Probabilistic curve-aligned clustering and prediction with regression mixture models

Clustering and prediction of sets of curves is an important problem in many areas of science and engineering. Most clustering algorithms operate on fixed-dimensional feature vectors, and as a result, curve analysis is often forced into this unnatural paradigm. Perhaps more importantly, curves tend to be misaligned from each other in a continuous manner, either in space (across the measurements) or in time. However, the notion of time within a feature-vector is very rigid corresponding only to the discrete dimensional setup of the space itself. In contrast to this, we develop a probabilistic framework that allows for the joint clustering and continuous alignment of sets of curves in curve space. Our proposed methodology integrates new probabilistic alignment models with model-based curve clustering algorithms. The probabilistic approach allows for the derivation of consistent EM-type learning algorithms for the joint clustering-alignment problem. Both simulated and real-world datasets are used for detailed experimentation, with two extensive applications to the clustering of cyclone trajectories presented.