I am examining the stability of a normal gait through a kinematic parameter (Displacement) analysis.But I am getting negative Lyapunov exponent to Z-direction (transverse plane) but positive in other directions for the same body position data.For displacement parameter, Z-direction is considered in the literature.I am using marker based standard data.Does negative Lyapunov exponent in z-direction means gait signal is non-chaotic? If it is, then what about other direction?
I guess you calculated the local dynamic stability in your study, so you just analysis the Maximum Lyapunov Exponent? If this value is negative, then it means all Lyapunov Exponents of this gait system is under 0, i.e. the system is stable in Z direction. In my opinion, the motion of a body segment can be chaotic in some directions while non-chaotic in other directions. Still, I feel it unsuitable for you to use displacement data in the analysis, as it may cause the effects of non-stationarity in the original position
data.
Thanks, Dr.Yu Wu for your valuable answer.
Can you please explain me, what do you mean by " effects of non-stationarity in the original position data"?
Will it be suitable to use Ground Reaction Force data instead of linear displacement data for this analysis?
@Saikat Chakraborty,
I suspect a confusion between the Lyapunov exponents assessed in a chaotic system along the dimensions of the attractor and the 3 dimensions of the body's movements.
In a 3D dissipative chaotic system, such as the Lorenz attractor, there is always a combination of one positive, one null and one negative Lyapunov exponent. Actually, the zero value means that along the flow, there is no divergence between consecutive points on a trajectory. The two other exponents are measured perpendicular to the flow: in one direction, the flow diverges exponentially and in the other one, converges exponentially.
Local dynamic stability of gait is based on Rosenstein or Wolf algorithms that can compute only the maximal Lyapunov exponent. For Rosenstein's algortihm, given the shape of the divergence curve (see for instance my publications), positive values are always observed. However, keep in mind that local dynamic stability of gait is only validated empirically. Theoretical ground is lacking to consider gait as a true chaotic system. Basic assumptions for a correct application of the attractor reconstruction are very likely not respected. See the recent publication by Ihlen et al. to see a discussion about that.
Finally be careful when using position to assess the gait dynamic. To reconstruct an attractor, you need a stationary and quasi-periodic dynamic. In a walking individual, this condition is violated, except if you use a treadmill.
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Even on treadmill, there is some trend in displacement data. But you could always effectively remove non-stationarity of your marker based signal by using velocity instead of displacement.
You can calculate the velocity by using the three-points method. For example, velocity (v) at the second sample would be:
v(2) = (p(3)-p(1))/(t(3)-t(1))
* p = displacement, t = time
This way you would end up with 2 less samples than you had in your displacement series, but you would have the corresponding velocity for each sample.
Good luck! My best to you!
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