We model each sub-trajectory using a linear statistical model, assuming a Gaussian distribution. Each sub-trajectory is described by a vector, which is a simple concatenation of the sub-trajectory points. A sub-trajectory group is a set of vectors, on which we can apply a principal component analysis. Each sub-trajectory 
is approximated by:
is a vector representing the mean sub-trajectory of the group, 
is a matrix computed by the principal component analysis and describing how the data varies, and 
is the vector of parameters for that particular sub-trajectory. The probability density function of the set of parameters is modelled by a Gaussian, by computing the mean and variance of for all sub-trajectories in the group.
In order to be able to perform the principal component analysis on a group of sub-trajectories, it is required that all the sub-trajectories are encoded with the same number of points. Therefore, we interpolate all the sub-trajectories by cubic splines and homogeneously re-sample them to a given number of points.
