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Figure 2 | Journal of NeuroEngineering and Rehabilitation

Figure 2

From: Wearing a safety harness during treadmill walking influences lower extremity kinematics mainly through changes in ankle regularity and local stability

Figure 2

Attractor reconstruction and calculation of the largest Lyapunov exponent (LyE) and sample entropy (SampEn). (A) The original x i i = 1 N angle time series and the time-delayed copies [x(i+ τ),...,x(i + (m-1)τ)] used for attractor reconstruction. (B) The attractors were composed of sets of m-dimensional vectors v(i) = [x(i), x(i+ τ),...,x(i + (m-1)τ)], with i = 1,...,N - (m-1)τ. The delay τ was obtained from the first minimum of the average mutual information function and the dimension m was selected where the percentage of the global false nearest neighbours approached zero. (C) The LyE algorithm tracked the divergence of nearest neighbours over time, focusing on a reference trajectory with a single nearest neighbour being followed and replaced when its separation L'(t k ) from the reference trajectory becomes large. The new neighbour was chosen to minimize the replacement length L(t k ) and the angular separation θ k . Once the reference trajectory has gone over the data sample, L y E = t M - t 0 - 1 k = 1 M log L t k / L t k - 1 was estimated, with M the total number of replacement steps [14, 15]. (D) For the SampEn, the first step consisted in calculating C i m τ , r = N - m τ - 1 number of  j  such that  d v i , v j r , where ji ranges from 1 to N - , and d v i , v j = max 0 k m - 1 x j + k - x i + k is the maximum difference between the scalar components of the vectors [v(i), v(j)]. The distance r was chosen as 0.2× standard deviation of x(i). The density Φ m τ , r = N - m τ - 1 i = 1 N - m τ C i m τ , r was obtained afterwards. (E) The procedure was repeated for an (m+1)-dimensional attractor, by computing Φm+1(τ, r). Finally, the negative log likelihood of the conditional probability that two close vectors (within r) in a m-dimensional attractor remain close in a (m+1)-dimensional attractor was obtained as SampEn = -τ-1 log (Φm+1(τ, r)/Φm(τ, r)) [16].

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