For each cluster, similar sub-trajectories will be found, so we can apply a PCA on these sub-trajectories to model the changes in the trajectory by a linear approximation. The sub-trajectories may have to be re-parametrized before this operation. A re-parameterization method has to be found. In order to do that, a once has to be made for the location of new points on the curve drawn by the trajectory. This certainly involves learning methods based on least square estimate. It can also be done by extrapolating the points that forms the trajectory by a spline and the re-sample the points on the spline. All this has to be investigated.
Another problem that can appear in way of modelling trajectories is the modelling of speed. One idea that can be tested is the use of flow vectors that have been introduced by Brand [8]. A flow vector is simply the concatenation of the position vector and a scaled speed vector. The space formed by these vectors is called configuration space. This would allow speed to be included in the framework and thus to be model in the same way the raw trajectory is. Unfortunately, this approach may model the same trajectories covered at different speeds with two different trajectories in the augmented configuration space. This would increase the number of sub-trajectories found.
Another possibility is to model the speed by a function of the length covered on the trajectory. This will separate the model of the trajectory and the model of the speed. The correlation between position and speed has to be study in order to know if this model is valid.
If the position and the speed are correlated, the first model is better since the dimensionality may decrease even more because the PCA will catch this correlation. If it is not the case, the second model is certainly better.