  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.
