Analysis of High-Dimensional Coordination in Human Movement Using Variance Spectrum Scaling and Intrinsic Dimensionality

Full-body movement involving multi-segmental coordination has been essential to our evolution as a species, but its study has been focused mostly on the analysis of one-dimensional data. The field is poised for a change by the availability of high-density recording and data sharing. New ideas are needed to revive classical theoretical questions such as the organization of the highly redundant biomechanical degrees of freedom and the optimal distribution of variability for efficiency and adaptiveness. In movement science, there are popular methods that up-dimensionalize: they start with one or a few recorded dimensions and make inferences about the properties of a higher-dimensional system. The opposite problem, dimensionality reduction, arises when making inferences about the properties of a low-dimensional manifold embedded inside a large number of kinematic degrees of freedom. We present an approach to quantify the smoothness and degree to which the kinematic manifold of full-body movement is distributed among embedding dimensions. The principal components of embedding dimensions are rank-ordered by variance. The power law scaling exponent of this variance spectrum is a function of the smoothness and dimensionality of the embedded manifold. It defines a threshold value below which the manifold becomes non-differentiable. We verified this approach by showing that the Kuramoto model obeys the threshold when approaching global synchronization. Next, we tested whether the scaling exponent was sensitive to participants' gait impairment in a full-body motion capture dataset containing short gait trials. Variance scaling was highest in healthy individuals, followed by osteoarthritis patients after hip replacement, and lastly, the same patients before surgery. Interestingly, in the same order of groups, the intrinsic dimensionality increased but the fractal dimension decreased, suggesting a more compact but complex manifold in the healthy group. Thinking about manifold dimensionality and smoothness could inform classic problems in movement science and the exploration of the biomechanics of full-body action.