false neighbors in the state space (Yang and Wu, 2010). As a result, the divergence-time relationship would not be linear in logarithmic scale. It could thus be concluded that in the
Conflict of interest
None.
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2021, Journal of BiomechanicsCitation Excerpt :This approach assumes that every cycle of force production could be identical, and difference in force production patterns from the other cycle is considered as small perturbations. MLE works best with data sets acquired over a short measurement time (i.e., 20 s; the error of MLE decreased with the data points ≤ 2,000) (Ekizos et al., 2018); (Mehdizadeh, 2019); (Mehdizadeh and Sanjari, 2017); (Rosenstein et al., 1993). The rate of change in the distance between nearest neighbors was estimated by fitting a linear region for these divergence curves over time (Fig. 2D; Equation 2 in supplementary file).
Filtering affects the calculation of the largest Lyapunov exponent
2020, Computers in Biology and MedicineCitation Excerpt :This would support the use of the former scripts and of unfiltered data or at least relatively high cut-off frequencies. This is agreement with a recent study by Mehdizadeh and Sanjari [20], who applied Gaussian white noise to a time series generated by a passive walker. These authors proposed that high cut-off frequencies rather than low frequencies should be applied to experimental data to purely eliminate unwanted high frequency noise.