Figure 2From: Wearing a safety harness during treadmill walking influences lower extremity kinematics mainly through changes in ankle regularity and local stabilityAttractor 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 j≠i ranges from 1 to N - mτ, 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].Back to article page