Variability, covariation, and invariance with respect to coordinate systems in motor control: reply to Smeets and Louw (2007)

In their comment on the tolerance-noise covariation (TNC) method for decomposing variability by H. Müller and D. Sternad (2003, 2004b), J. B. J. Smeets and S. Louw show that covariation (C), as defined within the TNC method, is not invariant with respect to coordinate transformations and contend that it is, therefore, meaningless. Although the observation is correct, their interpretation is misleading in the following ways: (a) They equate covariation C with the known statistical quantity covariance and noise (N) with standard deviations. The two quantities C and N are conceptually different statistical measures. (b) Dependency on the reference frame is not only a feature of C but of all 3 components. However, such dependency is ubiquitous in motor control. (c) As the frame of reference in biological systems is poorly understood, the TNC method may afford evaluation of different coordinates for control.