Estimating Velocities and Accelerations of Animal Locomotion: a Simulation Experiment Comparing Numerical Differentiation Algorithms

Functional biologists employ numerical differentiation for many purposes, including (1) estimation of maximum velocities and accelerations as measures of behavioral performance, (2) estimation of velocity and acceleration histories for biomechanical modeling, and (3) estimation of curvature, either of a structure during movement or of the path of movement itself. I used a computer simulation experiment to explore the efficacy of ten numerical differentiation algorithms to reconstruct velocities and accelerations accurately from displacement data. These algorithms include the quadratic moving regression (MR), two variants of an automated Butterworth filter (BF1–2), four variants of a method based on the signal’s power spectrum (PSA1–4), an approximation to the Wiener filter due to Kosarev and Pantos (KPF), and both a generalized cross-validatory (GCV) and predicted mean square error (MSE) quintic spline. The displacement data simulated the highly aperiodic escape responses of a rainbow trout Oncorhynchus mykiss and a Northern pike Esox lucius (published previously). I simulated the effects of video speed (60, 125, 250, 500 Hz) and magnification (0.25, 0.5, 1 and 2 screen widths per body length) on algorithmic performance. Four performance measures were compared: the per cent error of the estimated maximum velocity (Vmax) and acceleration (Amax) and the per cent root mean square error over the middle 80 % of the velocity (VRMSE) and acceleration (ARMSE) profiles. The results present a much more optimistic role for numerical differentiation than suggested previously. Overall, the two quintic spline algorithms performed best, although the rank order of the methods varied with video speed and magnification. The MSE quintic spline was extremely stable across the entire parameter space and can be generally recommended. When the MSE spline was outperformed by another algorithm, both the difference between the estimates and the errors from true values were very small. At high video speeds and low video magnification, the GCV quintic spline proved unstable. KPF and PSA2–4 performed well only at high video speeds. MR and BF1–2 methods, popular in animal locomotion studies, performed well when estimating velocities but poorly when estimating accelerations. Finally, the high variance of the estimates for some methods should be considered when choosing an algorithm.