A formal and data-based comparison of measures of motor-equivalent covariation

Different analysis methods have been developed for assessing motor-equivalent organization of movement variability. In the uncontrolled manifold (UCM) method, the structure of variability is analyzed by comparing goal-equivalent and non-goal-equivalent variability components at the level of elemental variables (e.g., joint angles). In contrast, in the covariation by randomization (CR) approach, motor-equivalent organization is assessed by comparing variability at the task level between empirical and decorrelated surrogate data. UCM effects can be due to both covariation among elemental variables and selective channeling of variability to elemental variables with low task sensitivity ("individual variation"), suggesting a link between the UCM and CR method. However, the precise relationship between the notion of covariation in the two approaches has not been analyzed in detail yet. Analysis of empirical and simulated data from a study on manual pointing shows that in general the two approaches are not equivalent, but the respective covariation measures are highly correlated (ρ > 0.7) for two proposed definitions of covariation in the UCM context. For one-dimensional task spaces, a formal comparison is possible and in fact the two notions of covariation are equivalent. In situations in which individual variation does not contribute to UCM effects, for which necessary and sufficient conditions are derived, this entails the equivalence of the UCM and CR analysis. Implications for the interpretation of UCM effects are discussed.