Elsevier

Journal of Biomechanics

Volume 64, 7 November 2017, Pages 236-239
Journal of Biomechanics

Short communication
Effect of noise and filtering on largest Lyapunov exponent of time series associated with human walking

https://doi.org/10.1016/j.jbiomech.2017.09.009Get rights and content

Abstract

This study aimed to determine the effect of added noise, filtering and time series length on the largest Lyapunov exponent (LyE) value calculated for time series obtained from a passive dynamic walker. The simplest passive dynamic walker model comprising of two massless legs connected by a frictionless hinge joint at the hip was adopted to generate walking time series. The generated time series was used to construct a state space with the embedding dimension of 3 and time delay of 100 samples. The LyE was calculated as the exponential rate of divergence of neighboring trajectories of the state space using Rosenstein’s algorithm. To determine the effect of noise on LyE values, seven levels of Gaussian white noise (SNR = 55–25 dB with 5 dB steps) were added to the time series. In addition, the filtering was performed using a range of cutoff frequencies from 3 Hz to 19 Hz with 2 Hz steps. The LyE was calculated for both noise-free and noisy time series with different lengths of 6, 50, 100 and 150 strides. Results demonstrated a high percent error in the presence of noise for LyE. Therefore, these observations suggest that Rosenstein’s algorithm might not perform well in the presence of added experimental noise. Furthermore, findings indicated that at least 50 walking strides are required to calculate LyE to account for the effect of noise. Finally, observations support that a conservative filtering of the time series with a high cutoff frequency might be more appropriate prior to calculating LyE.

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:θ¨st(t)-sin(θst(t)-γ)=0;θ¨st(t)-ϕ̇sw(t)-θ̇st(t)2sinϕsw(t)-cos(θst(t)-γ)sinϕsw(t)=0;where θst is the angle of the stance leg with respect to the perpendicular to γ and ϕsw 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.

References (26)

  • M.J. Toebes et al.

    Local dynamic stability and variability of gait are associated with fall history in elderly subjects

    Gait Posture

    (2012)
  • K.S. van Schooten et al.

    Assessing gait stability: the influence of state space reconstruction on inter- and intra-day reliability of local dynamic stability during over-ground walking

    J. Biomech.

    (2013)
  • S.M. Bruijn et al.

    Assessing the stability of human locomotion: a review of current measures

    J. R. Soc. Interf.

    (2013)
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