Comparative assessment of some algorithms for differentiating noisy biomechanical data

In this paper, a comparison is carried out between various algorithms for smoothing and differentiating noisy (non-exact) discrete time series, a problem frequently encountered in experimental movement studies. The algorithms compared are: the 'implicit procedure' of Anderssen and Bloomfield (Numer Math, 22 (1974) 157-182) with a refinement by Kosarev and Pantos (J Phys E Sci Instrum, 16 (1983) 537-543); the 'explicit procedure', a digital filter method in which the filter coefficients are computed starting from the measurements; the regularized Fourier series method (these two algorithms were also presented by Anderssen and Bloomfield); and, finally, natural B-splines regularized by the generalized cross-validation criterion. The comparison was performed mainly by analytical, noise-corrupted test sequences, whose derivatives were known a priori. The testing procedure proved capable of showing the different characteristics of the various algorithms, and providing criteria to choose that best suited to a given practical situation. In most cases, the regularized Fourier series method and the implicit procedure proved to have the best tradeoff between accuracy and speed.