A numerical solution to the Bayesian tracking problem using statistical manifolds☆

doi:10.1016/j.automatica.2024.112025

A numerical solution to the Bayesian tracking problem using statistical manifolds☆

2025-03-01

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DOI：10.1016/j.automatica.2024.112025

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In the general non-linear, non-Gaussian case, the Bayesian tracking formulas lack an analytic expression. For this reason a number of numerical approximate solutions are available. The most accurate solutions approximate the probability distributions (PDs) appearing in the Bayesian tracking recursions using weighted sums of elementary functions (e.g., Dirac impulses or Gaussian functions). This often leads to a large number of parameters needed to represent a PD in order to achieve a desired accuracy. This makes these approaches numerically complex as well as unsuitable for distributed applications where neighbor nodes need to exchange PD information. In order to tackle this limitation, we do the key observation that the large number of required parameters results from the fact that this kind of parameterization permit approximating any possible PD. However, the PDs appearing in a given Bayesian tracking problem can only belong to certain family of possible PDs induced by the problem. We propose a Bayesian tracking method exploiting this idea. We start by modeling the induced family as a manifold within the set of all possible PDs. We then build a coordinate system for this manifold, which permits representing each PD within the family using its coordinates along the manifold. We then represent the Bayesian tracking formulas as non-linear transformations of manifold coordinates representing PDs. In this sense, the proposed Bayesian tracking approach can be thought as the natural extension to the non-linear, non-Gaussian case of the Kalman filter, as the latter also uses a minimum set of coefficients (i.e., mean and correlation matrix) to represent each PD within the Gaussian family induced by a linear, Gaussian problem. We present a numerical experiment showing the massive reduction of parameters required to represent PDs in a benchmark example. (c) 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Keyword:

Estimation theory

Statistical analysis

Bayesian tracking

Statistical manifolds

Non-linear dimensionality reduction

A numerical solution to the Bayesian tracking problem using statistical manifolds☆

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