Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arithmetic Average Density Fusion -- Part I: Some Statistic and Information-theoretic Results

Tiancheng Li, Yan Song, Enbin Song, Hongqi Fan

TL;DR

This work analyzes Arithmetic Average (AA) density fusion for multi-sensor estimation, linking it to conservative fusion concepts like covariance union and covariance intersection. It develops both statistical and information-theoretic insights, proving covariance-consistency under certain conditions, and demonstrating mode preservation in AA fusion. A suboptimal, entropy/divergence-based weighting strategy is proposed and analyzed via a max-min optimization framework, with Gaussian-case approximations via moment matching. Simulations on linear and nonlinear tracking problems show AA fusion achieving competitive performance with CI and IC while offering robustness to sensor heterogeneity and potential computational advantages, establishing AA fusion as a practical foundation for distributed multi-sensor Kalman and particle filtering. This first part lays the theoretical groundwork and validates AA fusion's viability, guiding future work on distributed networks and complex sensor configurations.

Abstract

Finite mixture such as the Gaussian mixture is a flexible and powerful probabilistic modeling tool for representing the multimodal distribution widely involved in many estimation and learning problems. The core of it is representing the target distribution by the arithmetic average (AA) of a finite number of sub-distributions which constitute a mixture. While the mixture has been widely used for single sensor filter design, it is only recent that the AA fusion demonstrates compelling performance for multi-sensor filter design. In this paper, some statistic and information-theoretic results are given on the covariance consistency, mean square error, mode-preservation capacity, and the information divergence of the AA fusion approach. In particular, based on the concept of conservative fusion, the relationship of the AA fusion with the existing conservative fusion approaches such as covariance union and covariance intersection is exposed. A suboptimal weighting approach has been proposed, which jointly with the best mixture-fit property of the AA fusion leads to a max-min optimization problem. Linear Gaussian models are considered for algorithm illustration and simulation comparison, resulting in the first-ever AA fusion-based multi-sensor Kalman filter.

Arithmetic Average Density Fusion -- Part I: Some Statistic and Information-theoretic Results

TL;DR

This work analyzes Arithmetic Average (AA) density fusion for multi-sensor estimation, linking it to conservative fusion concepts like covariance union and covariance intersection. It develops both statistical and information-theoretic insights, proving covariance-consistency under certain conditions, and demonstrating mode preservation in AA fusion. A suboptimal, entropy/divergence-based weighting strategy is proposed and analyzed via a max-min optimization framework, with Gaussian-case approximations via moment matching. Simulations on linear and nonlinear tracking problems show AA fusion achieving competitive performance with CI and IC while offering robustness to sensor heterogeneity and potential computational advantages, establishing AA fusion as a practical foundation for distributed multi-sensor Kalman and particle filtering. This first part lays the theoretical groundwork and validates AA fusion's viability, guiding future work on distributed networks and complex sensor configurations.

Abstract

Finite mixture such as the Gaussian mixture is a flexible and powerful probabilistic modeling tool for representing the multimodal distribution widely involved in many estimation and learning problems. The core of it is representing the target distribution by the arithmetic average (AA) of a finite number of sub-distributions which constitute a mixture. While the mixture has been widely used for single sensor filter design, it is only recent that the AA fusion demonstrates compelling performance for multi-sensor filter design. In this paper, some statistic and information-theoretic results are given on the covariance consistency, mean square error, mode-preservation capacity, and the information divergence of the AA fusion approach. In particular, based on the concept of conservative fusion, the relationship of the AA fusion with the existing conservative fusion approaches such as covariance union and covariance intersection is exposed. A suboptimal weighting approach has been proposed, which jointly with the best mixture-fit property of the AA fusion leads to a max-min optimization problem. Linear Gaussian models are considered for algorithm illustration and simulation comparison, resulting in the first-ever AA fusion-based multi-sensor Kalman filter.

Paper Structure

This paper contains 20 sections, 7 theorems, 55 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Conservative Fusion and Statistics of AA Fusion
  3. Notations, Concepts and Definitions
  4. Comparison between CU and AA
  5. Some Other Conservative Fusion Approaches
  6. MSE of EAP estimator
  7. Multimodality and MAP estimator
  8. Information Divergence and Fusion Weights
  9. Error Covariance-based Fusion Weights
  10. Information-theoretic Suboptimal Fusion Weights
  11. Max-Min Optimization for suboptimal AA and GA Fusion
  12. Case Study: Gaussian Fusion
  13. Simulations
  14. Linear Scenario
  15. Nonlinear Scenario
  16. ...and 5 more sections

Key Result

Lemma 1

For a set of estimate pairs $(\hat{\mathbf{x}}_i,\mathbf{P}_i)$, $i \in \mathcal{I}$ in which at least one is conservative, a sufficient condition for the fused estimate pair ($\hat{\mathbf{x}}_\mathrm{AA},\mathbf{P}_\mathrm{CU}$) to be conservative is that for which a tight bound is given by We note that two matrixes may not be comparable for which we use $\mathbf{A} \succ \mathbf{B}$ if $\mathr

Figures (14)

  • Figure 1: Optimal AA fusion of four Gaussian densities (each given by a black dotted line) to best fit the target density (red solid line) in four different cases, resulting in the optimally merged Gaussian density (blue dashed line).
  • Figure 2: The real target trajectory, the estimates of the noncooperative filters and their two-sensor-fusion results when $R_2 =2R_1$ in one trial of the linear scenario.
  • Figure 3: The position RMSEs of noncooperative and two-sensor-fusion filters when $R_2 =2R_1$ in the linear scenario.
  • Figure 4: The velocity RMSEs of noncooperative and two-sensor-fusion filters when $R_2 =2R_1$ in the linear scenario.
  • Figure 5: The average and variance of the fusion weight assigned to sensor 1 in the KF-AA fusion when $R_2 =2R_1$ in the linear scenario.
  • ...and 9 more figures

Theorems & Definitions (11)

  • Lemma 1
  • Corollary 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 2
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 1 more