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

## TL;DR

This paper introduces a kernel density estimation method for modeling and forecasting trajectories of hidden dynamical systems on manifolds, effectively handling sparse and noisy data through energy minimization.

## Contribution

It presents a novel KDE-based approach for trajectory modeling that minimizes an energy function, addressing sparse sampling and noise in hidden dynamical systems.

## Key findings

- Estimator minimizes a specific energy function as data samples increase.
- Method effectively reconstructs trajectories from sparse, noisy data.
- Applicable to modeling recurrent trajectories on manifolds.

## Abstract

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.