A directional model for the statistical analysis of movement in three dimensions

Movement of an object in three dimensions involves rotation and translation. The data for analysis are the coordinates of landmarks on the object, recorded at several time points. Statistical models for describing the rotational component of the movement of an object are proposed in this work. Under the fixed‐axis model, the object rotates around an axis that does not change in time. The angular position of the object is then characterised by its rotation axis and an angle giving the extent of the rotation of the object with respect to a reference point. More complicated models occur when the rotation axis varies in time. Under the fixed‐angle model the quaternions for the time‐varying orientations of the object are shown to lie in a two‐dimensional great circle on the surface of the unit sphere in four dimensions. Simple estimators are given of the common rotation axis and of the angles characterising the time varying orientation of the object. A score statistic for testing the fit of the fixed‐axis model is constructed and methods for handling the autocorrelation of the errors at neighbouring data points are provided. The analysis of data, collected by an OPTOTRACK camera system on a rotating forearm, illustrates the methodology presented in this paper.