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

In this paper we develop a kernel density estimation (KDE) approach to modeling and forecasting recurrent trajectories on a compact manifold. For the purposes of this paper, a trajectory is a sequence of coordinates in a phase space defined by an underlying hidden dynamical system. Our work is inspired by earlier 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 earlier work in trajectory modeling. We focus specifically on the sparse, noisy trajectory reconstruction problem in which the data are (i) sparsely sampled and (ii) subject to an imperfect observer that introduces noise. Under certain regularity assumptions, we show that the constructed estimator minimizes a specific energy function defined over the trajectory as the number of samples obtained grows.