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Gaussian Process Upper Confidence Bounds in Distributed Point Target Tracking over Wireless Sensor Networks

Xingchi Liu, Lyudmila Mihaylova, Jemin George, Tien Pham

TL;DR

A distributed Gaussian process (DGP) approach for point target tracking and derives upper confidence bounds (UCBs) of the state estimates and a novel hybrid Bayesian filtering method is proposed to enhance the DGP approach by adopting a Poisson measurement likelihood model.

Abstract

Uncertainty quantification plays a key role in the development of autonomous systems, decision-making, and tracking over wireless sensor networks (WSNs). However, there is a need of providing uncertainty confidence bounds, especially for distributed machine learning-based tracking, dealing with different volumes of data collected by sensors. This paper aims to fill in this gap and proposes a distributed Gaussian process (DGP) approach for point target tracking and derives upper confidence bounds (UCBs) of the state estimates. A unique contribution of this paper includes the derived theoretical guarantees on the proposed approach and its maximum accuracy for tracking with and without clutter measurements. Particularly, the developed approaches with uncertainty bounds are generic and can provide trustworthy solutions with an increased level of reliability. A novel hybrid Bayesian filtering method is proposed to enhance the DGP approach by adopting a Poisson measurement likelihood model. The proposed approaches are validated over a WSN case study, where sensors have limited sensing ranges. Numerical results demonstrate the tracking accuracy and robustness of the proposed approaches. The derived UCBs constitute a tool for trustworthiness evaluation of DGP approaches. The simulation results reveal that the proposed UCBs successfully encompass the true target states with 88% and 42% higher probability in X and Y coordinates, respectively, when compared to the confidence interval-based method.

Gaussian Process Upper Confidence Bounds in Distributed Point Target Tracking over Wireless Sensor Networks

TL;DR

A distributed Gaussian process (DGP) approach for point target tracking and derives upper confidence bounds (UCBs) of the state estimates and a novel hybrid Bayesian filtering method is proposed to enhance the DGP approach by adopting a Poisson measurement likelihood model.

Abstract

Uncertainty quantification plays a key role in the development of autonomous systems, decision-making, and tracking over wireless sensor networks (WSNs). However, there is a need of providing uncertainty confidence bounds, especially for distributed machine learning-based tracking, dealing with different volumes of data collected by sensors. This paper aims to fill in this gap and proposes a distributed Gaussian process (DGP) approach for point target tracking and derives upper confidence bounds (UCBs) of the state estimates. A unique contribution of this paper includes the derived theoretical guarantees on the proposed approach and its maximum accuracy for tracking with and without clutter measurements. Particularly, the developed approaches with uncertainty bounds are generic and can provide trustworthy solutions with an increased level of reliability. A novel hybrid Bayesian filtering method is proposed to enhance the DGP approach by adopting a Poisson measurement likelihood model. The proposed approaches are validated over a WSN case study, where sensors have limited sensing ranges. Numerical results demonstrate the tracking accuracy and robustness of the proposed approaches. The derived UCBs constitute a tool for trustworthiness evaluation of DGP approaches. The simulation results reveal that the proposed UCBs successfully encompass the true target states with 88% and 42% higher probability in X and Y coordinates, respectively, when compared to the confidence interval-based method.
Paper Structure (25 sections, 5 theorems, 48 equations, 8 figures, 9 tables)

This paper contains 25 sections, 5 theorems, 48 equations, 8 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background Knowledge
  4. Standard Gaussian Process Method
  5. Distributed Gaussian Process Method
  6. Hyperparameter Learning
  7. DGP-based Point Target Tracking
  8. Temporal and Spatial-Temporal GP
  9. Sliding Window-based Tracking
  10. DGP-based Data Association
  11. Hyperparameter Online Learning in DGPT
  12. Complexity Analysis
  13. Theoretical Performance Analysis
  14. DGP-assisted Bayesian Filtering with Measurement Origin Uncertainty
  15. Measurement Likelihood Function
  16. ...and 10 more sections

Key Result

Lemma 1

(Lemma 5.1 of 105555) Given a trained local GP based on training data $D_t=\left\{\mathbf{X}_t,\mathbf{z}_t\right\}$ till $t$, for any input $\mathbf{x}_*\in \mathbf{X}_t$, the probability $\text{Pr}(.)$ that the predictive mean $\mu(\mathbf{x}_*)$ deviates from the true function value by more than where $\gamma$ is a positive constant.

Figures (8)

  • Figure 1: A distributed point tracking system with 4 sensors. The length of sliding window in this example is 5 time steps
  • Figure 2: Computation time for the log-marginal likelihood and its gradient versus the size of the training data and the number of local GPs
  • Figure 3: Target trajectories
  • Figure 4: UCB for DGPT-RBCM of S1 in X coordinate
  • Figure 5: Confidence interval for DGPT-RBCM of S1 in X coordinate
  • ...and 3 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Theorem 3