From Target Tracking to Targeting Track -- Part III: Stochastic Process Modeling and Online Learning

TL;DR

Online estimation of a target's continuous-time trajectory is tackled by modeling $f(t)$ as a stochastic process sample path and applying a deterministic-stochastic decomposition $f(t)=F(t;\mathbf{C})+\epsilon(t)$. The authors propose a two-stage learning pipeline: (i) fit a deterministic time-based trend $F(t;\mathbf{C})$ with a polynomial, and (ii) model the residual $\epsilon(t)$ with either a Gaussian process or a Student's-$t$ process to capture long-range temporal correlations and measurement-noise structure, yielding a Markov-free, uncertainty-aware trajectory. The GP-based and StP-based formulations include online hyperparameter learning (RGP*) and accommodate colored measurement noise via kernel-based covariances, with $\mathbf{y}_k=\mathbf{H}F(k;\mathbf{C})+\mathbf{H}\epsilon(k)+\mathbf{v}_k$ guiding the inference. Simulations across four maneuvering scenarios show the proposed T-FoT-GP and T-FoT-StP methods outperform a baseline T-FoT and a GP-motion tracker, with StP providing notable robustness under heavy-tailed noise. Collectively, the work delivers a practical, uncertainty-aware, online continuous-time trajectory estimation framework that can be extended to multisensor and nonlinear measurement settings.

Abstract

This is the third part of a series of studies that model the target trajectory, which describes the target state evolution over continuous time, as a sample path of a stochastic process (SP). By adopting a deterministic-stochastic decomposition framework, we decompose the learning of the trajectory SP into two sequential stages: the first fits the deterministic trend of the trajectory using a curve function of time, while the second estimates the residual stochastic component through parametric learning of either a Gaussian process (GP) or Student's-$t$ process (StP). This leads to a Markov-free data-driven tracking approach that produces the continuous-time trajectory with minimal prior knowledge of the target dynamics. Notably, our approach explicitly models both the temporal correlations of the state sequence and of measurement noises through the SP framework. It does not only take advantage of the smooth trend of the target but also makes use of the long-term temporal correlation of both the data noise and the model fitting error. Simulations in four maneuvering target tracking scenarios have demonstrated its effectiveness and superiority in comparison with existing approaches.

Figures (6)

• Figure 1: Comparison of our proposed approach with classic research
• Figure 2: Key idea of the DSD approach to T-FoT estimation.
• Figure 3: Sample trajectories in four scenarios.
• Figure 4: Trajectory estimates in one trial using GP colored measurement noise.
• Figure 5: RMSE of 100 trials in the presence of GP colored data noise.

Theorems & Definitions (20)

• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 2
• Lemma 4
• proof