Trajectory analysis using point distribution models

This thesis focuses on the analysis of the trajectories of a mobile agent. It presents different techniques to acquire a quantitative measure of the difference between two trajectories or two trajectory datasets. A novel approach is presented here, based on the Point Distribution Model (PDM). This model was developed by computer vision scientists to compare deformable shapes. This thesis presents the mathematical reformulation of the PDM to fit spatiotemporal data, such as trajectory information. The behavior of a mobile agent can rarely be represented by a unique trajectory, as its stochastic component will not be taken into account. Thus, the PDM focuses on the comparison of trajectory datasets. If the difference between datasets is greater than the variation within each dataset, it will be observable in the first few dimensions of the PDM. Moreover, this difference can also be quantified using the inter-cluster distance defined in this thesis. The resulting measure is much more efficient than visual comparisons of trajectories, as are often made in existing scientific literature. This thesis also compares the PDM with standard techniques, such as statistical tests, Hidden Markov Models (HMMs) or Correlated Random Walk (CRW) models. As a PDM is a linear transformation of space, it is much simpler to comprehend. Moreover, spatial representations of the deformation modes can easily be constructed in order to make the model more intuitive. This thesis also presents the limits of the PDM and offers other solutions when it is not adequate. From the different results obtained, it can be pointed out that no universal solution exists for the analysis of trajectories, however, solutions were found and described for all of the problems presented in this thesis. As the PDM requires that all the trajectories consist of the same number of points, techniques of resampling were studied. The main solution was developed for trajectories generated on a track, such as the trajectory of a car on a road or the trajectory of a pedestrian in a hallway. The different resampling techniques presented in this thesis provide solutions to all the experimental setups studied, and can easily be modified to fit other scenarios. It is however very important to understand how they work and to tune their parameters according to the characteristics of the experimental setup. The main principle of this thesis is that analysis techniques and data representations must be appropriately selected with respect to the fundamental goal. Even a simple tool such as the t-test can occasionally be sufficient to measure trajectory differences. However, if no dissimilarity can be observed, it does not necessarily mean that the trajectories are equal – it merely indicates that the analyzed feature is similar. Alternatively, other more complex methods could be used to highlight differences. Ultimately, two trajectories are equal if and only if they consist of the exact same sequence of points. Otherwise, a difference can always be found. Thus, it is important to know which trajectory features have to be compared. Finally, the diverse techniques used in this thesis offer a complete methodology to analyze trajectories.