Probabilistic models for joint clustering and time-warping of multidimensional curves

In this paper we present a family of algorithms that can simultaneously align and cluster sets of multidimensional curves measured on a discrete time grid. Our approach is based on a generative mixture model that allows non-linear time warping of the observed curves relative to the mean curves within the clusters. We also allow for arbitrary discrete-valued translation of the time axis, random real-valued offsets of the measured curves, and additive measurement noise. The resulting model can be viewed as a dynamic Bayesian network with a special transition structure that allows effective inference and learning. The Expectation-Maximization (EM) algorithm can be used to simultaneously recover both the curve models for each cluster, and the most likely time warping, translation, offset, and cluster membership for each curve. We demonstrate how Bayesian estimation methods improve the results for smaller sample sizes by enforcing smoothness in the cluster mean curves. We evaluate the methodology on two real-world data sets, and show that the DBN models provide systematic improvements in predictive power over competing approaches.