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In this paper we develop a kernel density estimation (KDE) approach tomodeling and forecasting recurrent trajectories on a compact manifold. For thepurposes of this paper, a trajectory is a sequence of coordinates in a phasespace defined by an underlying hidden dynamical system. Our work is inspired byearlier work on the use of KDE to detect shipping anomalies using high-density,high-quality automated information system (AIS) data as well as our own earlierwork in trajectory modeling. We focus specifically on the sparse, noisytrajectory reconstruction problem in which the data are (i) sparsely sampledand (ii) subject to an imperfect observer that introduces noise. Under certainregularity assumptions, we show that the constructed estimator minimizes aspecific energy function defined over the trajectory as the number of samplesobtained grows.

Modeling a Hidden Dynamical System Using Energy Minimization and Kernel Density Estimates