Estimating degrees of freedom in motor systems
  Studies of the degrees of freedom or "synergies" in musculoskeletal systems rely critically on algorithms to estimate the "dimension" of kinematic or neural data. Linear algorithms such as principal component analysis (PCA) are used almost exclusively for this purpose. However, biological systems tend to possess nonlinearities and operate at multiple spatial and temporal scales so that the set of reachable system states typically does not lie close to a single linear subspace. We compare the performance of PCA to two alternative nonlinear algorithms (Isomap and our novel pointwise dimension estimation (PD-E)) using synthetic and motion capture data from a robotic arm with known kinematic dimensions, as well as motion capture data from human hands. We find that consideration of the spectral properties of the singular value decomposition in PCA can lead to more accurate dimension estimates than the dominant practice of using a fixed variance capture threshold. We investigate methods for identifying a single integer dimension using PCA and Isomap. In contrast, PD-E provides a range of estimates of fractal dimension. This helps to identify heterogeneous geometric structure of data sets such as unions of manifolds of differing dimensions, to which Isomap is less sensitive. Contrary to common opinion regarding fractal dimension methods, PD-E yielded reasonable results with reasonable amounts of data. We conclude that it is necessary and feasible to complement PCA with other methods that take into consideration the nonlinear properties of biological systems for a more robust estimation of their degrees of freedom. 
Robert H. Clewley, John M. Guckenheimer, and Francisco J.   Valero-Cuevas