  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.
